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 complexOTHERauxiliary(3) LAPACK complexOTHERauxiliary(3)

# NAME¶

complexOTHERauxiliary

# SYNOPSIS¶

## Functions¶

subroutine clabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. subroutine clacgv (N, X, INCX)
CLACGV conjugates a complex vector. subroutine clacn2 (N, V, X, EST, KASE, ISAVE)
CLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutine clacon (N, V, X, EST, KASE)
CLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutine clacp2 (UPLO, M, N, A, LDA, B, LDB)
CLACP2 copies all or part of a real two-dimensional array to a complex array. subroutine clacpy (UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another. subroutine clacrm (M, N, A, LDA, B, LDB, C, LDC, RWORK)
CLACRM multiplies a complex matrix by a square real matrix. subroutine clacrt (N, CX, INCX, CY, INCY, C, S)
CLACRT performs a linear transformation of a pair of complex vectors. complex function cladiv (X, Y)
CLADIV performs complex division in real arithmetic, avoiding unnecessary overflow. subroutine claein (RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK, EPS3, SMLNUM, INFO)
CLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration. subroutine claev2 (A, B, C, RT1, RT2, CS1, SN1)
CLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix. subroutine clags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
CLAGS2 subroutine clagtm (TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)
CLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1. subroutine clahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. subroutine clahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. subroutine claic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
CLAIC1 applies one step of incremental condition estimation. real function clangt (NORM, N, DL, D, DU)
CLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix. real function clanhb (NORM, UPLO, N, K, AB, LDAB, WORK)
CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix. real function clanhp (NORM, UPLO, N, AP, WORK)
CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix supplied in packed form. real function clanhs (NORM, N, A, LDA, WORK)
CLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix. real function clanht (NORM, N, D, E)
CLANHT returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix. real function clansb (NORM, UPLO, N, K, AB, LDAB, WORK)
CLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix. real function clansp (NORM, UPLO, N, AP, WORK)
CLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form. real function clantb (NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
CLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix. real function clantp (NORM, UPLO, DIAG, N, AP, WORK)
CLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form. real function clantr (NORM, UPLO, DIAG, M, N, A, LDA, WORK)
CLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix. subroutine clapll (N, X, INCX, Y, INCY, SSMIN)
CLAPLL measures the linear dependence of two vectors. subroutine clapmr (FORWRD, M, N, X, LDX, K)
CLAPMR rearranges rows of a matrix as specified by a permutation vector. subroutine clapmt (FORWRD, M, N, X, LDX, K)
CLAPMT performs a forward or backward permutation of the columns of a matrix. subroutine claqhb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
CLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ. subroutine claqhp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
CLAQHP scales a Hermitian matrix stored in packed form. subroutine claqp2 (M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
CLAQP2 computes a QR factorization with column pivoting of the matrix block. subroutine claqps (M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
CLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3. subroutine claqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
CLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutine claqr1 (N, H, LDH, S1, S2, V)
CLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts. subroutine claqr2 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine claqr3 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine claqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutine claqr5 (WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)
CLAQR5 performs a single small-bulge multi-shift QR sweep. subroutine claqsb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
CLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ. subroutine claqsp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
CLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ. subroutine clar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. subroutine clar2v (N, X, Y, Z, INCX, C, S, INCC)
CLAR2V applies a vector of plane rotations with real cosines and complex sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices. subroutine clarcm (M, N, A, LDA, B, LDB, C, LDC, RWORK)
CLARCM copies all or part of a real two-dimensional array to a complex array. subroutine clarf (SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
CLARF applies an elementary reflector to a general rectangular matrix. subroutine clarfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
CLARFB applies a block reflector or its conjugate-transpose to a general rectangular matrix. subroutine clarfg (N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix). subroutine clarfgp (N, ALPHA, X, INCX, TAU)
CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta. subroutine clarft (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
CLARFT forms the triangular factor T of a block reflector H = I - vtvH subroutine clarfx (SIDE, M, N, V, TAU, C, LDC, WORK)
CLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10. subroutine clargv (N, X, INCX, Y, INCY, C, INCC)
CLARGV generates a vector of plane rotations with real cosines and complex sines. subroutine clarnv (IDIST, ISEED, N, X)
CLARNV returns a vector of random numbers from a uniform or normal distribution. subroutine clarrv (N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MINRGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO)
CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT. subroutine clartg (F, G, CS, SN, R)
CLARTG generates a plane rotation with real cosine and complex sine. subroutine clartv (N, X, INCX, Y, INCY, C, S, INCC)
CLARTV applies a vector of plane rotations with real cosines and complex sines to the elements of a pair of vectors. subroutine clascl (TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom. subroutine claset (UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values. subroutine clasr (SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA)
CLASR applies a sequence of plane rotations to a general rectangular matrix. subroutine classq (N, X, INCX, SCALE, SUMSQ)
CLASSQ updates a sum of squares represented in scaled form. subroutine claswp (N, A, LDA, K1, K2, IPIV, INCX)
CLASWP performs a series of row interchanges on a general rectangular matrix. subroutine clatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
CLATBS solves a triangular banded system of equations. subroutine clatdf (IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)
CLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. subroutine clatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
CLATPS solves a triangular system of equations with the matrix held in packed storage. subroutine clatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation. subroutine clatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
CLATRS solves a triangular system of equations with the scale factor set to prevent overflow. subroutine clauu2 (UPLO, N, A, LDA, INFO)
CLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm). subroutine clauum (UPLO, N, A, LDA, INFO)
CLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm). subroutine crot (N, CX, INCX, CY, INCY, C, S)
CROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors. subroutine cspmv (UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
CSPMV computes a matrix-vector product for complex vectors using a complex symmetric packed matrix subroutine cspr (UPLO, N, ALPHA, X, INCX, AP)
CSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix. subroutine csrscl (N, SA, SX, INCX)
CSRSCL multiplies a vector by the reciprocal of a real scalar. subroutine ctprfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
CTPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks. integer function icmax1 (N, CX, INCX)
ICMAX1 finds the index of the first vector element of maximum absolute value. integer function ilaclc (M, N, A, LDA)
ILACLC scans a matrix for its last non-zero column. integer function ilaclr (M, N, A, LDA)
ILACLR scans a matrix for its last non-zero row. integer function izmax1 (N, ZX, INCX)
IZMAX1 finds the index of the first vector element of maximum absolute value. real function scsum1 (N, CX, INCX)
SCSUM1 forms the 1-norm of the complex vector using the true absolute value.

# Detailed Description¶

This is the group of complex other auxiliary routines

# Function Documentation¶

## subroutine clabrd (integer M, integer N, integer NB, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ, complex, dimension( * ) TAUP, complex, dimension( ldx, * ) X, integer LDX, complex, dimension( ldy, * ) Y, integer LDY)¶

CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

``` CLABRD reduces the first NB rows and columns of a complex general
m by n matrix A to upper or lower real bidiagonal form by a unitary
transformation Q**H * A * P, and returns the matrices X and Y which
are needed to apply the transformation to the unreduced part of A.
If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
bidiagonal form.
This is an auxiliary routine called by CGEBRD
```

Parameters:

M

```          M is INTEGER
The number of rows in the matrix A.
```

N

```          N is INTEGER
The number of columns in the matrix A.
```

NB

```          NB is INTEGER
The number of leading rows and columns of A to be reduced.
```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the m by n general matrix to be reduced.
On exit, the first NB rows and columns of the matrix are
overwritten; the rest of the array is unchanged.
If m >= n, elements on and below the diagonal in the first NB
columns, with the array TAUQ, represent the unitary
matrix Q as a product of elementary reflectors; and
elements above the diagonal in the first NB rows, with the
array TAUP, represent the unitary matrix P as a product
of elementary reflectors.
If m < n, elements below the diagonal in the first NB
columns, with the array TAUQ, represent the unitary
matrix Q as a product of elementary reflectors, and
elements on and above the diagonal in the first NB rows,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors.
See Further Details.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).
```

D

```          D is REAL array, dimension (NB)
The diagonal elements of the first NB rows and columns of
the reduced matrix.  D(i) = A(i,i).
```

E

```          E is REAL array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of
the reduced matrix.
```

TAUQ

```          TAUQ is COMPLEX array, dimension (NB)
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.
```

TAUP

```          TAUP is COMPLEX array, dimension (NB)
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.
```

X

```          X is COMPLEX array, dimension (LDX,NB)
The m-by-nb matrix X required to update the unreduced part
of A.
```

LDX

```          LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,M).
```

Y

```          Y is COMPLEX array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part
of A.
```

LDY

```          LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Further Details:

```  The matrices Q and P are represented as products of elementary
reflectors:
Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
where tauq and taup are complex scalars, and v and u are complex
vectors.
If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
The elements of the vectors v and u together form the m-by-nb matrix
V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
the transformation to the unreduced part of the matrix, using a block
update of the form:  A := A - V*Y**H - X*U**H.
The contents of A on exit are illustrated by the following examples
with nb = 2:
m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
(  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
(  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
(  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )
where a denotes an element of the original matrix which is unchanged,
vi denotes an element of the vector defining H(i), and ui an element
of the vector defining G(i).
```

## subroutine clacgv (integer N, complex, dimension( * ) X, integer INCX)¶

CLACGV conjugates a complex vector.

Purpose:

``` CLACGV conjugates a complex vector of length N.
```

Parameters:

N

```          N is INTEGER
The length of the vector X.  N >= 0.
```

X

```          X is COMPLEX array, dimension
(1+(N-1)*abs(INCX))
On entry, the vector of length N to be conjugated.
On exit, X is overwritten with conjg(X).
```

INCX

```          INCX is INTEGER
The spacing between successive elements of X.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clacn2 (integer N, complex, dimension( * ) V, complex, dimension( * ) X, real EST, integer KASE, integer, dimension( 3 ) ISAVE)¶

CLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

Purpose:

``` CLACN2 estimates the 1-norm of a square, complex matrix A.
Reverse communication is used for evaluating matrix-vector products.
```

Parameters:

N

```          N is INTEGER
The order of the matrix.  N >= 1.
```

V

```          V is COMPLEX array, dimension (N)
On the final return, V = A*W,  where  EST = norm(V)/norm(W)
(W is not returned).
```

X

```          X is COMPLEX array, dimension (N)
On an intermediate return, X should be overwritten by
A * X,   if KASE=1,
A**H * X,  if KASE=2,
where A**H is the conjugate transpose of A, and CLACN2 must be
re-called with all the other parameters unchanged.
```

EST

```          EST is REAL
On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
unchanged from the previous call to CLACN2.
On exit, EST is an estimate (a lower bound) for norm(A).
```

KASE

```          KASE is INTEGER
On the initial call to CLACN2, KASE should be 0.
On an intermediate return, KASE will be 1 or 2, indicating
whether X should be overwritten by A * X  or A**H * X.
On the final return from CLACN2, KASE will again be 0.
```

ISAVE

```          ISAVE is INTEGER array, dimension (3)
ISAVE is used to save variables between calls to SLACN2
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  Originally named CONEST, dated March 16, 1988.
Last modified:  April, 1999
This is a thread safe version of CLACON, which uses the array ISAVE
in place of a SAVE statement, as follows:
CLACON     CLACN2
JUMP     ISAVE(1)
J        ISAVE(2)
ITER     ISAVE(3)
```

Contributors:

Nick Higham, University of Manchester

References:

N.J. Higham, 'FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

## subroutine clacon (integer N, complex, dimension( n ) V, complex, dimension( n ) X, real EST, integer KASE)¶

CLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

Purpose:

``` CLACON estimates the 1-norm of a square, complex matrix A.
Reverse communication is used for evaluating matrix-vector products.
```

Parameters:

N

```          N is INTEGER
The order of the matrix.  N >= 1.
```

V

```          V is COMPLEX array, dimension (N)
On the final return, V = A*W,  where  EST = norm(V)/norm(W)
(W is not returned).
```

X

```          X is COMPLEX array, dimension (N)
On an intermediate return, X should be overwritten by
A * X,   if KASE=1,
A**H * X,  if KASE=2,
where A**H is the conjugate transpose of A, and CLACON must be
re-called with all the other parameters unchanged.
```

EST

```          EST is REAL
On entry with KASE = 1 or 2 and JUMP = 3, EST should be
unchanged from the previous call to CLACON.
On exit, EST is an estimate (a lower bound) for norm(A).
```

KASE

```          KASE is INTEGER
On the initial call to CLACON, KASE should be 0.
On an intermediate return, KASE will be 1 or 2, indicating
whether X should be overwritten by A * X  or A**H * X.
On the final return from CLACON, KASE will again be 0.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

Originally named CONEST, dated March 16, 1988. Last modified: April, 1999

Contributors:

Nick Higham, University of Manchester

References:

N.J. Higham, 'FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

## subroutine clacp2 (character UPLO, integer M, integer N, real, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB)¶

CLACP2 copies all or part of a real two-dimensional array to a complex array.

Purpose:

``` CLACP2 copies all or part of a real two-dimensional matrix A to a
complex matrix B.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies the part of the matrix A to be copied to B.
= 'U':      Upper triangular part
= 'L':      Lower triangular part
Otherwise:  All of the matrix A
```

M

```          M is INTEGER
The number of rows of the matrix A.  M >= 0.
```

N

```          N is INTEGER
The number of columns of the matrix A.  N >= 0.
```

A

```          A is REAL array, dimension (LDA,N)
The m by n matrix A.  If UPLO = 'U', only the upper trapezium
is accessed; if UPLO = 'L', only the lower trapezium is
accessed.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).
```

B

```          B is COMPLEX array, dimension (LDB,N)
On exit, B = A in the locations specified by UPLO.
```

LDB

```          LDB is INTEGER
The leading dimension of the array B.  LDB >= max(1,M).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clacpy (character UPLO, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB)¶

CLACPY copies all or part of one two-dimensional array to another.

Purpose:

``` CLACPY copies all or part of a two-dimensional matrix A to another
matrix B.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies the part of the matrix A to be copied to B.
= 'U':      Upper triangular part
= 'L':      Lower triangular part
Otherwise:  All of the matrix A
```

M

```          M is INTEGER
The number of rows of the matrix A.  M >= 0.
```

N

```          N is INTEGER
The number of columns of the matrix A.  N >= 0.
```

A

```          A is COMPLEX array, dimension (LDA,N)
The m by n matrix A.  If UPLO = 'U', only the upper trapezium
is accessed; if UPLO = 'L', only the lower trapezium is
accessed.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).
```

B

```          B is COMPLEX array, dimension (LDB,N)
On exit, B = A in the locations specified by UPLO.
```

LDB

```          LDB is INTEGER
The leading dimension of the array B.  LDB >= max(1,M).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clacrm (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, real, dimension( * ) RWORK)¶

CLACRM multiplies a complex matrix by a square real matrix.

Purpose:

``` CLACRM performs a very simple matrix-matrix multiplication:
C := A * B,
where A is M by N and complex; B is N by N and real;
C is M by N and complex.
```

Parameters:

M

```          M is INTEGER
The number of rows of the matrix A and of the matrix C.
M >= 0.
```

N

```          N is INTEGER
The number of columns and rows of the matrix B and
the number of columns of the matrix C.
N >= 0.
```

A

```          A is COMPLEX array, dimension (LDA, N)
On entry, A contains the M by N matrix A.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >=max(1,M).
```

B

```          B is REAL array, dimension (LDB, N)
On entry, B contains the N by N matrix B.
```

LDB

```          LDB is INTEGER
The leading dimension of the array B. LDB >=max(1,N).
```

C

```          C is COMPLEX array, dimension (LDC, N)
On exit, C contains the M by N matrix C.
```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >=max(1,N).
```

RWORK

```          RWORK is REAL array, dimension (2*M*N)
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clacrt (integer N, complex, dimension( * ) CX, integer INCX, complex, dimension( * ) CY, integer INCY, complex C, complex S)¶

CLACRT performs a linear transformation of a pair of complex vectors.

