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- bookworm 3.11.0-2
| realGEcomputational(3) | LAPACK | realGEcomputational(3) |
NAME¶
realGEcomputational - real
SYNOPSIS¶
Functions¶
subroutine sgebak (JOB, SIDE, N, ILO, IHI, SCALE, M, V,
LDV, INFO)
SGEBAK subroutine sgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
SGEBAL subroutine sgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
INFO)
SGEBD2 reduces a general matrix to bidiagonal form using an unblocked
algorithm. subroutine sgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
LWORK, INFO)
SGEBRD subroutine sgecon (NORM, N, A, LDA, ANORM, RCOND, WORK,
IWORK, INFO)
SGECON subroutine sgeequ (M, N, A, LDA, R, C, ROWCND, COLCND,
AMAX, INFO)
SGEEQU subroutine sgeequb (M, N, A, LDA, R, C, ROWCND, COLCND,
AMAX, INFO)
SGEEQUB subroutine sgehd2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO)
SGEHD2 reduces a general square matrix to upper Hessenberg form using
an unblocked algorithm. subroutine sgehrd (N, ILO, IHI, A, LDA, TAU,
WORK, LWORK, INFO)
SGEHRD subroutine sgelq2 (M, N, A, LDA, TAU, WORK, INFO)
SGELQ2 computes the LQ factorization of a general rectangular matrix
using an unblocked algorithm. subroutine sgelqf (M, N, A, LDA, TAU,
WORK, LWORK, INFO)
SGELQF subroutine sgemqrt (SIDE, TRANS, M, N, K, NB, V, LDV, T,
LDT, C, LDC, WORK, INFO)
SGEMQRT subroutine sgeql2 (M, N, A, LDA, TAU, WORK, INFO)
SGEQL2 computes the QL factorization of a general rectangular matrix
using an unblocked algorithm. subroutine sgeqlf (M, N, A, LDA, TAU,
WORK, LWORK, INFO)
SGEQLF subroutine sgeqp3 (M, N, A, LDA, JPVT, TAU, WORK, LWORK,
INFO)
SGEQP3 subroutine sgeqr2 (M, N, A, LDA, TAU, WORK, INFO)
SGEQR2 computes the QR factorization of a general rectangular matrix
using an unblocked algorithm. subroutine sgeqr2p (M, N, A, LDA, TAU,
WORK, INFO)
SGEQR2P computes the QR factorization of a general rectangular matrix
with non-negative diagonal elements using an unblocked algorithm. subroutine
sgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF subroutine sgeqrfp (M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRFP subroutine sgeqrt (M, N, NB, A, LDA, T, LDT, WORK, INFO)
SGEQRT subroutine sgeqrt2 (M, N, A, LDA, T, LDT, INFO)
SGEQRT2 computes a QR factorization of a general real or complex matrix
using the compact WY representation of Q. recursive subroutine
sgeqrt3 (M, N, A, LDA, T, LDT, INFO)
SGEQRT3 recursively computes a QR factorization of a general real or
complex matrix using the compact WY representation of Q. subroutine
sgerfs (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR,
BERR, WORK, IWORK, INFO)
SGERFS subroutine sgerfsx (TRANS, EQUED, N, NRHS, A, LDA, AF,
LDAF, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
SGERFSX subroutine sgerq2 (M, N, A, LDA, TAU, WORK, INFO)
SGERQ2 computes the RQ factorization of a general rectangular matrix
using an unblocked algorithm. subroutine sgerqf (M, N, A, LDA, TAU,
WORK, LWORK, INFO)
SGERQF subroutine sgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA,
MV, V, LDV, WORK, LWORK, INFO)
SGESVJ subroutine sgetf2 (M, N, A, LDA, IPIV, INFO)
SGETF2 computes the LU factorization of a general m-by-n matrix using
partial pivoting with row interchanges (unblocked algorithm). subroutine
sgetrf (M, N, A, LDA, IPIV, INFO)
SGETRF recursive subroutine sgetrf2 (M, N, A, LDA, IPIV, INFO)
SGETRF2 subroutine sgetri (N, A, LDA, IPIV, WORK, LWORK, INFO)
SGETRI subroutine sgetrs (TRANS, N, NRHS, A, LDA, IPIV, B, LDB,
INFO)
SGETRS subroutine shgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H,
LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SHGEQZ subroutine sla_geamv (TRANS, M, N, ALPHA, A, LDA, X,
INCX, BETA, Y, INCY)
SLA_GEAMV computes a matrix-vector product using a general matrix to
calculate error bounds. real function sla_gercond (TRANS, N, A, LDA,
AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)
SLA_GERCOND estimates the Skeel condition number for a general matrix.
subroutine sla_gerfsx_extended (PREC_TYPE, TRANS_TYPE, N, NRHS, A,
LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERRS_N,
ERRS_C, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE,
INFO)
SLA_GERFSX_EXTENDED improves the computed solution to a system of
linear equations for general matrices by performing extra-precise iterative
refinement and provides error bounds and backward error estimates for the
solution. real function sla_gerpvgrw (N, NCOLS, A, LDA, AF, LDAF)
SLA_GERPVGRW subroutine slaorhr_col_getrfnp (M, N, A, LDA, D,
INFO)
SLAORHR_COL_GETRFNP recursive subroutine slaorhr_col_getrfnp2
(M, N, A, LDA, D, INFO)
SLAORHR_COL_GETRFNP2 subroutine stgevc (SIDE, HOWMNY, SELECT, N,
S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
STGEVC subroutine stgexc (WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO)
STGEXC
Detailed Description¶
This is the group of real computational functions for GE matrices
Function Documentation¶
subroutine sgebak (character JOB, character SIDE, integer N, integer ILO, integer IHI, real, dimension( * ) SCALE, integer M, real, dimension( ldv, * ) V, integer LDV, integer INFO)¶
SGEBAK
Purpose:
SGEBAK forms the right or left eigenvectors of a real general matrix
by backward transformation on the computed eigenvectors of the
balanced matrix output by SGEBAL.
Parameters
JOB
JOB is CHARACTER*1
Specifies the type of backward transformation required:
= 'N': do nothing, return immediately;
= 'P': do backward transformation for permutation only;
= 'S': do backward transformation for scaling only;
= 'B': do backward transformations for both permutation and
scaling.
JOB must be the same as the argument JOB supplied to SGEBAL.
SIDE
SIDE is CHARACTER*1
= 'R': V contains right eigenvectors;
= 'L': V contains left eigenvectors.
N
N is INTEGER
The number of rows of the matrix V. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
The integers ILO and IHI determined by SGEBAL.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
SCALE
SCALE is REAL array, dimension (N)
Details of the permutation and scaling factors, as returned
by SGEBAL.
M
M is INTEGER
The number of columns of the matrix V. M >= 0.
V
V is REAL array, dimension (LDV,M)
On entry, the matrix of right or left eigenvectors to be
transformed, as returned by SHSEIN or STREVC.
On exit, V is overwritten by the transformed eigenvectors.
LDV
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgebal (character JOB, integer N, real, dimension( lda, * ) A, integer LDA, integer ILO, integer IHI, real, dimension( * ) SCALE, integer INFO)¶
SGEBAL
Purpose:
SGEBAL balances a general real matrix A. This involves, first,
permuting A by a similarity transformation to isolate eigenvalues
in the first 1 to ILO-1 and last IHI+1 to N elements on the
diagonal; and second, applying a diagonal similarity transformation
to rows and columns ILO to IHI to make the rows and columns as
close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrix, and improve the
accuracy of the computed eigenvalues and/or eigenvectors.
