gbrfs(3)

NAME
gbrfs - gbrfs: iterative refinement

SYNOPSIS

  Functions
subroutine cgbrfs (trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CGBRFS
subroutine dgbrfs (trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DGBRFS
subroutine sgbrfs (trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SGBRFS
subroutine zgbrfs (trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZGBRFS

Detailed Description

Function Documentation

  subroutine cgbrfs (character trans, integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, 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)

CGBRFS

Purpose:

     CGBRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is banded, and provides
     error bounds and backward error estimates for the solution.

Parameters
    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B     (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose)

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 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 original band matrix A, stored in rows 1 to KL+KU+1.
              The j-th column of A is stored in the j-th column of the
              array AB as follows:
              AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KL+KU+1.

    AFB

              AFB is COMPLEX array, dimension (LDAFB,N)
              Details of the LU factorization of the band matrix A, as
              computed by CGBTRF.  U is stored as an upper triangular band
              matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
              the multipliers used during the factorization are stored in
              rows KL+KU+2 to 2*KL+KU+1.

    LDAFB

              LDAFB is INTEGER
              The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices from CGBTRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    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 CGBTRS.
              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 203 of file cgbrfs.f.

  subroutine dgbrfs (character trans, integer n, integer kl, integer ku, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldafb, * ) afb, integer ldafb, 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)

DGBRFS

Purpose:

     DGBRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is banded, and provides
     error bounds and backward error estimates for the solution.

Parameters
    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B     (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 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 original band matrix A, stored in rows 1 to KL+KU+1.
              The j-th column of A is stored in the j-th column of the
              array AB as follows:
              AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KL+KU+1.

    AFB

              AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
              Details of the LU factorization of the band matrix A, as
              computed by DGBTRF.  U is stored as an upper triangular band
              matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
              the multipliers used during the factorization are stored in
              rows KL+KU+2 to 2*KL+KU+1.

    LDAFB

              LDAFB is INTEGER
              The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices from DGBTRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    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 DGBTRS.
              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 202 of file dgbrfs.f.

  subroutine sgbrfs (character trans, integer n, integer kl, integer ku, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldafb, * ) afb, integer ldafb, 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)

SGBRFS

Purpose:

     SGBRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is banded, and provides
     error bounds and backward error estimates for the solution.

Parameters
    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B     (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 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 original band matrix A, stored in rows 1 to KL+KU+1.
              The j-th column of A is stored in the j-th column of the
              array AB as follows:
              AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KL+KU+1.

    AFB

              AFB is REAL array, dimension (LDAFB,N)
              Details of the LU factorization of the band matrix A, as
              computed by SGBTRF.  U is stored as an upper triangular band
              matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
              the multipliers used during the factorization are stored in
              rows KL+KU+2 to 2*KL+KU+1.

    LDAFB

              LDAFB is INTEGER
              The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices from SGBTRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    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 SGBTRS.
              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 202 of file sgbrfs.f.

  subroutine zgbrfs (character trans, integer n, integer kl, integer ku, integer nrhs, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldafb, * ) afb, integer ldafb, 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)

ZGBRFS

Purpose:

     ZGBRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is banded, and provides
     error bounds and backward error estimates for the solution.

Parameters
    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B     (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose)

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 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 original band matrix A, stored in rows 1 to KL+KU+1.
              The j-th column of A is stored in the j-th column of the
              array AB as follows:
              AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KL+KU+1.

    AFB

              AFB is COMPLEX*16 array, dimension (LDAFB,N)
              Details of the LU factorization of the band matrix A, as
              computed by ZGBTRF.  U is stored as an upper triangular band
              matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
              the multipliers used during the factorization are stored in
              rows KL+KU+2 to 2*KL+KU+1.

    LDAFB

              LDAFB is INTEGER
              The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices from ZGBTRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    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 ZGBTRS.
              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 203 of file zgbrfs.f.

Author

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