Purpose:

``` CLACRT performs the operation
(  c  s )( x )  ==> ( x )
( -s  c )( y )      ( y )
where c and s are complex and the vectors x and y are complex.
```

Parameters:

N

```          N is INTEGER
The number of elements in the vectors CX and CY.
```

CX

```          CX is COMPLEX array, dimension (N)
On input, the vector x.
On output, CX is overwritten with c*x + s*y.
```

INCX

```          INCX is INTEGER
The increment between successive values of CX.  INCX <> 0.
```

CY

```          CY is COMPLEX array, dimension (N)
On input, the vector y.
On output, CY is overwritten with -s*x + c*y.
```

INCY

```          INCY is INTEGER
The increment between successive values of CY.  INCY <> 0.
```

C

```          C is COMPLEX
```

S

```          S is COMPLEX
C and S define the matrix
[  C   S  ].
[ -S   C  ]
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## complex function cladiv (complex X, complex Y)¶

CLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.

Purpose:

``` CLADIV := X / Y, where X and Y are complex.  The computation of X / Y
will not overflow on an intermediary step unless the results
overflows.
```

Parameters:

X

```          X is COMPLEX
```

Y

```          Y is COMPLEX
The complex scalars X and Y.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine claein (logical RIGHTV, logical NOINIT, integer N, complex, dimension( ldh, * ) H, integer LDH, complex W, complex, dimension( * ) V, complex, dimension( ldb, * ) B, integer LDB, real, dimension( * ) RWORK, real EPS3, real SMLNUM, integer INFO)¶

CLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

Purpose:

``` CLAEIN uses inverse iteration to find a right or left eigenvector
corresponding to the eigenvalue W of a complex upper Hessenberg
matrix H.
```

Parameters:

RIGHTV

```          RIGHTV is LOGICAL
= .TRUE. : compute right eigenvector;
= .FALSE.: compute left eigenvector.
```

NOINIT

```          NOINIT is LOGICAL
= .TRUE. : no initial vector supplied in V
= .FALSE.: initial vector supplied in V.
```

N

```          N is INTEGER
The order of the matrix H.  N >= 0.
```

H

```          H is COMPLEX array, dimension (LDH,N)
The upper Hessenberg matrix H.
```

LDH

```          LDH is INTEGER
The leading dimension of the array H.  LDH >= max(1,N).
```

W

```          W is COMPLEX
The eigenvalue of H whose corresponding right or left
eigenvector is to be computed.
```

V

```          V is COMPLEX array, dimension (N)
On entry, if NOINIT = .FALSE., V must contain a starting
vector for inverse iteration; otherwise V need not be set.
On exit, V contains the computed eigenvector, normalized so
that the component of largest magnitude has magnitude 1; here
the magnitude of a complex number (x,y) is taken to be
|x| + |y|.
```

B

```          B is COMPLEX array, dimension (LDB,N)
```

LDB

```          LDB is INTEGER
The leading dimension of the array B.  LDB >= max(1,N).
```

RWORK

```          RWORK is REAL array, dimension (N)
```

EPS3

```          EPS3 is REAL
A small machine-dependent value which is used to perturb
close eigenvalues, and to replace zero pivots.
```

SMLNUM

```          SMLNUM is REAL
A machine-dependent value close to the underflow threshold.
```

INFO

```          INFO is INTEGER
= 0:  successful exit
= 1:  inverse iteration did not converge; V is set to the
last iterate.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine claev2 (complex A, complex B, complex C, real RT1, real RT2, real CS1, complex SN1)¶

CLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

Purpose:

``` CLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
[  A         B  ]
[  CONJG(B)  C  ].
On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
eigenvector for RT1, giving the decomposition
[ CS1  CONJG(SN1) ] [    A     B ] [ CS1 -CONJG(SN1) ] = [ RT1  0  ]
[-SN1     CS1     ] [ CONJG(B) C ] [ SN1     CS1     ]   [  0  RT2 ].
```

Parameters:

A

```          A is COMPLEX
The (1,1) element of the 2-by-2 matrix.
```

B

```          B is COMPLEX
The (1,2) element and the conjugate of the (2,1) element of
the 2-by-2 matrix.
```

C

```          C is COMPLEX
The (2,2) element of the 2-by-2 matrix.
```

RT1

```          RT1 is REAL
The eigenvalue of larger absolute value.
```

RT2

```          RT2 is REAL
The eigenvalue of smaller absolute value.
```

CS1

```          CS1 is REAL
```

SN1

```          SN1 is COMPLEX
The vector (CS1, SN1) is a unit right eigenvector for RT1.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  RT1 is accurate to a few ulps barring over/underflow.
RT2 may be inaccurate if there is massive cancellation in the
determinant A*C-B*B; higher precision or correctly rounded or
correctly truncated arithmetic would be needed to compute RT2
accurately in all cases.
CS1 and SN1 are accurate to a few ulps barring over/underflow.
Overflow is possible only if RT1 is within a factor of 5 of overflow.
Underflow is harmless if the input data is 0 or exceeds
underflow_threshold / macheps.
```

## subroutine clags2 (logical UPPER, real A1, complex A2, real A3, real B1, complex B2, real B3, real CSU, complex SNU, real CSV, complex SNV, real CSQ, complex SNQ)¶

CLAGS2

Purpose:

``` CLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
that if ( UPPER ) then
U**H *A*Q = U**H *( A1 A2 )*Q = ( x  0  )
( 0  A3 )     ( x  x  )
and
V**H*B*Q = V**H *( B1 B2 )*Q = ( x  0  )
( 0  B3 )     ( x  x  )
or if ( .NOT.UPPER ) then
U**H *A*Q = U**H *( A1 0  )*Q = ( x  x  )
( A2 A3 )     ( 0  x  )
and
V**H *B*Q = V**H *( B1 0  )*Q = ( x  x  )
( B2 B3 )     ( 0  x  )
where
U = (   CSU    SNU ), V = (  CSV    SNV ),
( -SNU**H  CSU )      ( -SNV**H CSV )
Q = (   CSQ    SNQ )
( -SNQ**H  CSQ )
The rows of the transformed A and B are parallel. Moreover, if the
input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
of A is not zero. If the input matrices A and B are both not zero,
then the transformed (2,2) element of B is not zero, except when the
first rows of input A and B are parallel and the second rows are
zero.
```

Parameters:

UPPER

```          UPPER is LOGICAL
= .TRUE.: the input matrices A and B are upper triangular.
= .FALSE.: the input matrices A and B are lower triangular.
```

A1

```          A1 is REAL
```

A2

```          A2 is COMPLEX
```

A3

```          A3 is REAL
On entry, A1, A2 and A3 are elements of the input 2-by-2
upper (lower) triangular matrix A.
```

B1

```          B1 is REAL
```

B2

```          B2 is COMPLEX
```

B3

```          B3 is REAL
On entry, B1, B2 and B3 are elements of the input 2-by-2
upper (lower) triangular matrix B.
```

CSU

```          CSU is REAL
```

SNU

```          SNU is COMPLEX
The desired unitary matrix U.
```

CSV

```          CSV is REAL
```

SNV

```          SNV is COMPLEX
The desired unitary matrix V.
```

CSQ

```          CSQ is REAL
```

SNQ

```          SNQ is COMPLEX
The desired unitary matrix Q.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clagtm (character TRANS, integer N, integer NRHS, real ALPHA, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( ldx, * ) X, integer LDX, real BETA, complex, dimension( ldb, * ) B, integer LDB)¶

CLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.

Purpose:

``` CLAGTM performs a matrix-vector product of the form
B := alpha * A * X + beta * B
where A is a tridiagonal matrix of order N, B and X are N by NRHS
matrices, and alpha and beta are real scalars, each of which may be
0., 1., or -1.
```

Parameters:

TRANS

```          TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N':  No transpose, B := alpha * A * X + beta * B
= 'T':  Transpose,    B := alpha * A**T * X + beta * B
= 'C':  Conjugate transpose, B := alpha * A**H * X + beta * B
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

NRHS

```          NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices X and B.
```

ALPHA

```          ALPHA is REAL
The scalar alpha.  ALPHA must be 0., 1., or -1.; otherwise,
it is assumed to be 0.
```

DL

```          DL is COMPLEX array, dimension (N-1)
The (n-1) sub-diagonal elements of T.
```

D

```          D is COMPLEX array, dimension (N)
The diagonal elements of T.
```

DU

```          DU is COMPLEX array, dimension (N-1)
The (n-1) super-diagonal elements of T.
```

X

```          X is COMPLEX array, dimension (LDX,NRHS)
The N by NRHS matrix X.
```

LDX

```          LDX is INTEGER
The leading dimension of the array X.  LDX >= max(N,1).
```

BETA

```          BETA is REAL
The scalar beta.  BETA must be 0., 1., or -1.; otherwise,
it is assumed to be 1.
```

B

```          B is COMPLEX array, dimension (LDB,NRHS)
On entry, the N by NRHS matrix B.
On exit, B is overwritten by the matrix expression
B := alpha * A * X + beta * B.
```

LDB

```          LDB is INTEGER
The leading dimension of the array B.  LDB >= max(N,1).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, integer INFO)¶

CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

```    CLAHQR is an auxiliary routine called by CHSEQR to update the
eigenvalues and Schur decomposition already computed by CHSEQR, by
dealing with the Hessenberg submatrix in rows and columns ILO to
IHI.
```

Parameters:

WANTT

```          WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.
```

WANTZ

```          WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.
```

N

```          N is INTEGER
The order of the matrix H.  N >= 0.
```

ILO

```          ILO is INTEGER
```

IHI

```          IHI is INTEGER
It is assumed that H is already upper triangular in rows and
columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
CLAHQR works primarily with the Hessenberg submatrix in rows
and columns ILO to IHI, but applies transformations to all of
H if WANTT is .TRUE..
1 <= ILO <= max(1,IHI); IHI <= N.
```

H

```          H is COMPLEX array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO is zero and if WANTT is .TRUE., then H
is upper triangular in rows and columns ILO:IHI.  If INFO
is zero and if WANTT is .FALSE., then the contents of H
are unspecified on exit.  The output state of H in case
INF is positive is below under the description of INFO.
```

LDH

```          LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
```

W

```          W is COMPLEX array, dimension (N)
The computed eigenvalues ILO to IHI are stored in the
corresponding elements of W. If WANTT is .TRUE., the
eigenvalues are stored in the same order as on the diagonal
of the Schur form returned in H, with W(i) = H(i,i).
```

ILOZ

```          ILOZ is INTEGER
```

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
```

Z

```          Z is COMPLEX array, dimension (LDZ,N)
If WANTZ is .TRUE., on entry Z must contain the current
matrix Z of transformations accumulated by CHSEQR, and on
exit Z has been updated; transformations are applied only to
the submatrix Z(ILOZ:IHIZ,ILO:IHI).
If WANTZ is .FALSE., Z is not referenced.
```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).
```

INFO

```          INFO is INTEGER
=   0: successful exit
.GT. 0: if INFO = i, CLAHQR failed to compute all the
eigenvalues ILO to IHI in a total of 30 iterations
per eigenvalue; elements i+1:ihi of W contain
those eigenvalues which have been successfully
computed.
If INFO .GT. 0 and WANTT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the
eigenvalues of the upper Hessenberg matrix
rows and columns ILO thorugh INFO of the final,
output value of H.
If INFO .GT. 0 and WANTT is .TRUE., then on exit
(*)       (initial value of H)*U  = U*(final value of H)
where U is an orthognal matrix.    The final
value of H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.
If INFO .GT. 0 and WANTZ is .TRUE., then on exit
(final value of Z)  = (initial value of Z)*U
where U is the orthogonal matrix in (*)
(regardless of the value of WANTT.)
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

```     02-96 Based on modifications by
David Day, Sandia National Laboratory, USA
12-04 Further modifications by
Ralph Byers, University of Kansas, USA
This is a modified version of CLAHQR from LAPACK version 3.0.
It is (1) more robust against overflow and underflow and
(2) adopts the more conservative Ahues & Tisseur stopping
criterion (LAWN 122, 1997).
```

## subroutine clahr2 (integer N, integer K, integer NB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( nb ) TAU, complex, dimension( ldt, nb ) T, integer LDT, complex, dimension( ldy, nb ) Y, integer LDY)¶

CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Purpose:

``` CLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
matrix A so that elements below the k-th subdiagonal are zero. The
reduction is performed by an unitary similarity transformation
Q**H * A * Q. The routine returns the matrices V and T which determine
Q as a block reflector I - V*T*v**H, and also the matrix Y = A * V * T.
This is an auxiliary routine called by CGEHRD.
```

Parameters:

N

```          N is INTEGER
The order of the matrix A.
```

K

```          K is INTEGER
The offset for the reduction. Elements below the k-th
subdiagonal in the first NB columns are reduced to zero.
K < N.
```

NB

```          NB is INTEGER
The number of columns to be reduced.
```

A

```          A is COMPLEX array, dimension (LDA,N-K+1)
On entry, the n-by-(n-k+1) general matrix A.
On exit, the elements on and above the k-th subdiagonal in
the first NB columns are overwritten with the corresponding
elements of the reduced matrix; the elements below the k-th
subdiagonal, with the array TAU, represent the matrix Q as a
product of elementary reflectors. The other columns of A are
unchanged. See Further Details.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).
```

TAU

```          TAU is COMPLEX array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.
```

T

```          T is COMPLEX array, dimension (LDT,NB)
The upper triangular matrix T.
```

LDT

```          LDT is INTEGER
The leading dimension of the array T.  LDT >= NB.
```

Y

```          Y is COMPLEX array, dimension (LDY,NB)
The n-by-nb matrix Y.
```

LDY

```          LDY is INTEGER
The leading dimension of the array Y. LDY >= N.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (n-k+1)-by-nb matrix
V which is needed, with T and Y, to apply the transformation to the
unreduced part of the matrix, using an update of the form:
A := (I - V*T*V**H) * (A - Y*V**H).
The contents of A on exit are illustrated by the following example
with n = 7, k = 3 and nb = 2:
( a   a   a   a   a )
( a   a   a   a   a )
( a   a   a   a   a )
( h   h   a   a   a )
( v1  h   a   a   a )
( v1  v2  a   a   a )
( v1  v2  a   a   a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).
This subroutine is a slight modification of LAPACK-3.0's DLAHRD
incorporating improvements proposed by Quintana-Orti and Van de
Gejin. Note that the entries of A(1:K,2:NB) differ from those
returned by the original LAPACK-3.0's DLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
```

References:

Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

## subroutine claic1 (integer JOB, integer J, complex, dimension( j ) X, real SEST, complex, dimension( j ) W, complex GAMMA, real SESTPR, complex S, complex C)¶

CLAIC1 applies one step of incremental condition estimation.

Purpose:

``` CLAIC1 applies one step of incremental condition estimation in
its simplest version:
Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
lower triangular matrix L, such that
twonorm(L*x) = sest
Then CLAIC1 computes sestpr, s, c such that
the vector
[ s*x ]
xhat = [  c  ]
is an approximate singular vector of
[ L      0  ]
Lhat = [ w**H gamma ]
in the sense that
twonorm(Lhat*xhat) = sestpr.
Depending on JOB, an estimate for the largest or smallest singular
value is computed.
Note that [s c]**H and sestpr**2 is an eigenpair of the system
diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
[ conjg(gamma) ]
where  alpha =  x**H*w.
```

Parameters:

JOB

```          JOB is INTEGER
= 1: an estimate for the largest singular value is computed.
= 2: an estimate for the smallest singular value is computed.
```

J

```          J is INTEGER
Length of X and W
```

X

```          X is COMPLEX array, dimension (J)
The j-vector x.
```

SEST

```          SEST is REAL
Estimated singular value of j by j matrix L
```

W

```          W is COMPLEX array, dimension (J)
The j-vector w.
```

GAMMA

```          GAMMA is COMPLEX
The diagonal element gamma.
```

SESTPR

```          SESTPR is REAL
Estimated singular value of (j+1) by (j+1) matrix Lhat.
```

S

```          S is COMPLEX
Sine needed in forming xhat.
```

C

```          C is COMPLEX
Cosine needed in forming xhat.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clangt (character NORM, integer N, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU)¶

CLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.

Purpose:

``` CLANGT  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the  element of  largest absolute value  of a
complex tridiagonal matrix A.
```

Returns:

CLANGT

```    CLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANGT as described
above.
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANGT is
set to zero.
```

DL

```          DL is COMPLEX array, dimension (N-1)
The (n-1) sub-diagonal elements of A.
```

D

```          D is COMPLEX array, dimension (N)
The diagonal elements of A.
```

DU

```          DU is COMPLEX array, dimension (N-1)
The (n-1) super-diagonal elements of A.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clanhb (character NORM, character UPLO, integer N, integer K, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)¶

CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix.

Purpose:

``` CLANHB  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the element of  largest absolute value  of an
n by n hermitian band matrix A,  with k super-diagonals.
```

Returns:

CLANHB

```    CLANHB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANHB as described
above.
```

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
band matrix A is supplied.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANHB is
set to zero.
```

K

```          K is INTEGER
The number of super-diagonals or sub-diagonals of the
band matrix A.  K >= 0.
```

AB

```          AB is COMPLEX array, dimension (LDAB,N)
The upper or lower triangle of the hermitian band matrix A,
stored in the first K+1 rows of AB.  The j-th column of A is
stored in the j-th column of the array AB as follows:
if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
Note that the imaginary parts of the diagonal elements need
not be set and are assumed to be zero.
```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= K+1.
```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clanhp (character NORM, character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) WORK)¶

CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix supplied in packed form.

Purpose:

``` CLANHP  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the  element of  largest absolute value  of a
complex hermitian matrix A,  supplied in packed form.
```

Returns:

CLANHP

```    CLANHP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANHP as described
above.
```

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
hermitian matrix A is supplied.
= 'U':  Upper triangular part of A is supplied
= 'L':  Lower triangular part of A is supplied
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANHP is
set to zero.
```

AP

```          AP is COMPLEX array, dimension (N*(N+1)/2)
The upper or lower triangle of the hermitian matrix A, packed
columnwise in a linear array.  The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
Note that the  imaginary parts of the diagonal elements need
not be set and are assumed to be zero.
```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clanhs (character NORM, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)¶

CLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.