Parameters
JOB
JOB is CHARACTER*1
Specifies the operations to be performed on A:
= 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
for i = 1,...,N;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the input matrix A.
On exit, A is overwritten by the balanced matrix.
If JOB = 'N', A is not referenced.
See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI are set to integers such that on exit
A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
If JOB = 'N' or 'S', ILO = 1 and IHI = N.
SCALE
SCALE is REAL array, dimension (N)
Details of the permutations and scaling factors applied to
A. If P(j) is the index of the row and column interchanged
with row and column j and D(j) is the scaling factor
applied to row and column j, then
SCALE(j) = P(j) for j = 1,...,ILO-1
= D(j) for j = ILO,...,IHI
= P(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N to IHI+1,
then 1 to ILO-1.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The permutations consist of row and column interchanges which put
the matrix in the form
( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )
where T1 and T2 are upper triangular matrices whose eigenvalues lie
along the diagonal. The column indices ILO and IHI mark the starting
and ending columns of the submatrix B. Balancing consists of applying
a diagonal similarity transformation inv(D) * B * D to make the
1-norms of each row of B and its corresponding column nearly equal.
The output matrix is
( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )
Information about the permutations P and the diagonal matrix D is
returned in the vector SCALE.
This subroutine is based on the EISPACK routine BALANC.
Modified by Tzu-Yi Chen, Computer Science Division, University of
California at Berkeley, USA
subroutine sgebd2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( * ) WORK, integer INFO)¶
SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
Purpose:
SGEBD2 reduces a real general m by n matrix A to upper or lower
bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
Parameters
M
M is INTEGER
The number of rows in the matrix A. M >= 0.
N
N is INTEGER
The number of columns in the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the m by n general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the orthogonal matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the orthogonal matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the orthogonal 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 (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
E
E is REAL array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix B:
if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
TAUQ
TAUQ is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details.
TAUP
TAUP is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details.
WORK
WORK is REAL array, dimension (max(M,N))
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrices Q and P are represented as products of elementary
reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
where tauq and taup are real scalars, and v and u are real vectors;
v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
where tauq and taup are real scalars, and v and u are real vectors;
v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) = 0, u(i) = 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).
The contents of A on exit are illustrated by the following examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).
subroutine sgebrd (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SGEBRD
Purpose:
SGEBRD reduces a general real M-by-N matrix A to upper or lower
bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
Parameters
M
M is INTEGER
The number of rows in the matrix A. M >= 0.
N
N is INTEGER
The number of columns in the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the orthogonal matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the orthogonal matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the orthogonal 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 (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
E
E is REAL array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix B:
if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
TAUQ
TAUQ is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details.
TAUP
TAUP is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,M,N).
For optimum performance LWORK >= (M+N)*NB, where NB
is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrices Q and P are represented as products of elementary
reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
where tauq and taup are real scalars, and v and u are real vectors;
v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
where tauq and taup are real scalars, and v and u are real vectors;
v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) = 0, u(i) = 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).
The contents of A on exit are illustrated by the following examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).
subroutine sgecon (character NORM, integer N, real, dimension( lda, * ) A, integer LDA, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶
SGECON
Purpose:
SGECON estimates the reciprocal of the condition number of a general
real matrix A, in either the 1-norm or the infinity-norm, using
the LU factorization computed by SGETRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).
Parameters
NORM
NORM is CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= '1' or 'O': 1-norm;
= 'I': Infinity-norm.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
The factors L and U from the factorization A = P*L*U
as computed by SGETRF.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
ANORM
ANORM is REAL
If NORM = '1' or 'O', the 1-norm of the original matrix A.
If NORM = 'I', the infinity-norm of the original matrix A.
RCOND
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK
WORK is REAL array, dimension (4*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgeequ (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)¶
SGEEQU
Purpose:
SGEEQU computes row and column scalings intended to equilibrate an
M-by-N matrix A and reduce its condition number. R returns the row
scale factors and C the column scale factors, chosen to try to make
the largest element in each row and column of the matrix B with
elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
R(i) and C(j) are restricted to be between SMLNUM = smallest safe
number and BIGNUM = largest safe number. Use of these scaling
factors is not guaranteed to reduce the condition number of A but
works well in practice.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
The M-by-N matrix whose equilibration factors are
to be computed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
R
R is REAL array, dimension (M)
If INFO = 0 or INFO > M, R contains the row scale factors
for A.
C
C is REAL array, dimension (N)
If INFO = 0, C contains the column scale factors for A.
ROWCND
ROWCND is REAL
If INFO = 0 or INFO > M, ROWCND contains the ratio of the
smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
AMAX is neither too large nor too small, it is not worth
scaling by R.
COLCND
COLCND is REAL
If INFO = 0, COLCND contains the ratio of the smallest
C(i) to the largest C(i). If COLCND >= 0.1, it is not
worth scaling by C.
AMAX
AMAX is REAL
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= M: the i-th row of A is exactly zero
> M: the (i-M)-th column of A is exactly zero
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgeequb (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)¶
SGEEQUB
Purpose:
SGEEQUB computes row and column scalings intended to equilibrate an
M-by-N matrix A and reduce its condition number. R returns the row
scale factors and C the column scale factors, chosen to try to make
the largest element in each row and column of the matrix B with
elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
the radix.
R(i) and C(j) are restricted to be a power of the radix between
SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
of these scaling factors is not guaranteed to reduce the condition
number of A but works well in practice.
This routine differs from SGEEQU by restricting the scaling factors
to a power of the radix. Barring over- and underflow, scaling by
these factors introduces no additional rounding errors. However, the
scaled entries' magnitudes are no longer approximately 1 but lie
between sqrt(radix) and 1/sqrt(radix).
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
The M-by-N matrix whose equilibration factors are
to be computed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
R
R is REAL array, dimension (M)
If INFO = 0 or INFO > M, R contains the row scale factors
for A.
C
C is REAL array, dimension (N)
If INFO = 0, C contains the column scale factors for A.
ROWCND
ROWCND is REAL
If INFO = 0 or INFO > M, ROWCND contains the ratio of the
smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
AMAX is neither too large nor too small, it is not worth
scaling by R.
COLCND
COLCND is REAL
If INFO = 0, COLCND contains the ratio of the smallest
C(i) to the largest C(i). If COLCND >= 0.1, it is not
worth scaling by C.
AMAX
AMAX is REAL
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= M: the i-th row of A is exactly zero
> M: the (i-M)-th column of A is exactly zero
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgehd2 (integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)¶
SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
Purpose:
SGEHD2 reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation: Q**T * A * Q = H .
Parameters
N
N is INTEGER
The order of the matrix A. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to SGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= max(1,N).
A
A is REAL array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the orthogonal 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).
TAU
TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( 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).
subroutine sgehrd (integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SGEHRD
Purpose:
SGEHRD reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation: Q**T * A * Q = H .
Parameters
N
N is INTEGER
The order of the matrix A. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to SGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A
A is REAL array, dimension (LDA,N)
On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the orthogonal 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).
TAU
TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
zero.
WORK
WORK is REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,N).
