# NAG Library Routine Document

## 1Purpose

f07fgf (dpocon) estimates the condition number of a real symmetric positive definite matrix $A$, where $A$ has been factorized by f07fdf (dpotrf).

## 2Specification

Fortran Interface
 Subroutine f07fgf ( uplo, n, a, lda, work, info)
 Integer, Intent (In) :: n, lda Integer, Intent (Out) :: iwork(n), info Real (Kind=nag_wp), Intent (In) :: a(lda,*), anorm Real (Kind=nag_wp), Intent (Out) :: rcond, work(3*n) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f07fgf_ (const char *uplo, const Integer *n, const double a[], const Integer *lda, const double *anorm, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dpocon.

## 3Description

f07fgf (dpocon) estimates the condition number (in the $1$-norm) of a real symmetric positive definite 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 f06rcf to compute ${‖A‖}_{1}$ and a call to f07fdf (dpotrf) to compute the Cholesky 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.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, 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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the Cholesky factor of $A$, as returned by f07fdf (dpotrf).
4:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07fgf (dpocon) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5:     $\mathbf{anorm}$ – Real (Kind=nag_wp)Input
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling f06rcf with its argument ${\mathbf{norm}}=\text{'1'}$. anorm must be computed either before calling f07fdf (dpotrf) 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:     $\mathbf{work}\left(3×{\mathbf{n}}\right)$ – 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

f07fgf (dpocon) 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 f07fgf (dpocon) 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 f07fef (dpotrs) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogue of this routine is f07fuf (zpocon).

## 10Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .$
Here $A$ is symmetric positive definite and must first be factorized by f07fdf (dpotrf). The true condition number in the $1$-norm is $97.32$.

### 10.1Program Text

Program Text (f07fgfe.f90)

### 10.2Program Data

Program Data (f07fgfe.d)

### 10.3Program Results

Program Results (f07fgfe.r)

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