Purpose:

``` CLANHS  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the  element of  largest absolute value  of a
Hessenberg matrix A.
```

Returns:

CLANHS

```    CLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANHS as described
above.
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANHS is
set to zero.
```

A

```          A is COMPLEX array, dimension (LDA,N)
The n by n upper Hessenberg matrix A; the part of A below the
first sub-diagonal is not referenced.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(N,1).
```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I'; otherwise, WORK is not
referenced.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clanht (character NORM, integer N, real, dimension( * ) D, complex, dimension( * ) E)¶

CLANHT returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix.

Purpose:

``` CLANHT  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the  element of  largest absolute value  of a
complex Hermitian tridiagonal matrix A.
```

Returns:

CLANHT

```    CLANHT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANHT as described
above.
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANHT is
set to zero.
```

D

```          D is REAL array, dimension (N)
The diagonal elements of A.
```

E

```          E is COMPLEX array, dimension (N-1)
The (n-1) sub-diagonal or super-diagonal elements of A.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clansb (character NORM, character UPLO, integer N, integer K, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)¶

CLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.

Purpose:

``` CLANSB  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the element of  largest absolute value  of an
n by n symmetric band matrix A,  with k super-diagonals.
```

Returns:

CLANSB

```    CLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANSB as described
above.
```

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
band matrix A is supplied.
= 'U':  Upper triangular part is supplied
= 'L':  Lower triangular part is supplied
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANSB is
set to zero.
```

K

```          K is INTEGER
The number of super-diagonals or sub-diagonals of the
band matrix A.  K >= 0.
```

AB

```          AB is COMPLEX array, dimension (LDAB,N)
The upper or lower triangle of the symmetric band matrix A,
stored in the first K+1 rows of AB.  The j-th column of A is
stored in the j-th column of the array AB as follows:
if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= K+1.
```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clansp (character NORM, character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) WORK)¶

CLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.

Purpose:

``` CLANSP  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the  element of  largest absolute value  of a
complex symmetric matrix A,  supplied in packed form.
```

Returns:

CLANSP

```    CLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANSP as described
above.
```

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is supplied.
= 'U':  Upper triangular part of A is supplied
= 'L':  Lower triangular part of A is supplied
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANSP is
set to zero.
```

AP

```          AP is COMPLEX array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array.  The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clantb (character NORM, character UPLO, character DIAG, integer N, integer K, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)¶

CLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.

Purpose:

``` CLANTB  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the element of  largest absolute value  of an
n by n triangular band matrix A,  with ( k + 1 ) diagonals.
```

Returns:

CLANTB

```    CLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANTB as described
above.
```

UPLO

```          UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

DIAG

```          DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N':  Non-unit triangular
= 'U':  Unit triangular
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANTB is
set to zero.
```

K

```          K is INTEGER
The number of super-diagonals of the matrix A if UPLO = 'U',
or the number of sub-diagonals of the matrix A if UPLO = 'L'.
K >= 0.
```

AB

```          AB is COMPLEX array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored in the
first k+1 rows of AB.  The j-th column of A is stored
in the j-th column of the array AB as follows:
if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
Note that when DIAG = 'U', the elements of the array AB
corresponding to the diagonal elements of the matrix A are
not referenced, but are assumed to be one.
```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= K+1.
```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I'; otherwise, WORK is not
referenced.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clantp (character NORM, character UPLO, character DIAG, integer N, complex, dimension( * ) AP, real, dimension( * ) WORK)¶

CLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.

Purpose:

``` CLANTP  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the  element of  largest absolute value  of a
triangular matrix A, supplied in packed form.
```

Returns:

CLANTP

```    CLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANTP as described
above.
```

UPLO

```          UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

DIAG

```          DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N':  Non-unit triangular
= 'U':  Unit triangular
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANTP is
set to zero.
```

AP

```          AP is COMPLEX array, dimension (N*(N+1)/2)
The upper or lower triangular matrix A, packed columnwise in
a linear array.  The j-th column of A is stored in the array
AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
Note that when DIAG = 'U', the elements of the array AP
corresponding to the diagonal elements of the matrix A are
not referenced, but are assumed to be one.
```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I'; otherwise, WORK is not
referenced.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## real function clantr (character NORM, character UPLO, character DIAG, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)¶

CLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.

Purpose:

``` CLANTR  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the  element of  largest absolute value  of a
trapezoidal or triangular matrix A.
```

Returns:

CLANTR

```    CLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'
where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
```

Parameters:

NORM

```          NORM is CHARACTER*1
Specifies the value to be returned in CLANTR as described
above.
```

UPLO

```          UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower trapezoidal.
= 'U':  Upper trapezoidal
= 'L':  Lower trapezoidal
Note that A is triangular instead of trapezoidal if M = N.
```

DIAG

```          DIAG is CHARACTER*1
Specifies whether or not the matrix A has unit diagonal.
= 'N':  Non-unit diagonal
= 'U':  Unit diagonal
```

M

```          M is INTEGER
The number of rows of the matrix A.  M >= 0, and if
UPLO = 'U', M <= N.  When M = 0, CLANTR is set to zero.
```

N

```          N is INTEGER
The number of columns of the matrix A.  N >= 0, and if
UPLO = 'L', N <= M.  When N = 0, CLANTR is set to zero.
```

A

```          A is COMPLEX array, dimension (LDA,N)
The trapezoidal matrix A (A is triangular if M = N).
If UPLO = 'U', the leading m by n upper trapezoidal part of
the array A contains the upper trapezoidal matrix, and the
strictly lower triangular part of A is not referenced.
If UPLO = 'L', the leading m by n lower trapezoidal part of
the array A contains the lower trapezoidal matrix, and the
strictly upper triangular part of A is not referenced.  Note
that when DIAG = 'U', the diagonal elements of A are not
referenced and are assumed to be one.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(M,1).
```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= M when NORM = 'I'; otherwise, WORK is not
referenced.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clapll (integer N, complex, dimension( * ) X, integer INCX, complex, dimension( * ) Y, integer INCY, real SSMIN)¶

CLAPLL measures the linear dependence of two vectors.

Purpose:

``` Given two column vectors X and Y, let
A = ( X Y ).
The subroutine first computes the QR factorization of A = Q*R,
and then computes the SVD of the 2-by-2 upper triangular matrix R.
The smaller singular value of R is returned in SSMIN, which is used
as the measurement of the linear dependency of the vectors X and Y.
```

Parameters:

N

```          N is INTEGER
The length of the vectors X and Y.
```

X

```          X is COMPLEX array, dimension (1+(N-1)*INCX)
On entry, X contains the N-vector X.
On exit, X is overwritten.
```

INCX

```          INCX is INTEGER
The increment between successive elements of X. INCX > 0.
```

Y

```          Y is COMPLEX array, dimension (1+(N-1)*INCY)
On entry, Y contains the N-vector Y.
On exit, Y is overwritten.
```

INCY

```          INCY is INTEGER
The increment between successive elements of Y. INCY > 0.
```

SSMIN

```          SSMIN is REAL
The smallest singular value of the N-by-2 matrix A = ( X Y ).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clapmr (logical FORWRD, integer M, integer N, complex, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)¶

CLAPMR rearranges rows of a matrix as specified by a permutation vector.

Purpose:

``` CLAPMR rearranges the rows of the M by N matrix X as specified
by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.
If FORWRD = .TRUE.,  forward permutation:
X(K(I),*) is moved X(I,*) for I = 1,2,...,M.
If FORWRD = .FALSE., backward permutation:
X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.
```

Parameters:

FORWRD

```          FORWRD is LOGICAL
= .TRUE., forward permutation
= .FALSE., backward permutation
```

M

```          M is INTEGER
The number of rows of the matrix X. M >= 0.
```

N

```          N is INTEGER
The number of columns of the matrix X. N >= 0.
```

X

```          X is COMPLEX array, dimension (LDX,N)
On entry, the M by N matrix X.
On exit, X contains the permuted matrix X.
```

LDX

```          LDX is INTEGER
The leading dimension of the array X, LDX >= MAX(1,M).
```

K

```          K is INTEGER array, dimension (M)
On entry, K contains the permutation vector. K is used as
internal workspace, but reset to its original value on
output.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clapmt (logical FORWRD, integer M, integer N, complex, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)¶

CLAPMT performs a forward or backward permutation of the columns of a matrix.

Purpose:

``` CLAPMT rearranges the columns of the M by N matrix X as specified
by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
If FORWRD = .TRUE.,  forward permutation:
X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.
If FORWRD = .FALSE., backward permutation:
X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.
```

Parameters:

FORWRD

```          FORWRD is LOGICAL
= .TRUE., forward permutation
= .FALSE., backward permutation
```

M

```          M is INTEGER
The number of rows of the matrix X. M >= 0.
```

N

```          N is INTEGER
The number of columns of the matrix X. N >= 0.
```

X

```          X is COMPLEX array, dimension (LDX,N)
On entry, the M by N matrix X.
On exit, X contains the permuted matrix X.
```

LDX

```          LDX is INTEGER
The leading dimension of the array X, LDX >= MAX(1,M).
```

K

```          K is INTEGER array, dimension (N)
On entry, K contains the permutation vector. K is used as
internal workspace, but reset to its original value on
output.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine claqhb (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)¶

CLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.

Purpose:

``` CLAQHB equilibrates an Hermitian band matrix A using the scaling
factors in the vector S.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

KD

```          KD is INTEGER
The number of super-diagonals of the matrix A if UPLO = 'U',
or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
```

AB

```          AB is COMPLEX array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array.  The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**H *U or A = L*L**H of the band
matrix A, in the same storage format as A.
```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.
```

S

```          S is REAL array, dimension (N)
The scale factors for A.
```

SCOND

```          SCOND is REAL
Ratio of the smallest S(i) to the largest S(i).
```

AMAX

```          AMAX is REAL
Absolute value of largest matrix entry.
```

EQUED

```          EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N':  No equilibration.
= 'Y':  Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
```

Internal Parameters:

```  THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors.  If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine claqhp (character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)¶

CLAQHP scales a Hermitian matrix stored in packed form.

Purpose:

``` CLAQHP equilibrates a Hermitian matrix A using the scaling factors
in the vector S.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

AP

```          AP is COMPLEX array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix
A, packed columnwise in a linear array.  The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
On exit, the equilibrated matrix:  diag(S) * A * diag(S), in
the same storage format as A.
```

S

```          S is REAL array, dimension (N)
The scale factors for A.
```

SCOND

```          SCOND is REAL
Ratio of the smallest S(i) to the largest S(i).
```

AMAX

```          AMAX is REAL
Absolute value of largest matrix entry.
```

EQUED

```          EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N':  No equilibration.
= 'Y':  Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
```

Internal Parameters:

```  THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors.  If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine claqp2 (integer M, integer N, integer OFFSET, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, complex, dimension( * ) TAU, real, dimension( * ) VN1, real, dimension( * ) VN2, complex, dimension( * ) WORK)¶

CLAQP2 computes a QR factorization with column pivoting of the matrix block.

Purpose:

``` CLAQP2 computes a QR factorization with column pivoting of
the block A(OFFSET+1:M,1:N).
The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
```

Parameters:

M

```          M is INTEGER
The number of rows of the matrix A. M >= 0.
```

N

```          N is INTEGER
The number of columns of the matrix A. N >= 0.
```

OFFSET

```          OFFSET is INTEGER
The number of rows of the matrix A that must be pivoted
but no factorized. OFFSET >= 0.
```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
the triangular factor obtained; the elements in block
A(OFFSET+1:M,1:N) below the diagonal, together with the
array TAU, represent the orthogonal matrix Q as a product of
elementary reflectors. Block A(1:OFFSET,1:N) has been
accordingly pivoted, but no factorized.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
```

JPVT

```          JPVT is INTEGER array, dimension (N)
On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
to the front of A*P (a leading column); if JPVT(i) = 0,
the i-th column of A is a free column.
On exit, if JPVT(i) = k, then the i-th column of A*P
was the k-th column of A.
```

TAU

```          TAU is COMPLEX array, dimension (min(M,N))
The scalar factors of the elementary reflectors.
```

VN1

```          VN1 is REAL array, dimension (N)
The vector with the partial column norms.
```

VN2

```          VN2 is REAL array, dimension (N)
The vector with the exact column norms.
```

WORK

```          WORK is COMPLEX array, dimension (N)
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

References:

LAPACK Working Note 176

## subroutine claqps (integer M, integer N, integer OFFSET, integer NB, integer KB, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, complex, dimension( * ) TAU, real, dimension( * ) VN1, real, dimension( * ) VN2, complex, dimension( * ) AUXV, complex, dimension( ldf, * ) F, integer LDF)¶

CLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.

Purpose:

``` CLAQPS computes a step of QR factorization with column pivoting
of a complex M-by-N matrix A by using Blas-3.  It tries to factorize
NB columns from A starting from the row OFFSET+1, and updates all
of the matrix with Blas-3 xGEMM.
In some cases, due to catastrophic cancellations, it cannot
factorize NB columns.  Hence, the actual number of factorized
columns is returned in KB.
Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
```

Parameters:

M

```          M is INTEGER
The number of rows of the matrix A. M >= 0.
```

N

```          N is INTEGER
The number of columns of the matrix A. N >= 0
```

OFFSET

```          OFFSET is INTEGER
The number of rows of A that have been factorized in
previous steps.
```

NB

```          NB is INTEGER
The number of columns to factorize.
```

KB

```          KB is INTEGER
The number of columns actually factorized.
```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, block A(OFFSET+1:M,1:KB) is the triangular
factor obtained and block A(1:OFFSET,1:N) has been
accordingly pivoted, but no factorized.
The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
been updated.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
```

JPVT

```          JPVT is INTEGER array, dimension (N)
JPVT(I) = K <==> Column K of the full matrix A has been
permuted into position I in AP.
```

TAU

```          TAU is COMPLEX array, dimension (KB)
The scalar factors of the elementary reflectors.
```

VN1

```          VN1 is REAL array, dimension (N)
The vector with the partial column norms.
```

VN2

```          VN2 is REAL array, dimension (N)
The vector with the exact column norms.
```

AUXV

```          AUXV is COMPLEX array, dimension (NB)
Auxiliar vector.
```

F

```          F is COMPLEX array, dimension (LDF,NB)
Matrix  F**H = L * Y**H * A.
```

LDF

```          LDF is INTEGER
The leading dimension of the array F. LDF >= max(1,N).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA

Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

References:

LAPACK Working Note 176

## subroutine claqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, integer INFO)¶

CLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

```    CLAQR0 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**H, where T is an upper triangular matrix (the
Schur form), and Z is the unitary matrix of Schur vectors.
Optionally Z may be postmultiplied into an input unitary
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
```

Parameters:

WANTT

```          WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.
```

WANTZ

```          WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.
```

N

```          N is INTEGER
The order of the matrix H.  N .GE. 0.
```

ILO

```          ILO is INTEGER
```

IHI

```          IHI is INTEGER
It is assumed that H is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
previous call to CGEBAL, and then passed to CGEHRD when the
matrix output by CGEBAL is reduced to Hessenberg form.
Otherwise, ILO and IHI should be set to 1 and N,
respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
If N = 0, then ILO = 1 and IHI = 0.
```

H

```          H is COMPLEX array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and WANTT is .TRUE., then H
contains the upper triangular matrix T from the Schur
decomposition (the Schur form). If INFO = 0 and WANT is
.FALSE., then the contents of H are unspecified on exit.
(The output value of H when INFO.GT.0 is given under the
description of INFO below.)
This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
```

LDH

```          LDH is INTEGER
The leading dimension of the array H. LDH .GE. max(1,N).
```

W

```          W is COMPLEX array, dimension (N)
The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
stored in the same order as on the diagonal of the Schur
form returned in H, with W(i) = H(i,i).
```

ILOZ

```          ILOZ is INTEGER
```

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
```

Z

```          Z is COMPLEX array, dimension (LDZ,IHI)
If WANTZ is .FALSE., then Z is not referenced.
If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
(The output value of Z when INFO.GT.0 is given under
the description of INFO below.)
```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z.  if WANTZ is .TRUE.
then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
```

WORK

