Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 11/07/2023
Public

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?ptrfs

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

Syntax

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 )

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

call ptrfs( d, df, e, ef, b, x [,uplo] [,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 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 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 ?pttrf

  • call the solver routine ?pttrs.

Input Parameters

uplo

CHARACTER*1. 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 UH*D*U.

If uplo = 'L', the array e stores the subdiagonal of A, and A is factored as L*D*LH.

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 2*n 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

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 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'.