table of contents
| latrs3(3) | LAPACK | latrs3(3) |
NAME¶
latrs3 - latrs3: triangular solve with robust scaling, level 3
SYNOPSIS¶
Functions¶
subroutine clatrs3 (uplo, trans, diag, normin, n, nrhs, a,
lda, x, ldx, scale, cnorm, work, lwork, info)
CLATRS3 solves a triangular system of equations with the scale factors
set to prevent overflow. subroutine dlatrs3 (uplo, trans, diag,
normin, n, nrhs, a, lda, x, ldx, scale, cnorm, work, lwork, info)
DLATRS3 solves a triangular system of equations with the scale factors
set to prevent overflow. subroutine slatrs3 (uplo, trans, diag,
normin, n, nrhs, a, lda, x, ldx, scale, cnorm, work, lwork, info)
SLATRS3 solves a triangular system of equations with the scale factors
set to prevent overflow. subroutine zlatrs3 (uplo, trans, diag,
normin, n, nrhs, a, lda, x, ldx, scale, cnorm, work, lwork, info)
ZLATRS3 solves a triangular system of equations with the scale factors
set to prevent overflow.
Detailed Description¶
Function Documentation¶
subroutine clatrs3 (character uplo, character trans, character diag, character normin, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldx, * ) x, integer ldx, real, dimension( * ) scale, real, dimension( * ) cnorm, real, dimension( * ) work, integer lwork, integer info)¶
CLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow.
Purpose:
CLATRS3 solves one of the triangular systems
A * X = B * diag(scale), A**T * X = B * diag(scale), or
A**H * X = B * diag(scale)
with scaling to prevent overflow. Here A is an upper or lower
triangular matrix, A**T denotes the transpose of A, A**H denotes the
conjugate transpose of A. X and B are n-by-nrhs matrices and scale
is an nrhs-element vector of scaling factors. A scaling factor scale(j)
is usually less than or equal to 1, chosen such that X(:,j) is less
than the overflow threshold. If the matrix A is singular (A(j,j) = 0
for some j), then a non-trivial solution to A*X = 0 is returned. If
the system is so badly scaled that the solution cannot be represented
as (1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned.
This is a BLAS-3 version of LATRS for solving several right
hand sides simultaneously.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
TRANS
TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N': Solve A * x = s*b (No transpose)
= 'T': Solve A**T* x = s*b (Transpose)
= 'C': Solve A**T* x = s*b (Conjugate transpose)
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
NORMIN
NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y': CNORM contains the column norms on entry
= 'N': CNORM is not set on entry. On exit, the norms will
be computed and stored in CNORM.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of columns of X. NRHS >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The triangular matrix A. If UPLO = 'U', the leading n by n
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If UPLO = 'L', the leading n by n lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If DIAG = 'U', the diagonal elements of A are
also not referenced and are assumed to be 1.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max (1,N).
X
X is COMPLEX array, dimension (LDX,NRHS)
On entry, the right hand side B of the triangular system.
On exit, X is overwritten by the solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max (1,N).
SCALE
SCALE is REAL array, dimension (NRHS)
The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k).
If SCALE = 0, the matrix A is singular or badly scaled.
If A(j,j) = 0 is encountered, a non-trivial vector x(:,k)
that is an exact or approximate solution to A*x(:,k) = 0
is returned. If the system so badly scaled that solution
cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0
is returned.
CNORM
CNORM is REAL array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A. If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
WORK
WORK is REAL array, dimension (LWORK).
On exit, if INFO = 0, WORK(1) returns the optimal size of
WORK.
LWORK LWORK is INTEGER LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N + NB - 1)/NB and NB is the optimal block size.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions 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.
Parameters
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
subroutine dlatrs3 (character uplo, character trans, character diag, character normin, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) scale, double precision, dimension( * ) cnorm, double precision, dimension( * ) work, integer lwork, integer info)¶
DLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow.
Purpose:
DLATRS3 solves one of the triangular systems
A * X = B * diag(scale) or A**T * X = B * diag(scale)
with scaling to prevent overflow. Here A is an upper or lower
triangular matrix, A**T denotes the transpose of A. X and B are
n by nrhs matrices and scale is an nrhs element vector of scaling
factors. A scaling factor scale(j) is usually less than or equal
to 1, chosen such that X(:,j) is less than the overflow threshold.
If the matrix A is singular (A(j,j) = 0 for some j), then
a non-trivial solution to A*X = 0 is returned. If the system is
so badly scaled that the solution cannot be represented as
(1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned.
This is a BLAS-3 version of LATRS for solving several right
hand sides simultaneously.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
TRANS
TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N': Solve A * x = s*b (No transpose)
= 'T': Solve A**T* x = s*b (Transpose)
= 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
NORMIN
NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y': CNORM contains the column norms on entry
= 'N': CNORM is not set on entry. On exit, the norms will
be computed and stored in CNORM.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of columns of X. NRHS >= 0.
A
A is DOUBLE PRECISION array, dimension (LDA,N)
The triangular matrix A. If UPLO = 'U', the leading n by n
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If UPLO = 'L', the leading n by n lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If DIAG = 'U', the diagonal elements of A are
also not referenced and are assumed to be 1.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max (1,N).
X
X is DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the right hand side B of the triangular system.
On exit, X is overwritten by the solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max (1,N).
SCALE
SCALE is DOUBLE PRECISION array, dimension (NRHS)
The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k).