```          WORK is COMPLEX array, dimension LWORK
On exit, if LWORK = -1, WORK(1) returns an estimate of
the optimal value for LWORK.
```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK .GE. max(1,N)
is sufficient, but LWORK typically as large as 6*N may
be required for optimal performance.  A workspace query
to determine the optimal workspace size is recommended.
If LWORK = -1, then CLAQR0 does a workspace query.
In this case, CLAQR0 checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI.  The estimate is returned
in WORK(1).  No error message related to LWORK is
issued by XERBLA.  Neither H nor Z are accessed.
```

INFO

```          INFO is INTEGER
=  0:  successful exit
.GT. 0:  if INFO = i, CLAQR0 failed to compute all of
the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
and WI contain those eigenvalues which have been
successfully computed.  (Failures are rare.)
If INFO .GT. 0 and WANT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.
If INFO .GT. 0 and WANTT is .TRUE., then on exit
(*)  (initial value of H)*U  = U*(final value of H)
where U is a unitary matrix.  The final
value of  H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.
If INFO .GT. 0 and WANTZ is .TRUE., then on exit
(final value of Z(ILO:IHI,ILOZ:IHIZ)
=  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
where U is the unitary matrix in (*) (regard-
less of the value of WANTT.)
If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
accessed.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

References:

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

## subroutine claqr1 (integer N, complex, dimension( ldh, * ) H, integer LDH, complex S1, complex S2, complex, dimension( * ) V)¶

CLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.

Purpose:

```      Given a 2-by-2 or 3-by-3 matrix H, CLAQR1 sets v to a
scalar multiple of the first column of the product
(*)  K = (H - s1*I)*(H - s2*I)
scaling to avoid overflows and most underflows.
This is useful for starting double implicit shift bulges
in the QR algorithm.
```

Parameters:

N

```          N is INTEGER
Order of the matrix H. N must be either 2 or 3.
```

H

```          H is COMPLEX array, dimension (LDH,N)
The 2-by-2 or 3-by-3 matrix H in (*).
```

LDH

```          LDH is INTEGER
The leading dimension of H as declared in
the calling procedure.  LDH.GE.N
```

S1

```          S1 is COMPLEX
```

S2

```          S2 is COMPLEX
S1 and S2 are the shifts defining K in (*) above.
```

V

```          V is COMPLEX array, dimension (N)
A scalar multiple of the first column of the
matrix K in (*).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

## subroutine claqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, complex, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex, dimension( * ) SH, complex, dimension( ldv, * ) V, integer LDV, integer NH, complex, dimension( ldt, * ) T, integer LDT, integer NV, complex, dimension( ldwv, * ) WV, integer LDWV, complex, dimension( * ) WORK, integer LWORK)¶

CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

Purpose:

```    CLAQR2 is identical to CLAQR3 except that it avoids
recursion by calling CLAHQR instead of CLAQR4.
Aggressive early deflation:
This subroutine accepts as input an upper Hessenberg matrix
H and performs an unitary similarity transformation
designed to detect and deflate fully converged eigenvalues from
a trailing principal submatrix.  On output H has been over-
written by a new Hessenberg matrix that is a perturbation of
an unitary similarity transformation of H.  It is to be
hoped that the final version of H has many zero subdiagonal
entries.
```

Parameters:

WANTT

```          WANTT is LOGICAL
If .TRUE., then the Hessenberg matrix H is fully updated
so that the triangular Schur factor may be
computed (in cooperation with the calling subroutine).
If .FALSE., then only enough of H is updated to preserve
the eigenvalues.
```

WANTZ

```          WANTZ is LOGICAL
If .TRUE., then the unitary matrix Z is updated so
so that the unitary Schur factor may be computed
(in cooperation with the calling subroutine).
If .FALSE., then Z is not referenced.
```

N

```          N is INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the unitary matrix Z.
```

KTOP

```          KTOP is INTEGER
It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
KBOT and KTOP together determine an isolated block
along the diagonal of the Hessenberg matrix.
```

KBOT

```          KBOT is INTEGER
It is assumed without a check that either
KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
determine an isolated block along the diagonal of the
Hessenberg matrix.
```

NW

```          NW is INTEGER
Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).
```

H

```          H is COMPLEX array, dimension (LDH,N)
On input the initial N-by-N section of H stores the
Hessenberg matrix undergoing aggressive early deflation.
On output H has been transformed by a unitary
similarity transformation, perturbed, and the returned
to Hessenberg form that (it is to be hoped) has some
zero subdiagonal entries.
```

LDH

```          LDH is INTEGER
Leading dimension of H just as declared in the calling
subroutine.  N .LE. LDH
```

ILOZ

```          ILOZ is INTEGER
```

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
```

Z

```          Z is COMPLEX array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the unitary
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
```

LDZ

```          LDZ is INTEGER
The leading dimension of Z just as declared in the
calling subroutine.  1 .LE. LDZ.
```

NS

```          NS is INTEGER
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
```

ND

```          ND is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.
```

SH

```          SH is COMPLEX array, dimension (KBOT)
On output, approximate eigenvalues that may
be used for shifts are stored in SH(KBOT-ND-NS+1)
through SR(KBOT-ND).  Converged eigenvalues are
stored in SH(KBOT-ND+1) through SH(KBOT).
```

V

```          V is COMPLEX array, dimension (LDV,NW)
An NW-by-NW work array.
```

LDV

```          LDV is INTEGER
The leading dimension of V just as declared in the
calling subroutine.  NW .LE. LDV
```

NH

```          NH is INTEGER
The number of columns of T.  NH.GE.NW.
```

T

```          T is COMPLEX array, dimension (LDT,NW)
```

LDT

```          LDT is INTEGER
The leading dimension of T just as declared in the
calling subroutine.  NW .LE. LDT
```

NV

```          NV is INTEGER
The number of rows of work array WV available for
workspace.  NV.GE.NW.
```

WV

```          WV is COMPLEX array, dimension (LDWV,NW)
```

LDWV

```          LDWV is INTEGER
The leading dimension of W just as declared in the
calling subroutine.  NW .LE. LDV
```

WORK

```          WORK is COMPLEX array, dimension (LWORK)
On exit, WORK(1) is set to an estimate of the optimal value
of LWORK for the given values of N, NW, KTOP and KBOT.
```

LWORK

```          LWORK is INTEGER
The dimension of the work array WORK.  LWORK = 2*NW
suffices, but greater efficiency may result from larger
values of LWORK.
If LWORK = -1, then a workspace query is assumed; CLAQR2
only estimates the optimal workspace size for the given
values of N, NW, KTOP and KBOT.  The estimate is returned
in WORK(1).  No error message related to LWORK is issued
by XERBLA.  Neither H nor Z are accessed.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

## subroutine claqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, complex, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex, dimension( * ) SH, complex, dimension( ldv, * ) V, integer LDV, integer NH, complex, dimension( ldt, * ) T, integer LDT, integer NV, complex, dimension( ldwv, * ) WV, integer LDWV, complex, dimension( * ) WORK, integer LWORK)¶

CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

Purpose:

```    Aggressive early deflation:
CLAQR3 accepts as input an upper Hessenberg matrix
H and performs an unitary similarity transformation
designed to detect and deflate fully converged eigenvalues from
a trailing principal submatrix.  On output H has been over-
written by a new Hessenberg matrix that is a perturbation of
an unitary similarity transformation of H.  It is to be
hoped that the final version of H has many zero subdiagonal
entries.
```

Parameters:

WANTT

```          WANTT is LOGICAL
If .TRUE., then the Hessenberg matrix H is fully updated
so that the triangular Schur factor may be
computed (in cooperation with the calling subroutine).
If .FALSE., then only enough of H is updated to preserve
the eigenvalues.
```

WANTZ

```          WANTZ is LOGICAL
If .TRUE., then the unitary matrix Z is updated so
so that the unitary Schur factor may be computed
(in cooperation with the calling subroutine).
If .FALSE., then Z is not referenced.
```

N

```          N is INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the unitary matrix Z.
```

KTOP

```          KTOP is INTEGER
It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
KBOT and KTOP together determine an isolated block
along the diagonal of the Hessenberg matrix.
```

KBOT

```          KBOT is INTEGER
It is assumed without a check that either
KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
determine an isolated block along the diagonal of the
Hessenberg matrix.
```

NW

```          NW is INTEGER
Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).
```

H

```          H is COMPLEX array, dimension (LDH,N)
On input the initial N-by-N section of H stores the
Hessenberg matrix undergoing aggressive early deflation.
On output H has been transformed by a unitary
similarity transformation, perturbed, and the returned
to Hessenberg form that (it is to be hoped) has some
zero subdiagonal entries.
```

LDH

```          LDH is INTEGER
Leading dimension of H just as declared in the calling
subroutine.  N .LE. LDH
```

ILOZ

```          ILOZ is INTEGER
```

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
```

Z

```          Z is COMPLEX array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the unitary
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
```

LDZ

```          LDZ is INTEGER
The leading dimension of Z just as declared in the
calling subroutine.  1 .LE. LDZ.
```

NS

```          NS is INTEGER
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
```

ND

```          ND is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.
```

SH

```          SH is COMPLEX array, dimension (KBOT)
On output, approximate eigenvalues that may
be used for shifts are stored in SH(KBOT-ND-NS+1)
through SR(KBOT-ND).  Converged eigenvalues are
stored in SH(KBOT-ND+1) through SH(KBOT).
```

V

```          V is COMPLEX array, dimension (LDV,NW)
An NW-by-NW work array.
```

LDV

```          LDV is INTEGER
The leading dimension of V just as declared in the
calling subroutine.  NW .LE. LDV
```

NH

```          NH is INTEGER
The number of columns of T.  NH.GE.NW.
```

T

```          T is COMPLEX array, dimension (LDT,NW)
```

LDT

```          LDT is INTEGER
The leading dimension of T just as declared in the
calling subroutine.  NW .LE. LDT
```

NV

```          NV is INTEGER
The number of rows of work array WV available for
workspace.  NV.GE.NW.
```

WV

```          WV is COMPLEX array, dimension (LDWV,NW)
```

LDWV

```          LDWV is INTEGER
The leading dimension of W just as declared in the
calling subroutine.  NW .LE. LDV
```

WORK

```          WORK is COMPLEX array, dimension (LWORK)
On exit, WORK(1) is set to an estimate of the optimal value
of LWORK for the given values of N, NW, KTOP and KBOT.
```

LWORK

```          LWORK is INTEGER
The dimension of the work array WORK.  LWORK = 2*NW
suffices, but greater efficiency may result from larger
values of LWORK.
If LWORK = -1, then a workspace query is assumed; CLAQR3
only estimates the optimal workspace size for the given
values of N, NW, KTOP and KBOT.  The estimate is returned
in WORK(1).  No error message related to LWORK is issued
by XERBLA.  Neither H nor Z are accessed.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

## subroutine claqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, integer INFO)¶

CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

```    CLAQR4 implements one level of recursion for CLAQR0.
It is a complete implementation of the small bulge multi-shift
QR algorithm.  It may be called by CLAQR0 and, for large enough
deflation window size, it may be called by CLAQR3.  This
subroutine is identical to CLAQR0 except that it calls CLAQR2
instead of CLAQR3.
CLAQR4 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**H, where T is an upper triangular matrix (the
Schur form), and Z is the unitary matrix of Schur vectors.
Optionally Z may be postmultiplied into an input unitary
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
```

Parameters:

WANTT

```          WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.
```

WANTZ

```          WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.
```

N

```          N is INTEGER
The order of the matrix H.  N .GE. 0.
```

ILO

```          ILO is INTEGER
```

IHI

```          IHI is INTEGER
It is assumed that H is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
previous call to CGEBAL, and then passed to CGEHRD when the
matrix output by CGEBAL is reduced to Hessenberg form.
Otherwise, ILO and IHI should be set to 1 and N,
respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
If N = 0, then ILO = 1 and IHI = 0.
```

H

```          H is COMPLEX array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and WANTT is .TRUE., then H
contains the upper triangular matrix T from the Schur
decomposition (the Schur form). If INFO = 0 and WANT is
.FALSE., then the contents of H are unspecified on exit.
(The output value of H when INFO.GT.0 is given under the
description of INFO below.)
This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
```

LDH

```          LDH is INTEGER
The leading dimension of the array H. LDH .GE. max(1,N).
```

W

```          W is COMPLEX array, dimension (N)
The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
stored in the same order as on the diagonal of the Schur
form returned in H, with W(i) = H(i,i).
```

ILOZ

```          ILOZ is INTEGER
```

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
```

Z

```          Z is COMPLEX array, dimension (LDZ,IHI)
If WANTZ is .FALSE., then Z is not referenced.
If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
(The output value of Z when INFO.GT.0 is given under
the description of INFO below.)
```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z.  if WANTZ is .TRUE.
then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
```

WORK

```          WORK is COMPLEX array, dimension LWORK
On exit, if LWORK = -1, WORK(1) returns an estimate of
the optimal value for LWORK.
```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK .GE. max(1,N)
is sufficient, but LWORK typically as large as 6*N may
be required for optimal performance.  A workspace query
to determine the optimal workspace size is recommended.
If LWORK = -1, then CLAQR4 does a workspace query.
In this case, CLAQR4 checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI.  The estimate is returned
in WORK(1).  No error message related to LWORK is
issued by XERBLA.  Neither H nor Z are accessed.
```

INFO

```          INFO is INTEGER
=  0:  successful exit
.GT. 0:  if INFO = i, CLAQR4 failed to compute all of
the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
and WI contain those eigenvalues which have been
successfully computed.  (Failures are rare.)
If INFO .GT. 0 and WANT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.
If INFO .GT. 0 and WANTT is .TRUE., then on exit
(*)  (initial value of H)*U  = U*(final value of H)
where U is a unitary matrix.  The final
value of  H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.
If INFO .GT. 0 and WANTZ is .TRUE., then on exit
(final value of Z(ILO:IHI,ILOZ:IHIZ)
=  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
where U is the unitary matrix in (*) (regard-
less of the value of WANTT.)
If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
accessed.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

References:

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

## subroutine claqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP, integer KBOT, integer NSHFTS, complex, dimension( * ) S, complex, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldu, * ) U, integer LDU, integer NV, complex, dimension( ldwv, * ) WV, integer LDWV, integer NH, complex, dimension( ldwh, * ) WH, integer LDWH)¶

CLAQR5 performs a single small-bulge multi-shift QR sweep.

Purpose:

```    CLAQR5 called by CLAQR0 performs a
single small-bulge multi-shift QR sweep.
```

Parameters:

WANTT

```          WANTT is LOGICAL
WANTT = .true. if the triangular Schur factor
is being computed.  WANTT is set to .false. otherwise.
```

WANTZ

```          WANTZ is LOGICAL
WANTZ = .true. if the unitary Schur factor is being
computed.  WANTZ is set to .false. otherwise.
```

KACC22

```          KACC22 is INTEGER with value 0, 1, or 2.
Specifies the computation mode of far-from-diagonal
orthogonal updates.
= 0: CLAQR5 does not accumulate reflections and does not
use matrix-matrix multiply to update far-from-diagonal
matrix entries.
= 1: CLAQR5 accumulates reflections and uses matrix-matrix
multiply to update the far-from-diagonal matrix entries.
= 2: CLAQR5 accumulates reflections, uses matrix-matrix
multiply to update the far-from-diagonal matrix entries,
and takes advantage of 2-by-2 block structure during
matrix multiplies.
```

N

```          N is INTEGER
N is the order of the Hessenberg matrix H upon which this
subroutine operates.
```

KTOP

```          KTOP is INTEGER
```

KBOT

```          KBOT is INTEGER
These are the first and last rows and columns of an
isolated diagonal block upon which the QR sweep is to be
applied. It is assumed without a check that
either KTOP = 1  or   H(KTOP,KTOP-1) = 0
and
either KBOT = N  or   H(KBOT+1,KBOT) = 0.
```

NSHFTS

```          NSHFTS is INTEGER
NSHFTS gives the number of simultaneous shifts.  NSHFTS
must be positive and even.
```

S

```          S is COMPLEX array, dimension (NSHFTS)
S contains the shifts of origin that define the multi-
shift QR sweep.  On output S may be reordered.
```

H

```          H is COMPLEX array, dimension (LDH,N)
On input H contains a Hessenberg matrix.  On output a
multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
to the isolated diagonal block in rows and columns KTOP
through KBOT.
```

LDH

```          LDH is INTEGER
LDH is the leading dimension of H just as declared in the
calling procedure.  LDH.GE.MAX(1,N).
```

ILOZ

```          ILOZ is INTEGER
```

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N
```

Z

```          Z is COMPLEX array, dimension (LDZ,IHIZ)
If WANTZ = .TRUE., then the QR Sweep unitary
similarity transformation is accumulated into
Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ = .FALSE., then Z is unreferenced.
```

LDZ

```          LDZ is INTEGER
LDA is the leading dimension of Z just as declared in
the calling procedure. LDZ.GE.N.
```

V

```          V is COMPLEX array, dimension (LDV,NSHFTS/2)
```

LDV

```          LDV is INTEGER
LDV is the leading dimension of V as declared in the
calling procedure.  LDV.GE.3.
```

U

```          U is COMPLEX array, dimension (LDU,3*NSHFTS-3)
```

LDU

```          LDU is INTEGER
LDU is the leading dimension of U just as declared in the
in the calling subroutine.  LDU.GE.3*NSHFTS-3.
```

NH

```          NH is INTEGER
NH is the number of columns in array WH available for
workspace. NH.GE.1.
```

WH

```          WH is COMPLEX array, dimension (LDWH,NH)
```

LDWH

```          LDWH is INTEGER
Leading dimension of WH just as declared in the
calling procedure.  LDWH.GE.3*NSHFTS-3.
```

NV

```          NV is INTEGER
NV is the number of rows in WV agailable for workspace.
NV.GE.1.
```

WV

```          WV is COMPLEX array, dimension (LDWV,3*NSHFTS-3)
```

LDWV

```          LDWV is INTEGER
LDWV is the leading dimension of WV as declared in the
in the calling subroutine.  LDWV.GE.NV.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

References:

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

## subroutine claqsb (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)¶

CLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.

