?pbrfs

Refines the solution of a system of linear equations with a band symmetric (Hermitian) positive-definite matrix and estimates its error.

Syntax

FORTRAN 77:

call spbrfs( uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call dpbrfs( uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call cpbrfs( uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call zpbrfs( uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, rwork, info )

FORTRAN 95:

call pbrfs( ab, afb, b, x [,uplo] [,ferr] [,berr] [,info] )

lapack_int LAPACKE_spbrfs( int matrix_order, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, const float* ab, lapack_int ldab, const float* afb, lapack_int ldafb, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_dpbrfs( int matrix_order, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, const double* ab, lapack_int ldab, const double* afb, lapack_int ldafb, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );

lapack_int LAPACKE_cpbrfs( int matrix_order, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, const lapack_complex_float* ab, lapack_int ldab, const lapack_complex_float* afb, lapack_int ldafb, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_zpbrfs( int matrix_order, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, const lapack_complex_double* ab, lapack_int ldab, const lapack_complex_double* afb, lapack_int ldafb, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );

Include Files

Fortran: mkl_lapack.fi and mkl_lapack.h
Fortran 95: lapack.f90
C: mkl_lapacke.h

Description

The routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a symmetric (Hermitian) positive definite band matrix A, with multiple right-hand sides. For each computed solution vector x, the routine computes the component-wise backward error β. 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 ?pbtrf
call the solver routine ?pbtrs.

Input Parameters

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

uplo	CHARACTER*1. Must be 'U' or 'L'. Indicates how the input matrix `A` has been factored: If `uplo = 'U'`, the upper triangle of `A` is stored. If `uplo = 'L'`, the lower triangle of `A` is stored.
n	INTEGER. The order of the matrix `A`; `n` ≥ 0.
kd	INTEGER. The number of superdiagonals or subdiagonals in the matrix `A`; `kd` ≥ 0.
nrhs	INTEGER. The number of right-hand sides; `nrhs` ≥ 0.
ab,afb,b,x,work	REAL for spbrfs DOUBLE PRECISION for dpbrfs COMPLEX for cpbrfs DOUBLE COMPLEX for zpbrfs. Arrays: `ab(ldab,)` contains the original band matrix `A`, as supplied to ?pbtrf. `afb(ldafb,)` contains the factored band matrix `A`, as returned by ?pbtrf. `b(ldb,)` contains the right-hand side matrix `B`. `x(ldx,)` 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`; `ldab` ≥ `kd` + 1.
ldafb	INTEGER. The leading dimension of `afb`; `ldafb` ≥ `kd` + 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, DIMENSION at least `max(1, n)`.
rwork	REAL for cpbrfs DOUBLE PRECISION for zpbrfs. Workspace array, DIMENSION 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, DIMENSION 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.

Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine pbrfs interface are as follows:

ab	Holds the array `A` of size (`kd`+1, `n`).
afb	Holds the array `AF` 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'.

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 8n*kd floating-point operations (for real flavors) or 32n*kd operations (for complex flavors). In addition, each step of iterative refinement involves 12n*kd operations (for real flavors) or 48n*kd 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 4n*kd floating-point operations for real flavors or 16n*kd for complex flavors.