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NAG Toolbox: nag_lapack_ztbcon (f07vu)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_ztbcon (f07vu) estimates the condition number of a complex triangular band matrix.

Syntax

[rcond, info] = f07vu(norm_p, uplo, diag, kd, ab, 'n', n)
[rcond, info] = nag_lapack_ztbcon(norm_p, uplo, diag, kd, ab, 'n', n)

Description

nag_lapack_ztbcon (f07vu) estimates the condition number of a complex triangular 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 computes A1 or A exactly, and 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:     uplo – string (length ≥ 1)
Specifies whether A is upper or lower triangular.
uplo='U'
A is upper triangular.
uplo='L'
A is lower triangular.
Constraint: uplo='U' or 'L'.
3:     diag – string (length ≥ 1)
Indicates whether A is a nonunit or unit triangular matrix.
diag='N'
A is a nonunit triangular matrix.
diag='U'
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
4:     kd int64int32nag_int scalar
kd, the number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'.
Constraint: kd0.
5:     abldab: – complex 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 n by n triangular band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abkd+1+i-jj​ for ​max1,j-kdij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,j+kd.
If diag='U', the diagonal elements of A are assumed to be 1, and are not referenced.

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_ztbcon (f07vu) involves solving a number of systems of linear equations of the form Ax=b or AHx=b; the number is usually 5 and never more than 11. Each solution involves approximately 8nk real floating-point operations (assuming nk) but takes considerably longer than a call to nag_lapack_ztbtrs (f07vs) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The real analogue of this function is nag_lapack_dtbcon (f07vg).

Example

This example estimates the condition number in the 1-norm of the matrix A, where
A= -1.94+4.43i 0.00+0.00i 0.00+0.00i 0.00+0.00i -3.39+3.44i 4.12-4.27i 0.00+0.00i 0.00+0.00i 1.62+3.68i -1.84+5.53i 0.43-2.66i 0.00+0.00i 0.00+0.00i -2.77-1.93i 1.74-0.04i 0.44+0.10i .  
Here A is treated as a lower triangular band matrix with two subdiagonals. The true condition number in the 1-norm is 71.51.
function f07vu_example


fprintf('f07vu example results\n\n');

% Condition number of A, where A is complex lower triangular banded
% and stored in triangular/symmetric banded format 
kd = int64(2);
ab = [-1.94 + 4.43i,   4.12 - 4.27i,  0.43 - 2.66i,  0.44 + 0.10i;
      -3.39 + 3.44i,  -1.84 + 5.53i,  1.74 - 0.04i,  0    + 0i;
       1.62 + 3.68i,  -2.77 - 1.93i,  0    + 0i,     0    + 0i];
b = [ -8.86 -  3.88i, -24.09 -  5.27i;
     -15.57 - 23.41i, -57.97 +  8.14i;
      -7.63 + 22.78i,  19.09 - 29.51i;
     -14.74 -  2.40i,  19.17 + 21.33i];

% Reciprocal condition number
norm_p = '1';
uplo = 'L';
diag = 'N';
[rcond, info] = f07vu( ...
                       norm_p, uplo, diag, kd, ab);

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


f07vu example results

Estimate of condition number =  3.35e+01

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