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

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).

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  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).

 CLACGV conjugates a complex vector of length N.

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.

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 CLACN2 estimates the 1-norm of a square, complex matrix A.


 Reverse communication is used for evaluating matrix-vector products.

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

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  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)

Nick Higham, University of Manchester

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.

 CLACON estimates the 1-norm of a square, complex matrix A.


 Reverse communication is used for evaluating matrix-vector products.

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.

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

Nick Higham, University of Manchester

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.

 CLACP2 copies all or part of a real two-dimensional matrix A to a


 complex matrix B.

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).

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


 matrix B.

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).

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 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.

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)

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 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.

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  ]

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 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.

X

          X is COMPLEX

Y

          Y is COMPLEX


          The complex scalars X and Y.

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 CLAEIN uses inverse iteration to find a right or left eigenvector


 corresponding to the eigenvalue W of a complex upper Hessenberg


 matrix H.

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.

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 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 ].

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.

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  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.

 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.

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.

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 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.

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).

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    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.

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


           > 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 > 0 and WANTT is .FALSE., then on exit,


                  the remaining unconverged eigenvalues are the


                  eigenvalues of the upper Hessenberg matrix


                  rows and columns ILO through INFO of the final,


                  output value of H.


                  If INFO > 0 and WANTT is .TRUE., then on exit


          (*)       (initial value of H)*U  = U*(final value of H)


                  where U is an orthogonal matrix.    The final


                  value of H is upper Hessenberg and triangular in


                  rows and columns INFO+1 through IHI.


                  If INFO > 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.)

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     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).

 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.

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.

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  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 CLAHRD


  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 CLAHRD routine. (This


  subroutine is not backward compatible with LAPACK-3.0's CLAHRD.)

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.

 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.

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.

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 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.

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.

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.

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 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.

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.

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.

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 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.

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.

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.

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 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.

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.

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.

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 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.

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.

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.

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 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.

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.

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.

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 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.

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.

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.

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 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.

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.

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.

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 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.

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.

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.

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 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.

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.

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.

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 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.

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 ).

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 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.

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.

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 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.

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.

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 CLAQHB equilibrates an Hermitian band matrix A using the scaling


 factors in the vector S.

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).

  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.

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 CLAQHP equilibrates a Hermitian matrix A using the scaling factors


 in the vector S.

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).

  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.

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 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.

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)

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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.

LAPACK Working Note 176

 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.

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)


          Auxiliary 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).

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LAPACK Working Note 176

    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.

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 1:ILO-1 and IHI+1:N and, if ILO > 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 > 0, then 1 <= ILO <= IHI <= 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 > 0 is given under the


           description of INFO below.)


           This subroutine may explicitly set H(i,j) = 0 for i > 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 >= 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 <= ILOZ <= ILO; IHI <= IHIZ <= 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 > 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 >= MAX(1,IHIZ).  Otherwise, LDZ >= 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 >= 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


             > 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not


                accessed.

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  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.

      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.

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 >= 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 (*).

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    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.

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 <= NW <= (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 <= 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 <= ILOZ <= IHIZ <= 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 <= 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 <= LDV

NH

          NH is INTEGER


          The number of columns of T.  NH >= 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 <= LDT

NV

          NV is INTEGER


          The number of rows of work array WV available for


          workspace.  NV >= 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 <= 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.

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    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.

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 <= NW <= (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 <= 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 <= ILOZ <= IHIZ <= 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 <= 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 <= LDV

NH

          NH is INTEGER


          The number of columns of T.  NH >= 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 <= LDT

NV

          NV is INTEGER


          The number of rows of work array WV available for


          workspace.  NV >= 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 <= 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.

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

    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.

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 1:ILO-1 and IHI+1:N and, if ILO > 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 > 0, then 1 <= ILO <= IHI <= 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 > 0 is given under the


           description of INFO below.)


           This subroutine may explicitly set H(i,j) = 0 for i > 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 >= 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 <= ILOZ <= ILO; IHI <= IHIZ <= 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 > 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 >= MAX(1,IHIZ).  Otherwise, LDZ >= 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 >= 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


             > 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not


                accessed.

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

  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.

    CLAQR5 called by CLAQR0 performs a


    single small-bulge multi-shift QR sweep.

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: Same as KACC22 = 1. This option used to enable exploiting


             the 2-by-2 structure during matrix multiplications, but


             this is no longer supported.