For good performance, LWORK should generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( 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 file is a slight modification of LAPACK-3.0's SGEHRD
subroutine incorporating improvements proposed by Quintana-Orti and
Van de Geijn (2006). (See SLAHR2.)
subroutine sgelq2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)¶
SGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
Purpose:
SGELQ2 computes an LQ factorization of a real m-by-n matrix A:
A = ( L 0 ) * Q
where:
Q is a n-by-n orthogonal matrix;
L is a lower-triangular m-by-m matrix;
0 is a m-by-(n-m) zero matrix, if m < n.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and below the diagonal of the array
contain the m by min(m,n) lower trapezoidal matrix L (L is
lower triangular if m <= n); the elements above the diagonal,
with the array TAU, represent the orthogonal 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,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
and tau in TAU(i).
subroutine sgelqf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SGELQF
Purpose:
SGELQF computes an LQ factorization of a real M-by-N matrix A:
A = ( L 0 ) * Q
where:
Q is a N-by-N orthogonal matrix;
L is a lower-triangular M-by-M matrix;
0 is a M-by-(N-M) zero matrix, if M < N.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and below the diagonal of the array
contain the m-by-min(m,n) lower trapezoidal matrix L (L is
lower triangular if m <= n); the elements above the diagonal,
with the array TAU, represent the orthogonal 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,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
and tau in TAU(i).
subroutine sgemqrt (character SIDE, character TRANS, integer M, integer N, integer K, integer NB, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer INFO)¶
SGEMQRT
Purpose:
SGEMQRT overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q C C Q
TRANS = 'T': Q**T C C Q**T
where Q is a real orthogonal matrix defined as the product of K
elementary reflectors:
Q = H(1) H(2) . . . H(K) = I - V T V**T
generated using the compact WY representation as returned by SGEQRT.
Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0.
NB
NB is INTEGER
The block size used for the storage of T. K >= NB >= 1.
This must be the same value of NB used to generate T
in SGEQRT.
V
V is REAL array, dimension (LDV,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGEQRT in the first K columns of its array argument A.
LDV
LDV is INTEGER
The leading dimension of the array V.
If SIDE = 'L', LDA >= max(1,M);
if SIDE = 'R', LDA >= max(1,N).
T
T is REAL array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by SGEQRT, stored as a NB-by-N matrix.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= NB.
C
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL array. The dimension of WORK is
N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgeql2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)¶
SGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
Purpose:
SGEQL2 computes a QL factorization of a real m by n matrix A:
A = Q * L.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, if m >= n, the lower triangle of the subarray
A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
if m <= n, the elements on and below the (n-m)-th
superdiagonal contain the m by n lower trapezoidal matrix L;
the remaining elements, with the array TAU, represent the
orthogonal 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,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i).
subroutine sgeqlf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SGEQLF
Purpose:
SGEQLF computes a QL factorization of a real M-by-N matrix A:
A = Q * L.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
if m >= n, the lower triangle of the subarray
A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
if m <= n, the elements on and below the (n-m)-th
superdiagonal contain the M-by-N lower trapezoidal matrix L;
the remaining elements, with the array TAU, represent the
orthogonal 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,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i).
subroutine sgeqp3 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SGEQP3
Purpose:
SGEQP3 computes a QR factorization with column pivoting of a
matrix A: A*P = Q*R using Level 3 BLAS.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the upper triangle of the array contains the
min(M,N)-by-N upper trapezoidal matrix R; the elements below
the diagonal, together with the array TAU, represent the
orthogonal matrix Q as a product of min(M,N) elementary
reflectors.
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(J).ne.0, the J-th column of A is permuted
to the front of A*P (a leading column); if JPVT(J)=0,
the J-th column of A is a free column.
On exit, if JPVT(J)=K, then the J-th column of A*P was the
the K-th column of A.
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO=0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 3*N+1.
For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real/complex vector
with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in TAU(i).
Contributors:
G. Quintana-Orti, Depto. de Informatica, Universidad
Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA
subroutine sgeqr2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)¶
SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
Purpose:
SGEQR2 computes a QR factorization of a real m-by-n matrix A:
A = Q * ( R ),
( 0 )
where:
Q is a m-by-m orthogonal matrix;
R is an upper-triangular n-by-n matrix;
0 is a (m-n)-by-n zero matrix, if m > n.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(m,n) by n upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal 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,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
subroutine sgeqr2p (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)¶
SGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
Purpose:
SGEQR2P computes a QR factorization of a real m-by-n matrix A:
A = Q * ( R ),
( 0 )
where:
Q is a m-by-m orthogonal matrix;
R is an upper-triangular n-by-n matrix with nonnegative diagonal
entries;
0 is a (m-n)-by-n zero matrix, if m > n.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(m,n) by n upper trapezoidal matrix R (R is
upper triangular if m >= n). The diagonal entries of R
are nonnegative; the elements below the diagonal,
with the array TAU, represent the orthogonal 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,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
See Lapack Working Note 203 for details
subroutine sgeqrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SGEQRF
Purpose:
SGEQRF computes a QR factorization of a real M-by-N matrix A:
A = Q * ( R ),
( 0 )
where:
Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix;
0 is a (M-N)-by-N zero matrix, if M > N.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
For optimum performance LWORK >= N*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
subroutine sgeqrfp (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SGEQRFP
Purpose:
SGEQR2P computes a QR factorization of a real M-by-N matrix A:
A = Q * ( R ),
( 0 )
where:
Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix with nonnegative diagonal
entries;
0 is a (M-N)-by-N zero matrix, if M > N.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n). The diagonal entries of R
are nonnegative; the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
See Lapack Working Note 203 for details
subroutine sgeqrt (integer M, integer N, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) WORK, integer INFO)¶
SGEQRT
Purpose:
SGEQRT computes a blocked QR factorization of a real M-by-N matrix A
using the compact WY representation of Q.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
NB
NB is INTEGER
The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if M >= N); the elements below the diagonal
are the columns of V.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is REAL array, dimension (LDT,MIN(M,N))
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See below
for further details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= NB.
WORK
WORK is REAL array, dimension (NB*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is
V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )
where the vi's represent the vectors which define H(i), which are returned
in the matrix A. The 1's along the diagonal of V are not stored in A.
Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each
block is of order NB except for the last block, which is of order
IB = K - (B-1)*NB. For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
for the last block) T's are stored in the NB-by-K matrix T as
T = (T1 T2 ... TB).
subroutine sgeqrt2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)¶
SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
Purpose:
SGEQRT2 computes a QR factorization of a real M-by-N matrix A,
using the compact WY representation of Q.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= N.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the real M-by-N matrix A. On exit, the elements on and
above the diagonal contain the N-by-N upper triangular matrix R; the
elements below the diagonal are the columns of V. See below for
further details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is REAL array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is
V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )
where the vi's represent the vectors which define H(i), which are returned
in the matrix A. The 1's along the diagonal of V are not stored in A. The
block reflector H is then given by
H = I - V * T * V**T
where V**T is the transpose of V.
recursive subroutine sgeqrt3 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)¶
SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
Purpose:
SGEQRT3 recursively computes a QR factorization of a real M-by-N
matrix A, using the compact WY representation of Q.
Based on the algorithm of Elmroth and Gustavson,
IBM J. Res. Develop. Vol 44 No. 4 July 2000.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= N.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the real M-by-N matrix A. On exit, the elements on and
above the diagonal contain the N-by-N upper triangular matrix R; the
elements below the diagonal are the columns of V. See below for
further details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is REAL array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is
V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )
where the vi's represent the vectors which define H(i), which are returned
in the matrix A. The 1's along the diagonal of V are not stored in A. The
block reflector H is then given by
H = I - V * T * V**T
where V**T is the transpose of V.
