Man page - hesvx(3)

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Manual

hesvx

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine chesvx (character fact, character uplo, integer n, integer nrhs,complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf,* ) af, integer ldaf, integer, dimension( * ) ipiv, complex, dimension(ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx,real rcond, real, dimension( * ) ferr, real, dimension( * ) berr,complex, dimension( * ) work, integer lwork, real, dimension( * )rwork, integer info)
subroutine csysvx (character fact, character uplo, integer n, integer nrhs,complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf,* ) af, integer ldaf, integer, dimension( * ) ipiv, complex, dimension(ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx,real rcond, real, dimension( * ) ferr, real, dimension( * ) berr,complex, dimension( * ) work, integer lwork, real, dimension( * )rwork, integer info)
subroutine dsysvx (character fact, character uplo, integer n, integer nrhs,double precision, dimension( lda, * ) a, integer lda, double precision,dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv,double precision, dimension( ldb, * ) b, integer ldb, double precision,dimension( ldx, * ) x, integer ldx, double precision rcond, doubleprecision, dimension( * ) ferr, double precision, dimension( * ) berr,double precision, dimension( * ) work, integer lwork, integer,dimension( * ) iwork, integer info)
subroutine ssysvx (character fact, character uplo, 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 rcond,real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension(* ) work, integer lwork, integer, dimension( * ) iwork, integer info)
subroutine zhesvx (character fact, character uplo, integer n, integer nrhs,complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex*16,dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x,integer ldx, double precision rcond, double precision, dimension( * )ferr, double precision, dimension( * ) berr, complex*16, dimension( * )work, integer lwork, double precision, dimension( * ) rwork, integerinfo)
subroutine zsysvx (character fact, character uplo, integer n, integer nrhs,complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex*16,dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x,integer ldx, double precision rcond, double precision, dimension( * )ferr, double precision, dimension( * ) berr, complex*16, dimension( * )work, integer lwork, double precision, dimension( * ) rwork, integerinfo)
Author

NAME

hesvx - {he,sy}svx: rook (v1, expert)

SYNOPSIS

Functions

subroutine chesvx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info)
CHESVX computes the solution to system of linear equations A * X = B for HE matrices
subroutine csysvx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info)
CSYSVX computes the solution to system of linear equations A * X = B for SY matrices
subroutine dsysvx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, iwork, info)
DSYSVX computes the solution to system of linear equations A * X = B for SY matrices
subroutine ssysvx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, iwork, info)
SSYSVX computes the solution to system of linear equations A * X = B for SY matrices
subroutine zhesvx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info)
ZHESVX computes the solution to system of linear equations A * X = B for HE matrices
subroutine zsysvx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info)
ZSYSVX computes the solution to system of linear equations A * X = B for SY matrices

Detailed Description

Function Documentation

subroutine chesvx (character fact, character uplo, integer n, integer nrhs,complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf,* ) af, integer ldaf, integer, dimension( * ) ipiv, complex, dimension(ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx,real rcond, real, dimension( * ) ferr, real, dimension( * ) berr,complex, dimension( * ) work, integer lwork, real, dimension( * )rwork, integer info)

CHESVX computes the solution to system of linear equations A * X = B for HE matrices

Purpose:

CHESVX uses the diagonal pivoting factorization to compute the
solution to a complex system of linear equations A * X = B,
where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
matrices.

Error bounds on the solution and a condition estimate are also
provided.

Description:

The following steps are performed:

1. If FACT = ā€™Nā€™, the diagonal pivoting method is used to factor A.
The form of the factorization is
A = U * D * U**H, if UPLO = ā€™Uā€™, or
A = L * D * L**H, if UPLO = ā€™Lā€™,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is Hermitian and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D is exactly singular, then the routine
returns with INFO = i. Otherwise, the factored form of A is used
to estimate the condition number of the matrix A. If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form
of A.

4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

Parameters

FACT

FACT is CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= ā€™Fā€™: On entry, AF and IPIV contain the factored form
of A. A, AF and IPIV will not be modified.
= ā€™Nā€™: The matrix A will be copied to AF and factored.

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

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 matrices B and X. NRHS >= 0.

A

A is COMPLEX array, dimension (LDA,N)
The Hermitian matrix A. If UPLO = ā€™Uā€™, 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 = ā€™Lā€™, 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.

LDA

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

AF

AF is COMPLEX array, dimension (LDAF,N)
If FACT = ā€™Fā€™, then AF is an input argument and on entry
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**H or A = L*D*L**H as computed by CHETRF.