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 >= 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 <= ILOZ <= IHIZ <= 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 >= 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 >= 3.

U

          U is COMPLEX array, dimension (LDU,2*NSHFTS)

LDU

          LDU is INTEGER


             LDU is the leading dimension of U just as declared in the


             in the calling subroutine.  LDU >= 2*NSHFTS.

NV

          NV is INTEGER


             NV is the number of rows in WV agailable for workspace.


             NV >= 1.

WV

          WV is COMPLEX array, dimension (LDWV,2*NSHFTS)

LDWV

          LDWV is INTEGER


             LDWV is the leading dimension of WV as declared in the


             in the calling subroutine.  LDWV >= NV.

NH

          NH is INTEGER


             NH is the number of columns in array WH available for


             workspace. NH >= 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 >= 2*NSHFTS.

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NAG Ltd.

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

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.

 CLAQSB equilibrates a symmetric band matrix A using the scaling


 factors in the vector S.

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).

  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.

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 CLAQSP equilibrates a symmetric matrix A using the scaling factors


 in the vector S.

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).

  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.

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 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.

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)

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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

 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)  )

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.

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 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.

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)

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 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.

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'

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 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.

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).


          If SIDE = 'L', M >= K >= 0;


          if SIDE = 'R', N >= K >= 0.

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).

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  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 )

 CLARFB_GETT applies a complex Householder block reflector H from the


 left to a complex (K+M)-by-N  'triangular-pentagonal' matrix


 composed of two block matrices: an upper trapezoidal K-by-N matrix A


 stored in the array A, and a rectangular M-by-(N-K) matrix B, stored


 in the array B. The block reflector H is stored in a compact


 WY-representation, where the elementary reflectors are in the


 arrays A, B and T. See Further Details section.

IDENT

          IDENT is CHARACTER*1


          If IDENT = not 'I', or not 'i', then V1 is unit


             lower-triangular and stored in the left K-by-K block of


             the input matrix A,


          If IDENT = 'I' or 'i', then  V1 is an identity matrix and


             not stored.


          See Further Details section.

M

          M is INTEGER


          The number of rows of the matrix B.


          M >= 0.

N

          N is INTEGER


          The number of columns of the matrices A and B.


          N >= 0.

K

          K is INTEGER


          The number or rows of the matrix A.


          K is also order of the matrix T, i.e. the number of


          elementary reflectors whose product defines the block


          reflector. 0 <= K <= N.

T

          T is COMPLEX array, dimension (LDT,K)


          The upper-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)


          On entry:


           a) In the K-by-N upper-trapezoidal part A: input matrix A.


           b) In the columns below the diagonal: columns of V1


              (ones are not stored on the diagonal).


          On exit:


            A is overwritten by rectangular K-by-N product H*A.


          See Further Details section.

LDA

          LDB is INTEGER


          The leading dimension of the array A. LDA >= max(1,K).

B

          B is COMPLEX array, dimension (LDB,N)


          On entry:


            a) In the M-by-(N-K) right block: input matrix B.


            b) In the M-by-N left block: columns of V2.


          On exit:


            B is overwritten by rectangular M-by-N product H*B.


          See Further Details section.

LDB

          LDB is INTEGER


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

WORK

          WORK is COMPLEX array,


          dimension (LDWORK,max(K,N-K))

LDWORK

          LDWORK is INTEGER


          The leading dimension of the array WORK. LDWORK>=max(1,K).

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                University of California, Berkeley

    (1) Description of the Algebraic Operation.


    The matrix A is a K-by-N matrix composed of two column block


    matrices, A1, which is K-by-K, and A2, which is K-by-(N-K):


    A = ( A1, A2 ).


    The matrix B is an M-by-N matrix composed of two column block


    matrices, B1, which is M-by-K, and B2, which is M-by-(N-K):


    B = ( B1, B2 ).


    Perform the operation:


       ( A_out ) := H * ( A_in ) = ( I - V * T * V**H ) * ( A_in ) =


       ( B_out )        ( B_in )                          ( B_in )


                  = ( I - ( V1 ) * T * ( V1**H, V2**H ) ) * ( A_in )


                          ( V2 )                            ( B_in )


     On input:


    a) ( A_in )  consists of two block columns:


       ( B_in )


       ( A_in ) = (( A1_in ) ( A2_in )) = (( A1_in ) ( A2_in ))


       ( B_in )   (( B1_in ) ( B2_in ))   ((     0 ) ( B2_in )),


       where the column blocks are:


       (  A1_in )  is a K-by-K upper-triangular matrix stored in the


                   upper triangular part of the array A(1:K,1:K).