For details of the algorithm, see Elmroth and Gustavson (cited above).
subroutine sgerfs (character TRANS, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶
SGERFS
Purpose:
SGERFS improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates for
the solution.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
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 B and X. NRHS >= 0.
A
A is REAL array, dimension (LDA,N)
The original N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by SGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
B
B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
On exit, the improved solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR
FERR is REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK
WORK is REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgerfsx (character TRANS, character EQUED, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx , * ) X, integer LDX, real RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶
SGERFSX
Purpose:
SGERFSX improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates
for the solution. In addition to normwise error bound, the code
provides maximum componentwise error bound if possible. See
comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
error bounds.
The original system of linear equations may have been equilibrated
before calling this routine, as described by arguments EQUED, R
and C below. In this case, the solution and error bounds returned
are for the original unequilibrated system.
Some optional parameters are bundled in the PARAMS array. These
settings determine how refinement is performed, but often the
defaults are acceptable. If the defaults are acceptable, users
can pass NPARAMS = 0 which prevents the source code from accessing
the PARAMS argument.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
EQUED
EQUED is CHARACTER*1
Specifies the form of equilibration that was done to A
before calling this routine. This is needed to compute
the solution and error bounds correctly.
= 'N': No equilibration
= 'R': Row equilibration, i.e., A has been premultiplied by
diag(R).
= 'C': Column equilibration, i.e., A has been postmultiplied
by diag(C).
= 'B': Both row and column equilibration, i.e., A has been
replaced by diag(R) * A * diag(C).
The right hand side B has been changed accordingly.
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 B and X. NRHS >= 0.
A
A is REAL array, dimension (LDA,N)
The original N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by SGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
R
R is REAL array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is
multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
is not accessed.
If R is accessed, each element of R should be a power of the radix
to ensure a reliable solution and error estimates. Scaling by
powers of the radix does not cause rounding errors unless the
result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.
C
C is REAL array, dimension (N)
The column scale factors for A. If EQUED = 'C' or 'B', A is
multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
is not accessed.
If C is accessed, each element of C should be a power of the radix
to ensure a reliable solution and error estimates. Scaling by
powers of the radix does not cause rounding errors unless the
result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.
B
B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
On exit, the improved solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND
RCOND is REAL
Reciprocal scaled condition number. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done). If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.
BERR
BERR is REAL array, dimension (NRHS)
Componentwise relative backward error. This is the
componentwise relative backward error of each solution vector X(j)
(i.e., the smallest relative change in any element of A or B that
makes X(j) an exact solution).
N_ERR_BNDS
N_ERR_BNDS is INTEGER
Number of error bounds to return for each right hand side
and each type (normwise or componentwise). See ERR_BNDS_NORM and
ERR_BNDS_COMP below.
ERR_BNDS_NORM
ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 'Trust/don't trust' boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is 'guaranteed'. These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.
ERR_BNDS_COMP
ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 'Trust/don't trust' boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is 'guaranteed'. These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.
NPARAMS
NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS. If <= 0, the
PARAMS array is never referenced and default values are used.
PARAMS
PARAMS is REAL array, dimension NPARAMS
Specifies algorithm parameters. If an entry is < 0.0, then
that entry will be filled with default value used for that
parameter. Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.
PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
refinement or not.
Default: 1.0
= 0.0: No refinement is performed, and no error bounds are
computed.
= 1.0: Use the double-precision refinement algorithm,
possibly with doubled-single computations if the
compilation environment does not support DOUBLE
PRECISION.
(other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
computations allowed for refinement.
Default: 10
Aggressive: Set to 100 to permit convergence using approximate
factorizations or factorizations other than LU. If
the factorization uses a technique other than
Gaussian elimination, the guarantees in
err_bnds_norm and err_bnds_comp may no longer be
trustworthy.
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
will attempt to find a solution with small componentwise
relative error in the double-precision algorithm. Positive
is true, 0.0 is false.
Default: 1.0 (attempt componentwise convergence)
WORK
WORK is REAL array, dimension (4*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: Successful exit. The solution to every right-hand side is
guaranteed.
< 0: If INFO = -i, the i-th argument had an illegal value
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
has been completed, but the factor U is exactly singular, so
the solution and error bounds could not be computed. RCOND = 0
is returned.
= N+J: The solution corresponding to the Jth right-hand side is
not guaranteed. The solutions corresponding to other right-
hand sides K with K > J may not be guaranteed as well, but
only the first such right-hand side is reported. If a small
componentwise error is not requested (PARAMS(3) = 0.0) then
the Jth right-hand side is the first with a normwise error
bound that is not guaranteed (the smallest J such
that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
the Jth right-hand side is the first with either a normwise or
componentwise error bound that is not guaranteed (the smallest
J such that either ERR_BNDS_NORM(J,1) = 0.0 or
ERR_BNDS_COMP(J,1) = 0.0). See the definition of
ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
about all of the right-hand sides check ERR_BNDS_NORM or
ERR_BNDS_COMP.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgerq2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)¶
SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
Purpose:
SGERQ2 computes an RQ factorization of a real m by n matrix A:
A = R * Q.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the m by n upper trapezoidal matrix R; the remaining
elements, with the array TAU, represent the orthogonal 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,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and tau in TAU(i).
subroutine sgerqf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SGERQF
Purpose:
SGERQF computes an RQ factorization of a real M-by-N matrix A:
A = R * Q.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the M-by-N upper trapezoidal matrix R;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of min(m,n) elementary
reflectors (see Further Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
For optimum performance LWORK >= M*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and tau in TAU(i).
subroutine sgesvj (character*1 JOBA, character*1 JOBU, character*1 JOBV, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( n ) SVA, integer MV, real, dimension( ldv, * ) V, integer LDV, real, dimension( lwork ) WORK, integer LWORK, integer INFO)¶
SGESVJ
Purpose:
SGESVJ computes the singular value decomposition (SVD) of a real
M-by-N matrix A, where M >= N. The SVD of A is written as
[++] [xx] [x0] [xx]
A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx]
[++] [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively.
SGESVJ can sometimes compute tiny singular values and their singular vectors much
more accurately than other SVD routines, see below under Further Details.
Parameters
JOBA
JOBA is CHARACTER*1
Specifies the structure of A.
= 'L': The input matrix A is lower triangular;
= 'U': The input matrix A is upper triangular;
= 'G': The input matrix A is general M-by-N matrix, M >= N.
JOBU
JOBU is CHARACTER*1
Specifies whether to compute the left singular vectors
(columns of U):
= 'U': The left singular vectors corresponding to the nonzero
singular values are computed and returned in the leading
columns of A. See more details in the description of A.
The default numerical orthogonality threshold is set to
approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
= 'C': Analogous to JOBU='U', except that user can control the
level of numerical orthogonality of the computed left
singular vectors. TOL can be set to TOL = CTOL*EPS, where
CTOL is given on input in the array WORK.
No CTOL smaller than ONE is allowed. CTOL greater
than 1 / EPS is meaningless. The option 'C'
can be used if M*EPS is satisfactory orthogonality
of the computed left singular vectors, so CTOL=M could
save few sweeps of Jacobi rotations.
See the descriptions of A and WORK(1).
= 'N': The matrix U is not computed. However, see the
description of A.
JOBV
JOBV is CHARACTER*1
Specifies whether to compute the right singular vectors, that
is, the matrix V:
= 'V': the matrix V is computed and returned in the array V
= 'A': the Jacobi rotations are applied to the MV-by-N
array V. In other words, the right singular vector
matrix V is not computed explicitly; instead it is
applied to an MV-by-N matrix initially stored in the
first MV rows of V.