If FACT = ā€™Nā€™, then AF is an output argument and on exit
returns the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**H or A = L*D*L**H.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
If FACT = ā€™Fā€™, then IPIV is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by CHETRF.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = ā€™Uā€™ and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = ā€™Lā€™ and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = ā€™Nā€™, then IPIV is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by CHETRF.

B

B is COMPLEX array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.

LDB

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

X

X is COMPLEX array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

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

RCOND

RCOND is REAL
The estimate of the reciprocal condition number of the matrix
A. If RCOND is less than the machine precision (in
particular, if RCOND = 0), the matrix is singular to working
precision. This condition is indicated by a return code of
INFO > 0.

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 COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= max(1,2*N), and for best
performance, when FACT = ā€™Nā€™, LWORK >= max(1,2*N,N*NB), where
NB is the optimal blocksize for CHETRF.

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.

RWORK

RWORK is REAL array, dimension (N)

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
<= N: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned.
= N+1: D is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine csysvx (character fact, character uplo, integer n, integer nrhs,complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf,* ) af, integer ldaf, integer, dimension( * ) ipiv, complex, dimension(ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx,real rcond, real, dimension( * ) ferr, real, dimension( * ) berr,complex, dimension( * ) work, integer lwork, real, dimension( * )rwork, integer info)

CSYSVX computes the solution to system of linear equations A * X = B for SY matrices

Purpose:

CSYSVX uses the diagonal pivoting factorization to compute the
solution to a complex system of linear equations A * X = B,
where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.

Error bounds on the solution and a condition estimate are also
provided.

Description:

The following steps are performed:

1. If FACT = ā€™Nā€™, the diagonal pivoting method is used to factor A.
The form of the factorization is
A = U * D * U**T, if UPLO = ā€™Uā€™, or
A = L * D * L**T, if UPLO = ā€™Lā€™,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D is exactly singular, then the routine
returns with INFO = i. Otherwise, the factored form of A is used
to estimate the condition number of the matrix A. If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form
of A.

4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

Parameters

FACT

FACT is CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= ā€™Fā€™: On entry, AF and IPIV contain the factored form
of A. A, AF and IPIV will not be modified.
= ā€™Nā€™: The matrix A will be copied to AF and factored.

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

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 matrices B and X. NRHS >= 0.

A

A is COMPLEX array, dimension (LDA,N)
The symmetric matrix A. If UPLO = ā€™Uā€™, 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 = ā€™Lā€™, 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.

LDA

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

AF

AF is COMPLEX array, dimension (LDAF,N)
If FACT = ā€™Fā€™, then AF is an input argument and on entry
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T as computed by CSYTRF.

If FACT = ā€™Nā€™, then AF is an output argument and on exit
returns the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
If FACT = ā€™Fā€™, then IPIV is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by CSYTRF.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = ā€™Uā€™ and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = ā€™Lā€™ and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = ā€™Nā€™, then IPIV is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by CSYTRF.

B

B is COMPLEX array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.

LDB

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

X

X is COMPLEX array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

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

RCOND

RCOND is REAL
The estimate of the reciprocal condition number of the matrix
A. If RCOND is less than the machine precision (in
particular, if RCOND = 0), the matrix is singular to working
precision. This condition is indicated by a return code of
INFO > 0.

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 COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= max(1,2*N), and for best
performance, when FACT = ā€™Nā€™, LWORK >= max(1,2*N,N*NB), where
NB is the optimal blocksize for CSYTRF.

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.

RWORK

RWORK is REAL array, dimension (N)

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
<= N: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned.
= N+1: D is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsysvx (character fact, character uplo, integer n, integer nrhs,double precision, dimension( lda, * ) a, integer lda, double precision,dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv,double precision, dimension( ldb, * ) b, integer ldb, double precision,dimension( ldx, * ) x, integer ldx, double precision rcond, doubleprecision, dimension( * ) ferr, double precision, dimension( * ) berr,double precision, dimension( * ) work, integer lwork, integer,dimension( * ) iwork, integer info)

DSYSVX computes the solution to system of linear equations A * X = B for SY matrices

Purpose:

DSYSVX uses the diagonal pivoting factorization to compute the
solution to a real system of linear equations A * X = B,
where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.

Error bounds on the solution and a condition estimate are also
provided.

Description:

The following steps are performed:

1. If FACT = ā€™Nā€™, the diagonal pivoting method is used to factor A.
The form of the factorization is
A = U * D * U**T, if UPLO = ā€™Uā€™, or
A = L * D * L**T, if UPLO = ā€™Lā€™,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D is exactly singular, then the routine
returns with INFO = i. Otherwise, the factored form of A is used
to estimate the condition number of the matrix A. If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form
of A.