       (  B1_in )  is an M-by-K rectangular ZERO matrix and not stored.


       ( A2_in )  is a K-by-(N-K) rectangular matrix stored


                  in the array A(1:K,K+1:N).


       ( B2_in )  is an M-by-(N-K) rectangular matrix stored


                  in the array B(1:M,K+1:N).


    b) V = ( V1 )


           ( V2 )


       where:


       1) if IDENT == 'I',V1 is a K-by-K identity matrix, not stored;


       2) if IDENT != 'I',V1 is a K-by-K unit lower-triangular matrix,


          stored in the lower-triangular part of the array


          A(1:K,1:K) (ones are not stored),


       and V2 is an M-by-K rectangular stored the array B(1:M,1:K),


                 (because on input B1_in is a rectangular zero


                  matrix that is not stored and the space is


                  used to store V2).


    c) T is a K-by-K upper-triangular matrix stored


       in the array T(1:K,1:K).


    On output:


    a) ( A_out ) consists of two  block columns:


       ( B_out )


       ( A_out ) = (( A1_out ) ( A2_out ))


       ( B_out )   (( B1_out ) ( B2_out )),


       where the column blocks are:


       ( A1_out )  is a K-by-K square matrix, or a K-by-K


                   upper-triangular matrix, if V1 is an


                   identity matrix. AiOut is stored in


                   the array A(1:K,1:K).


       ( B1_out )  is an M-by-K rectangular matrix stored


                   in the array B(1:M,K:N).


       ( A2_out )  is a K-by-(N-K) rectangular matrix stored


                   in the array A(1:K,K+1:N).


       ( B2_out )  is an M-by-(N-K) rectangular matrix stored


                   in the array B(1:M,K+1:N).


    The operation above can be represented as the same operation


    on each block column:


       ( A1_out ) := H * ( A1_in ) = ( I - V * T * V**H ) * ( A1_in )


       ( B1_out )        (     0 )                          (     0 )


       ( A2_out ) := H * ( A2_in ) = ( I - V * T * V**H ) * ( A2_in )


       ( B2_out )        ( B2_in )                          ( B2_in )


    If IDENT != 'I':


       The computation for column block 1:


       A1_out: = A1_in - V1*T*(V1**H)*A1_in


       B1_out: = - V2*T*(V1**H)*A1_in


       The computation for column block 2, which exists if N > K:


       A2_out: = A2_in - V1*T*( (V1**H)*A2_in + (V2**H)*B2_in )


       B2_out: = B2_in - V2*T*( (V1**H)*A2_in + (V2**H)*B2_in )


    If IDENT == 'I':


       The operation for column block 1:


       A1_out: = A1_in - V1*T*A1_in


       B1_out: = - V2*T*A1_in


       The computation for column block 2, which exists if N > K:


       A2_out: = A2_in - T*( A2_in + (V2**H)*B2_in )


       B2_out: = B2_in - V2*T*( A2_in + (V2**H)*B2_in )


    (2) Description of the Algorithmic Computation.


    In the first step, we compute column block 2, i.e. A2 and B2.


    Here, we need to use the K-by-(N-K) rectangular workspace


    matrix W2 that is of the same size as the matrix A2.


    W2 is stored in the array WORK(1:K,1:(N-K)).


    In the second step, we compute column block 1, i.e. A1 and B1.


    Here, we need to use the K-by-K square workspace matrix W1


    that is of the same size as the as the matrix A1.


    W1 is stored in the array WORK(1:K,1:K).


    NOTE: Hence, in this routine, we need the workspace array WORK


    only of size WORK(1:K,1:max(K,N-K)) so it can hold both W2 from


    the first step and W1 from the second step.


