?ptrfs

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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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:

|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

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, 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'.

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

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?ptrfs

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