?ptrfs

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

Syntax

FORTRAN 77:

call sptrfs( n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info )

call dptrfs( n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info )

call cptrfs( uplo, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call zptrfs( uplo, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, rwork, info )

FORTRAN 95:

call ptrfs( d, df, e, ef, b, x [,ferr] [,berr] [,info] )

call ptrfs( d, df, e, ef, b, x [,uplo] [,ferr] [,berr] [,info] )

lapack_int LAPACKE_sptrfs( int matrix_order, lapack_int n, lapack_int nrhs, const float* d, const float* e, const float* df, const float* ef, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_dptrfs( int matrix_order, lapack_int n, lapack_int nrhs, const double* d, const double* e, const double* df, const double* ef, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );

lapack_int LAPACKE_cptrfs( int matrix_order, char uplo, lapack_int n, lapack_int nrhs, const float* d, const lapack_complex_float* e, const float* df, const lapack_complex_float* ef, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_zptrfs( int matrix_order, char uplo, lapack_int n, lapack_int nrhs, const double* d, const lapack_complex_double* e, const double* df, const lapack_complex_double* ef, 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 tridiagonal 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 ?pttrf
call the solver routine ?pttrs.

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	CHARACTER1. Used for complex flavors only. Must be 'U' or 'L'. Specifies whether the superdiagonal or the subdiagonal of the tridiagonal matrix `A` is stored and how `A` is factored: If `uplo = 'U'`, the array `e` stores the superdiagonal of `A`, and `A` is factored as `U^HDU`. If `uplo = 'L'`, the array `e` stores the subdiagonal of `A`, and `A` is factored as `LD*L^H`.
n	INTEGER. The order of the matrix `A`; `n` ≥ 0.
nrhs	INTEGER. The number of right-hand sides; `nrhs` ≥ 0.
d, df, rwork	REAL for single precision flavors DOUBLE PRECISION for double precision flavors Arrays: `d(n)`, `df(n)`, `rwork(n)`. The array `d` contains the `n` diagonal elements of the tridiagonal matrix `A`. The array `df` contains the `n` diagonal elements of the diagonal matrix `D` from the factorization of `A` as computed by ?pttrf. The array `rwork` is a workspace array used for complex flavors only.
e,ef,b,x,work	REAL for sptrfs DOUBLE PRECISION for dptrfs COMPLEX for cptrfs DOUBLE COMPLEX for zptrfs. Arrays: `e(n -1)`, `ef(n -1)`, `b(ldb,nrhs)`, `x(ldx,nrhs)`, `work()`. The array `e` contains the (`n` - 1) off-diagonal elements of the tridiagonal matrix `A` (see `uplo`). The array `ef` contains the (`n` - 1) off-diagonal elements of the unit bidiagonal factor `U` or `L` from the factorization computed by ?pttrf (see `uplo`). The array `b` contains the matrix `B` whose columns are the right-hand sides for the systems of equations. The array `x` contains the solution matrix `X` as computed by ?pttrs. The array `work` is a workspace array. The dimension of `work` must be at least `2n` for real flavors, and at least `n` for complex flavors.
ldb	INTEGER. The leading dimension of `b`; `ldb ≥ max(1, n)`.
ldx	INTEGER. The leading dimension of `x`; `ldx ≥ 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 ptrfs interface are as follows:

d	Holds the vector of length `n`.
df	Holds the vector of length `n`.
e	Holds the vector of length (`n`-1).
ef	Holds the vector of length (`n`-1).
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	Used in complex flavors only. Must be 'U' or 'L'. The default value is 'U'.