= 'N': the matrix V is not computed and the array V is not
referenced
M
M is INTEGER
The number of rows of the input matrix A. 1/SLAMCH('E') > M >= 0.
N
N is INTEGER
The number of columns of the input matrix A.
M >= N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
If JOBU = 'U' .OR. JOBU = 'C':
If INFO = 0:
RANKA orthonormal columns of U are returned in the
leading RANKA columns of the array A. Here RANKA <= N
is the number of computed singular values of A that are
above the underflow threshold SLAMCH('S'). The singular
vectors corresponding to underflowed or zero singular
values are not computed. The value of RANKA is returned
in the array WORK as RANKA=NINT(WORK(2)). Also see the
descriptions of SVA and WORK. The computed columns of U
are mutually numerically orthogonal up to approximately
TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
see the description of JOBU.
If INFO > 0,
the procedure SGESVJ did not converge in the given number
of iterations (sweeps). In that case, the computed
columns of U may not be orthogonal up to TOL. The output
U (stored in A), SIGMA (given by the computed singular
values in SVA(1:N)) and V is still a decomposition of the
input matrix A in the sense that the residual
||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
If JOBU = 'N':
If INFO = 0:
Note that the left singular vectors are 'for free' in the
one-sided Jacobi SVD algorithm. However, if only the
singular values are needed, the level of numerical
orthogonality of U is not an issue and iterations are
stopped when the columns of the iterated matrix are
numerically orthogonal up to approximately M*EPS. Thus,
on exit, A contains the columns of U scaled with the
corresponding singular values.
If INFO > 0:
the procedure SGESVJ did not converge in the given number
of iterations (sweeps).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
SVA
SVA is REAL array, dimension (N)
On exit,
If INFO = 0 :
depending on the value SCALE = WORK(1), we have:
If SCALE = ONE:
SVA(1:N) contains the computed singular values of A.
During the computation SVA contains the Euclidean column
norms of the iterated matrices in the array A.
If SCALE .NE. ONE:
The singular values of A are SCALE*SVA(1:N), and this
factored representation is due to the fact that some of the
singular values of A might underflow or overflow.
If INFO > 0 :
the procedure SGESVJ did not converge in the given number of
iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
MV
MV is INTEGER
If JOBV = 'A', then the product of Jacobi rotations in SGESVJ
is applied to the first MV rows of V. See the description of JOBV.
V
V is REAL array, dimension (LDV,N)
If JOBV = 'V', then V contains on exit the N-by-N matrix of
the right singular vectors;
If JOBV = 'A', then V contains the product of the computed right
singular vector matrix and the initial matrix in
the array V.
If JOBV = 'N', then V is not referenced.
LDV
LDV is INTEGER
The leading dimension of the array V, LDV >= 1.
If JOBV = 'V', then LDV >= max(1,N).
If JOBV = 'A', then LDV >= max(1,MV) .
WORK
WORK is REAL array, dimension (LWORK)
On entry,
If JOBU = 'C' :
WORK(1) = CTOL, where CTOL defines the threshold for convergence.
The process stops if all columns of A are mutually
orthogonal up to CTOL*EPS, EPS=SLAMCH('E').
It is required that CTOL >= ONE, i.e. it is not
allowed to force the routine to obtain orthogonality
below EPSILON.
On exit,
WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
are the computed singular vcalues of A.
(See description of SVA().)
WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
singular values.
WORK(3) = NINT(WORK(3)) is the number of the computed singular
values that are larger than the underflow threshold.
WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
rotations needed for numerical convergence.
WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
This is useful information in cases when SGESVJ did
not converge, as it can be used to estimate whether
the output is still useful and for post festum analysis.
WORK(6) = the largest absolute value over all sines of the
Jacobi rotation angles in the last sweep. It can be
useful for a post festum analysis.
LWORK
LWORK is INTEGER
length of WORK, WORK >= MAX(6,M+N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, then the i-th argument had an illegal value
> 0: SGESVJ did not converge in the maximal allowed number (30)
of sweeps. The output may still be useful. See the
description of WORK.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The orthogonal N-by-N matrix V is obtained as a product
of Jacobi plane rotations. The rotations are implemented as fast scaled
rotations of Anda and Park [1]. In the case of underflow of the Jacobi angle,
a modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
column interchanges of de Rijk [2]. The relative accuracy of the computed
singular values and the accuracy of the computed singular vectors (in angle
metric) is as guaranteed by the theory of Demmel and Veselic [3]. The
condition number that determines the accuracy in the full rank case is
essentially min_{D=diag} kappa(A*D), where kappa(.) is the spectral condition
number. The best performance of this Jacobi SVD procedure is achieved if used
in an accelerated version of Drmac and Veselic [5,6], and it is the kernel
routine in the SIGMA library [7]. Some tuning parameters (marked with [TP])
are available for the implementer.
The computational range for the nonzero singular values is the machine number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even denormalized singular values can be computed with the corresponding gradual loss of accurate digits.
The computational range for the nonzero singular values is the machine number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even denormalized singular values can be computed with the corresponding gradual loss of accurate digits.
Contributors:
Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic
(Hagen, Germany)
References:
[1] A. A. Anda and H. Park: Fast plane rotations with
dynamic scaling.
SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
[2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the singular
value decomposition on a vector computer.
SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
[3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
[4] Z. Drmac: Implementation of Jacobi rotations for accurate singular value
computation in floating point arithmetic.
SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
[5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
LAPACK Working note 169.
[6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
LAPACK Working note 170.
[7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
QSVD, (H,K)-SVD computations.
Department of Mathematics, University of Zagreb, 2008.
Bugs, Examples and Comments:
Please report all bugs and send interesting test examples
and comments to drmac@math.hr. Thank you.
subroutine sgetf2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶
SGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).
Purpose:
SGETF2 computes an LU factorization of a general m-by-n matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This is the right-looking Level 2 BLAS version of the algorithm.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the m by n matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
IPIV
IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, U(k,k) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgetrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶
SGETRF SGETRF VARIANT: iterative version of Sivan Toledo's recursive LU algorithm
SGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.
Purpose:
SGETRF computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This is the right-looking Level 3 BLAS version of the algorithm.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
IPIV
IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Purpose:
SGETRF computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This is the left-looking Level 3 BLAS version of the algorithm.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
IPIV
IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016
Purpose:
SGETRF computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This code implements an iterative version of Sivan Toledo's recursive
LU algorithm[1]. For square matrices, this iterative versions should
be within a factor of two of the optimum number of memory transfers.
The pattern is as follows, with the large blocks of U being updated
in one call to STRSM, and the dotted lines denoting sections that
have had all pending permutations applied:
1 2 3 4 5 6 7 8
+-+-+---+-------+------
| |1| | |
|.+-+ 2 | |
| | | | |
|.|.+-+-+ 4 |
| | | |1| |
| | |.+-+ |
| | | | | |
|.|.|.|.+-+-+---+ 8
| | | | | |1| |
| | | | |.+-+ 2 |
| | | | | | | |
| | | | |.|.+-+-+
| | | | | | | |1|
| | | | | | |.+-+
| | | | | | | | |
|.|.|.|.|.|.|.|.+-----
| | | | | | | | |
The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
the binary expansion of the current column. Each Schur update is
applied as soon as the necessary portion of U is available.
