F07QSF (ZSPTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


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

F07QSF (ZSPTRS) solves a complex symmetric system of linear equations with multiple right-hand sides,
AX=B ,
where A has been factorized by F07QRF (ZSPTRF), using packed storage.

2  Specification

COMPLEX (KIND=nag_wp)  AP(*), B(LDB,*)
The routine may be called by its LAPACK name zsptrs.

3  Description

F07QSF (ZSPTRS) is used to solve a complex symmetric system of linear equations AX=B, the routine must be preceded by a call to F07QRF (ZSPTRF) which computes the Bunch–Kaufman factorization of A, using packed storage.
If UPLO='U', A=PUDUTPT, where P is a permutation matrix, U is an upper triangular matrix and D is a symmetric block diagonal matrix with 1 by 1 and 2 by 2 blocks; the solution X is computed by solving PUDY=B and then UTPTX=Y.
If UPLO='L', A=PLDLTPT, where L is a lower triangular matrix; the solution X is computed by solving PLDY=B and then LTPTX=Y.

4  References

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

5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: specifies how A has been factorized.
A=PUDUTPT, where U is upper triangular.
A=PLDLTPT, where L is lower triangular.
Constraint: UPLO='U' or 'L'.
2:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
3:     NRHS – INTEGERInput
On entry: r, the number of right-hand sides.
Constraint: NRHS0.
4:     AP(*) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: the factorization of A stored in packed form, as returned by F07QRF (ZSPTRF).
5:     IPIV(*) – INTEGER arrayInput
Note: the dimension of the array IPIV must be at least max1,N.
On entry: details of the interchanges and the block structure of D, as returned by F07QRF (ZSPTRF).
6:     B(LDB,*) – COMPLEX (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.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07QSF (ZSPTRS) is called.
Constraint: LDBmax1,N.
8:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

7  Accuracy

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 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
where condA,x=A-1Ax/xcondA=A-1AκA.
Note that condA,x can be much smaller than condA.
Forward and backward error bounds can be computed by calling F07QVF (ZSPRFS), and an estimate for κA (=κ1A) can be obtained by calling F07QUF (ZSPCON).

8  Further Comments

The total number of real floating point operations is approximately 8n2r.
This routine may be followed by a call to F07QVF (ZSPRFS) to refine the solution and return an error estimate.
The real analogue of this routine is F07PEF (DSPTRS).

9  Example

This example solves the system of equations AX=B, where
A= -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i
B= -55.64+41.22i -19.09-35.97i -48.18+66.00i -12.08-27.02i -0.49-01.47i 6.95+20.49i -6.43+19.24i -4.59-35.53i .
Here A is symmetric, stored in packed form, and must first be factorized by F07QRF (ZSPTRF).

9.1  Program Text

Program Text (f07qsfe.f90)

9.2  Program Data

Program Data (f07qsfe.d)

9.3  Program Results

Program Results (f07qsfe.r)

F07QSF (ZSPTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012