4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

Parameters

FACT

FACT is CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= ā€™Fā€™: On entry, AF and IPIV contain the factored form of
A. AF and IPIV will not be modified.
= ā€™Nā€™: The matrix A will be copied to AF and factored.

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

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 matrices B and X. NRHS >= 0.

A

A is DOUBLE PRECISION array, dimension (LDA,N)
The symmetric matrix A. If UPLO = ā€™Uā€™, 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 = ā€™Lā€™, 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.

LDA

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

AF

AF is DOUBLE PRECISION array, dimension (LDAF,N)
If FACT = ā€™Fā€™, then AF is an input argument and on entry
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.

If FACT = ā€™Nā€™, then AF is an output argument and on exit
returns the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
If FACT = ā€™Fā€™, then IPIV is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by DSYTRF.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = ā€™Uā€™ and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = ā€™Lā€™ and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = ā€™Nā€™, then IPIV is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by DSYTRF.

B

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.

LDB

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

X

X is DOUBLE PRECISION array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

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

RCOND

RCOND is DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix
A. If RCOND is less than the machine precision (in
particular, if RCOND = 0), the matrix is singular to working
precision. This condition is indicated by a return code of
INFO > 0.

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.

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).

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= max(1,3*N), and for best
performance, when FACT = ā€™Nā€™, LWORK >= max(1,3*N,N*NB), where
NB is the optimal blocksize for DSYTRF.

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.

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
> 0: if INFO = i, and i is
<= N: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned.
= N+1: D is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine ssysvx (character fact, character uplo, 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 rcond,real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension(* ) work, integer lwork, integer, dimension( * ) iwork, integer info)

SSYSVX computes the solution to system of linear equations A * X = B for SY matrices

Purpose:

SSYSVX uses the diagonal pivoting factorization to compute the
solution to a real system of linear equations A * X = B,
where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.

Error bounds on the solution and a condition estimate are also
provided.

Description:

The following steps are performed:

1. If FACT = ā€™Nā€™, the diagonal pivoting method is used to factor A.
The form of the factorization is
A = U * D * U**T, if UPLO = ā€™Uā€™, or
A = L * D * L**T, if UPLO = ā€™Lā€™,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D is exactly singular, then the routine
returns with INFO = i. Otherwise, the factored form of A is used
to estimate the condition number of the matrix A. If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form
of A.

4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

Parameters

FACT

FACT is CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= ā€™Fā€™: On entry, AF and IPIV contain the factored form of
A. AF and IPIV will not be modified.
= ā€™Nā€™: The matrix A will be copied to AF and factored.

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

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 matrices B and X. NRHS >= 0.

A

A is REAL array, dimension (LDA,N)
The symmetric matrix A. If UPLO = ā€™Uā€™, 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 = ā€™Lā€™, 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.

LDA

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

AF

AF is REAL array, dimension (LDAF,N)
If FACT = ā€™Fā€™, then AF is an input argument and on entry
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T as computed by SSYTRF.

If FACT = ā€™Nā€™, then AF is an output argument and on exit
returns the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
If FACT = ā€™Fā€™, then IPIV is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by SSYTRF.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = ā€™Uā€™ and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = ā€™Lā€™ and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = ā€™Nā€™, then IPIV is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by SSYTRF.

B

B is REAL array, dimension (LDB,NRHS)
The N-by-NRHS 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)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

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

RCOND

RCOND is REAL
The estimate of the reciprocal condition number of the matrix
A. If RCOND is less than the machine precision (in
particular, if RCOND = 0), the matrix is singular to working
precision. This condition is indicated by a return code of
INFO > 0.

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 (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= max(1,3*N), and for best
performance, when FACT = ā€™Nā€™, LWORK >= max(1,3*N,N*NB), where
NB is the optimal blocksize for SSYTRF.

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.

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
> 0: if INFO = i, and i is
<= N: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned.
= N+1: D is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zhesvx (character fact, character uplo, integer n, integer nrhs,complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex*16,dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x,integer ldx, double precision rcond, double precision, dimension( * )ferr, double precision, dimension( * ) berr, complex*16, dimension( * )work, integer lwork, double precision, dimension( * ) rwork, integerinfo)

ZHESVX computes the solution to system of linear equations A * X = B for HE matrices

Purpose:

ZHESVX uses the diagonal pivoting factorization to compute the
solution to a complex system of linear equations A * X = B,
where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
matrices.

Error bounds on the solution and a condition estimate are also
provided.