[1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
1065-1081. http://dx.doi.org/10.1137/S0895479896297744
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
IPIV
IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016
recursive subroutine sgetrf2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶
SGETRF2
Purpose:
SGETRF2 computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This is the recursive version of the algorithm. It divides
the matrix into four submatrices:
[ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2
A = [ -----|----- ] with n1 = min(m,n)/2
[ A21 | A22 ] n2 = n-n1
[ A11 ]
The subroutine calls itself to factor [ --- ],
[ A12 ]
[ A12 ]
do the swaps on [ --- ], solve A12, update A22,
[ A22 ]
then calls itself to factor A22 and do the swaps on A21.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
IPIV
IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgetri (integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SGETRI
Purpose:
SGETRI computes the inverse of a matrix using the LU factorization
computed by SGETRF.
This method inverts U and then computes inv(A) by solving the system
inv(A)*L = inv(U) for inv(A).
Parameters
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the factors L and U from the factorization
A = P*L*U as computed by SGETRF.
On exit, if INFO = 0, the inverse of the original matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO=0, then WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimal performance LWORK >= N*NB, where NB is
the optimal blocksize returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero; the matrix is
singular and its inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgetrs (character TRANS, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶
SGETRS
Purpose:
SGETRS solves a system of linear equations
A * X = B or A**T * X = B
with a general N-by-N matrix A using the LU factorization computed
by SGETRF.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T* X = B (Transpose)
= 'C': A**T* X = B (Conjugate transpose = Transpose)
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 matrix B. NRHS >= 0.
A
A is REAL array, dimension (LDA,N)
The factors L and U from the factorization A = P*L*U
as computed by SGETRF.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
B
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine shgeqz (character JOB, character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
SHGEQZ
Purpose:
SHGEQZ computes the eigenvalues of a real matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the double-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a real matrix pair (A,B):
A = Q1*H*Z1**T, B = Q1*T*Z1**T,
as computed by SGGHRD.
If JOB='S', then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,
H = Q*S*Z**T, T = Q*P*Z**T,
where Q and Z are orthogonal matrices, P is an upper triangular
matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
diagonal blocks.
The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
(H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
eigenvalues.
Additionally, the 2-by-2 upper triangular diagonal blocks of P
corresponding to 2-by-2 blocks of S are reduced to positive diagonal
form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
P(j,j) > 0, and P(j+1,j+1) > 0.
Optionally, the orthogonal matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
orthogonal matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
the matrix pair (A,B) to generalized upper Hessenberg form, then the
output matrices Q1*Q and Z1*Z are the orthogonal factors from the
generalized Schur factorization of (A,B):
A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
complex and beta real.
If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
generalized nonsymmetric eigenvalue problem (GNEP)
A*x = lambda*B*x
and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
alternate form of the GNEP
mu*A*y = B*y.
Real eigenvalues can be read directly from the generalized Schur
form:
alpha = S(i,i), beta = P(i,i).
Ref: C.B. Moler & G.W. Stewart, 'An Algorithm for Generalized Matrix
Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
Parameters
JOB
JOB is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Compute eigenvalues and the Schur form.
COMPQ
COMPQ is CHARACTER*1
= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (H,T) is returned;
= 'V': Q must contain an orthogonal matrix Q1 on entry and
the product Q1*Q is returned.
COMPZ
COMPZ is CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Z is initialized to the unit matrix and the matrix Z
of right Schur vectors of (H,T) is returned;
= 'V': Z must contain an orthogonal matrix Z1 on entry and
the product Z1*Z is returned.
N
N is INTEGER
The order of the matrices H, T, Q, and Z. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of H which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H
H is REAL array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H.
On exit, if JOB = 'S', H contains the upper quasi-triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal blocks of H match those of S, but
the rest of H is unspecified.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max( 1, N ).
T
T is REAL array, dimension (LDT, N)
On entry, the N-by-N upper triangular matrix T.
On exit, if JOB = 'S', T contains the upper triangular
matrix P from the generalized Schur factorization;
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
are reduced to positive diagonal form, i.e., if H(j+1,j) is
non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
T(j+1,j+1) > 0.
If JOB = 'E', the diagonal blocks of T match those of P, but
the rest of T is unspecified.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max( 1, N ).
ALPHAR
ALPHAR is REAL array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.
ALPHAI
ALPHAI is REAL array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of GNEP.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
BETA
BETA is REAL array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the matrix
pair (A,B), in one of the forms lambda = alpha/beta or
mu = beta/alpha. Since either lambda or mu may overflow,
they should not, in general, be computed.
Q
Q is REAL array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
of left Schur vectors of (A,B).
Not referenced if COMPQ = 'N'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
Z
Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the orthogonal matrix of
right Schur vectors of (H,T), and if COMPZ = 'V', the
orthogonal matrix of right Schur vectors of (A,B).
Not referenced if COMPZ = 'N'.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO+1,...,N should be correct.
= N+1,...,2*N: the shift calculation failed. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO-N+1,...,N should be correct.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Iteration counters:
JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last
changed. This is therefore reset only when a 1-by-1 or
2-by-2 block deflates off the bottom.
subroutine sla_geamv (integer TRANS, integer M, integer N, real ALPHA, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) X, integer INCX, real BETA, real, dimension( * ) Y, integer INCY)¶
SLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.
Purpose:
SLA_GEAMV performs one of the matrix-vector operations
y := alpha*abs(A)*abs(x) + beta*abs(y),
or y := alpha*abs(A)**T*abs(x) + beta*abs(y),
where alpha and beta are scalars, x and y are vectors and A is an
m by n matrix.
This function is primarily used in calculating error bounds.
To protect against underflow during evaluation, components in
the resulting vector are perturbed away from zero by (N+1)
times the underflow threshold. To prevent unnecessarily large
errors for block-structure embedded in general matrices,
'symbolically' zero components are not perturbed. A zero
entry is considered 'symbolic' if all multiplications involved
in computing that entry have at least one zero multiplicand.
Parameters
TRANS
TRANS is INTEGER
On entry, TRANS specifies the operation to be performed as
follows:
BLAS_NO_TRANS y := alpha*abs(A)*abs(x) + beta*abs(y)
BLAS_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)
BLAS_CONJ_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)
Unchanged on exit.
M
M is INTEGER
On entry, M specifies the number of rows of the matrix A.
M must be at least zero.
Unchanged on exit.
N
N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.
Unchanged on exit.
ALPHA
ALPHA is REAL
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
A
A is REAL array, dimension ( LDA, n )
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients.
Unchanged on exit.
LDA
LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, m ).
Unchanged on exit.
X
X is REAL array, dimension
( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
Before entry, the incremented array X must contain the
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 REAL
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 REAL array,
dimension at least
( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
Before entry with BETA non-zero, the incremented array Y
must contain the 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.
Level 2 Blas routine.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
real function sla_gercond (character TRANS, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, integer CMODE, real, dimension( * ) C, integer INFO, real, dimension( * ) WORK, integer, dimension( * ) IWORK)¶
SLA_GERCOND estimates the Skeel condition number for a general matrix.
Purpose:
SLA_GERCOND estimates the Skeel condition number of op(A) * op2(C)
where op2 is determined by CMODE as follows
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C)
The Skeel condition number cond(A) = norminf( |inv(A)||A| )
is computed by computing scaling factors R such that
diag(R)*A*op2(C) is row equilibrated and computing the standard
infinity-norm condition number.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate Transpose = Transpose)
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by SGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by SGETRF; row i of the matrix was interchanged
with row IPIV(i).
