F04BDF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F04BDF

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

F04BDF computes the solution to a real system of linear equations AX=B, where A is an n by n symmetric positive definite matrix 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

SUBROUTINE F04BDF ( UPLO, N, NRHS, A, LDA, B, LDB, RCOND, ERRBND, IFAIL)
INTEGER  N, NRHS, LDA, LDB, IFAIL
REAL (KIND=nag_wp)  A(LDA,*), B(LDB,*), RCOND, ERRBND
CHARACTER(1)  UPLO

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:     ALDA* – REAL (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 symmetric matrix A.
If UPLO='U', the leading N by N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced.
If UPLO='L', the leading N by N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
On exit: if IFAIL=0 or N+1, the factor U or L from the Cholesky factorization A=UTU or A=LLT.
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F04BDF is called.
Constraint: LDAmax1,N.
6:     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 matrix of right-hand sides B.
On exit: if IFAIL=0 or N+1, 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 F04BDF is called.
Constraint: LDBmax1,N.
8:     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.
9:     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.
10:   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<0andIFAIL-999
If IFAIL=-i, the ith argument had an illegal value.
IFAIL>0andIFAILN
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.
IFAIL=N+1
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.
IFAIL=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.
IFAIL=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
IFAIL=-999
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

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. F04BDF uses the approximation E1=εA1 to estimate ERRBND. See Section 4.4 of Anderson et al. (1999) for further details.

8  Parallelism and Performance

F04BDF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F04BDF 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 integer allocatable memory required is N, and the real allocatable memory required is 3×N. Allocation failed before the solution could be computed.
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 F04BDF is F04CDF.

10  Example

This example solves the equations
AX=B,  
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.

10.1  Program Text

Program Text (f04bdfe.f90)

10.2  Program Data

Program Data (f04bdfe.d)

10.3  Program Results

Program Results (f04bdfe.r)


F04BDF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

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