# NAG Library Routine Document

## 1Purpose

f07pgf (dspcon) estimates the condition number of a real symmetric indefinite matrix $A$, where $A$ has been factorized by f07pdf (dsptrf), using packed storage.

## 2Specification

Fortran Interface
 Subroutine f07pgf ( uplo, n, ap, ipiv, work, info)
 Integer, Intent (In) :: n, ipiv(*) Integer, Intent (Out) :: iwork(n), info Real (Kind=nag_wp), Intent (In) :: ap(*), anorm Real (Kind=nag_wp), Intent (Out) :: rcond, work(2*n) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f07pgf_ ( const char *uplo, const Integer *n, const double ap[], const Integer ipiv[], const double *anorm, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dspcon.

## 3Description

f07pgf (dspcon) estimates the condition number (in the $1$-norm) of a real symmetric indefinite matrix $A$:
 $κ1A=A1A-11 .$
Since $A$ is symmetric, ${\kappa }_{1}\left(A\right)={\kappa }_{\infty }\left(A\right)={‖A‖}_{\infty }{‖{A}^{-1}‖}_{\infty }$.
Because ${\kappa }_{1}\left(A\right)$ is infinite if $A$ is singular, the routine actually returns an estimate of the reciprocal of ${\kappa }_{1}\left(A\right)$.
The routine should be preceded by a call to f06rdf to compute ${‖A‖}_{1}$ and a call to f07pdf (dsptrf) to compute the Bunch–Kaufman factorization of $A$. The routine then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$.

## 4References

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

## 5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the factorization of $A$ stored in packed form, as returned by f07pdf (dsptrf).
4:     $\mathbf{ipiv}\left(*\right)$ – Integer arrayInput
Note: the dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: details of the interchanges and the block structure of $D$, as returned by f07pdf (dsptrf).
5:     $\mathbf{anorm}$ – Real (Kind=nag_wp)Input
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling f06rdf with its argument ${\mathbf{norm}}=\text{'1'}$. anorm must be computed either before calling f07pdf (dsptrf) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.
6:     $\mathbf{rcond}$ – Real (Kind=nag_wp)Output
On exit: 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.
7:     $\mathbf{work}\left(2×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
8:     $\mathbf{iwork}\left({\mathbf{n}}\right)$ – Integer arrayWorkspace
9:     $\mathbf{info}$ – 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 estimate rcond is never less than the true value $\rho$, and in practice is nearly always less than $10\rho$, although examples can be constructed where rcond is much larger.

## 8Parallelism and Performance

f07pgf (dspcon) 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.

A call to f07pgf (dspcon) involves solving a number of systems of linear equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2{n}^{2}$ floating-point operations but takes considerably longer than a call to f07pef (dsptrs) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogues of this routine are f07puf (zhpcon) for Hermitian matrices and f07quf (zspcon) for symmetric matrices.

## 10Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 .$
Here $A$ is symmetric indefinite, stored in packed form, and must first be factorized by f07pdf (dsptrf). The true condition number in the $1$-norm is $75.68$.

### 10.1Program Text

Program Text (f07pgfe.f90)

### 10.2Program Data

Program Data (f07pgfe.d)

### 10.3Program Results

Program Results (f07pgfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017