CMODE
CMODE is INTEGER
Determines op2(C) in the formula op(A) * op2(C) as follows:
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C)
C
C is REAL array, dimension (N)
The vector C in the formula op(A) * op2(C).
INFO
INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid.
WORK
WORK is REAL array, dimension (3*N).
Workspace.
IWORK
IWORK is INTEGER array, dimension (N).
Workspace.2
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sla_gerfsx_extended (integer PREC_TYPE, integer TRANS_TYPE, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldy, * ) Y, integer LDY, real, dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * ) ERRS_N, real, dimension( nrhs, * ) ERRS_C, real, dimension( * ) RES, real, dimension( * ) AYB, real, dimension( * ) DY, real, dimension( * ) Y_TAIL, real RCOND, integer ITHRESH, real RTHRESH, real DZ_UB, logical IGNORE_CWISE, integer INFO)¶
SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
Purpose:
SLA_GERFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by SGERFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERRS_N
and ERRS_C for details of the error bounds. Note that this
subroutine is only responsible for setting the second fields of
ERRS_N and ERRS_C.
Parameters
PREC_TYPE
PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
= 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X' or 'E': Extra
TRANS_TYPE
TRANS_TYPE is INTEGER
Specifies the transposition operation on A.
The value is defined by ILATRANS(T) where T is a CHARACTER and T
= 'N': No transpose
= 'T': Transpose
= 'C': Conjugate transpose
N
N is INTEGER
The number of linear equations, i.e., 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
matrix B.
A
A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by SGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by SGETRF; row i of the matrix was interchanged
with row IPIV(i).
COLEQU
COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.
C
C is REAL array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. If C is input, each element of C should be a power
of the radix to ensure a reliable solution and error estimates.
Scaling by powers of the radix does not cause rounding errors unless
the result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.
B
B is REAL array, dimension (LDB,NRHS)
The right-hand-side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Y
Y is REAL array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
On exit, the improved solution matrix Y.
LDY
LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT
BERR_OUT is REAL array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by SLA_LIN_BERR.
N_NORMS
N_NORMS is INTEGER
Determines which error bounds to return (see ERRS_N
and ERRS_C).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.
ERRS_N
ERRS_N is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERRS_N(i,:) corresponds to the ith
right-hand side.
The second index in ERRS_N(:,err) contains the following
three fields:
err = 1 'Trust/don't trust' boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is 'guaranteed'. These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
ERRS_C
ERRS_C is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERRS_C(i,:) corresponds to the ith
right-hand side.
The second index in ERRS_C(:,err) contains the following
three fields:
err = 1 'Trust/don't trust' boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is 'guaranteed'. These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
RES
RES is REAL array, dimension (N)
Workspace to hold the intermediate residual.
AYB
AYB is REAL array, dimension (N)
Workspace. This can be the same workspace passed for Y_TAIL.
DY
DY is REAL array, dimension (N)
Workspace to hold the intermediate solution.
Y_TAIL
Y_TAIL is REAL array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.
RCOND
RCOND is REAL
Reciprocal scaled condition number. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done). If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.
ITHRESH
ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For 'aggressive' set to 100 to
permit convergence using approximate factorizations or
factorizations other than LU. If the factorization uses a
technique other than Gaussian elimination, the guarantees in
ERRS_N and ERRS_C may no longer be trustworthy.
RTHRESH
RTHRESH is REAL
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For 'aggressive' set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.
DZ_UB
DZ_UB is REAL
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we define as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.
IGNORE_CWISE
IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..
INFO
INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to SGETRS had an illegal
value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
real function sla_gerpvgrw (integer N, integer NCOLS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF)¶
SLA_GERPVGRW
Purpose:
SLA_GERPVGRW computes the reciprocal pivot growth factor
norm(A)/norm(U). The 'max absolute element' norm is used. If this is
much less than 1, the stability of the LU factorization of the
(equilibrated) matrix A could be poor. This also means that the
solution X, estimated condition numbers, and error bounds could be
unreliable.
Parameters
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
NCOLS
NCOLS is INTEGER
The number of columns of the matrix A. NCOLS >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by SGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine slaorhr_col_getrfnp (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, integer INFO)¶
SLAORHR_COL_GETRFNP
Purpose:
SLAORHR_COL_GETRFNP computes the modified LU factorization without
pivoting of a real general M-by-N matrix A. The factorization has
the form:
A - S = L * U,
where:
S is a m-by-n diagonal sign matrix with the diagonal D, so that
D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
i-1 steps of Gaussian elimination. This means that the diagonal
element at each step of 'modified' Gaussian elimination is
at least one in absolute value (so that division-by-zero not
not possible during the division by the diagonal element);
L is a M-by-N lower triangular matrix with unit diagonal elements
(lower trapezoidal if M > N);
and U is a M-by-N upper triangular matrix
(upper trapezoidal if M < N).
This routine is an auxiliary routine used in the Householder
reconstruction routine SORHR_COL. In SORHR_COL, this routine is
applied to an M-by-N matrix A with orthonormal columns, where each
element is bounded by one in absolute value. With the choice of
the matrix S above, one can show that the diagonal element at each
step of Gaussian elimination is the largest (in absolute value) in
the column on or below the diagonal, so that no pivoting is required
for numerical stability [1].
For more details on the Householder reconstruction algorithm,
including the modified LU factorization, see [1].
This is the blocked right-looking version of the algorithm,
calling Level 3 BLAS to update the submatrix. To factorize a block,
this routine calls the recursive routine SLAORHR_COL_GETRFNP2.
[1] 'Reconstructing Householder vectors from tall-skinny QR',
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A-S=L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D
D is REAL array, dimension min(M,N)
The diagonal elements of the diagonal M-by-N sign matrix S,
D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
be only plus or minus one.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
recursive subroutine slaorhr_col_getrfnp2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, integer INFO)¶
SLAORHR_COL_GETRFNP2
Purpose:
SLAORHR_COL_GETRFNP2 computes the modified LU factorization without
pivoting of a real general M-by-N matrix A. The factorization has
the form:
A - S = L * U,
where:
S is a m-by-n diagonal sign matrix with the diagonal D, so that
D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
i-1 steps of Gaussian elimination. This means that the diagonal
element at each step of 'modified' Gaussian elimination is at
least one in absolute value (so that division-by-zero not
possible during the division by the diagonal element);
L is a M-by-N lower triangular matrix with unit diagonal elements
(lower trapezoidal if M > N);
and U is a M-by-N upper triangular matrix
(upper trapezoidal if M < N).
This routine is an auxiliary routine used in the Householder
reconstruction routine SORHR_COL. In SORHR_COL, this routine is
applied to an M-by-N matrix A with orthonormal columns, where each
element is bounded by one in absolute value. With the choice of
the matrix S above, one can show that the diagonal element at each
step of Gaussian elimination is the largest (in absolute value) in
the column on or below the diagonal, so that no pivoting is required
for numerical stability [1].
For more details on the Householder reconstruction algorithm,
including the modified LU factorization, see [1].
This is the recursive version of the LU factorization algorithm.
Denote A - S by B. The algorithm divides the matrix B into four
submatrices:
[ B11 | B12 ] where B11 is n1 by n1,
B = [ -----|----- ] B21 is (m-n1) by n1,
[ B21 | B22 ] B12 is n1 by n2,
B22 is (m-n1) by n2,
with n1 = min(m,n)/2, n2 = n-n1.
The subroutine calls itself to factor B11, solves for B21,
solves for B12, updates B22, then calls itself to factor B22.
