Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 6/24/2024
Public

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?gbrfs

Refines the solution of a system of linear equations with a general band coefficient matrix and estimates its error.

Syntax

call sgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call dgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call cgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call zgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call gbrfs( ab, afb, ipiv, b, x [,kl] [,trans] [,ferr] [,berr] [,info] )

Include Files

  • mkl.fi, lapack.f90

Description

The routine performs an iterative refinement of the solution to a system of linear equations A*X = B or AT*X = B or AH*X = B with a band matrix A, with multiple right-hand sides. For each computed solution vector x, the routine computes the component-wise backward errorβ. This error is the smallest relative perturbation in elements of A and b such that x is the exact solution of the perturbed system:

|δaij| β|aij|, |δbi| β|bi| such that (A + δA)x = (b + δb).

Finally, the routine estimates the component-wise forward error in the computed solution ||x - xe||/||x|| (here xe is the exact solution).

Before calling this routine:

  • call the factorization routine ?gbtrf

  • call the solver routine ?gbtrs.

Input Parameters

trans

CHARACTER*1. Must be 'N' or 'T' or 'C'.

Indicates the form of the equations:

If trans = 'N', the system has the form A*X = B.

If trans = 'T', the system has the form AT*X = B.

If trans = 'C', the system has the form AH*X = B.

n

INTEGER. The order of the matrix A; n 0.

kl

INTEGER. The number of sub-diagonals within the band of A; kl 0.

ku

INTEGER. The number of super-diagonals within the band of A; ku 0.

nrhs

INTEGER. The number of right-hand sides; nrhs 0.

ab,afb,b,x,work

REAL for sgbrfs

DOUBLE PRECISION for dgbrfs

COMPLEX for cgbrfs

DOUBLE COMPLEX for zgbrfs.

Arrays:

ab(size ldab by *) contains the original band matrix A, as supplied to ?gbtrf, but stored in rows from 1 to kl + ku + 1.

afb(size ldafb by *) contains the factored band matrix A, as returned by ?gbtrf.

b(size ldb by *) contains the right-hand side matrix B.

x(size ldx by *) contains the solution matrix X.

work(*) is a workspace array.

The second dimension of ab and afb must be at least max(1, n); the second dimension of b and x must be at least max(1, nrhs); the dimension of work must be at least max(1, 3*n) for real flavors and max(1, 2*n) for complex flavors.

ldab

INTEGER. The leading dimension of ab.

ldafb

INTEGER. The leading dimension of afb.

ldb

INTEGER. The leading dimension of b; ldb max(1, n).

ldx

INTEGER. The leading dimension of x; ldx max(1, n).

ipiv

INTEGER.

Array, size at least max(1, n). The ipiv array, as returned by ?gbtrf.

iwork

INTEGER. Workspace array, size at least max(1, n).

rwork

REAL for cgbrfs

DOUBLE PRECISION for zgbrfs.

Workspace array, size at least max(1, n).

Output Parameters

x

The refined solution matrix X.

ferr, berr

REAL for single precision flavors

DOUBLE PRECISION for double precision flavors.

Arrays, size at least max(1, nrhs). Contain the component-wise forward and backward errors, respectively, for each solution vector.

info

INTEGER. If info =0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine gbrfs interface are as follows:

ab

Holds the array A of size (kl+ku+1,n).

afb

Holds the array AF of size (2*kl*ku+1,n).

ipiv

Holds the vector of length n.

b

Holds the matrix B of size (n, nrhs).

x

Holds the matrix X of size (n, nrhs).

ferr

Holds the vector of length (nrhs).

berr

Holds the vector of length (nrhs).

trans

Must be 'N', 'C', or 'T'. The default value is 'N'.

kl

If omitted, assumed kl = ku.

ku

Restored as ku = lda-kl-1.

Application Notes

The bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual error.

For each right-hand side, computation of the backward error involves a minimum of 4n(kl + ku) floating-point operations (for real flavors) or 16n(kl + ku) operations (for complex flavors). In addition, each step of iterative refinement involves 2n(4kl + 3ku) operations (for real flavors) or 8n(4kl + 3ku) operations (for complex flavors); the number of iterations may range from 1 to 5. Estimating the forward error involves solving a number of systems of linear equations A*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2 floating-point operations for real flavors or 8n2 for complex flavors.