Purpose:

``` CLAQSB equilibrates a symmetric band matrix A using the scaling
factors in the vector S.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

KD

```          KD is INTEGER
The number of super-diagonals of the matrix A if UPLO = 'U',
or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
```

AB

```          AB is COMPLEX array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array.  The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**H *U or A = L*L**H of the band
matrix A, in the same storage format as A.
```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.
```

S

```          S is REAL array, dimension (N)
The scale factors for A.
```

SCOND

```          SCOND is REAL
Ratio of the smallest S(i) to the largest S(i).
```

AMAX

```          AMAX is REAL
Absolute value of largest matrix entry.
```

EQUED

```          EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N':  No equilibration.
= 'Y':  Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
```

Internal Parameters:

```  THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors.  If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine claqsp (character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)¶

CLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.

Purpose:

``` CLAQSP equilibrates a symmetric matrix A using the scaling factors
in the vector S.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

AP

```          AP is COMPLEX array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array.  The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
On exit, the equilibrated matrix:  diag(S) * A * diag(S), in
the same storage format as A.
```

S

```          S is REAL array, dimension (N)
The scale factors for A.
```

SCOND

```          SCOND is REAL
Ratio of the smallest S(i) to the largest S(i).
```

AMAX

```          AMAX is REAL
Absolute value of largest matrix entry.
```

EQUED

```          EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N':  No equilibration.
= 'Y':  Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
```

Internal Parameters:

```  THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors.  If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clar1v (integer N, integer B1, integer BN, real LAMBDA, real, dimension( * ) D, real, dimension( * ) L, real, dimension( * ) LD, real, dimension( * ) LLD, real PIVMIN, real GAPTOL, complex, dimension( * ) Z, logical WANTNC, integer NEGCNT, real ZTZ, real MINGMA, integer R, integer, dimension( * ) ISUPPZ, real NRMINV, real RESID, real RQCORR, real, dimension( * ) WORK)¶

CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

Purpose:

``` CLAR1V computes the (scaled) r-th column of the inverse of
the sumbmatrix in rows B1 through BN of the tridiagonal matrix
L D L**T - sigma I. When sigma is close to an eigenvalue, the
computed vector is an accurate eigenvector. Usually, r corresponds
to the index where the eigenvector is largest in magnitude.
The following steps accomplish this computation :
(a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,
(b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
(c) Computation of the diagonal elements of the inverse of
L D L**T - sigma I by combining the above transforms, and choosing
r as the index where the diagonal of the inverse is (one of the)
largest in magnitude.
(d) Computation of the (scaled) r-th column of the inverse using the
twisted factorization obtained by combining the top part of the
the stationary and the bottom part of the progressive transform.
```

Parameters:

N

```          N is INTEGER
The order of the matrix L D L**T.
```

B1

```          B1 is INTEGER
First index of the submatrix of L D L**T.
```

BN

```          BN is INTEGER
Last index of the submatrix of L D L**T.
```

LAMBDA

```          LAMBDA is REAL
The shift. In order to compute an accurate eigenvector,
LAMBDA should be a good approximation to an eigenvalue
of L D L**T.
```

L

```          L is REAL array, dimension (N-1)
The (n-1) subdiagonal elements of the unit bidiagonal matrix
L, in elements 1 to N-1.
```

D

```          D is REAL array, dimension (N)
The n diagonal elements of the diagonal matrix D.
```

LD

```          LD is REAL array, dimension (N-1)
The n-1 elements L(i)*D(i).
```

LLD

```          LLD is REAL array, dimension (N-1)
The n-1 elements L(i)*L(i)*D(i).
```

PIVMIN

```          PIVMIN is REAL
The minimum pivot in the Sturm sequence.
```

GAPTOL

```          GAPTOL is REAL
Tolerance that indicates when eigenvector entries are negligible
w.r.t. their contribution to the residual.
```

Z

```          Z is COMPLEX array, dimension (N)
On input, all entries of Z must be set to 0.
On output, Z contains the (scaled) r-th column of the
inverse. The scaling is such that Z(R) equals 1.
```

WANTNC

```          WANTNC is LOGICAL
Specifies whether NEGCNT has to be computed.
```

NEGCNT

```          NEGCNT is INTEGER
If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.
```

ZTZ

```          ZTZ is REAL
The square of the 2-norm of Z.
```

MINGMA

```          MINGMA is REAL
The reciprocal of the largest (in magnitude) diagonal
element of the inverse of L D L**T - sigma I.
```

R

```          R is INTEGER
The twist index for the twisted factorization used to
compute Z.
On input, 0 <= R <= N. If R is input as 0, R is set to
the index where (L D L**T - sigma I)^{-1} is largest
in magnitude. If 1 <= R <= N, R is unchanged.
On output, R contains the twist index used to compute Z.
Ideally, R designates the position of the maximum entry in the
eigenvector.
```

ISUPPZ

```          ISUPPZ is INTEGER array, dimension (2)
The support of the vector in Z, i.e., the vector Z is
nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).
```

NRMINV

```          NRMINV is REAL
NRMINV = 1/SQRT( ZTZ )
```

RESID

```          RESID is REAL
The residual of the FP vector.
RESID = ABS( MINGMA )/SQRT( ZTZ )
```

RQCORR

```          RQCORR is REAL
The Rayleigh Quotient correction to LAMBDA.
RQCORR = MINGMA*TMP
```

WORK

```          WORK is REAL array, dimension (4*N)
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USA

## subroutine clar2v (integer N, complex, dimension( * ) X, complex, dimension( * ) Y, complex, dimension( * ) Z, integer INCX, real, dimension( * ) C, complex, dimension( * ) S, integer INCC)¶

CLAR2V applies a vector of plane rotations with real cosines and complex sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.

Purpose:

``` CLAR2V applies a vector of complex plane rotations with real cosines
from both sides to a sequence of 2-by-2 complex Hermitian matrices,
defined by the elements of the vectors x, y and z. For i = 1,2,...,n
(       x(i)  z(i) ) :=
( conjg(z(i)) y(i) )
(  c(i) conjg(s(i)) ) (       x(i)  z(i) ) ( c(i) -conjg(s(i)) )
( -s(i)       c(i)  ) ( conjg(z(i)) y(i) ) ( s(i)        c(i)  )
```

Parameters:

N

```          N is INTEGER
The number of plane rotations to be applied.
```

X

```          X is COMPLEX array, dimension (1+(N-1)*INCX)
The vector x; the elements of x are assumed to be real.
```

Y

```          Y is COMPLEX array, dimension (1+(N-1)*INCX)
The vector y; the elements of y are assumed to be real.
```

Z

```          Z is COMPLEX array, dimension (1+(N-1)*INCX)
The vector z.
```

INCX

```          INCX is INTEGER
The increment between elements of X, Y and Z. INCX > 0.
```

C

```          C is REAL array, dimension (1+(N-1)*INCC)
The cosines of the plane rotations.
```

S

```          S is COMPLEX array, dimension (1+(N-1)*INCC)
The sines of the plane rotations.
```

INCC

```          INCC is INTEGER
The increment between elements of C and S. INCC > 0.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clarcm (integer M, integer N, real, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, real, dimension( * ) RWORK)¶

CLARCM copies all or part of a real two-dimensional array to a complex array.

Purpose:

``` CLARCM performs a very simple matrix-matrix multiplication:
C := A * B,
where A is M by M and real; B is M by N and complex;
C is M by N and complex.
```

Parameters:

M

```          M is INTEGER
The number of rows of the matrix A and of the matrix C.
M >= 0.
```

N

```          N is INTEGER
The number of columns and rows of the matrix B and
the number of columns of the matrix C.
N >= 0.
```

A

```          A is REAL array, dimension (LDA, M)
On entry, A contains the M by M matrix A.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >=max(1,M).
```

B

```          B is COMPLEX array, dimension (LDB, N)
On entry, B contains the M by N matrix B.
```

LDB

```          LDB is INTEGER
The leading dimension of the array B. LDB >=max(1,M).
```

C

```          C is COMPLEX array, dimension (LDC, N)
On exit, C contains the M by N matrix C.
```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >=max(1,M).
```

RWORK

```          RWORK is REAL array, dimension (2*M*N)
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

## subroutine clarf (character SIDE, integer M, integer N, complex, dimension( * ) V, integer INCV, complex TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK)¶

CLARF applies an elementary reflector to a general rectangular matrix.

Purpose:

``` CLARF applies a complex elementary reflector H to a complex M-by-N
matrix C, from either the left or the right. H is represented in the
form
H = I - tau * v * v**H
where tau is a complex scalar and v is a complex vector.
If tau = 0, then H is taken to be the unit matrix.
To apply H**H (the conjugate transpose of H), supply conjg(tau) instead
tau.
```

Parameters:

SIDE

```          SIDE is CHARACTER*1
= 'L': form  H * C
= 'R': form  C * H
```

M

```          M is INTEGER
The number of rows of the matrix C.
```

N

```          N is INTEGER
The number of columns of the matrix C.
```

V

```          V is COMPLEX array, dimension
(1 + (M-1)*abs(INCV)) if SIDE = 'L'
or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
The vector v in the representation of H. V is not used if
TAU = 0.
```

INCV

```          INCV is INTEGER
The increment between elements of v. INCV <> 0.
```

TAU

```          TAU is COMPLEX
The value tau in the representation of H.
```

C

```          C is COMPLEX array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by the matrix H * C if SIDE = 'L',
or C * H if SIDE = 'R'.
```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
```

WORK

```          WORK is COMPLEX array, dimension
(N) if SIDE = 'L'
or (M) if SIDE = 'R'
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clarfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( ldwork, * ) WORK, integer LDWORK)¶

CLARFB applies a block reflector or its conjugate-transpose to a general rectangular matrix.

Purpose:

``` CLARFB applies a complex block reflector H or its transpose H**H to a
complex M-by-N matrix C, from either the left or the right.
```

Parameters:

SIDE

```          SIDE is CHARACTER*1
= 'L': apply H or H**H from the Left
= 'R': apply H or H**H from the Right
```

TRANS

```          TRANS is CHARACTER*1
= 'N': apply H (No transpose)
= 'C': apply H**H (Conjugate transpose)
```

DIRECT

```          DIRECT is CHARACTER*1
Indicates how H is formed from a product of elementary
reflectors
= 'F': H = H(1) H(2) . . . H(k) (Forward)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
```

STOREV

```          STOREV is CHARACTER*1
Indicates how the vectors which define the elementary
reflectors are stored:
= 'C': Columnwise
= 'R': Rowwise
```

M

```          M is INTEGER
The number of rows of the matrix C.
```

N

```          N is INTEGER
The number of columns of the matrix C.
```

K

```          K is INTEGER
The order of the matrix T (= the number of elementary
reflectors whose product defines the block reflector).
```

V

```          V is COMPLEX array, dimension
(LDV,K) if STOREV = 'C'
(LDV,M) if STOREV = 'R' and SIDE = 'L'
(LDV,N) if STOREV = 'R' and SIDE = 'R'
The matrix V. See Further Details.
```

LDV

```          LDV is INTEGER
The leading dimension of the array V.
If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
if STOREV = 'R', LDV >= K.
```

T

```          T is COMPLEX array, dimension (LDT,K)
The triangular K-by-K matrix T in the representation of the
block reflector.
```

LDT

```          LDT is INTEGER
The leading dimension of the array T. LDT >= K.
```

C

```          C is COMPLEX array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by H*C or H**H*C or C*H or C*H**H.
```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
```

WORK

```          WORK is COMPLEX array, dimension (LDWORK,K)
```

LDWORK

```          LDWORK is INTEGER
The leading dimension of the array WORK.
If SIDE = 'L', LDWORK >= max(1,N);
if SIDE = 'R', LDWORK >= max(1,M).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2013

Further Details:

```  The shape of the matrix V and the storage of the vectors which define
the H(i) is best illustrated by the following example with n = 5 and
k = 3. The elements equal to 1 are not stored; the corresponding
array elements are modified but restored on exit. The rest of the
array is not used.
DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
V = (  1       )                 V = (  1 v1 v1 v1 v1 )
( v1  1    )                     (     1 v2 v2 v2 )
( v1 v2  1 )                     (        1 v3 v3 )
( v1 v2 v3 )
( v1 v2 v3 )
DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
( v1 v2 v3 )                     ( v2 v2 v2  1    )
(  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
(     1 v3 )
(        1 )
```

## subroutine clarfg (integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex TAU)¶

CLARFG generates an elementary reflector (Householder matrix).

Purpose:

``` CLARFG generates a complex elementary reflector H of order n, such
that
H**H * ( alpha ) = ( beta ),   H**H * H = I.
(   x   )   (   0  )
where alpha and beta are scalars, with beta real, and x is an
(n-1)-element complex vector. H is represented in the form
H = I - tau * ( 1 ) * ( 1 v**H ) ,
( v )
where tau is a complex scalar and v is a complex (n-1)-element
vector. Note that H is not hermitian.
If the elements of x are all zero and alpha is real, then tau = 0
and H is taken to be the unit matrix.
Otherwise  1 <= real(tau) <= 2  and  abs(tau-1) <= 1 .
```

Parameters:

N

```          N is INTEGER
The order of the elementary reflector.
```

ALPHA

```          ALPHA is COMPLEX
On entry, the value alpha.
On exit, it is overwritten with the value beta.
```

X

```          X is COMPLEX array, dimension
(1+(N-2)*abs(INCX))
On entry, the vector x.
On exit, it is overwritten with the vector v.
```

INCX

```          INCX is INTEGER
The increment between elements of X. INCX > 0.
```

TAU

```          TAU is COMPLEX
The value tau.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2017

## subroutine clarfgp (integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex TAU)¶

CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

Purpose:

``` CLARFGP generates a complex elementary reflector H of order n, such
that
H**H * ( alpha ) = ( beta ),   H**H * H = I.
(   x   )   (   0  )
where alpha and beta are scalars, beta is real and non-negative, and
x is an (n-1)-element complex vector.  H is represented in the form
H = I - tau * ( 1 ) * ( 1 v**H ) ,
( v )
where tau is a complex scalar and v is a complex (n-1)-element
vector. Note that H is not hermitian.
If the elements of x are all zero and alpha is real, then tau = 0
and H is taken to be the unit matrix.
```

Parameters:

N

```          N is INTEGER
The order of the elementary reflector.
```

ALPHA

```          ALPHA is COMPLEX
On entry, the value alpha.
On exit, it is overwritten with the value beta.
```

X

```          X is COMPLEX array, dimension
(1+(N-2)*abs(INCX))
On entry, the vector x.
On exit, it is overwritten with the vector v.
```

INCX

```          INCX is INTEGER
The increment between elements of X. INCX > 0.
```

TAU

```          TAU is COMPLEX
The value tau.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2017

## subroutine clarft (character DIRECT, character STOREV, integer N, integer K, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( * ) TAU, complex, dimension( ldt, * ) T, integer LDT)¶

CLARFT forms the triangular factor T of a block reflector H = I - vtvH

Purpose:

``` CLARFT forms the triangular factor T of a complex block reflector H
of order n, which is defined as a product of k elementary reflectors.
If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
If STOREV = 'C', the vector which defines the elementary reflector
H(i) is stored in the i-th column of the array V, and
H  =  I - V * T * V**H
If STOREV = 'R', the vector which defines the elementary reflector
H(i) is stored in the i-th row of the array V, and
H  =  I - V**H * T * V
```

Parameters:

DIRECT

```          DIRECT is CHARACTER*1
Specifies the order in which the elementary reflectors are
multiplied to form the block reflector:
= 'F': H = H(1) H(2) . . . H(k) (Forward)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
```

STOREV

```          STOREV is CHARACTER*1
Specifies how the vectors which define the elementary
reflectors are stored (see also Further Details):
= 'C': columnwise
= 'R': rowwise
```

N

