# NAG FL Interfacef07ggf (dppcon)

## 1Purpose

f07ggf estimates the condition number of a real symmetric positive definite matrix $A$, where $A$ has been factorized by f07gdf, using packed storage.

## 2Specification

Fortran Interface
 Subroutine f07ggf ( uplo, n, ap, work, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: iwork(n), info Real (Kind=nag_wp), Intent (In) :: ap(*), anorm Real (Kind=nag_wp), Intent (Out) :: rcond, work(3*n) Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
 void f07ggf_ (const char *uplo, const Integer *n, const double ap[], const double *anorm, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f07ggf, nagf_lapacklin_dppcon or its LAPACK name dppcon.

## 3Description

f07ggf 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 f06rdf to compute ${‖A‖}_{1}$ and a call to f07gdf 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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{ap}\left(*\right)$Real (Kind=nag_wp) array Input
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 Cholesky factor of $A$ stored in packed form, as returned by f07gdf.
4: $\mathbf{anorm}$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 f07gdf 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: 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.
6: $\mathbf{work}\left(3×{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
7: $\mathbf{iwork}\left({\mathbf{n}}\right)$Integer array Workspace
8: $\mathbf{info}$Integer Output
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

f07ggf 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

A call to f07ggf 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 f07gef 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 f07guf.

## 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, stored in packed form, and must first be factorized by f07gdf. The true condition number in the $1$-norm is $97.32$.

### 10.1Program Text

Program Text (f07ggfe.f90)

### 10.2Program Data

Program Data (f07ggfe.d)

### 10.3Program Results

Program Results (f07ggfe.r)