 LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
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Functions/Subroutines

DOUBLE PRECISION function dla_porcond (UPLO, N, A, LDA, AF, LDAF, CMODE, C, INFO, WORK, IWORK)
DLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite matrix.
subroutine dla_porfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
DOUBLE PRECISION function dla_porpvgrw (UPLO, NCOLS, A, LDA, AF, LDAF, WORK)
DLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
subroutine dpocon (UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
DPOCON
subroutine dpoequ (N, A, LDA, S, SCOND, AMAX, INFO)
DPOEQU
subroutine dpoequb (N, A, LDA, S, SCOND, AMAX, INFO)
DPOEQUB
subroutine dporfs (UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DPORFS
subroutine dporfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
DPORFSX
subroutine dpotf2 (UPLO, N, A, LDA, INFO)
DPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
subroutine dpotrf (UPLO, N, A, LDA, INFO)
DPOTRF
subroutine dpotri (UPLO, N, A, LDA, INFO)
DPOTRI
subroutine dpotrs (UPLO, N, NRHS, A, LDA, B, LDB, INFO)
DPOTRS

Detailed Description

This is the group of double computational functions for PO matrices

Function/Subroutine Documentation

 DOUBLE PRECISION function dla_porcond ( character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer CMODE, double precision, dimension( * ) C, integer INFO, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK )

DLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite matrix.

Purpose:
DLA_PORCOND 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:
 [in] UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. [in] N N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. [in] A A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N matrix A. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [in] AF AF is DOUBLE PRECISION array, dimension (LDAF,N) The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF. [in] LDAF LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). [in] 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) [in] C C is DOUBLE PRECISION array, dimension (N) The vector C in the formula op(A) * op2(C). [out] INFO INFO is INTEGER = 0: Successful exit. i > 0: The ith argument is invalid. [in] WORK WORK is DOUBLE PRECISION array, dimension (3*N). Workspace. [in] IWORK IWORK is INTEGER array, dimension (N). Workspace.
Date:
September 2012

Definition at line 141 of file dla_porcond.f.

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 subroutine dla_porfsx_extended ( integer PREC_TYPE, character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, logical COLEQU, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldy, * ) Y, integer LDY, double precision, dimension( * ) BERR_OUT, integer N_NORMS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, double precision, dimension( * ) RES, double precision, dimension(*) AYB, double precision, dimension( * ) DY, double precision, dimension( * ) Y_TAIL, double precision RCOND, integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO )

DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:
DLA_PORFSX_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 DPORFSX to perform iterative refinement.
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. Note that this
subroutine is only resonsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.
Parameters:
Date:
September 2012

Definition at line 385 of file dla_porfsx_extended.f.

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 DOUBLE PRECISION function dla_porpvgrw ( character*1 UPLO, integer NCOLS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, double precision, dimension( * ) WORK )

DLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.

Purpose:
DLA_PORPVGRW 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:
 [in] UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. [in] NCOLS NCOLS is INTEGER The number of columns of the matrix A. NCOLS >= 0. [in] A A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N matrix A. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [in] AF AF is DOUBLE PRECISION array, dimension (LDAF,N) The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF. [in] LDAF LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). [in] WORK WORK is DOUBLE PRECISION array, dimension (2*N)
Date:
September 2012

Definition at line 106 of file dla_porpvgrw.f.

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 subroutine dpocon ( character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

DPOCON

Purpose:
DPOCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite matrix using the
Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
Parameters:
 [in] UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. [in] N N is INTEGER The order of the matrix A. N >= 0. [in] A A is DOUBLE PRECISION array, dimension (LDA,N) The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [in] ANORM ANORM is DOUBLE PRECISION The 1-norm (or infinity-norm) of the symmetric matrix A. [out] RCOND RCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine. [out] WORK WORK is DOUBLE PRECISION array, dimension (3*N) [out] IWORK IWORK is INTEGER array, dimension (N) [out] INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Date:
November 2011

Definition at line 121 of file dpocon.f.

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 subroutine dpoequ ( integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO )

DPOEQU

Purpose:
DPOEQU computes row and column scalings intended to equilibrate a
symmetric positive definite matrix A and reduce its condition number
(with respect to the two-norm).  S contains the scale factors,
S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal
scalings.
Parameters:
 [in] N N is INTEGER The order of the matrix A. N >= 0. [in] A A is DOUBLE PRECISION array, dimension (LDA,N) The N-by-N symmetric positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [out] S S is DOUBLE PRECISION array, dimension (N) If INFO = 0, S contains the scale factors for A. [out] SCOND SCOND is DOUBLE PRECISION If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S. [out] AMAX AMAX is DOUBLE PRECISION Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. [out] INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the i-th diagonal element is nonpositive.
Date:
November 2011

Definition at line 113 of file dpoequ.f.

