pbrfs(3)

Library Functions Manual

pbrfs(3)

NAME

pbrfs - pbrfs: iterative refinement

SYNOPSIS

Functions

subroutine cpbrfs (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CPBRFS subroutine dpbrfs (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DPBRFS subroutine spbrfs (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SPBRFS subroutine zpbrfs (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZPBRFS

Detailed Description

Function Documentation

subroutine cpbrfs (character uplo, integer n, integer kd, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, 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)

CPBRFS

Purpose:

!>
!> CPBRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is Hermitian positive definite
!> and banded, 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.
!>

!>          KD is INTEGER
!>          The number of superdiagonals of the matrix A if UPLO = 'U',
!>          or the number of subdiagonals if UPLO = 'L'.  KD >= 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.
!>

!>          AB is COMPLEX array, dimension (LDAB,N)
!>          The upper or lower triangle of the Hermitian band matrix A,
!>          stored in the first KD+1 rows of the array.  The j-th column
!>          of A is stored in the j-th column of the array AB as follows:
!>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
!>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
!>

LDAB

!>          LDAB is INTEGER
!>          The leading dimension of the array AB.  LDAB >= KD+1.
!>

AFB

!>          AFB is COMPLEX array, dimension (LDAFB,N)
!>          The triangular factor U or L from the Cholesky factorization
!>          A = U**H*U or A = L*L**H of the band matrix A as computed by
!>          CPBTRF, in the same storage format as A (see AB).
!>

LDAFB

!>          LDAFB is INTEGER
!>          The leading dimension of the array AFB.  LDAFB >= KD+1.
!>

!>          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 CPBTRS.
!>          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 187 of file cpbrfs.f.

subroutine dpbrfs (character uplo, integer n, integer kd, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldafb, * ) afb, integer ldafb, 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)

DPBRFS

Purpose:

!>
!> DPBRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric positive definite
!> and banded, 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.
!>

!>          KD is INTEGER
!>          The number of superdiagonals of the matrix A if UPLO = 'U',
!>          or the number of subdiagonals if UPLO = 'L'.  KD >= 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.
!>

!>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
!>          The upper or lower triangle of the symmetric band matrix A,
!>          stored in the first KD+1 rows of the array.  The j-th column
!>          of A is stored in the j-th column of the array AB as follows:
!>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
!>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
!>

LDAB

!>          LDAB is INTEGER
!>          The leading dimension of the array AB.  LDAB >= KD+1.
!>

AFB

!>          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
!>          The triangular factor U or L from the Cholesky factorization
!>          A = U**T*U or A = L*L**T of the band matrix A as computed by
!>          DPBTRF, in the same storage format as A (see AB).
!>

LDAFB

!>          LDAFB is INTEGER
!>          The leading dimension of the array AFB.  LDAFB >= KD+1.
!>

!>          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 DPBTRS.
!>          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 187 of file dpbrfs.f.

subroutine spbrfs (character uplo, integer n, integer kd, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldafb, * ) afb, integer ldafb, 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)

SPBRFS

Purpose:

!>
!> SPBRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric positive definite
!> and banded, 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.
!>

!>          KD is INTEGER
!>          The number of superdiagonals of the matrix A if UPLO = 'U',
!>          or the number of subdiagonals if UPLO = 'L'.  KD >= 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.
!>

!>          AB is REAL array, dimension (LDAB,N)
!>          The upper or lower triangle of the symmetric band matrix A,
!>          stored in the first KD+1 rows of the array.  The j-th column
!>          of A is stored in the j-th column of the array AB as follows:
!>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
!>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
!>

LDAB

!>          LDAB is INTEGER
!>          The leading dimension of the array AB.  LDAB >= KD+1.
!>

AFB

!>          AFB is REAL array, dimension (LDAFB,N)
!>          The triangular factor U or L from the Cholesky factorization
!>          A = U**T*U or A = L*L**T of the band matrix A as computed by
!>          SPBTRF, in the same storage format as A (see AB).
!>

LDAFB

!>          LDAFB is INTEGER
!>          The leading dimension of the array AFB.  LDAFB >= KD+1.
!>

!>          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 SPBTRS.
!>          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 187 of file spbrfs.f.

subroutine zpbrfs (character uplo, integer n, integer kd, integer nrhs, complex16, dimension( ldab, ) ab, integer ldab, complex16, dimension( ldafb, ) afb, integer ldafb, 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)

ZPBRFS

Purpose:

!>
!> ZPBRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is Hermitian positive definite
!> and banded, 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.
!>

!>          KD is INTEGER
!>          The number of superdiagonals of the matrix A if UPLO = 'U',
!>          or the number of subdiagonals if UPLO = 'L'.  KD >= 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.
!>

!>          AB is COMPLEX*16 array, dimension (LDAB,N)
!>          The upper or lower triangle of the Hermitian band matrix A,
!>          stored in the first KD+1 rows of the array.  The j-th column
!>          of A is stored in the j-th column of the array AB as follows:
!>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
!>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
!>

LDAB

!>          LDAB is INTEGER
!>          The leading dimension of the array AB.  LDAB >= KD+1.
!>

AFB

!>          AFB is COMPLEX*16 array, dimension (LDAFB,N)
!>          The triangular factor U or L from the Cholesky factorization
!>          A = U**H*U or A = L*L**H of the band matrix A as computed by
!>          ZPBTRF, in the same storage format as A (see AB).
!>

LDAFB

!>          LDAFB is INTEGER
!>          The leading dimension of the array AFB.  LDAFB >= KD+1.
!>

!>          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 ZPBTRS.
!>          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 187 of file zpbrfs.f.

Author

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