    Case (A), when V1 is unit lower-triangular, i.e. IDENT != 'I',


    more computations than in the Case (B).


    if( IDENT != 'I' ) then


     if ( N > K ) then


       (First Step - column block 2)


       col2_(1) W2: = A2


       col2_(2) W2: = (V1**H) * W2 = (unit_lower_tr_of_(A1)**H) * W2


       col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2


       col2_(4) W2: = T * W2


       col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2


       col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2


       col2_(7) A2: = A2 - W2


     else


       (Second Step - column block 1)


       col1_(1) W1: = A1


       col1_(2) W1: = (V1**H) * W1 = (unit_lower_tr_of_(A1)**H) * W1


       col1_(3) W1: = T * W1


       col1_(4) B1: = - V2 * W1 = - B1 * W1


       col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1


       col1_(6) square A1: = A1 - W1


     end if


    end if


    Case (B), when V1 is an identity matrix, i.e. IDENT == 'I',


    less computations than in the Case (A)


    if( IDENT == 'I' ) then


     if ( N > K ) then


       (First Step - column block 2)


       col2_(1) W2: = A2


       col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2


       col2_(4) W2: = T * W2


       col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2


       col2_(7) A2: = A2 - W2


     else


       (Second Step - column block 1)


       col1_(1) W1: = A1


       col1_(3) W1: = T * W1


       col1_(4) B1: = - V2 * W1 = - B1 * W1


       col1_(6) upper-triangular_of_(A1): = A1 - W1


     end if


    end if


    Combine these cases (A) and (B) together, this is the resulting


    algorithm:


    if ( N > K ) then


      (First Step - column block 2)


      col2_(1)  W2: = A2


      if( IDENT != 'I' ) then


        col2_(2)  W2: = (V1**H) * W2


                      = (unit_lower_tr_of_(A1)**H) * W2


      end if


      col2_(3)  W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2]


      col2_(4)  W2: = T * W2


      col2_(5)  B2: = B2 - V2 * W2 = B2 - B1 * W2


      if( IDENT != 'I' ) then


        col2_(6)    W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2


      end if


      col2_(7) A2: = A2 - W2


    else


    (Second Step - column block 1)


      col1_(1) W1: = A1


      if( IDENT != 'I' ) then


        col1_(2) W1: = (V1**H) * W1


                    = (unit_lower_tr_of_(A1)**H) * W1


      end if


      col1_(3) W1: = T * W1


      col1_(4) B1: = - V2 * W1 = - B1 * W1


      if( IDENT != 'I' ) then


        col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1


        col1_(6_a) below_diag_of_(A1): =  - below_diag_of_(W1)


      end if


      col1_(6_b) up_tr_of_(A1): = up_tr_of_(A1) - up_tr_of_(W1)


    end if

 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 .

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.

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 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.

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.

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 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

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.

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  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 )

 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.

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. LDC >= 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.

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 CLARFY applies an elementary reflector, or Householder matrix, H,


 to an n x n Hermitian matrix C, from both the left and the right.


 H is represented in the form


    H = I - tau * v * v'


 where  tau  is a scalar and  v  is a vector.


 If  tau  is  zero, then  H  is taken to be the unit matrix.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the upper or lower triangular part of the


          Hermitian matrix C is stored.


          = 'U':  Upper triangle


          = 'L':  Lower triangle

N

          N is INTEGER


          The number of rows and columns of the matrix C.  N >= 0.

V

          V is COMPLEX array, dimension


                  (1 + (N-1)*abs(INCV))


          The vector v as described above.

INCV

          INCV is INTEGER


          The increment between successive elements of v.  INCV must


          not be zero.

TAU

          TAU is COMPLEX


          The value tau as described above.

C

          C is COMPLEX array, dimension (LDC, N)


          On entry, the matrix C.


          On exit, C is overwritten by H * C * H'.

LDC

          LDC is INTEGER


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

WORK

          WORK is COMPLEX array, dimension (N)

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 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.

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.

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  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.

 CLARNV returns a vector of n random complex numbers from a uniform or


 normal distribution.

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.

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  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.

 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.

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 < 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.

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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

 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) )

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.

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 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.

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.

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 CLASET initializes a 2-D array A to BETA on the diagonal and


 ALPHA on the offdiagonals.

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).

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 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.

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).

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 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.

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.

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  Modified by


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

 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.

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

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  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).

 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.

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).

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

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

[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.

 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.

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

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  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).

 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.

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).

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  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).

 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.

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

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  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).

 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.

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

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 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.

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

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 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.

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 -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.

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 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.

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.

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 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.

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.

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 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.

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

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 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.

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', LDA >= max(1,K);


          If SIDE = 'R', LDA >= 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.