?gbrfs

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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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 A^T*X = B or A^H*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:

|δa_ij| ≤β|a_ij|, |δb_i| ≤β|b_i| such that (A + δA)x = (b + δb).

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

Before calling this routine:

call the factorization routine ?gbtrf
call the solver routine ?gbtrs.

Input Parameters

trans	CHARACTER1. Must be 'N' or 'T' or 'C'. Indicates the form of the equations: If `trans = 'N'`, the system has the form `AX = B`. If `trans = 'T'`, the system has the form `A`^T`X` = `B`. If `trans = 'C'`, the system has the form `A`^H`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, 3n)` 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

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 (`2klku+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.

Parent topic: Refining the Solution and Estimating Its Error: LAPACK Computational Routines

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