NAG Library Routine Document

f07fnf  (zposv)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

f07fnf (zposv) computes the solution to a complex system of linear equations
AX=B ,  
where A is an n by n Hermitian positive definite matrix and X and B are n by r matrices.

2
Specification

Fortran Interface
Subroutine f07fnf ( uplo, n, nrhs, a, lda, b, ldb, info)
Integer, Intent (In):: n, nrhs, lda, ldb
Integer, Intent (Out):: info
Complex (Kind=nag_wp), Intent (Inout):: a(lda,*), b(ldb,*)
Character (1), Intent (In):: uplo
C Header Interface
#include nagmk26.h
void  f07fnf_ ( const char *uplo, const Integer *n, const Integer *nrhs, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zposv.

3
Description

f07fnf (zposv) uses the Cholesky decomposition to factor A as A=UHU if uplo='U' or A=LLH if uplo='L', where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations AX=B.

4
References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
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: if uplo='U', the upper triangle of A is stored.
If uplo='L', the lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2:     n – IntegerInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
3:     nrhs – IntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4:     alda* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least max1,n.
On entry: the n by n Hermitian matrix A.
  • If uplo='U', the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo='L', the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if info=0, the factor U or L from the Cholesky factorization A=UHU or A=LLH.
5:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07fnf (zposv) is called.
Constraint: ldamax1,n.
6:     bldb* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least max1,nrhs.
To solve the equations Ax=b, where b is a single right-hand side, b may be supplied as a one-dimensional array with length ldb=max1,n.
On entry: the n by r right-hand side matrix B.
On exit: if info=0, 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 f07fnf (zposv) 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

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
The leading minor of order value of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

7
Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,  
where
E1 = Oε A1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,  
where κA = A-11 A1 , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
f07fpf (zposvx) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04cdf solves Ax=b  and returns a forward error bound and condition estimate. f04cdf calls f07fnf (zposv) to solve the equations.

8
Parallelism and Performance

f07fnf (zposv) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07fnf (zposv) 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 43 n3 + 8n2r , where r  is the number of right-hand sides.
The real analogue of this routine is f07faf (dposv).

10
Example

This example solves the equations
Ax=b ,  
where A  is the symmetric positive definite matrix
A = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00  
and
b = 3.93-06.14i 6.17+09.42i -7.17-21.83i 1.99-14.38i .  
Details of the Cholesky factorization of A  are also output.

10.1
Program Text

Program Text (f07fnfe.f90)

10.2
Program Data

Program Data (f07fnfe.d)

10.3
Program Results

Program Results (f07fnfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017