?tprfs

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?tprfs

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

Syntax

call stprfs( uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call dtprfs( uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call ctprfs( uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call ztprfs( uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call tprfs( ap, 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 A^T*X = B or A^H*X = B with a packed triangular 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).

The routine also 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 solver routine ?tptrs.

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	CHARACTER1. Must be 'N' or 'T' or 'C'. Indicates the form of the equations: If `trans = 'N'`, the system has the form `AX = B`. If `trans = 'T'`, the system has the form `A`^T`X` = `B`. If `trans = 'C'`, the system has the form `A`^H`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 `ap`.
n	INTEGER. The order of the matrix `A`; `n`≥ 0.
nrhs	INTEGER. The number of right-hand sides; `nrhs`≥ 0.
ap, b, x, work	REAL for stprfs DOUBLE PRECISION for dtprfs COMPLEX for ctprfs DOUBLE COMPLEX for ztprfs. Arrays: `ap` (size ) contains the upper or lower triangular matrix `A`, as specified by `uplo`. `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 dimension of `ap` must be at least `max(1,n(n+1)/2)`; 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(1,2n)` 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)`.
iwork	INTEGER. Workspace array, size at least `max(1, n)`.
rwork	REAL for ctprfs DOUBLE PRECISION for ztprfs. 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 tprfs interface are as follows:

ap	Holds the array `A` of size (`n*(n+1)/2`).
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 n² floating-point operations for real flavors or 4n² for complex flavors.

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

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