```          N is INTEGER
The order of the block reflector H. N >= 0.
```

K

```          K is INTEGER
The order of the triangular factor T (= the number of
elementary reflectors). K >= 1.
```

V

```          V is COMPLEX array, dimension
(LDV,K) if STOREV = 'C'
(LDV,N) if STOREV = 'R'
The matrix V. See further details.
```

LDV

```          LDV is INTEGER
The leading dimension of the array V.
If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
```

TAU

```          TAU is COMPLEX array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i).
```

T

```          T is COMPLEX array, dimension (LDT,K)
The k by k triangular factor T of the block reflector.
If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
lower triangular. The rest of the array is not used.
```

LDT

```          LDT is INTEGER
The leading dimension of the array T. LDT >= K.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  The shape of the matrix V and the storage of the vectors which define
the H(i) is best illustrated by the following example with n = 5 and
k = 3. The elements equal to 1 are not stored.
DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
V = (  1       )                 V = (  1 v1 v1 v1 v1 )
( v1  1    )                     (     1 v2 v2 v2 )
( v1 v2  1 )                     (        1 v3 v3 )
( v1 v2 v3 )
( v1 v2 v3 )
DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
( v1 v2 v3 )                     ( v2 v2 v2  1    )
(  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
(     1 v3 )
(        1 )
```

## subroutine clarfx (character SIDE, integer M, integer N, complex, dimension( * ) V, complex TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK)¶

CLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.

Purpose:

``` CLARFX applies a complex elementary reflector H to a complex m by n
matrix C, from either the left or the right. H is represented in the
form
H = I - tau * v * v**H
where tau is a complex scalar and v is a complex vector.
If tau = 0, then H is taken to be the unit matrix
This version uses inline code if H has order < 11.
```

Parameters:

SIDE

```          SIDE is CHARACTER*1
= 'L': form  H * C
= 'R': form  C * H
```

M

```          M is INTEGER
The number of rows of the matrix C.
```

N

```          N is INTEGER
The number of columns of the matrix C.
```

V

```          V is COMPLEX array, dimension (M) if SIDE = 'L'
or (N) if SIDE = 'R'
The vector v in the representation of H.
```

TAU

```          TAU is COMPLEX
The value tau in the representation of H.
```

C

```          C is COMPLEX array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by the matrix H * C if SIDE = 'L',
or C * H if SIDE = 'R'.
```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDA >= max(1,M).
```

WORK

```          WORK is COMPLEX array, dimension (N) if SIDE = 'L'
or (M) if SIDE = 'R'
WORK is not referenced if H has order < 11.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clargv (integer N, complex, dimension( * ) X, integer INCX, complex, dimension( * ) Y, integer INCY, real, dimension( * ) C, integer INCC)¶

CLARGV generates a vector of plane rotations with real cosines and complex sines.

Purpose:

``` CLARGV generates a vector of complex plane rotations with real
cosines, determined by elements of the complex vectors x and y.
For i = 1,2,...,n
(        c(i)   s(i) ) ( x(i) ) = ( r(i) )
( -conjg(s(i))  c(i) ) ( y(i) ) = (   0  )
where c(i)**2 + ABS(s(i))**2 = 1
The following conventions are used (these are the same as in CLARTG,
but differ from the BLAS1 routine CROTG):
If y(i)=0, then c(i)=1 and s(i)=0.
If x(i)=0, then c(i)=0 and s(i) is chosen so that r(i) is real.
```

Parameters:

N

```          N is INTEGER
The number of plane rotations to be generated.
```

X

```          X is COMPLEX array, dimension (1+(N-1)*INCX)
On entry, the vector x.
On exit, x(i) is overwritten by r(i), for i = 1,...,n.
```

INCX

```          INCX is INTEGER
The increment between elements of X. INCX > 0.
```

Y

```          Y is COMPLEX array, dimension (1+(N-1)*INCY)
On entry, the vector y.
On exit, the sines of the plane rotations.
```

INCY

```          INCY is INTEGER
The increment between elements of Y. INCY > 0.
```

C

```          C is REAL array, dimension (1+(N-1)*INCC)
The cosines of the plane rotations.
```

INCC

```          INCC is INTEGER
The increment between elements of C. INCC > 0.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  6-6-96 - Modified with a new algorithm by W. Kahan and J. Demmel
This version has a few statements commented out for thread safety
(machine parameters are computed on each entry). 10 feb 03, SJH.
```

## subroutine clarnv (integer IDIST, integer, dimension( 4 ) ISEED, integer N, complex, dimension( * ) X)¶

CLARNV returns a vector of random numbers from a uniform or normal distribution.

Purpose:

``` CLARNV returns a vector of n random complex numbers from a uniform or
normal distribution.
```

Parameters:

IDIST

```          IDIST is INTEGER
Specifies the distribution of the random numbers:
= 1:  real and imaginary parts each uniform (0,1)
= 2:  real and imaginary parts each uniform (-1,1)
= 3:  real and imaginary parts each normal (0,1)
= 4:  uniformly distributed on the disc abs(z) < 1
= 5:  uniformly distributed on the circle abs(z) = 1
```

ISEED

```          ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator; the array
elements must be between 0 and 4095, and ISEED(4) must be
odd.
On exit, the seed is updated.
```

N

```          N is INTEGER
The number of random numbers to be generated.
```

X

```          X is COMPLEX array, dimension (N)
The generated random numbers.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  This routine calls the auxiliary routine SLARUV to generate random
real numbers from a uniform (0,1) distribution, in batches of up to
128 using vectorisable code. The Box-Muller method is used to
transform numbers from a uniform to a normal distribution.
```

## subroutine clarrv (integer N, real VL, real VU, real, dimension( * ) D, real, dimension( * ) L, real PIVMIN, integer, dimension( * ) ISPLIT, integer M, integer DOL, integer DOU, real MINRGP, real RTOL1, real RTOL2, real, dimension( * ) W, real, dimension( * ) WERR, real, dimension( * ) WGAP, integer, dimension( * ) IBLOCK, integer, dimension( * ) INDEXW, real, dimension( * ) GERS, complex, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

Purpose:

``` CLARRV computes the eigenvectors of the tridiagonal matrix
T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
The input eigenvalues should have been computed by SLARRE.
```

Parameters:

N

```          N is INTEGER
The order of the matrix.  N >= 0.
```

VL

```          VL is REAL
Lower bound of the interval that contains the desired
eigenvalues. VL < VU. Needed to compute gaps on the left or right
end of the extremal eigenvalues in the desired RANGE.
```

VU

```          VU is REAL
Upper bound of the interval that contains the desired
eigenvalues. VL < VU. Needed to compute gaps on the left or right
end of the extremal eigenvalues in the desired RANGE.
```

D

```          D is REAL array, dimension (N)
On entry, the N diagonal elements of the diagonal matrix D.
On exit, D may be overwritten.
```

L

```          L is REAL array, dimension (N)
On entry, the (N-1) subdiagonal elements of the unit
bidiagonal matrix L are in elements 1 to N-1 of L
(if the matrix is not split.) At the end of each block
is stored the corresponding shift as given by SLARRE.
On exit, L is overwritten.
```

PIVMIN

```          PIVMIN is REAL
The minimum pivot allowed in the Sturm sequence.
```

ISPLIT

```          ISPLIT is INTEGER array, dimension (N)
The splitting points, at which T breaks up into blocks.
The first block consists of rows/columns 1 to
ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
through ISPLIT( 2 ), etc.
```

M

```          M is INTEGER
The total number of input eigenvalues.  0 <= M <= N.
```

DOL

```          DOL is INTEGER
```

DOU

```          DOU is INTEGER
If the user wants to compute only selected eigenvectors from all
the eigenvalues supplied, he can specify an index range DOL:DOU.
Or else the setting DOL=1, DOU=M should be applied.
Note that DOL and DOU refer to the order in which the eigenvalues
are stored in W.
If the user wants to compute only selected eigenpairs, then
the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
computed eigenvectors. All other columns of Z are set to zero.
```

MINRGP

```          MINRGP is REAL
```

RTOL1

```          RTOL1 is REAL
```

RTOL2

```          RTOL2 is REAL
Parameters for bisection.
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
```

W

```          W is REAL array, dimension (N)
The first M elements of W contain the APPROXIMATE eigenvalues for
which eigenvectors are to be computed.  The eigenvalues
should be grouped by split-off block and ordered from
smallest to largest within the block ( The output array
W from SLARRE is expected here ). Furthermore, they are with
respect to the shift of the corresponding root representation
for their block. On exit, W holds the eigenvalues of the
UNshifted matrix.
```

WERR

```          WERR is REAL array, dimension (N)
The first M elements contain the semiwidth of the uncertainty
interval of the corresponding eigenvalue in W
```

WGAP

```          WGAP is REAL array, dimension (N)
The separation from the right neighbor eigenvalue in W.
```

IBLOCK

```          IBLOCK is INTEGER array, dimension (N)
The indices of the blocks (submatrices) associated with the
corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
W(i) belongs to the first block from the top, =2 if W(i)
belongs to the second block, etc.
```

INDEXW

```          INDEXW is INTEGER array, dimension (N)
The indices of the eigenvalues within each block (submatrix);
for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.
```

GERS

```          GERS is REAL array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval
is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
be computed from the original UNshifted matrix.
```

Z

```          Z is COMPLEX array, dimension (LDZ, max(1,M) )
If INFO = 0, the first M columns of Z contain the
orthonormal eigenvectors of the matrix T
corresponding to the input eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z.
```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z.  LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).
```

ISUPPZ

```          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The I-th eigenvector
is nonzero only in elements ISUPPZ( 2*I-1 ) through
ISUPPZ( 2*I ).
```

WORK

```          WORK is REAL array, dimension (12*N)
```

IWORK

```          IWORK is INTEGER array, dimension (7*N)
```

INFO

```          INFO is INTEGER
= 0:  successful exit
> 0:  A problem occurred in CLARRV.
< 0:  One of the called subroutines signaled an internal problem.
Needs inspection of the corresponding parameter IINFO
for further information.
=-1:  Problem in SLARRB when refining a child's eigenvalues.
=-2:  Problem in SLARRF when computing the RRR of a child.
When a child is inside a tight cluster, it can be difficult
to find an RRR. A partial remedy from the user's point of
view is to make the parameter MINRGP smaller and recompile.
However, as the orthogonality of the computed vectors is
proportional to 1/MINRGP, the user should be aware that
he might be trading in precision when he decreases MINRGP.
=-3:  Problem in SLARRB when refining a single eigenvalue
after the Rayleigh correction was rejected.
= 5:  The Rayleigh Quotient Iteration failed to converge to
full accuracy in MAXITR steps.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Contributors:

Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USA

## subroutine clartg (complex F, complex G, real CS, complex SN, complex R)¶

CLARTG generates a plane rotation with real cosine and complex sine.

Purpose:

``` CLARTG generates a plane rotation so that
[  CS  SN  ]     [ F ]     [ R ]
[  __      ]  .  [   ]  =  [   ]   where CS**2 + |SN|**2 = 1.
[ -SN  CS  ]     [ G ]     [ 0 ]
This is a faster version of the BLAS1 routine CROTG, except for
the following differences:
F and G are unchanged on return.
If G=0, then CS=1 and SN=0.
If F=0, then CS=0 and SN is chosen so that R is real.
```

Parameters:

F

```          F is COMPLEX
The first component of vector to be rotated.
```

G

```          G is COMPLEX
The second component of vector to be rotated.
```

CS

```          CS is REAL
The cosine of the rotation.
```

SN

```          SN is COMPLEX
The sine of the rotation.
```

R

```          R is COMPLEX
The nonzero component of the rotated vector.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  3-5-96 - Modified with a new algorithm by W. Kahan and J. Demmel
This version has a few statements commented out for thread safety
(machine parameters are computed on each entry). 10 feb 03, SJH.
```

## subroutine clartv (integer N, complex, dimension( * ) X, integer INCX, complex, dimension( * ) Y, integer INCY, real, dimension( * ) C, complex, dimension( * ) S, integer INCC)¶

CLARTV applies a vector of plane rotations with real cosines and complex sines to the elements of a pair of vectors.

Purpose:

``` CLARTV applies a vector of complex plane rotations with real cosines
to elements of the complex vectors x and y. For i = 1,2,...,n
( x(i) ) := (        c(i)   s(i) ) ( x(i) )
( y(i) )    ( -conjg(s(i))  c(i) ) ( y(i) )
```

Parameters:

N

```          N is INTEGER
The number of plane rotations to be applied.
```

X

```          X is COMPLEX array, dimension (1+(N-1)*INCX)
The vector x.
```

INCX

```          INCX is INTEGER
The increment between elements of X. INCX > 0.
```

Y

```          Y is COMPLEX array, dimension (1+(N-1)*INCY)
The vector y.
```

INCY

```          INCY is INTEGER
The increment between elements of Y. INCY > 0.
```

C

```          C is REAL array, dimension (1+(N-1)*INCC)
The cosines of the plane rotations.
```

S

```          S is COMPLEX array, dimension (1+(N-1)*INCC)
The sines of the plane rotations.
```

INCC

```          INCC is INTEGER
The increment between elements of C and S. INCC > 0.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clascl (character TYPE, integer KL, integer KU, real CFROM, real CTO, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)¶

CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.

Purpose:

``` CLASCL multiplies the M by N complex matrix A by the real scalar
CTO/CFROM.  This is done without over/underflow as long as the final
result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
A may be full, upper triangular, lower triangular, upper Hessenberg,
or banded.
```

Parameters:

TYPE

```          TYPE is CHARACTER*1
TYPE indices the storage type of the input matrix.
= 'G':  A is a full matrix.
= 'L':  A is a lower triangular matrix.
= 'U':  A is an upper triangular matrix.
= 'H':  A is an upper Hessenberg matrix.
= 'B':  A is a symmetric band matrix with lower bandwidth KL
and upper bandwidth KU and with the only the lower
half stored.
= 'Q':  A is a symmetric band matrix with lower bandwidth KL
and upper bandwidth KU and with the only the upper
half stored.
= 'Z':  A is a band matrix with lower bandwidth KL and upper
bandwidth KU. See CGBTRF for storage details.
```

KL

```          KL is INTEGER
The lower bandwidth of A.  Referenced only if TYPE = 'B',
'Q' or 'Z'.
```

KU

```          KU is INTEGER
The upper bandwidth of A.  Referenced only if TYPE = 'B',
'Q' or 'Z'.
```

CFROM

```          CFROM is REAL
```

CTO

```          CTO is REAL
The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
without over/underflow if the final result CTO*A(I,J)/CFROM
can be represented without over/underflow.  CFROM must be
nonzero.
```

M

```          M is INTEGER
The number of rows of the matrix A.  M >= 0.
```

N

```          N is INTEGER
The number of columns of the matrix A.  N >= 0.
```

A

```          A is COMPLEX array, dimension (LDA,N)
The matrix to be multiplied by CTO/CFROM.  See TYPE for the
storage type.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.
If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M);
TYPE = 'B', LDA >= KL+1;
TYPE = 'Q', LDA >= KU+1;
TYPE = 'Z', LDA >= 2*KL+KU+1.
```

INFO

```          INFO is INTEGER
0  - successful exit
<0 - if INFO = -i, the i-th argument had an illegal value.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

## subroutine claset (character UPLO, integer M, integer N, complex ALPHA, complex BETA, complex, dimension( lda, * ) A, integer LDA)¶

CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.

Purpose:

``` CLASET initializes a 2-D array A to BETA on the diagonal and
ALPHA on the offdiagonals.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies the part of the matrix A to be set.
= 'U':      Upper triangular part is set. The lower triangle
is unchanged.
= 'L':      Lower triangular part is set. The upper triangle
is unchanged.
Otherwise:  All of the matrix A is set.
```

M

```          M is INTEGER
On entry, M specifies the number of rows of A.
```

N

```          N is INTEGER
On entry, N specifies the number of columns of A.
```

ALPHA

```          ALPHA is COMPLEX
All the offdiagonal array elements are set to ALPHA.
```

BETA

```          BETA is COMPLEX
All the diagonal array elements are set to BETA.
```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, A(i,j) = ALPHA, 1 <= i <= m, 1 <= j <= n, i.ne.j;
A(i,i) = BETA , 1 <= i <= min(m,n)
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clasr (character SIDE, character PIVOT, character DIRECT, integer M, integer N, real, dimension( * ) C, real, dimension( * ) S, complex, dimension( lda, * ) A, integer LDA)¶

CLASR applies a sequence of plane rotations to a general rectangular matrix.

Purpose:

