F04BEF (PDF version)
F04 Chapter Contents
F04 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

F04BEF computes the solution to a real system of linear equations AX=B, where A is an n by n symmetric positive definite matrix, stored in packed format, and X and B are n by r matrices. An estimate of the condition number of A and an error bound for the computed solution are also returned.

2  Specification

REAL (KIND=nag_wp)  AP(*), B(LDB,*), RCOND, ERRBND

3  Description

The Cholesky factorization is used to factor A as A=UTU, if UPLO='U', or A=LLT, 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: if UPLO='U', the upper triangle of the matrix A is stored.
If UPLO='L', the lower triangle of the matrix A is stored.
Constraint: UPLO='U' or 'L'.
2:     N – INTEGERInput
On entry: the number of linear equations n, i.e., the order of the matrix A.
Constraint: N0.
3:     NRHS – INTEGERInput
On entry: the number of right-hand sides r, i.e., the number of columns of the matrix B.
Constraint: NRHS0.
4:     AP(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: the n by n symmetric matrix A. The upper or lower triangular part of the symmetric matrix is packed column-wise in a linear array. The jth column of A is stored in the array AP as follows:
  • if UPLO='U', APi+j-1j/2=aij for 1ij;
  • if UPLO='L', APi+j-12n-j/2=aij for jin.
See Section 8 below for further details.
On exit: if IFAIL=0 or N+1, the factor U or L from the Cholesky factorization A=UTU or A=LLT, in the same storage format as A.
5:     B(LDB,*) – 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 matrix of right-hand sides B.
On exit: if IFAIL=0 or N+1, the n by r solution matrix X.
6:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F04BEF is called.
Constraint: LDBmax1,N.
7:     RCOND – REAL (KIND=nag_wp)Output
On exit: if IFAIL=0 or N+1, an estimate of the reciprocal of the condition number of the matrix A, computed as RCOND=1/A1A-11.
8:     ERRBND – REAL (KIND=nag_wp)Output
On exit: if IFAIL=0 or N+1, an estimate of the forward error bound for a computed solution x^, such that x^-x1/x1ERRBND, where x^ is a column of the computed solution returned in the array B and x is the corresponding column of the exact solution X. If RCOND is less than machine precision, then ERRBND is returned as unity.
9:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
IFAIL<0 and IFAIL-999
If IFAIL=-i, the ith argument had an illegal value.
Allocation of memory failed. The integer allocatable memory required is N, and the real allocatable memory required is 3×N. Allocation failed before the solution could be computed.
If IFAIL=i, the leading minor of order i of A is not positive definite. The factorization could not be completed, and the solution has not been computed.
RCOND is less than machine precision, so that the matrix A is numerically singular. A solution to the equations AX=B has nevertheless been computed.

7  Accuracy

The computed solution for a single right-hand side, x^, satisfies an equation of the form
A+E x^=b,
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-11A1, the condition number of A with respect to the solution of the linear equations. F04BEF uses the approximation E1=εA1 to estimate ERRBND. See Section 4.4 of Anderson et al. (1999) for further details.

8  Further Comments

The packed storage scheme is illustrated by the following example when n=4 and UPLO='U'. Two-dimensional storage of the symmetric matrix A:
a11 a12 a13 a14 a22 a23 a24 a33 a34 a44 aij = aji
Packed storage of the upper triangle of A:
AP= a11, a12, a22, a13, a23, a33, a14, a24, a34, a44
The total number of floating point operations required to solve the equations AX=B is proportional to 13n3+n2r. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The complex analogue of F04BEF is F04CEF.

9  Example

This example solves the equations
where A is the symmetric positive definite matrix
A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18   and   B= 8.70 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 .
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.

9.1  Program Text

Program Text (f04befe.f90)

9.2  Program Data

Program Data (f04befe.d)

9.3  Program Results

Program Results (f04befe.r)

F04BEF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

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