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 subroutine dpoequb ( integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO )

DPOEQUB

Purpose:
DPOEQU computes row and column scalings intended to equilibrate a
symmetric positive definite matrix A and reduce its condition number
(with respect to the two-norm).  S contains the scale factors,
S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal
scalings.
Parameters:
 [in] N N is INTEGER The order of the matrix A. N >= 0. [in] A A is DOUBLE PRECISION array, dimension (LDA,N) The N-by-N symmetric positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [out] S S is DOUBLE PRECISION array, dimension (N) If INFO = 0, S contains the scale factors for A. [out] SCOND SCOND is DOUBLE PRECISION If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S. [out] AMAX AMAX is DOUBLE PRECISION Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. [out] INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the i-th diagonal element is nonpositive.
Date:
November 2011

Definition at line 113 of file dpoequb.f.

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 subroutine dporfs ( character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

DPORFS

Purpose:
DPORFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive definite,
and provides error bounds and backward error estimates for the
solution.
Parameters:
 [in] UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. [in] N N is INTEGER The order of the matrix A. N >= 0. [in] NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. [in] A A is DOUBLE PRECISION array, dimension (LDA,N) The symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [in] AF AF is DOUBLE PRECISION array, dimension (LDAF,N) The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF. [in] LDAF LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). [in] B B is DOUBLE PRECISION array, dimension (LDB,NRHS) The right hand side matrix B. [in] LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). [in,out] X X is DOUBLE PRECISION array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by DPOTRS. On exit, the improved solution matrix X. [in] LDX LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). [out] FERR FERR is DOUBLE PRECISION 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. [out] BERR BERR is DOUBLE PRECISION 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). [out] WORK WORK is DOUBLE PRECISION array, dimension (3*N) [out] IWORK IWORK is INTEGER array, dimension (N) [out] 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.
Date:
November 2011

Definition at line 183 of file dporfs.f.

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 subroutine dporfsx ( character UPLO, character EQUED, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, double precision, dimension( * ) S, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) BERR, integer N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double precision, dimension( * ) PARAMS, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

DPORFSX

Purpose:
DPORFSX improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive
definite, 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 and S
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:
Date:
April 2012

Definition at line 392 of file dporfsx.f.

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 subroutine dpotf2 ( character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO )

DPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).

Purpose:
DPOTF2 computes the Cholesky factorization of a real symmetric
positive definite matrix A.

The factorization has the form
A = U**T * U ,  if UPLO = 'U', or
A = L  * L**T,  if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.

This is the unblocked version of the algorithm, calling Level 2 BLAS.
Parameters:
 [in] UPLO UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular [in] N N is INTEGER The order of the matrix A. N >= 0. [in,out] A A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading n by n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if INFO = 0, the factor U or L from the Cholesky factorization A = U**T *U or A = L*L**T. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [out] INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, the leading minor of order k is not positive definite, and the factorization could not be completed.
Date:
September 2012

Definition at line 110 of file dpotf2.f.

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 subroutine dpotrf ( character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO )

DPOTRF

Purpose:
DPOTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A.

The factorization has the form
A = U**T * U,  if UPLO = 'U', or
A = L  * L**T,  if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.

This is the block version of the algorithm, calling Level 3 BLAS.
Parameters:
 [in] UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. [in] N N is INTEGER The order of the matrix A. N >= 0. [in,out] A A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if INFO = 0, the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [out] INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.
Date:
November 2011

Definition at line 108 of file dpotrf.f.

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 subroutine dpotri ( character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO )

DPOTRI

Purpose:
DPOTRI computes the inverse of a real symmetric positive definite
matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
computed by DPOTRF.
Parameters:
 [in] UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. [in] N N is INTEGER The order of the matrix A. N >= 0. [in,out] A A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF. On exit, the upper or lower triangle of the (symmetric) inverse of A, overwriting the input factor U or L. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [out] INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed.
Date:
November 2011

Definition at line 96 of file dpotri.f.

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 subroutine dpotrs ( character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer INFO )

DPOTRS

Purpose:
DPOTRS solves a system of linear equations A*X = B with a symmetric
positive definite matrix A using the Cholesky factorization
A = U**T*U or A = L*L**T computed by DPOTRF.
Parameters:
 [in] UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. [in] N N is INTEGER The order of the matrix A. N >= 0. [in] NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. [in] A A is DOUBLE PRECISION array, dimension (LDA,N) The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [in,out] B B is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X. [in] LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). [out] INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Date:
November 2011

Definition at line 111 of file dpotrs.f.

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