For more details on the recursive LU algorithm, see [2].
SLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
routine SLAORHR_COL_GETRFNP, which uses blocked code calling
Level 3 BLAS to update the submatrix. However, SLAORHR_COL_GETRFNP2
is self-sufficient and can be used without SLAORHR_COL_GETRFNP.
[1] 'Reconstructing Householder vectors from tall-skinny QR',
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.
[2] 'Recursion leads to automatic variable blocking for dense linear
algebra algorithms', F. Gustavson, IBM J. of Res. and Dev.,
vol. 41, no. 6, pp. 737-755, 1997.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A-S=L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D
D is REAL array, dimension min(M,N)
The diagonal elements of the diagonal M-by-N sign matrix S,
D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
be only plus or minus one.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
subroutine stgevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, real, dimension( lds, * ) S, integer LDS, real, dimension( ldp, * ) P, integer LDP, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, real, dimension( * ) WORK, integer INFO)¶
STGEVC
Purpose:
STGEVC computes some or all of the right and/or left eigenvectors of
a pair of real matrices (S,P), where S is a quasi-triangular matrix
and P is upper triangular. Matrix pairs of this type are produced by
the generalized Schur factorization of a matrix pair (A,B):
A = Q*S*Z**T, B = Q*P*Z**T
as computed by SGGHRD + SHGEQZ.
The right eigenvector x and the left eigenvector y of (S,P)
corresponding to an eigenvalue w are defined by:
S*x = w*P*x, (y**H)*S = w*(y**H)*P,
where y**H denotes the conjugate tranpose of y.
The eigenvalues are not input to this routine, but are computed
directly from the diagonal blocks of S and P.
This routine returns the matrices X and/or Y of right and left
eigenvectors of (S,P), or the products Z*X and/or Q*Y,
where Z and Q are input matrices.
If Q and Z are the orthogonal factors from the generalized Schur
factorization of a matrix pair (A,B), then Z*X and Q*Y
are the matrices of right and left eigenvectors of (A,B).
Parameters
SIDE
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S': compute selected right and/or left eigenvectors,
specified by the logical array SELECT.
SELECT
SELECT is LOGICAL array, dimension (N)
If HOWMNY='S', SELECT specifies the eigenvectors to be
computed. If w(j) is a real eigenvalue, the corresponding
real eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector
is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
set to .FALSE..
Not referenced if HOWMNY = 'A' or 'B'.
N
N is INTEGER
The order of the matrices S and P. N >= 0.
S
S is REAL array, dimension (LDS,N)
The upper quasi-triangular matrix S from a generalized Schur
factorization, as computed by SHGEQZ.
LDS
LDS is INTEGER
The leading dimension of array S. LDS >= max(1,N).
P
P is REAL array, dimension (LDP,N)
The upper triangular matrix P from a generalized Schur
factorization, as computed by SHGEQZ.
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
of S must be in positive diagonal form.
LDP
LDP is INTEGER
The leading dimension of array P. LDP >= max(1,N).
VL
VL is REAL array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of left Schur vectors returned by SHGEQZ).
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VL, in the same order as their eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
Not referenced if SIDE = 'R'.
LDVL
LDVL is INTEGER
The leading dimension of array VL. LDVL >= 1, and if
SIDE = 'L' or 'B', LDVL >= N.
VR
VR is REAL array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Z (usually the orthogonal matrix Z
of right Schur vectors returned by SHGEQZ).
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
if HOWMNY = 'B' or 'b', the matrix Z*X;
if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
specified by SELECT, stored consecutively in the
columns of VR, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = 'L'.
LDVR
LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
SIDE = 'R' or 'B', LDVR >= N.
MM
MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
is set to N. Each selected real eigenvector occupies one
column and each selected complex eigenvector occupies two
columns.
WORK
WORK is REAL array, dimension (6*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
eigenvalue.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Allocation of workspace:
---------- -- ---------
WORK( j ) = 1-norm of j-th column of A, above the diagonal
WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
WORK( 2*N+1:3*N ) = real part of eigenvector
WORK( 3*N+1:4*N ) = imaginary part of eigenvector
WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
Rowwise vs. columnwise solution methods:
------- -- ---------- -------- -------
Finding a generalized eigenvector consists basically of solving the
singular triangular system
(A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)
Consider finding the i-th right eigenvector (assume all eigenvalues
are real). The equation to be solved is:
n i
0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1
k=j k=j
where C = (A - w B) (The components v(i+1:n) are 0.)
The 'rowwise' method is:
(1) v(i) := 1
for j = i-1,. . .,1:
i
(2) compute s = - sum C(j,k) v(k) and
k=j+1
(3) v(j) := s / C(j,j)
Step 2 is sometimes called the 'dot product' step, since it is an
inner product between the j-th row and the portion of the eigenvector
that has been computed so far.
The 'columnwise' method consists basically in doing the sums
for all the rows in parallel. As each v(j) is computed, the
contribution of v(j) times the j-th column of C is added to the
partial sums. Since FORTRAN arrays are stored columnwise, this has
the advantage that at each step, the elements of C that are accessed
are adjacent to one another, whereas with the rowwise method, the
elements accessed at a step are spaced LDS (and LDP) words apart.
When finding left eigenvectors, the matrix in question is the
transpose of the one in storage, so the rowwise method then
actually accesses columns of A and B at each step, and so is the
preferred method.
subroutine stgexc (logical WANTQ, logical WANTZ, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, integer IFST, integer ILST, real, dimension( * ) WORK, integer LWORK, integer INFO)¶
STGEXC
Purpose:
STGEXC reorders the generalized real Schur decomposition of a real
matrix pair (A,B) using an orthogonal equivalence transformation
(A, B) = Q * (A, B) * Z**T,
so that the diagonal block of (A, B) with row index IFST is moved
to row ILST.
(A, B) must be in generalized real Schur canonical form (as returned
by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
diagonal blocks. B is upper triangular.
Optionally, the matrices Q and Z of generalized Schur vectors are
updated.
Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
Parameters
WANTQ
WANTQ is LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.
WANTZ
WANTZ is LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the matrix A in generalized real Schur canonical
form.
On exit, the updated matrix A, again in generalized
real Schur canonical form.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is REAL array, dimension (LDB,N)
On entry, the matrix B in generalized real Schur canonical
form (A,B).
On exit, the updated matrix B, again in generalized
real Schur canonical form (A,B).
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q
Q is REAL array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
On exit, the updated matrix Q.
If WANTQ = .FALSE., Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= N.
Z
Z is REAL array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., the orthogonal matrix Z.
On exit, the updated matrix Z.
If WANTZ = .FALSE., Z is not referenced.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= N.
IFST
IFST is INTEGER
ILST
ILST is INTEGER
Specify the reordering of the diagonal blocks of (A, B).
The block with row index IFST is moved to row ILST, by a
sequence of swapping between adjacent blocks.
On exit, if IFST pointed on entry to the second row of
a 2-by-2 block, it is changed to point to the first row;
ILST always points to the first row of the block in its
final position (which may differ from its input value by
+1 or -1). 1 <= IFST, ILST <= N.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
=0: successful exit.
<0: if INFO = -i, the i-th argument had an illegal value.
=1: The transformed matrix pair (A, B) would be too far
from generalized Schur form; the problem is ill-
conditioned. (A, B) may have been partially reordered,
and ILST points to the first row of the current
position of the block being moved.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing
Science, Umea University, S-901 87 Umea, Sweden.
References:
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
Author¶
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