Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 12/16/2022
Public

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?tbrfs

Estimates the error in the solution of a system of linear equations with a triangular band coefficient matrix.

Syntax

call stbrfs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call dtbrfs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call ctbrfs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call ztbrfs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call tbrfs( ab, b, x [,uplo] [,trans] [,diag] [,ferr] [,berr] [,info] )

Include Files
  • mkl.fi, lapack.f90
Description

The routine estimates the errors in the solution to a system of linear equations A*X = B or AT*X = B or AH*X = B with a triangular 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).

The routine also 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 solver routine ?tbtrs.

Input Parameters

uplo

CHARACTER*1. Must be 'U' or 'L'.

Indicates whether A is upper or lower triangular:

If uplo = 'U', then A is upper triangular.

If uplo = 'L', then A is lower triangular.

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.

diag

CHARACTER*1. Must be 'N' or 'U'.

If diag = 'N', A is not a unit triangular matrix.

If diag = 'U', A is unit triangular: diagonal elements of A are assumed to be 1 and not referenced in the array ab.

n

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

kd

INTEGER. The number of super-diagonals or sub-diagonals in the matrix A; kd 0.

nrhs

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

ab, b, x, work

REAL for stbrfs

DOUBLE PRECISION for dtbrfs

COMPLEX for ctbrfs

DOUBLE COMPLEX for ztbrfs.

Arrays:

ab(size ldab by *) contains the upper or lower triangular matrix A, as specified by uplo, in band storage format.

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 a 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 the array ab; ldabkd +1.

ldb

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

ldx

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

iwork

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

rwork

REAL for ctbrfs

DOUBLE PRECISION for ztbrfs.

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

Output Parameters

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 tbrfs interface are as follows:

ab

Holds the array A of size (kd+1,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).

uplo

Must be 'U' or 'L'. The default value is 'U'.

trans

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

diag

Must be 'N' or 'U'. The default value is 'N'.

Application Notes

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

A call to this routine involves, for each right-hand side, solving a number of systems of linear equations A*x = b; the number of systems is usually 4 or 5 and never more than 11. Each solution requires approximately 2n*kd floating-point operations for real flavors or 8n*kd operations for complex flavors.