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NAG Toolbox

NAG Toolbox: nag_lapack_dpbcon (f07hg)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dpbcon (f07hg) estimates the condition number of a real symmetric positive definite band matrix A, where A has been factorized by nag_lapack_dpbtrf (f07hd).

Syntax

[rcond, info] = f07hg(uplo, kd, ab, anorm, 'n', n)
[rcond, info] = nag_lapack_dpbcon(uplo, kd, ab, anorm, 'n', n)

Description

nag_lapack_dpbcon (f07hg) estimates the condition number (in the 1-norm) of a real symmetric positive definite band 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_dpbtrf (f07hd) 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:     kd int64int32nag_int scalar
kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
3:     abldab: – double array
The first dimension of the array ab must be at least kd+1.
The second dimension of the array ab must be at least max1,n.
The Cholesky factor of A, as returned by nag_lapack_dpbtrf (f07hd).
4:     anorm – double scalar
The 1-norm of the original matrix A. anorm must be computed either before calling nag_lapack_dpbtrf (f07hd) or else from a copy of the original matrix A.
Constraint: anorm0.0.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the second dimension of the array ab.
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_dpbcon (f07hg) 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 4nk floating-point operations (assuming nk) but takes considerably longer than a call to nag_lapack_dpbtrs (f07he) 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_zpbcon (f07hu).

Example

This example estimates the condition number in the 1-norm (or -norm) of the matrix A, where
A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 .  
Here A is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_lapack_dpbtrf (f07hd). The true condition number in the 1-norm is 74.15.
function f07hg_example


fprintf('f07hg example results\n\n');

% Symmetric banded matrix A in ab.
uplo = 'L';
kd  = int64(1);
ab  = [5.49,  5.63,  2.60, 5.17;
       2.68, -2.39, -2.22, 0.00];

% To calculate 1-norm here, need to add superdiagonal
abn = [0.00,  2.68, -2.39, -2.22;
       ab];
% 1-norm of A = 1-norm of abn
anorm = norm(abn,1);

% Factorize A
[abf, info] = f07hd( ...
                     uplo, kd, ab);

% Get reciprocal condition number
[rcond, info] = f07hg( ...
                       uplo, kd, abf, anorm);

fprintf('Condition number of A = %7.1f\n',1/rcond);


f07hg example results

Condition number of A =    74.2

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