f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Documentnag_zptcon (f07juc)

1  Purpose

nag_zptcon (f07juc) computes the reciprocal condition number of a complex $n$ by $n$ Hermitian positive definite tridiagonal matrix $A$, using the $LD{L}^{\mathrm{H}}$ factorization returned by nag_zpttrf (f07jrc).

2  Specification

 #include #include
 void nag_zptcon (Integer n, const double d[], const Complex e[], double anorm, double *rcond, NagError *fail)

3  Description

nag_zptcon (f07juc) should be preceded by a call to nag_zpttrf (f07jrc), which computes a modified Cholesky factorization of the matrix $A$ as
 $A=LDLH ,$
where $L$ is a unit lower bidiagonal matrix and $D$ is a diagonal matrix, with positive diagonal elements. nag_zptcon (f07juc) then utilizes the factorization to compute ${‖{A}^{-1}‖}_{1}$ by a direct method, from which the reciprocal of the condition number of $A$, $1/\kappa \left(A\right)$ is computed as
 $1/κ1A=1 / A1 A-11 .$
$1/\kappa \left(A\right)$ is returned, rather than $\kappa \left(A\right)$, since when $A$ is singular $\kappa \left(A\right)$ is infinite.

4  References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5  Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{d}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{H}}$ factorization of $A$.
3:    $\mathbf{e}\left[\mathit{dim}\right]$const ComplexInput
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix $L$. (e can also be regarded as the superdiagonal of the unit upper bidiagonal matrix $U$ from the ${U}^{\mathrm{H}}DU$ factorization of $A$.)
4:    $\mathbf{anorm}$doubleInput
On entry: the $1$-norm of the original matrix $A$, which may be computed as shown in Section 10. anorm must be computed either before calling nag_zpttrf (f07jrc) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.
5:    $\mathbf{rcond}$double *Output
On exit: the reciprocal condition number, $1/{\kappa }_{1}\left(A\right)=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{anorm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.

7  Accuracy

The computed condition number will be the exact condition number for a closely neighbouring matrix.

8  Parallelism and Performance

nag_zptcon (f07juc) is not threaded in any implementation.

The condition number estimation requires $\mathit{O}\left(n\right)$ floating-point operations.
See Section 15.6 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.
The real analogue of this function is nag_dptcon (f07jgc).

10  Example

This example computes the condition number of the Hermitian positive definite tridiagonal matrix $A$ given by
 $A = 16.0i+00.0 16.0-16.0i 0.0i+0.0 0.0i+0.0 16.0+16.0i 41.0i+00.0 18.0+9.0i 0.0i+0.0 0.0i+00.0 18.0-09.0i 46.0i+0.0 1.0+4.0i 0.0i+00.0 0.0i+00.0 1.0-4.0i 21.0i+0.0 .$

10.1  Program Text

Program Text (f07juce.c)

10.2  Program Data

Program Data (f07juce.d)

10.3  Program Results

Program Results (f07juce.r)