If SCALE = 0, the matrix A is singular or badly scaled.
If A(j,j) = 0 is encountered, a non-trivial vector x(:,k)
that is an exact or approximate solution to A*x(:,k) = 0
is returned. If the system so badly scaled that solution
cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0
is returned.
CNORM
CNORM is DOUBLE PRECISION array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A. If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
WORK
WORK is DOUBLE PRECISION array, dimension (LWORK).
On exit, if INFO = 0, WORK(1) returns the optimal size of
WORK.
LWORK LWORK is INTEGER LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N + NB - 1)/NB and NB is the optimal block size.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions 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.
Parameters
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
subroutine slatrs3 (character uplo, character trans, character diag, character normin, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) scale, real, dimension( * ) cnorm, real, dimension( * ) work, integer lwork, integer info)¶
SLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow.
Purpose:
SLATRS3 solves one of the triangular systems
A * X = B * diag(scale) or A**T * X = B * diag(scale)
with scaling to prevent overflow. Here A is an upper or lower
triangular matrix, A**T denotes the transpose of A. X and B are
n by nrhs matrices and scale is an nrhs element vector of scaling
factors. A scaling factor scale(j) is usually less than or equal
to 1, chosen such that X(:,j) is less than the overflow threshold.
If the matrix A is singular (A(j,j) = 0 for some j), then
a non-trivial solution to A*X = 0 is returned. If the system is
so badly scaled that the solution cannot be represented as
(1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned.
This is a BLAS-3 version of LATRS for solving several right
hand sides simultaneously.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
TRANS
TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N': Solve A * x = s*b (No transpose)
= 'T': Solve A**T* x = s*b (Transpose)
= 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
NORMIN
NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y': CNORM contains the column norms on entry
= 'N': CNORM is not set on entry. On exit, the norms will
be computed and stored in CNORM.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of columns of X. NRHS >= 0.
A
A is REAL array, dimension (LDA,N)
The triangular matrix A. If UPLO = 'U', the leading n by n
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If UPLO = 'L', the leading n by n lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If DIAG = 'U', the diagonal elements of A are
also not referenced and are assumed to be 1.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max (1,N).
X
X is REAL array, dimension (LDX,NRHS)
On entry, the right hand side B of the triangular system.
On exit, X is overwritten by the solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max (1,N).
SCALE
SCALE is REAL array, dimension (NRHS)
The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k).
If SCALE = 0, the matrix A is singular or badly scaled.
If A(j,j) = 0 is encountered, a non-trivial vector x(:,k)
that is an exact or approximate solution to A*x(:,k) = 0
is returned. If the system so badly scaled that solution
cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0
is returned.
CNORM
CNORM is REAL array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A. If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
WORK
WORK is REAL array, dimension (LWORK).
On exit, if INFO = 0, WORK(1) returns the optimal size of
WORK.
LWORK LWORK is INTEGER LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N + NB - 1)/NB and NB is the optimal block size.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions 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.
Parameters
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
subroutine zlatrs3 (character uplo, character trans, character diag, character normin, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) scale, double precision, dimension( * ) cnorm, double precision, dimension( * ) work, integer lwork, integer info)¶
ZLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow.
Purpose:
ZLATRS3 solves one of the triangular systems
A * X = B * diag(scale), A**T * X = B * diag(scale), or
A**H * X = B * diag(scale)
with scaling to prevent overflow. Here A is an upper or lower
triangular matrix, A**T denotes the transpose of A, A**H denotes the
conjugate transpose of A. X and B are n-by-nrhs matrices and scale
is an nrhs-element vector of scaling factors. A scaling factor scale(j)
is usually less than or equal to 1, chosen such that X(:,j) is less
than the overflow threshold. If the matrix A is singular (A(j,j) = 0
for some j), then a non-trivial solution to A*X = 0 is returned. If
the system is so badly scaled that the solution cannot be represented
as (1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned.
This is a BLAS-3 version of LATRS for solving several right
hand sides simultaneously.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
TRANS
TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N': Solve A * x = s*b (No transpose)
= 'T': Solve A**T* x = s*b (Transpose)
= 'C': Solve A**T* x = s*b (Conjugate transpose)
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
NORMIN
NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y': CNORM contains the column norms on entry
= 'N': CNORM is not set on entry. On exit, the norms will
be computed and stored in CNORM.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of columns of X. NRHS >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
The triangular matrix A. If UPLO = 'U', the leading n by n
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If UPLO = 'L', the leading n by n lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If DIAG = 'U', the diagonal elements of A are
also not referenced and are assumed to be 1.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max (1,N).
X
X is COMPLEX*16 array, dimension (LDX,NRHS)
On entry, the right hand side B of the triangular system.
On exit, X is overwritten by the solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max (1,N).
SCALE
SCALE is DOUBLE PRECISION array, dimension (NRHS)
The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k).
If SCALE = 0, the matrix A is singular or badly scaled.
If A(j,j) = 0 is encountered, a non-trivial vector x(:,k)
that is an exact or approximate solution to A*x(:,k) = 0
is returned. If the system so badly scaled that solution
cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0
is returned.
CNORM
CNORM is DOUBLE PRECISION array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A. If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
WORK
WORK is DOUBLE PRECISION array, dimension (LWORK).
On exit, if INFO = 0, WORK(1) returns the optimal size of
WORK.
LWORK LWORK is INTEGER LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N + NB - 1)/NB and NB is the optimal block size.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions 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.
Parameters
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Author¶
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