NAG Toolbox: nag_lapack_ztbcon (f07vu)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{norm\_p}$ – string (length ≥ 1)

Indicates whether ${\kappa }_1\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.

$\mathbf{norm\_p}=\text{'1'}$ or $\text{'O'}$

${\kappa }_1\left(A\right)$ is estimated.

$\mathbf{norm\_p}=\text{'I'}$

${\kappa }_{\infty }\left(A\right)$ is estimated.

Constraint: $\mathbf{norm\_p}=\text{'1'}$, $\text{'O'}$ or $\text{'I'}$.

2:     $\mathrm{uplo}$ – string (length ≥ 1)

Specifies whether $A$ is upper or lower triangular.

$\mathbf{uplo}=\text{'U'}$

$A$ is upper triangular.

$\mathbf{uplo}=\text{'L'}$

$A$ is lower triangular.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

3:     $\mathrm{diag}$ – string (length ≥ 1)

Indicates whether $A$ is a nonunit or unit triangular matrix.

$\mathbf{diag}=\text{'N'}$

$A$ is a nonunit triangular matrix.

$\mathbf{diag}=\text{'U'}$

$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.

Constraint: $\mathbf{diag}=\text{'N'}$ or $\text{'U'}$.

4:     $\mathrm{kd}$ – int64int32nag_int scalar

$k_d$, the number of superdiagonals of the matrix $A$ if $\mathbf{uplo}=\text{'U'}$, or the number of subdiagonals if $\mathbf{uplo}=\text{'L'}$.

Constraint: $\mathbf{kd}\ge 0$.

5:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array

The first dimension of the array ab must be at least $\mathbf{kd}+1$.

The second dimension of the array ab must be at least $\max \left(1,\mathbf{n}\right)$.

The $n$ by $n$ triangular band matrix $A$.

The matrix is stored in rows $1$ to $k_d+1$, more precisely,

• if $\mathbf{uplo}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(k_d+1+i-j,j\right)\text{ for }\max \left(1,j-k_d\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

If $\mathbf{diag}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the second dimension of the array ab.

$n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

Output Parameters

1:     $\mathrm{rcond}$ – 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:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f07vu_example

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