NAG FL Interface
f07fgf (dpocon)

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1 Purpose

f07fgf estimates the condition number of a real symmetric positive definite matrix A, where A has been factorized by f07fdf.

2 Specification

Fortran Interface
Subroutine f07fgf ( uplo, n, a, lda, anorm, rcond, work, iwork, info)
Integer, Intent (In) :: n, lda
Integer, Intent (Out) :: iwork(n), info
Real (Kind=nag_wp), Intent (In) :: a(lda,*), anorm
Real (Kind=nag_wp), Intent (Out) :: rcond, work(3*n)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07fgf_ (const char *uplo, const Integer *n, const double a[], const Integer *lda, const double *anorm, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f07fgf, nagf_lapacklin_dpocon or its LAPACK name dpocon.

3 Description

f07fgf estimates the condition number (in the 1-norm) of a real symmetric positive definite matrix A:
κ1(A)=A1A-11 .  
Since A is symmetric, κ1(A)=κ(A)=AA-1.
Because κ1(A) is infinite if A is singular, the routine actually returns an estimate of the reciprocal of κ1(A).
The routine should be preceded by a call to f06rcf to compute A1 and a call to f07fdf to compute the Cholesky factorization of A. The routine then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11.

4 References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

5 Arguments

1: uplo Character(1) Input
On entry: specifies how A has been factorized.
uplo='U'
A=UTU, where U is upper triangular.
uplo='L'
A=LLT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: a(lda,*) Real (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least max(1,n).
On entry: the Cholesky factor of A, as returned by f07fdf.
4: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07fgf is called.
Constraint: ldamax(1,n).
5: anorm Real (Kind=nag_wp) Input
On entry: the 1-norm of the original matrix A, which may be computed by calling f06rcf with its argument norm='1'. anorm must be computed either before calling f07fdf or else from a copy of the original matrix A.
Constraint: anorm0.0.
6: rcond Real (Kind=nag_wp) Output
On exit: an estimate of the reciprocal of the condition number of A. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, A is singular to working precision.
7: work(3×n) Real (Kind=nag_wp) array Workspace
8: iwork(n) Integer array Workspace
9: info Integer Output
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.

7 Accuracy

The computed estimate rcond is never less than the true value ρ, and in practice is nearly always less than 10ρ, although examples can be constructed where rcond is much larger.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07fgf 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

A call to f07fgf involves solving a number of systems of linear equations of the form Ax=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n2 floating-point operations but takes considerably longer than a call to f07fef with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The complex analogue of this routine is f07fuf.

10 Example

This example estimates the condition number in the 1-norm (or -norm) of the matrix A, where
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 ) .  
Here A is symmetric positive definite and must first be factorized by f07fdf. The true condition number in the 1-norm is 97.32.

10.1 Program Text

Program Text (f07fgfe.f90)

10.2 Program Data

Program Data (f07fgfe.d)

10.3 Program Results

Program Results (f07fgfe.r)