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NAG Toolbox: nag_lapack_dtbcon (f07vg)

Purpose

nag_lapack_dtbcon (f07vg) estimates the condition number of a real triangular band matrix.

Syntax

[rcond, info] = f07vg(norm_p, uplo, diag, kd, ab, 'n', n)
[rcond, info] = nag_lapack_dtbcon(norm_p, uplo, diag, kd, ab, 'n', n)

Description

nag_lapack_dtbcon (f07vg) estimates the condition number of a real triangular 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.
The function computes A1A1 or AA exactly, and uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A11A-11 or A1A-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 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:     uplo – string (length ≥ 1)
Specifies whether AA is upper or lower triangular.
uplo = 'U'uplo='U'
AA is upper triangular.
uplo = 'L'uplo='L'
AA is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
3:     diag – string (length ≥ 1)
Indicates whether AA is a nonunit or unit triangular matrix.
diag = 'N'diag='N'
AA is a nonunit triangular matrix.
diag = 'U'diag='U'
AA is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 11.
Constraint: diag = 'N'diag='N' or 'U''U'.
4:     kd – int64int32nag_int scalar
kdkd, the number of superdiagonals of the matrix AA if uplo = 'U'uplo='U', or the number of subdiagonals if uplo = 'L'uplo='L'.
Constraint: kd0kd0.
5:     ab(ldab, : :) – double array
The first dimension of the array ab must be at least kd + 1kd+1
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn triangular band matrix AA.
The matrix is stored in rows 11 to kd + 1kd+1, more precisely,
  • if uplo = 'U'uplo='U', the elements of the upper triangle of AA within the band must be stored with element AijAij in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ijabkd+1+i-jj​ for ​max(1,j-kd)ij;
  • if uplo = 'L'uplo='L', the elements of the lower triangle of AA within the band must be stored with element AijAij in ab(1 + ij,j)​ for ​jimin (n,j + kd).ab1+i-jj​ for ​jimin(n,j+kd).
If diag = 'U'diag='U', the diagonal elements of AA are assumed to be 11, and are not referenced.

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: uplo, 3: diag, 4: n, 5: kd, 6: ab, 7: ldab, 8: rcond, 9: work, 10: iwork, 11: 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_dtbcon (f07vg) 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 2nk2nk floating point operations (assuming nknk) but takes considerably longer than a call to nag_lapack_dtbtrs (f07ve) 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_ztbcon (f07vu).

Example

function nag_lapack_dtbcon_example
norm_p = '1';
uplo = 'L';
diag = 'N';
kd = int64(1);
ab = [-4.16, 4.78, 6.32, 0.16;
     -2.25, 5.86, -4.82, 0];
[rcond, info] = nag_lapack_dtbcon(norm_p, uplo, diag, kd, ab)
 

rcond =

    0.0144


info =

                    0


function f07vg_example
norm_p = '1';
uplo = 'L';
diag = 'N';
kd = int64(1);
ab = [-4.16, 4.78, 6.32, 0.16;
     -2.25, 5.86, -4.82, 0];
[rcond, info] = f07vg(norm_p, uplo, diag, kd, ab)
 

rcond =

    0.0144


info =

                    0



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