Description:

The following steps are performed:

1. If FACT = ā€™Nā€™, the diagonal pivoting method is used to factor A.
The form of the factorization is
A = U * D * U**H, if UPLO = ā€™Uā€™, or
A = L * D * L**H, if UPLO = ā€™Lā€™,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is Hermitian and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D is exactly singular, then the routine
returns with INFO = i. Otherwise, the factored form of A is used
to estimate the condition number of the matrix A. If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form
of A.

4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

Parameters

FACT

FACT is CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= ā€™Fā€™: On entry, AF and IPIV contain the factored form
of A. A, AF and IPIV will not be modified.
= ā€™Nā€™: The matrix A will be copied to AF and factored.

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

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 matrices B and X. NRHS >= 0.

A

A is COMPLEX*16 array, dimension (LDA,N)
The Hermitian matrix A. If UPLO = ā€™Uā€™, 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 = ā€™Lā€™, 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.

LDA

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

AF

AF is COMPLEX*16 array, dimension (LDAF,N)
If FACT = ā€™Fā€™, then AF is an input argument and on entry
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**H or A = L*D*L**H as computed by ZHETRF.

If FACT = ā€™Nā€™, then AF is an output argument and on exit
returns the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**H or A = L*D*L**H.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
If FACT = ā€™Fā€™, then IPIV is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by ZHETRF.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = ā€™Uā€™ and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = ā€™Lā€™ and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = ā€™Nā€™, then IPIV is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by ZHETRF.

B

B is COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.

LDB

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

X

X is COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

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

RCOND

RCOND is DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix
A. If RCOND is less than the machine precision (in
particular, if RCOND = 0), the matrix is singular to working
precision. This condition is indicated by a return code of
INFO > 0.

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.

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).

WORK

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= MAX(1,2*N), and for best
performance, when FACT = ā€™Nā€™, LWORK >= MAX(1,2*N,N*NB), where
NB is the optimal blocksize for ZHETRF.

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.

RWORK

RWORK is DOUBLE PRECISION array, dimension (N)

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
<= N: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned.
= N+1: D is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zsysvx (character fact, character uplo, integer n, integer nrhs,complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex*16,dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x,integer ldx, double precision rcond, double precision, dimension( * )ferr, double precision, dimension( * ) berr, complex*16, dimension( * )work, integer lwork, double precision, dimension( * ) rwork, integerinfo)

ZSYSVX computes the solution to system of linear equations A * X = B for SY matrices

Purpose:

ZSYSVX uses the diagonal pivoting factorization to compute the
solution to a complex system of linear equations A * X = B,
where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.

Error bounds on the solution and a condition estimate are also
provided.

Description:

The following steps are performed:

1. If FACT = ā€™Nā€™, the diagonal pivoting method is used to factor A.
The form of the factorization is
A = U * D * U**T, if UPLO = ā€™Uā€™, or
A = L * D * L**T, if UPLO = ā€™Lā€™,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D is exactly singular, then the routine
returns with INFO = i. Otherwise, the factored form of A is used
to estimate the condition number of the matrix A. If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form
of A.

4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

Parameters

FACT

FACT is CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= ā€™Fā€™: On entry, AF and IPIV contain the factored form
of A. A, AF and IPIV will not be modified.
= ā€™Nā€™: The matrix A will be copied to AF and factored.

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

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 matrices B and X. NRHS >= 0.

A

A is COMPLEX*16 array, dimension (LDA,N)
The symmetric matrix A. If UPLO = ā€™Uā€™, 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 = ā€™Lā€™, 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.

LDA

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

AF

AF is COMPLEX*16 array, dimension (LDAF,N)
If FACT = ā€™Fā€™, then AF is an input argument and on entry
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T as computed by ZSYTRF.

If FACT = ā€™Nā€™, then AF is an output argument and on exit
returns the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
If FACT = ā€™Fā€™, then IPIV is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by ZSYTRF.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = ā€™Uā€™ and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = ā€™Lā€™ and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = ā€™Nā€™, then IPIV is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by ZSYTRF.

B

B is COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.

LDB

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

X

X is COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

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

RCOND

RCOND is DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix
A. If RCOND is less than the machine precision (in
particular, if RCOND = 0), the matrix is singular to working
precision. This condition is indicated by a return code of
INFO > 0.

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.

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).

WORK

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= max(1,2*N), and for best
performance, when FACT = ā€™Nā€™, LWORK >= max(1,2*N,N*NB), where
NB is the optimal blocksize for ZSYTRF.

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.

RWORK

RWORK is DOUBLE PRECISION array, dimension (N)

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
<= N: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned.
= N+1: D is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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