NAG FL Interface
f07uvf (ztprfs)

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1 Purpose

f07uvf returns error bounds for the solution of a complex triangular system of linear equations with multiple right-hand sides, AX=B, ATX=B or AHX=B, using packed storage.

2 Specification

Fortran Interface
Subroutine f07uvf ( uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, rwork, info)
Integer, Intent (In) :: n, nrhs, ldb, ldx
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Out) :: ferr(nrhs), berr(nrhs), rwork(n)
Complex (Kind=nag_wp), Intent (In) :: ap(*), b(ldb,*), x(ldx,*)
Complex (Kind=nag_wp), Intent (Out) :: work(2*n)
Character (1), Intent (In) :: uplo, trans, diag
C Header Interface
#include <nag.h>
void  f07uvf_ (const char *uplo, const char *trans, const char *diag, const Integer *n, const Integer *nrhs, const Complex ap[], const Complex b[], const Integer *ldb, const Complex x[], const Integer *ldx, double ferr[], double berr[], Complex work[], double rwork[], Integer *info, const Charlen length_uplo, const Charlen length_trans, const Charlen length_diag)
The routine may be called by the names f07uvf, nagf_lapacklin_ztprfs or its LAPACK name ztprfs.

3 Description

f07uvf returns the backward errors and estimated bounds on the forward errors for the solution of a complex triangular system of linear equations with multiple right-hand sides AX=B, ATX=B or AHX=B, using packed storage. The routine handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of f07uvf in terms of a single right-hand side b and solution x.
Given a computed solution x, the routine computes the component-wise backward error β. This is the size of the smallest relative perturbation in each element of A and b such that x is the exact solution of a perturbed system
(A+δA)x=b+δb |δaij|β|aij|   and   |δbi|β|bi| .  
Then the routine estimates a bound for the component-wise forward error in the computed solution, defined by:
maxi|xi-x^i|/maxi|xi|  
where x^ is the true solution.
For details of the method, see the F07 Chapter Introduction.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: uplo Character(1) Input
On entry: specifies whether A is upper or lower triangular.
uplo='U'
A is upper triangular.
uplo='L'
A is lower triangular.
Constraint: uplo='U' or 'L'.
2: trans Character(1) Input
On entry: indicates the form of the equations.
trans='N'
The equations are of the form AX=B.
trans='T'
The equations are of the form ATX=B.
trans='C'
The equations are of the form AHX=B.
Constraint: trans='N', 'T' or 'C'.
3: diag Character(1) Input
On entry: indicates whether A is a nonunit or unit triangular matrix.
diag='N'
A is a nonunit triangular matrix.
diag='U'
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
4: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
5: nrhs Integer Input
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
6: ap(*) Complex (Kind=nag_wp) array Input
Note: the dimension of the array ap must be at least max(1,n×(n+1)/2).
On entry: the n×n triangular matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in ap(i+j(j-1)/2) for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in ap(i+(2n-j)(j-1)/2) for ij.
If diag='U', the diagonal elements of A are assumed to be 1, and are not referenced; the same storage scheme is used whether diag='N' or ‘U’.
7: b(ldb,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least max(1,nrhs).
On entry: the n×r right-hand side matrix B.
8: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07uvf is called.
Constraint: ldbmax(1,n).
9: x(ldx,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array x must be at least max(1,nrhs).
On entry: the n×r solution matrix X, as returned by f07usf.
10: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which f07uvf is called.
Constraint: ldxmax(1,n).
11: ferr(nrhs) Real (Kind=nag_wp) array Output
On exit: ferr(j) contains an estimated error bound for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
12: berr(nrhs) Real (Kind=nag_wp) array Output
On exit: berr(j) contains the component-wise backward error bound β for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
13: work(2×n) Complex (Kind=nag_wp) array Workspace
14: rwork(n) Real (Kind=nag_wp) array Workspace
15: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07uvf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07uvf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

A call to f07uvf, for each right-hand side, involves solving a number of systems of linear equations of the form Ax=b or AHx=b; the number is usually 5 and never more than 11. Each solution involves approximately 4n2 real floating-point operations.
The real analogue of this routine is f07uhf.

10 Example

This example solves the system of equations AX=B and to compute forward and backward error bounds, where
A= ( 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i )  
and
B= ( -14.78-32.36i -18.02+28.46i 2.98-02.14i 14.22+15.42i -20.96+17.06i 5.62+35.89i 9.54+09.91i -16.46-01.73i ) ,  
using packed storage for A.

10.1 Program Text

Program Text (f07uvfe.f90)

10.2 Program Data

Program Data (f07uvfe.d)

10.3 Program Results

Program Results (f07uvfe.r)