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NAG Toolbox

NAG Toolbox: nag_lapack_dgbcon (f07bg)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgbcon (f07bg) estimates the condition number of a real band matrix A, where A has been factorized by nag_lapack_dgbtrf (f07bd).

Syntax

[rcond, info] = f07bg(norm_p, kl, ku, ab, ipiv, anorm, 'n', n)
[rcond, info] = nag_lapack_dgbcon(norm_p, kl, ku, ab, ipiv, anorm, 'n', n)

Description

nag_lapack_dgbcon (f07bg) estimates the condition number of a real band matrix A, in either the 1-norm or the -norm:
κ1A=A1A-11   or   κA=AA-1 .  
Note that κA=κ1AT.
Because the condition number is infinite if A is singular, the function actually returns an estimate of the reciprocal of the condition number.
The function should be preceded by a computation of A1 or A, and a call to nag_lapack_dgbtrf (f07bd) to compute the LU factorization of A. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11 or A-1.

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:     norm_p – string (length ≥ 1)
Indicates whether κ1A or κA is estimated.
norm_p='1' or 'O'
κ1A is estimated.
norm_p='I'
κA is estimated.
Constraint: norm_p='1', 'O' or 'I'.
2:     kl int64int32nag_int scalar
kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
3:     ku int64int32nag_int scalar
ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
4:     abldab: – double array
The first dimension of the array ab must be at least 2×kl+ku+1.
The second dimension of the array ab must be at least max1,n.
The LU factorization of A, as returned by nag_lapack_dgbtrf (f07bd).
5:     ipiv: int64int32nag_int array
The dimension of the array ipiv must be at least max1,n
The pivot indices, as returned by nag_lapack_dgbtrf (f07bd).
6:     anorm – double scalar
If norm_p='1' or 'O', the 1-norm of the original matrix A.
If norm_p='I', the -norm of the original matrix A.
anorm must be computed either before calling nag_lapack_dgbtrf (f07bd) or else from a copy of the original matrix A (see Example).
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_dgbcon (f07bg) involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n2kl+ku floating-point operations (assuming nkl and nku) but takes considerably longer than a call to nag_lapack_dgbtrs (f07be) 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_zgbcon (f07bu).

Example

This example estimates the condition number in the 1-norm of the matrix A, where
A= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 .  
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_lapack_dgbtrf (f07bd). The true condition number in the 1-norm is 56.40.
function f07bg_example


fprintf('f07bg example results\n\n');

m  = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0,    0,    -3.66, -2.13;
       0,    2.54, -2.73,  4.07;
      -0.23, 2.46,  2.46, -3.82;
      -6.98, 2.56, -4.78,  0];

norm_p = 'O';
anorm = f16rb(norm_p, m, kl, ku, ab);

ab = [zeros(kl,m); ab];
  
% Factorize A
[abf, ipiv, info] = f07bd( ...
                           m, kl, ku, ab);

% Estimate condition number
[rcond, info] = f07bg( ...
                       norm_p, kl, ku, abf, ipiv, anorm);

if rcond > x02aj
  fprintf('Estimate of condition number = %10.2e\n', 1/rcond);
else
  fprintf('A is singular to working precision\n');
end


f07bg example results

Estimate of condition number =   5.64e+01

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