Library Functions Manual

NAME¶

hprfs - {hp,sp}rfs: iterative refinement

SYNOPSIS¶

Functions¶

subroutine CHPRFS (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CHPRFS subroutine CSPRFS (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CSPRFS subroutine DSPRFS (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DSPRFS subroutine SSPRFS (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SSPRFS subroutine ZHPRFS (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZHPRFS subroutine ZSPRFS (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZSPRFS

Detailed Description¶

Function Documentation¶

subroutine CHPRFS (character uplo, integer n, integer nrhs, complex, dimension( * ) ap, complex, dimension( * ) afp, 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)¶

CHPRFS

Purpose:

!>
!> CHPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is Hermitian indefinite
!> and packed, 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 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.
!>

!>          AP is COMPLEX array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the Hermitian matrix A, packed
!>          columnwise in a linear array.  The j-th column of A is stored
!>          in the array AP as follows:
!>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
!>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
!>

AFP

!>          AFP is COMPLEX array, dimension (N*(N+1)/2)
!>          The factored form of the matrix A.  AFP 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 CHPTRF, stored as a packed
!>          triangular matrix.
!>

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D
!>          as determined by CHPTRF.
!>

!>          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 is COMPLEX array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by CHPTRS.
!>          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.

Definition at line 178 of file chprfs.f.

subroutine CSPRFS (character uplo, integer n, integer nrhs, complex, dimension( * ) ap, complex, dimension( * ) afp, 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)¶

CSPRFS

Purpose:

!>
!> CSPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric indefinite
!> and packed, 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 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.
!>

!>          AP is COMPLEX array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the symmetric matrix A, packed
!>          columnwise in a linear array.  The j-th column of A is stored
!>          in the array AP as follows:
!>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
!>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
!>

AFP

!>          AFP is COMPLEX array, dimension (N*(N+1)/2)
!>          The factored form of the matrix A.  AFP 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 CSPTRF, stored as a packed
!>          triangular matrix.
!>

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D
!>          as determined by CSPTRF.
!>

!>          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 is COMPLEX array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by CSPTRS.
!>          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.

Definition at line 178 of file csprfs.f.

subroutine DSPRFS (character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( * ) afp, 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)¶

DSPRFS

Purpose:

!>
!> DSPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric indefinite
!> and packed, 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 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.
!>

!>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the symmetric matrix A, packed
!>          columnwise in a linear array.  The j-th column of A is stored
!>          in the array AP as follows:
!>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
!>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
!>

AFP

!>          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
!>          The factored form of the matrix A.  AFP 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 DSPTRF, stored as a packed
!>          triangular matrix.
!>

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D
!>          as determined by DSPTRF.
!>

!>          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 is DOUBLE PRECISION array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by DSPTRS.
!>          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.

Definition at line 177 of file dsprfs.f.

subroutine SSPRFS (character uplo, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( * ) afp, 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)¶

SSPRFS

Purpose:

!>
!> SSPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric indefinite
!> and packed, 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 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.
!>

!>          AP is REAL array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the symmetric matrix A, packed
!>          columnwise in a linear array.  The j-th column of A is stored
!>          in the array AP as follows:
!>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
!>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
!>

AFP

!>          AFP is REAL array, dimension (N*(N+1)/2)
!>          The factored form of the matrix A.  AFP 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 SSPTRF, stored as a packed
!>          triangular matrix.
!>

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D
!>          as determined by SSPTRF.
!>

!>          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 is REAL array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by SSPTRS.
!>          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.

Definition at line 177 of file ssprfs.f.

subroutine ZHPRFS (character uplo, integer n, integer nrhs, complex16, dimension( ) ap, complex16, dimension( ) afp, integer, dimension( * ) ipiv, complex16, dimension( ldb, ) b, integer ldb, complex16, dimension( ldx, ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex16, dimension( ) work, double precision, dimension( * ) rwork, integer info)¶

ZHPRFS

Purpose:

!>
!> ZHPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is Hermitian indefinite
!> and packed, 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 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.
!>

!>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the Hermitian matrix A, packed
!>          columnwise in a linear array.  The j-th column of A is stored
!>          in the array AP as follows:
!>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
!>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
!>

AFP

!>          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
!>          The factored form of the matrix A.  AFP 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 ZHPTRF, stored as a packed
!>          triangular matrix.
!>

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D
!>          as determined by ZHPTRF.
!>

!>          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 is COMPLEX*16 array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by ZHPTRS.
!>          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.

Definition at line 178 of file zhprfs.f.

subroutine ZSPRFS (character uplo, integer n, integer nrhs, complex16, dimension( ) ap, complex16, dimension( ) afp, integer, dimension( * ) ipiv, complex16, dimension( ldb, ) b, integer ldb, complex16, dimension( ldx, ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex16, dimension( ) work, double precision, dimension( * ) rwork, integer info)¶

ZSPRFS

Purpose:

!>
!> ZSPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric indefinite
!> and packed, 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 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.
!>

!>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the symmetric matrix A, packed
!>          columnwise in a linear array.  The j-th column of A is stored
!>          in the array AP as follows:
!>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
!>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
!>

AFP

!>          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
!>          The factored form of the matrix A.  AFP 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 ZSPTRF, stored as a packed
!>          triangular matrix.
!>

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D
!>          as determined by ZSPTRF.
!>

!>          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 is COMPLEX*16 array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by ZSPTRS.
!>          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.

Definition at line 178 of file zsprfs.f.

Author¶

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