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NAG Toolbox: nag_lapack_dpocon (f07fg)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dpocon (f07fg) estimates the condition number of a real symmetric positive definite matrix A, where A has been factorized by nag_lapack_dpotrf (f07fd).

Syntax

[rcond, info] = f07fg(uplo, a, anorm, 'n', n)
[rcond, info] = nag_lapack_dpocon(uplo, a, anorm, 'n', n)

Description

nag_lapack_dpocon (f07fg) estimates the condition number (in the 1-norm) of a real symmetric positive definite matrix A:
κ1A=A1A-11 .  
Since A is symmetric, κ1A=κA=AA-1.
Because κ1A is infinite if A is singular, the function actually returns an estimate of the reciprocal of κ1A.
The function should be preceded by a computation of A1 and a call to nag_lapack_dpotrf (f07fd) to compute the Cholesky factorization of A. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11.

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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
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:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The Cholesky factor of A, as returned by nag_lapack_dpotrf (f07fd).
3:     anorm – double scalar
The 1-norm of the original matrix A. anorm must be computed either before calling nag_lapack_dpotrf (f07fd) or else from a copy of the original matrix A.
Constraint: anorm0.0.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     rcond – double scalar
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.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

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.

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.

Further Comments

A call to nag_lapack_dpocon (f07fg) 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 nag_lapack_dpotrs (f07fe) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The complex analogue of this function is nag_lapack_zpocon (f07fu).

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 nag_lapack_dpotrf (f07fd). The true condition number in the 1-norm is 97.32.
function f07fg_example


fprintf('f07fg example results\n\n');

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];

% Factorize
uplo = 'L';
[af, info] = f07fd(uplo, a);

% Estimate condition number
anorm = norm(a, 1);
[rcond, info] = f07fg( ...
                       uplo, af, anorm);

fprintf('Estimate of condition number = %9.2e\n', 1/rcond);


f07fg example results

Estimate of condition number =  9.73e+01

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