``` CLASR applies a sequence of real plane rotations to a complex matrix
A, from either the left or the right.
When SIDE = 'L', the transformation takes the form
A := P*A
and when SIDE = 'R', the transformation takes the form
A := A*P**T
where P is an orthogonal matrix consisting of a sequence of z plane
rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
and P**T is the transpose of P.
When DIRECT = 'F' (Forward sequence), then
P = P(z-1) * ... * P(2) * P(1)
and when DIRECT = 'B' (Backward sequence), then
P = P(1) * P(2) * ... * P(z-1)
where P(k) is a plane rotation matrix defined by the 2-by-2 rotation
R(k) = (  c(k)  s(k) )
= ( -s(k)  c(k) ).
When PIVOT = 'V' (Variable pivot), the rotation is performed
for the plane (k,k+1), i.e., P(k) has the form
P(k) = (  1                                            )
(       ...                                     )
(              1                                )
(                   c(k)  s(k)                  )
(                  -s(k)  c(k)                  )
(                                1              )
(                                     ...       )
(                                            1  )
where R(k) appears as a rank-2 modification to the identity matrix in
rows and columns k and k+1.
When PIVOT = 'T' (Top pivot), the rotation is performed for the
plane (1,k+1), so P(k) has the form
P(k) = (  c(k)                    s(k)                 )
(         1                                     )
(              ...                              )
(                     1                         )
( -s(k)                    c(k)                 )
(                                 1             )
(                                      ...      )
(                                             1 )
where R(k) appears in rows and columns 1 and k+1.
Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
performed for the plane (k,z), giving P(k) the form
P(k) = ( 1                                             )
(      ...                                      )
(             1                                 )
(                  c(k)                    s(k) )
(                         1                     )
(                              ...              )
(                                     1         )
(                 -s(k)                    c(k) )
where R(k) appears in rows and columns k and z.  The rotations are
performed without ever forming P(k) explicitly.
```

Parameters:

SIDE

```          SIDE is CHARACTER*1
Specifies whether the plane rotation matrix P is applied to
A on the left or the right.
= 'L':  Left, compute A := P*A
= 'R':  Right, compute A:= A*P**T
```

PIVOT

```          PIVOT is CHARACTER*1
Specifies the plane for which P(k) is a plane rotation
matrix.
= 'V':  Variable pivot, the plane (k,k+1)
= 'T':  Top pivot, the plane (1,k+1)
= 'B':  Bottom pivot, the plane (k,z)
```

DIRECT

```          DIRECT is CHARACTER*1
Specifies whether P is a forward or backward sequence of
plane rotations.
= 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
= 'B':  Backward, P = P(1)*P(2)*...*P(z-1)
```

M

```          M is INTEGER
The number of rows of the matrix A.  If m <= 1, an immediate
return is effected.
```

N

```          N is INTEGER
The number of columns of the matrix A.  If n <= 1, an
immediate return is effected.
```

C

```          C is REAL array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The cosines c(k) of the plane rotations.
```

S

```          S is REAL array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The sines s(k) of the plane rotations.  The 2-by-2 plane
rotation part of the matrix P(k), R(k), has the form
R(k) = (  c(k)  s(k) )
( -s(k)  c(k) ).
```

A

```          A is COMPLEX array, dimension (LDA,N)
The M-by-N matrix A.  On exit, A is overwritten by P*A if
SIDE = 'R' or by A*P**T if SIDE = 'L'.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine classq (integer N, complex, dimension( * ) X, integer INCX, real SCALE, real SUMSQ)¶

CLASSQ updates a sum of squares represented in scaled form.

Purpose:

``` CLASSQ returns the values scl and ssq such that
( scl**2 )*ssq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq,
where x( i ) = abs( X( 1 + ( i - 1 )*INCX ) ). The value of sumsq is
assumed to be at least unity and the value of ssq will then satisfy
1.0 .le. ssq .le. ( sumsq + 2*n ).
scale is assumed to be non-negative and scl returns the value
scl = max( scale, abs( real( x( i ) ) ), abs( aimag( x( i ) ) ) ),
i
scale and sumsq must be supplied in SCALE and SUMSQ respectively.
SCALE and SUMSQ are overwritten by scl and ssq respectively.
The routine makes only one pass through the vector X.
```

Parameters:

N

```          N is INTEGER
The number of elements to be used from the vector X.
```

X

```          X is COMPLEX array, dimension (N)
The vector x as described above.
x( i )  = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.
```

INCX

```          INCX is INTEGER
The increment between successive values of the vector X.
INCX > 0.
```

SCALE

```          SCALE is REAL
On entry, the value  scale  in the equation above.
On exit, SCALE is overwritten with the value  scl .
```

SUMSQ

```          SUMSQ is REAL
On entry, the value  sumsq  in the equation above.
On exit, SUMSQ is overwritten with the value  ssq .
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine claswp (integer N, complex, dimension( lda, * ) A, integer LDA, integer K1, integer K2, integer, dimension( * ) IPIV, integer INCX)¶

CLASWP performs a series of row interchanges on a general rectangular matrix.

Purpose:

``` CLASWP performs a series of row interchanges on the matrix A.
One row interchange is initiated for each of rows K1 through K2 of A.
```

Parameters:

N

```          N is INTEGER
The number of columns of the matrix A.
```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the matrix of column dimension N to which the row
interchanges will be applied.
On exit, the permuted matrix.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.
```

K1

```          K1 is INTEGER
The first element of IPIV for which a row interchange will
be done.
```

K2

```          K2 is INTEGER
(K2-K1+1) is the number of elements of IPIV for which a row
interchange will be done.
```

IPIV

```          IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
The vector of pivot indices. Only the elements in positions
K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.
IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be
interchanged.
```

INCX

```          INCX is INTEGER
The increment between successive values of IPIV. If INCX
is negative, the pivots are applied in reverse order.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Further Details:

```  Modified by
R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA
```

## subroutine clatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO)¶

CLATBS solves a triangular banded system of equations.

Purpose:

``` CLATBS solves one of the triangular systems
A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,
with scaling to prevent overflow, where A is an upper or lower
triangular band matrix.  Here A**T denotes the transpose of A, x and b
are n-element vectors, and s is a scaling factor, usually less than
or equal to 1, chosen so that the components of x will be less than
the overflow threshold.  If the unscaled problem will not cause
overflow, the Level 2 BLAS routine CTBSV is called.  If the matrix A
is singular (A(j,j) = 0 for some j), then s is set to 0 and a
non-trivial solution to A*x = 0 is returned.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

TRANS

```          TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N':  Solve A * x = s*b     (No transpose)
= 'T':  Solve A**T * x = s*b  (Transpose)
= 'C':  Solve A**H * x = s*b  (Conjugate transpose)
```

DIAG

```          DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N':  Non-unit triangular
= 'U':  Unit triangular
```

NORMIN

```          NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y':  CNORM contains the column norms on entry
= 'N':  CNORM is not set on entry.  On exit, the norms will
be computed and stored in CNORM.
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

KD

```          KD is INTEGER
The number of subdiagonals or superdiagonals in the
triangular matrix A.  KD >= 0.
```

AB

```          AB is COMPLEX array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored in the
first KD+1 rows of the array. The j-th column of A is stored
in the j-th column of the array AB as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.
```

X

```          X is COMPLEX array, dimension (N)
On entry, the right hand side b of the triangular system.
On exit, X is overwritten by the solution vector x.
```

SCALE

```          SCALE is REAL
The scaling factor s for the triangular system
A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
If SCALE = 0, the matrix A is singular or badly scaled, and
the vector x is an exact or approximate solution to A*x = 0.
```

CNORM

```          CNORM is REAL array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -k, the k-th argument had an illegal value
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  A rough bound on x is computed; if that is less than overflow, CTBSV
is called, otherwise, specific code is used which checks for possible
overflow or divide-by-zero at every operation.
A columnwise scheme is used for solving A*x = b.  The basic algorithm
if A is lower triangular is
x[1:n] := b[1:n]
for j = 1, ..., n
x(j) := x(j) / A(j,j)
x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
end
Define bounds on the components of x after j iterations of the loop:
M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]
Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have
M(j+1) <= G(j) / | A(j+1,j+1) |
G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
<= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
where CNORM(j+1) is greater than or equal to the infinity-norm of
column j+1 of A, not counting the diagonal.  Hence
G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
1<=i<=j
and
|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
1<=i< j
Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTBSV if the
reciprocal of the largest M(j), j=1,..,n, is larger than
max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the
columnwise method can be performed without fear of overflow.  If
the computed bound is greater than a large constant, x is scaled to
prevent overflow, but if the bound overflows, x is set to 0, x(j) to
1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
Similarly, a row-wise scheme is used to solve A**T *x = b  or
A**H *x = b.  The basic algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
Then the bound on x(j) is
M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
1<=i<=j
and we can safely call CTBSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
```

## subroutine clatdf (integer IJOB, integer N, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) RHS, real RDSUM, real RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV)¶

CLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.

Purpose:

``` CLATDF computes the contribution to the reciprocal Dif-estimate
by solving for x in Z * x = b, where b is chosen such that the norm
of x is as large as possible. It is assumed that LU decomposition
of Z has been computed by CGETC2. On entry RHS = f holds the
contribution from earlier solved sub-systems, and on return RHS = x.
The factorization of Z returned by CGETC2 has the form
Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
triangular with unit diagonal elements and U is upper triangular.
```

Parameters:

IJOB

```          IJOB is INTEGER
IJOB = 2: First compute an approximative null-vector e
of Z using CGECON, e is normalized and solve for
Zx = +-e - f with the sign giving the greater value of
2-norm(x).  About 5 times as expensive as Default.
IJOB .ne. 2: Local look ahead strategy where
all entries of the r.h.s. b is chosen as either +1 or
-1.  Default.
```

N

```          N is INTEGER
The number of columns of the matrix Z.
```

Z

```          Z is COMPLEX array, dimension (LDZ, N)
On entry, the LU part of the factorization of the n-by-n
matrix Z computed by CGETC2:  Z = P * L * U * Q
```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z.  LDA >= max(1, N).
```

RHS

```          RHS is COMPLEX array, dimension (N).
On entry, RHS contains contributions from other subsystems.
On exit, RHS contains the solution of the subsystem with
entries according to the value of IJOB (see above).
```

RDSUM

```          RDSUM is REAL
On entry, the sum of squares of computed contributions to
the Dif-estimate under computation by CTGSYL, where the
scaling factor RDSCAL (see below) has been factored out.
On exit, the corresponding sum of squares updated with the
contributions from the current sub-system.
If TRANS = 'T' RDSUM is not touched.
NOTE: RDSUM only makes sense when CTGSY2 is called by CTGSYL.
```

RDSCAL

```          RDSCAL is REAL
On entry, scaling factor used to prevent overflow in RDSUM.
On exit, RDSCAL is updated w.r.t. the current contributions
in RDSUM.
If TRANS = 'T', RDSCAL is not touched.
NOTE: RDSCAL only makes sense when CTGSY2 is called by
CTGSYL.
```

IPIV

```          IPIV is INTEGER array, dimension (N).
The pivot indices; for 1 <= i <= N, row i of the
matrix has been interchanged with row IPIV(i).
```

JPIV

```          JPIV is INTEGER array, dimension (N).
The pivot indices; for 1 <= j <= N, column j of the
matrix has been interchanged with column JPIV(j).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Further Details:

This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

[1] Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

[2] Peter Poromaa, On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. Report UMINF-95.05, Department of Computing Science, Umea University, S-901 87 Umea, Sweden,

1995.

## subroutine clatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, complex, dimension( * ) AP, complex, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO)¶

CLATPS solves a triangular system of equations with the matrix held in packed storage.

Purpose:

``` CLATPS solves one of the triangular systems
A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,
with scaling to prevent overflow, where A is an upper or lower
triangular matrix stored in packed form.  Here A**T denotes the
transpose of A, A**H denotes the conjugate transpose of A, x and b
are n-element vectors, and s is a scaling factor, usually less than
or equal to 1, chosen so that the components of x will be less than
the overflow threshold.  If the unscaled problem will not cause
overflow, the Level 2 BLAS routine CTPSV is called. If the matrix A
is singular (A(j,j) = 0 for some j), then s is set to 0 and a
non-trivial solution to A*x = 0 is returned.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

TRANS

```          TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N':  Solve A * x = s*b     (No transpose)
= 'T':  Solve A**T * x = s*b  (Transpose)
= 'C':  Solve A**H * x = s*b  (Conjugate transpose)
```

DIAG

```          DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N':  Non-unit triangular
= 'U':  Unit triangular
```

NORMIN

```          NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y':  CNORM contains the column norms on entry
= 'N':  CNORM is not set on entry.  On exit, the norms will
be computed and stored in CNORM.
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

AP

```          AP is COMPLEX array, dimension (N*(N+1)/2)
The upper or lower triangular matrix A, packed columnwise in
a linear array.  The j-th column of A is stored in the array
AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
```

X

```          X is COMPLEX array, dimension (N)
On entry, the right hand side b of the triangular system.
On exit, X is overwritten by the solution vector x.
```

SCALE

```          SCALE is REAL
The scaling factor s for the triangular system
A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
If SCALE = 0, the matrix A is singular or badly scaled, and
the vector x is an exact or approximate solution to A*x = 0.
```

CNORM

```          CNORM is REAL array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -k, the k-th argument had an illegal value
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  A rough bound on x is computed; if that is less than overflow, CTPSV
is called, otherwise, specific code is used which checks for possible
overflow or divide-by-zero at every operation.
A columnwise scheme is used for solving A*x = b.  The basic algorithm
if A is lower triangular is
x[1:n] := b[1:n]
for j = 1, ..., n
x(j) := x(j) / A(j,j)
x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
end
Define bounds on the components of x after j iterations of the loop:
M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]
Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have
M(j+1) <= G(j) / | A(j+1,j+1) |
G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
<= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
where CNORM(j+1) is greater than or equal to the infinity-norm of
column j+1 of A, not counting the diagonal.  Hence
G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
1<=i<=j
and
|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
1<=i< j
Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTPSV if the
reciprocal of the largest M(j), j=1,..,n, is larger than
max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the
columnwise method can be performed without fear of overflow.  If
the computed bound is greater than a large constant, x is scaled to
prevent overflow, but if the bound overflows, x is set to 0, x(j) to
1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
Similarly, a row-wise scheme is used to solve A**T *x = b  or
A**H *x = b.  The basic algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
Then the bound on x(j) is
M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
1<=i<=j
and we can safely call CTPSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
```

## subroutine clatrd (character UPLO, integer N, integer NB, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) E, complex, dimension( * ) TAU, complex, dimension( ldw, * ) W, integer LDW)¶

CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Purpose:

``` CLATRD reduces NB rows and columns of a complex Hermitian matrix A to
Hermitian tridiagonal form by a unitary similarity
transformation Q**H * A * Q, and returns the matrices V and W which are
needed to apply the transformation to the unreduced part of A.
If UPLO = 'U', CLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by CHETRD.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
```

N

```          N is INTEGER
The order of the matrix A.
```

NB

```          NB is INTEGER
The number of rows and columns to be reduced.
```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced.  If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit:
if UPLO = 'U', the last NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements above the diagonal
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors;
if UPLO = 'L', the first NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements below the diagonal
with the array TAU, represent the  unitary matrix Q as a
product of elementary reflectors.
See Further Details.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).
```

E

```          E is REAL array, dimension (N-1)
If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
elements of the last NB columns of the reduced matrix;
if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
the first NB columns of the reduced matrix.
```

TAU

```          TAU is COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors, stored in
TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
See Further Details.
```

W

```          W is COMPLEX array, dimension (LDW,NB)
The n-by-nb matrix W required to update the unreduced part
of A.
```

LDW

```          LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n) H(n-1) . . . H(n-nb+1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
and tau in TAU(i-1).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(nb).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and tau in TAU(i).
The elements of the vectors v together form the n-by-nb matrix V
which is needed, with W, to apply the transformation to the unreduced
part of the matrix, using a Hermitian rank-2k update of the form:
A := A - V*W**H - W*V**H.
The contents of A on exit are illustrated by the following examples
with n = 5 and nb = 2:
if UPLO = 'U':                       if UPLO = 'L':
(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )
where d denotes a diagonal element of the reduced matrix, a denotes
an element of the original matrix that is unchanged, and vi denotes
an element of the vector defining H(i).
```

## subroutine clatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO)¶

CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Purpose:

``` CLATRS solves one of the triangular systems
A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,
with scaling to prevent overflow.  Here A is an upper or lower
triangular matrix, A**T denotes the transpose of A, A**H denotes the
conjugate transpose of A, x and b are n-element vectors, and s is a
scaling factor, usually less than or equal to 1, chosen so that the
components of x will be less than the overflow threshold.  If the
unscaled problem will not cause overflow, the Level 2 BLAS routine
CTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U':  Upper triangular
= 'L':  Lower triangular
```

TRANS

```          TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N':  Solve A * x = s*b     (No transpose)
= 'T':  Solve A**T * x = s*b  (Transpose)
= 'C':  Solve A**H * x = s*b  (Conjugate transpose)
```

