F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07FGF (DPOCON)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07FGF (DPOCON) estimates the condition number of a real symmetric positive definite matrix $A$, where $A$ has been factorized by F07FDF (DPOTRF).

## 2  Specification

 SUBROUTINE F07FGF ( UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
 INTEGER N, LDA, IWORK(N), INFO REAL (KIND=nag_wp) A(LDA,*), ANORM, RCOND, WORK(3*N) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dpocon.

## 3  Description

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 ${‖{A}^{-1}‖}_{1}$.

## 4  References

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

## 5  Parameters

1:     $\mathrm{UPLO}$ – 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:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – 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:     $\mathrm{LDA}$ – 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:     $\mathrm{ANORM}$ – REAL (KIND=nag_wp)Input
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling F06RCF with its parameter ${\mathbf{NORM}}=\text{'1'}$. 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:     $\mathrm{RCOND}$ – 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:     $\mathrm{WORK}\left(3×{\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace
8:     $\mathrm{IWORK}\left({\mathbf{N}}\right)$ – INTEGER arrayWorkspace
9:     $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error 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.

## 7  Accuracy

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.

## 8  Parallelism and Performance

F07FGF (DPOCON) is not threaded by NAG in any implementation.
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).

## 10  Example

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 $97.32$.

### 10.1  Program Text

Program Text (f07fgfe.f90)

### 10.2  Program Data

Program Data (f07fgfe.d)

### 10.3  Program Results

Program Results (f07fgfe.r)