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pbrfs(3) LAPACK 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



          N is INTEGER


          The order of the matrix A.  N >= 0.

KD



          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



          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



          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 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.

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



          N is INTEGER


          The order of the matrix A.  N >= 0.

KD



          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



          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



          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 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.

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



          N is INTEGER


          The order of the matrix A.  N >= 0.

KD



          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



          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



          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 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.

subroutine zpbrfs (character uplo, integer n, integer kd, integer nrhs, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldafb, * ) afb, integer ldafb, 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)

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



          N is INTEGER


          The order of the matrix A.  N >= 0.

KD



          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



          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



          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 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.

Author

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