f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dppcon (f07ggc)

## 1  Purpose

nag_dppcon (f07ggc) estimates the condition number of a real symmetric positive definite matrix $A$, where $A$ has been factorized by nag_dpptrf (f07gdc), using packed storage.

## 2  Specification

 #include #include
 void nag_dppcon (Nag_OrderType order, Nag_UploType uplo, Integer n, const double ap[], double anorm, double *rcond, NagError *fail)

## 3  Description

nag_dppcon (f07ggc) 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 function actually returns an estimate of the reciprocal of ${\kappa }_{1}\left(A\right)$.
The function should be preceded by a call to nag_dsp_norm (f16rdc) to compute ${‖A‖}_{1}$ and a call to nag_dpptrf (f07gdc) to compute the Cholesky factorization of $A$. The function 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  Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{ap}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, 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 nag_dpptrf (f07gdc).
5:    $\mathbf{anorm}$doubleInput
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling nag_dsp_norm (f16rdc) with its argument ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$. anorm must be computed either before calling nag_dpptrf (f07gdc) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.
6:    $\mathbf{rcond}$double *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:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{anorm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.

## 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

nag_dppcon (f07ggc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9  Further Comments

A call to nag_dppcon (f07ggc) 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 nag_dpptrs (f07gec) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogue of this function is nag_zppcon (f07guc).

## 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, stored in packed form, and must first be factorized by nag_dpptrf (f07gdc). The true condition number in the $1$-norm is $97.32$.

### 10.1  Program Text

Program Text (f07ggce.c)

### 10.2  Program Data

Program Data (f07ggce.d)

### 10.3  Program Results

Program Results (f07ggce.r)