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NAG Toolbox: nag_lapack_dgbcon (f07bg)

Purpose

nag_lapack_dgbcon (f07bg) estimates the condition number of a real band matrix AA, where AA 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 AA, in either the 11-norm or the -norm:
κ1(A) = A1A11   or   κ(A) = AA1 .
κ1(A)=A1A-11   or   κ(A)=AA-1 .
Note that κ(A) = κ1(AT)κ(A)=κ1(AT).
Because the condition number is infinite if AA is singular, the function actually returns an estimate of the reciprocal of the condition number.

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 κ1(A)κ1(A) or κ(A)κ(A) is estimated.
norm = '1'norm='1' or 'O''O'
κ1(A)κ1(A) is estimated.
norm = 'I'norm='I'
κ(A)κ(A) is estimated.
Constraint: norm = '1'norm='1', 'O''O' or 'I''I'.
2:     kl – int64int32nag_int scalar
klkl, the number of subdiagonals within the band of the matrix AA.
Constraint: kl0kl0.
3:     ku – int64int32nag_int scalar
kuku, the number of superdiagonals within the band of the matrix AA.
Constraint: ku0ku0.
4:     ab(ldab, : :) – double array
The first dimension of the array ab must be at least 2 × kl + ku + 12×kl+ku+1
The second dimension of the array must be at least max (1,n)max(1,n)
The LULU factorization of AA, as returned by nag_lapack_dgbtrf (f07bd).
5:     ipiv( : :) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)max(1,n).
The pivot indices, as returned by nag_lapack_dgbtrf (f07bd).
6:     anorm – double scalar
If norm = '1'norm='1' or 'O''O', the 11-norm of the original matrix AA.
If norm = 'I'norm='I', the -norm of the original matrix AA.
anorm may be computed by calling nag_blas_dlangb (f06rb) with the same value for the parameter norm_p.
anorm must be computed either before calling nag_lapack_dgbtrf (f07bd) or else from a copy of the original matrix AA (see Section [Example]).
Constraint: anorm0.0anorm0.0.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The second dimension of the array ab.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

ldab work iwork

Output Parameters

1:     rcond – double scalar
An estimate of the reciprocal of the condition number of AA. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, AA is singular to working precision.
2:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: norm_p, 2: n, 3: kl, 4: ku, 5: ab, 6: ldab, 7: ipiv, 8: anorm, 9: rcond, 10: work, 11: iwork, 12: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed estimate rcond is never less than the true value ρρ, and in practice is nearly always less than 10ρ10ρ, 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 = bAx=b or ATx = bATx=b; the number is usually 44 or 55 and never more than 1111. Each solution involves approximately 2n(2kl + ku)2n(2kl+ku) floating point operations (assuming nklnkl and nkunku) 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 AA is approximately singular.
The complex analogue of this function is nag_lapack_zgbcon (f07bu).

Example

function nag_lapack_dgbcon_example
norm_p = '1';
m = int64(4);
kl = int64(1);
ku = int64(2);
ab = [0, 0, 0, 0;
     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];
anorm = 13.63;
[ab, ipiv, info] = nag_lapack_dgbtrf(m, kl, ku, ab);
[rcond, info] = nag_lapack_dgbcon(norm_p, kl, ku, ab, ipiv, anorm)
 

rcond =

    0.0177


info =

                    0


function f07bg_example
norm_p = '1';
m = int64(4);
kl = int64(1);
ku = int64(2);
ab = [0, 0, 0, 0;
     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];
anorm = 13.63;
[ab, ipiv, info] = f07bd(m, kl, ku, ab);
[rcond, info] = f07bg(norm_p, kl, ku, ab, ipiv, anorm)
 

rcond =

    0.0177


info =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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