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 is INTEGER


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



          KL is INTEGER


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



          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 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 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 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 is INTEGER


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



          KL is INTEGER


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



          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 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 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 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 is INTEGER


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



          KL is INTEGER


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



          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 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 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 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, complex16, dimension( ldab, ) ab, integer ldab, complex16, dimension( ldafb, ) afb, integer ldafb, 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)

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 is INTEGER


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



          KL is INTEGER


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



          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 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 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 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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