F07UEF (DTPTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07UEF (DTPTRS)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07UEF (DTPTRS) solves a real triangular system of linear equations with multiple right-hand sides, AX=B or ATX=B, using packed storage.

2  Specification

SUBROUTINE F07UEF ( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO)
INTEGER  N, NRHS, LDB, INFO
REAL (KIND=nag_wp)  AP(*), B(LDB,*)
CHARACTER(1)  UPLO, TRANS, DIAG
The routine may be called by its LAPACK name dtptrs.

3  Description

F07UEF (DTPTRS) solves a real triangular system of linear equations AX=B or ATX=B, using packed storage.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

5  Parameters

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' or 'C'
The equations are of the form ATX=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 – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
5:     NRHS – INTEGERInput
On entry: r, the number of right-hand sides.
Constraint: NRHS0.
6:     AP* – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: the n by n triangular matrix A, packed by columns.
More precisely,
  • if UPLO='U', the upper triangle of A must be stored with element Aij in APi+jj-1/2 for ij;
  • if UPLO='L', the lower triangle of A must be stored with element Aij in APi+2n-jj-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:     BLDB* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least max1,NRHS.
On entry: the n by r right-hand side matrix B.
On exit: the n by r solution matrix X.
8:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07UEF (DTPTRS) is called.
Constraint: LDBmax1,N.
9:     INFO – INTEGEROutput
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.
INFO>0
Element value of the diagonal is exactly zero. A is singular and the solution has not been computed.

7  Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where
EcnεA ,  
cn is a modest linear function of n, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x cncondA,xε ,   provided   cncondA,xε<1 ,  
where condA,x=A-1Ax/x.
Note that condA,xcondA=A-1AκA; condA,x can be much smaller than condA and it is also possible for condAT to be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling F07UHF (DTPRFS), and an estimate for κA can be obtained by calling F07UGF (DTPCON) with NORM='I'.

8  Parallelism and Performance

F07UEF (DTPTRS) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F07UEF (DTPTRS) 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

The total number of floating-point operations is approximately n2r.
The complex analogue of this routine is F07USF (ZTPTRS).

10  Example

This example solves the system of equations AX=B, where
A= 4.30 0.00 0.00 0.00 -3.96 -4.87 0.00 0.00 0.40 0.31 -8.02 0.00 -0.27 0.07 -5.95 0.12   and   B= -12.90 -21.50 16.75 14.93 -17.55 6.33 -11.04 8.09 ,  
using packed storage for A.

10.1  Program Text

Program Text (f07uefe.f90)

10.2  Program Data

Program Data (f07uefe.d)

10.3  Program Results

Program Results (f07uefe.r)


F07UEF (DTPTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015