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herfs(3) LAPACK herfs(3)

NAME

herfs - {he,sy}rfs: iterative refinement

SYNOPSIS

Functions


subroutine cherfs (uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CHERFS subroutine csyrfs (uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CSYRFS subroutine dsyrfs (uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DSYRFS subroutine ssyrfs (uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SSYRFS subroutine zherfs (uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZHERFS subroutine zsyrfs (uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZSYRFS

Detailed Description

Function Documentation

subroutine cherfs (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, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CHERFS

Purpose:



 CHERFS improves the computed solution to a system of linear


 equations when the coefficient matrix is Hermitian indefinite, and


 provides error bounds and backward error estimates for the solution.

Parameters

UPLO 



          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

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 COMPLEX array, dimension (LDA,N)


          The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N


          upper triangular part of A contains the upper triangular part


          of the matrix A, and the strictly lower triangular part of A


          is not referenced.  If UPLO = 'L', the leading N-by-N lower


          triangular part of A contains the lower triangular part of


          the matrix A, and the strictly upper triangular part of A is


          not referenced.

LDA



          LDA is INTEGER


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

AF



          AF is COMPLEX array, dimension (LDAF,N)


          The factored form of the matrix A.  AF 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.

LDAF



          LDAF is INTEGER


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

IPIV



          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D


          as determined by CHETRF.

B



          B is COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)


          On entry, the solution matrix X, as computed by CHETRS.


          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 COMPLEX array, dimension (2*N)

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

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 csyrfs (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, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CSYRFS

Purpose:



 CSYRFS improves the computed solution to a system of linear


 equations when the coefficient matrix is symmetric indefinite, and


 provides error bounds and backward error estimates for the solution.

Parameters

UPLO 



          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

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 COMPLEX 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.

LDA



          LDA is INTEGER


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

AF



          AF is COMPLEX array, dimension (LDAF,N)


          The factored form of the matrix A.  AF 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.

LDAF



          LDAF is INTEGER


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

IPIV



          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D


          as determined by CSYTRF.

B



          B is COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)


          On entry, the solution matrix X, as computed by CSYTRS.


          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 COMPLEX array, dimension (2*N)

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

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 dsyrfs (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, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DSYRFS

Purpose:



 DSYRFS improves the computed solution to a system of linear


 equations when the coefficient matrix is symmetric indefinite, and


 provides error bounds and backward error estimates for the solution.

Parameters

UPLO 



          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

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 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.

LDA



          LDA is INTEGER


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

AF



          AF is DOUBLE PRECISION array, dimension (LDAF,N)


          The factored form of the matrix A.  AF 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.

LDAF



          LDAF is INTEGER


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

IPIV



          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D


          as determined by DSYTRF.

B



          B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)


          On entry, the solution matrix X, as computed by DSYTRS.


          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 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 (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 ssyrfs (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, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSYRFS

Purpose:



 SSYRFS improves the computed solution to a system of linear


 equations when the coefficient matrix is symmetric indefinite, and


 provides error bounds and backward error estimates for the solution.

Parameters

UPLO 



          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

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 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.

LDA



          LDA is INTEGER


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

AF



          AF is REAL array, dimension (LDAF,N)


          The factored form of the matrix A.  AF 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.

LDAF



          LDAF is INTEGER


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

IPIV



          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D


          as determined by SSYTRF.

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 SSYTRS.


          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 zherfs (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, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZHERFS

Purpose:



 ZHERFS improves the computed solution to a system of linear


 equations when the coefficient matrix is Hermitian indefinite, and


 provides error bounds and backward error estimates for the solution.

Parameters

UPLO 



          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

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 COMPLEX*16 array, dimension (LDA,N)


          The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N


          upper triangular part of A contains the upper triangular part


          of the matrix A, and the strictly lower triangular part of A


          is not referenced.  If UPLO = 'L', the leading N-by-N lower


          triangular part of A contains the lower triangular part of


          the matrix A, and the strictly upper triangular part of A is


          not referenced.

LDA



          LDA is INTEGER


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

AF



          AF is COMPLEX*16 array, dimension (LDAF,N)


          The factored form of the matrix A.  AF 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.

LDAF



          LDAF is INTEGER


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

IPIV



          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D


          as determined by ZHETRF.

B



          B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)


          On entry, the solution matrix X, as computed by ZHETRS.


          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 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 (2*N)

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

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 zsyrfs (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, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZSYRFS

Purpose:



 ZSYRFS improves the computed solution to a system of linear


 equations when the coefficient matrix is symmetric indefinite, and


 provides error bounds and backward error estimates for the solution.

Parameters

UPLO 



          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

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 COMPLEX*16 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.

LDA



          LDA is INTEGER


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

AF



          AF is COMPLEX*16 array, dimension (LDAF,N)


          The factored form of the matrix A.  AF 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.

LDAF



          LDAF is INTEGER


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

IPIV



          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D


          as determined by ZSYTRF.

B



          B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)


          On entry, the solution matrix X, as computed by ZSYTRS.


          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 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 (2*N)

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

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.

Author

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