# NAG Library Routine Document

## 1Purpose

f07juf (zptcon) 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 f07jrf (zpttrf).

## 2Specification

Fortran Interface
 Subroutine f07juf ( n, d, e, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: d(*), anorm Real (Kind=nag_wp), Intent (Out) :: rcond, rwork(n) Complex (Kind=nag_wp), Intent (In) :: e(*)
C Header Interface
#include <nagmk26.h>
 void f07juf_ (const Integer *n, const double d[], const Complex e[], const double *anorm, double *rcond, double rwork[], Integer *info)
The routine may be called by its LAPACK name zptcon.

## 3Description

f07juf (zptcon) should be preceded by a call to f07jrf (zpttrf), 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. f07juf (zptcon) 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.
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension 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:     $\mathbf{e}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension 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:     $\mathbf{anorm}$ – Real (Kind=nag_wp)Input
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling f06upf with its argument ${\mathbf{norm}}=\text{'1'}$. anorm must be computed either before calling f07jrf (zpttrf) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.
5:     $\mathbf{rcond}$ – Real (Kind=nag_wp)Output
On exit: the reciprocal condition number, $1/{\kappa }_{1}\left(A\right)=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
6:     $\mathbf{rwork}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
7:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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.

## 7Accuracy

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

## 8Parallelism and Performance

f07juf (zptcon) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

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 routine is f07jgf (dptcon).

## 10Example

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

### 10.1Program Text

Program Text (f07jufe.f90)

### 10.2Program Data

Program Data (f07jufe.d)

### 10.3Program Results

Program Results (f07jufe.r)