DIAG

```          DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N':  Non-unit triangular
= 'U':  Unit triangular
```

NORMIN

```          NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y':  CNORM contains the column norms on entry
= 'N':  CNORM is not set on entry.  On exit, the norms will
be computed and stored in CNORM.
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

A

```          A is COMPLEX array, dimension (LDA,N)
The triangular matrix A.  If UPLO = 'U', the leading n by n
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced.  If UPLO = 'L', the leading n by n lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced.  If DIAG = 'U', the diagonal elements of A are
also not referenced and are assumed to be 1.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max (1,N).
```

X

```          X is COMPLEX array, dimension (N)
On entry, the right hand side b of the triangular system.
On exit, X is overwritten by the solution vector x.
```

SCALE

```          SCALE is REAL
The scaling factor s for the triangular system
A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
If SCALE = 0, the matrix A is singular or badly scaled, and
the vector x is an exact or approximate solution to A*x = 0.
```

CNORM

```          CNORM is REAL array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -k, the k-th argument had an illegal value
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  A rough bound on x is computed; if that is less than overflow, CTRSV
is called, otherwise, specific code is used which checks for possible
overflow or divide-by-zero at every operation.
A columnwise scheme is used for solving A*x = b.  The basic algorithm
if A is lower triangular is
x[1:n] := b[1:n]
for j = 1, ..., n
x(j) := x(j) / A(j,j)
x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
end
Define bounds on the components of x after j iterations of the loop:
M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]
Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have
M(j+1) <= G(j) / | A(j+1,j+1) |
G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
<= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
where CNORM(j+1) is greater than or equal to the infinity-norm of
column j+1 of A, not counting the diagonal.  Hence
G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
1<=i<=j
and
|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
1<=i< j
Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTRSV if the
reciprocal of the largest M(j), j=1,..,n, is larger than
max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the
columnwise method can be performed without fear of overflow.  If
the computed bound is greater than a large constant, x is scaled to
prevent overflow, but if the bound overflows, x is set to 0, x(j) to
1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
Similarly, a row-wise scheme is used to solve A**T *x = b  or
A**H *x = b.  The basic algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
Then the bound on x(j) is
M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
1<=i<=j
and we can safely call CTRSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
```

## subroutine clauu2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)¶

CLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).

Purpose:

``` CLAUU2 computes the product U * U**H or L**H * L, where the triangular
factor U or L is stored in the upper or lower triangular part of
the array A.
If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
overwriting the factor U in A.
If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
overwriting the factor L in A.
This is the unblocked form of the algorithm, calling Level 2 BLAS.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the triangular factor stored in the array A
is upper or lower triangular:
= 'U':  Upper triangular
= 'L':  Lower triangular
```

N

```          N is INTEGER
The order of the triangular factor U or L.  N >= 0.
```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the triangular factor U or L.
On exit, if UPLO = 'U', the upper triangle of A is
overwritten with the upper triangle of the product U * U**H;
if UPLO = 'L', the lower triangle of A is overwritten with
the lower triangle of the product L**H * L.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).
```

INFO

```          INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine clauum (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)¶

CLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).

Purpose:

``` CLAUUM computes the product U * U**H or L**H * L, where the triangular
factor U or L is stored in the upper or lower triangular part of
the array A.
If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
overwriting the factor U in A.
If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
overwriting the factor L in A.
This is the blocked form of the algorithm, calling Level 3 BLAS.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the triangular factor stored in the array A
is upper or lower triangular:
= 'U':  Upper triangular
= 'L':  Lower triangular
```

N

```          N is INTEGER
The order of the triangular factor U or L.  N >= 0.
```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the triangular factor U or L.
On exit, if UPLO = 'U', the upper triangle of A is
overwritten with the upper triangle of the product U * U**H;
if UPLO = 'L', the lower triangle of A is overwritten with
the lower triangle of the product L**H * L.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).
```

INFO

```          INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine crot (integer N, complex, dimension( * ) CX, integer INCX, complex, dimension( * ) CY, integer INCY, real C, complex S)¶

CROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors.

Purpose:

``` CROT   applies a plane rotation, where the cos (C) is real and the
sin (S) is complex, and the vectors CX and CY are complex.
```

Parameters:

N

```          N is INTEGER
The number of elements in the vectors CX and CY.
```

CX

```          CX is COMPLEX array, dimension (N)
On input, the vector X.
On output, CX is overwritten with C*X + S*Y.
```

INCX

```          INCX is INTEGER
The increment between successive values of CY.  INCX <> 0.
```

CY

```          CY is COMPLEX array, dimension (N)
On input, the vector Y.
On output, CY is overwritten with -CONJG(S)*X + C*Y.
```

INCY

```          INCY is INTEGER
The increment between successive values of CY.  INCX <> 0.
```

C

```          C is REAL
```

S

```          S is COMPLEX
C and S define a rotation
[  C          S  ]
[ -conjg(S)   C  ]
where C*C + S*CONJG(S) = 1.0.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine cspmv (character UPLO, integer N, complex ALPHA, complex, dimension( * ) AP, complex, dimension( * ) X, integer INCX, complex BETA, complex, dimension( * ) Y, integer INCY)¶

CSPMV computes a matrix-vector product for complex vectors using a complex symmetric packed matrix

Purpose:

``` CSPMV  performs the matrix-vector operation
y := alpha*A*x + beta*y,
where alpha and beta are scalars, x and y are n element vectors and
A is an n by n symmetric matrix, supplied in packed form.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the matrix A is supplied in the packed
array AP as follows:
UPLO = 'U' or 'u'   The upper triangular part of A is
supplied in AP.
UPLO = 'L' or 'l'   The lower triangular part of A is
supplied in AP.
Unchanged on exit.
```

N

```          N is INTEGER
On entry, N specifies the order of the matrix A.
N must be at least zero.
Unchanged on exit.
```

ALPHA

```          ALPHA is COMPLEX
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
```

AP

```          AP is COMPLEX array, dimension at least
( ( N*( N + 1 ) )/2 ).
Before entry, with UPLO = 'U' or 'u', the array AP must
contain the upper triangular part of the symmetric matrix
packed sequentially, column by column, so that AP( 1 )
contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
and a( 2, 2 ) respectively, and so on.
Before entry, with UPLO = 'L' or 'l', the array AP must
contain the lower triangular part of the symmetric matrix
packed sequentially, column by column, so that AP( 1 )
contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
and a( 3, 1 ) respectively, and so on.
Unchanged on exit.
```

X

```          X is COMPLEX array, dimension at least
( 1 + ( N - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the N-
element vector x.
Unchanged on exit.
```

INCX

```          INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
```

BETA

```          BETA is COMPLEX
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.
Unchanged on exit.
```

Y

```          Y is COMPLEX array, dimension at least
( 1 + ( N - 1 )*abs( INCY ) ).
Before entry, the incremented array Y must contain the n
element vector y. On exit, Y is overwritten by the updated
vector y.
```

INCY

```          INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
Unchanged on exit.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine cspr (character UPLO, integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex, dimension( * ) AP)¶

CSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.

Purpose:

``` CSPR    performs the symmetric rank 1 operation
A := alpha*x*x**H + A,
where alpha is a complex scalar, x is an n element vector and A is an
n by n symmetric matrix, supplied in packed form.
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the matrix A is supplied in the packed
array AP as follows:
UPLO = 'U' or 'u'   The upper triangular part of A is
supplied in AP.
UPLO = 'L' or 'l'   The lower triangular part of A is
supplied in AP.
Unchanged on exit.
```

N

```          N is INTEGER
On entry, N specifies the order of the matrix A.
N must be at least zero.
Unchanged on exit.
```

ALPHA

```          ALPHA is COMPLEX
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
```

X

```          X is COMPLEX array, dimension at least
( 1 + ( N - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the N-
element vector x.
Unchanged on exit.
```

INCX

```          INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
```

AP

```          AP is COMPLEX array, dimension at least
( ( N*( N + 1 ) )/2 ).
Before entry, with  UPLO = 'U' or 'u', the array AP must
contain the upper triangular part of the symmetric matrix
packed sequentially, column by column, so that AP( 1 )
contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
and a( 2, 2 ) respectively, and so on. On exit, the array
AP is overwritten by the upper triangular part of the
updated matrix.
Before entry, with UPLO = 'L' or 'l', the array AP must
contain the lower triangular part of the symmetric matrix
packed sequentially, column by column, so that AP( 1 )
contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
and a( 3, 1 ) respectively, and so on. On exit, the array
AP is overwritten by the lower triangular part of the
updated matrix.
Note that the imaginary parts of the diagonal elements need
not be set, they are assumed to be zero, and on exit they
are set to zero.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine csrscl (integer N, real SA, complex, dimension( * ) SX, integer INCX)¶

CSRSCL multiplies a vector by the reciprocal of a real scalar.

Purpose:

``` CSRSCL multiplies an n-element complex vector x by the real scalar
1/a.  This is done without overflow or underflow as long as
the final result x/a does not overflow or underflow.
```

Parameters:

N

```          N is INTEGER
The number of components of the vector x.
```

SA

```          SA is REAL
The scalar a which is used to divide each component of x.
SA must be >= 0, or the subroutine will divide by zero.
```

SX

```          SX is COMPLEX array, dimension
(1+(N-1)*abs(INCX))
The n-element vector x.
```

INCX

```          INCX is INTEGER
The increment between successive values of the vector SX.
> 0:  SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i),     1< i<= n
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## subroutine ctprfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldwork, * ) WORK, integer LDWORK)¶

CTPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks.

Purpose:

``` CTPRFB applies a complex "triangular-pentagonal" block reflector H or its
conjugate transpose H**H to a complex matrix C, which is composed of two
blocks A and B, either from the left or right.
```

Parameters:

SIDE

```          SIDE is CHARACTER*1
= 'L': apply H or H**H from the Left
= 'R': apply H or H**H from the Right
```

TRANS

```          TRANS is CHARACTER*1
= 'N': apply H (No transpose)
= 'C': apply H**H (Conjugate transpose)
```

DIRECT

```          DIRECT is CHARACTER*1
Indicates how H is formed from a product of elementary
reflectors
= 'F': H = H(1) H(2) . . . H(k) (Forward)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
```

STOREV

```          STOREV is CHARACTER*1
Indicates how the vectors which define the elementary
reflectors are stored:
= 'C': Columns
= 'R': Rows
```

M

```          M is INTEGER
The number of rows of the matrix B.
M >= 0.
```

N

```          N is INTEGER
The number of columns of the matrix B.
N >= 0.
```

K

```          K is INTEGER
The order of the matrix T, i.e. the number of elementary
reflectors whose product defines the block reflector.
K >= 0.
```

L

```          L is INTEGER
The order of the trapezoidal part of V.
K >= L >= 0.  See Further Details.
```

V

```          V is COMPLEX array, dimension
(LDV,K) if STOREV = 'C'
(LDV,M) if STOREV = 'R' and SIDE = 'L'
(LDV,N) if STOREV = 'R' and SIDE = 'R'
The pentagonal matrix V, which contains the elementary reflectors
H(1), H(2), ..., H(K).  See Further Details.
```

LDV

```          LDV is INTEGER
The leading dimension of the array V.
If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
if STOREV = 'R', LDV >= K.
```

T

```          T is COMPLEX array, dimension (LDT,K)
The triangular K-by-K matrix T in the representation of the
block reflector.
```

LDT

```          LDT is INTEGER
The leading dimension of the array T.
LDT >= K.
```

A

```          A is COMPLEX array, dimension
(LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'
On entry, the K-by-N or M-by-K matrix A.
On exit, A is overwritten by the corresponding block of
H*C or H**H*C or C*H or C*H**H.  See Further Details.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.
If SIDE = 'L', LDC >= max(1,K);
If SIDE = 'R', LDC >= max(1,M).
```

B

```          B is COMPLEX array, dimension (LDB,N)
On entry, the M-by-N matrix B.
On exit, B is overwritten by the corresponding block of
H*C or H**H*C or C*H or C*H**H.  See Further Details.
```

LDB

```          LDB is INTEGER
The leading dimension of the array B.
LDB >= max(1,M).
```

WORK

```          WORK is COMPLEX array, dimension
(LDWORK,N) if SIDE = 'L',
(LDWORK,K) if SIDE = 'R'.
```

LDWORK

```          LDWORK is INTEGER
The leading dimension of the array WORK.
If SIDE = 'L', LDWORK >= K;
if SIDE = 'R', LDWORK >= M.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

```  The matrix C is a composite matrix formed from blocks A and B.
The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K,
and if SIDE = 'L', A is of size K-by-N.
If SIDE = 'R' and DIRECT = 'F', C = [A B].
If SIDE = 'L' and DIRECT = 'F', C = [A]
[B].
If SIDE = 'R' and DIRECT = 'B', C = [B A].
If SIDE = 'L' and DIRECT = 'B', C = [B]
[A].
The pentagonal matrix V is composed of a rectangular block V1 and a
trapezoidal block V2.  The size of the trapezoidal block is determined by
the parameter L, where 0<=L<=K.  If L=K, the V2 block of V is triangular;
if L=0, there is no trapezoidal block, thus V = V1 is rectangular.
If DIRECT = 'F' and STOREV = 'C':  V = [V1]
[V2]
- V2 is upper trapezoidal (first L rows of K-by-K upper triangular)
If DIRECT = 'F' and STOREV = 'R':  V = [V1 V2]
- V2 is lower trapezoidal (first L columns of K-by-K lower triangular)
If DIRECT = 'B' and STOREV = 'C':  V = [V2]
[V1]
- V2 is lower trapezoidal (last L rows of K-by-K lower triangular)
If DIRECT = 'B' and STOREV = 'R':  V = [V2 V1]
- V2 is upper trapezoidal (last L columns of K-by-K upper triangular)
If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.
If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.
If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.
If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.
```

## integer function icmax1 (integer N, complex, dimension(*) CX, integer INCX)¶

ICMAX1 finds the index of the first vector element of maximum absolute value.

Purpose:

``` ICMAX1 finds the index of the first vector element of maximum absolute value.
Based on ICAMAX from Level 1 BLAS.
The change is to use the 'genuine' absolute value.
```

Parameters:

N

```          N is INTEGER
The number of elements in the vector CX.
```

CX

```          CX is COMPLEX array, dimension (N)
The vector CX. The ICMAX1 function returns the index of its first
element of maximum absolute value.
```

INCX

```          INCX is INTEGER
The spacing between successive values of CX.  INCX >= 1.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

February 2014

Contributors:

Nick Higham for use with CLACON.

## integer function ilaclc (integer M, integer N, complex, dimension( lda, * ) A, integer LDA)¶

ILACLC scans a matrix for its last non-zero column.

Purpose:

``` ILACLC scans A for its last non-zero column.
```

Parameters:

M

```          M is INTEGER
The number of rows of the matrix A.
```

N

```          N is INTEGER
The number of columns of the matrix A.
```

A

```          A is COMPLEX array, dimension (LDA,N)
The m by n matrix A.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

## integer function ilaclr (integer M, integer N, complex, dimension( lda, * ) A, integer LDA)¶

ILACLR scans a matrix for its last non-zero row.

Purpose:

``` ILACLR scans A for its last non-zero row.
```

Parameters:

M

```          M is INTEGER
The number of rows of the matrix A.
```

N

```          N is INTEGER
The number of columns of the matrix A.
```

A

```          A is COMPLEX array, dimension (LDA,N)
The m by n matrix A.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

## integer function izmax1 (integer N, complex*16, dimension(*) ZX, integer INCX)¶

IZMAX1 finds the index of the first vector element of maximum absolute value.

Purpose:

``` IZMAX1 finds the index of the first vector element of maximum absolute value.
Based on IZAMAX from Level 1 BLAS.
The change is to use the 'genuine' absolute value.
```

Parameters:

N

```          N is INTEGER
The number of elements in the vector ZX.
```

ZX

```          ZX is COMPLEX*16 array, dimension (N)
The vector ZX. The IZMAX1 function returns the index of its first
element of maximum absolute value.
```

INCX

```          INCX is INTEGER
The spacing between successive values of ZX.  INCX >= 1.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

February 2014

Contributors:

Nick Higham for use with ZLACON.

## real function scsum1 (integer N, complex, dimension( * ) CX, integer INCX)¶

SCSUM1 forms the 1-norm of the complex vector using the true absolute value.

Purpose:

``` SCSUM1 takes the sum of the absolute values of a complex
vector and returns a single precision result.
Based on SCASUM from the Level 1 BLAS.
The change is to use the 'genuine' absolute value.
```

Parameters:

N

```          N is INTEGER
The number of elements in the vector CX.
```

CX

```          CX is COMPLEX array, dimension (N)
The vector whose elements will be summed.
```

INCX

```          INCX is INTEGER
The spacing between successive values of CX.  INCX > 0.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

Nick Higham for use with CLACON.

# Author¶

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