f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dpbcon (f07hgc)

## 1  Purpose

nag_dpbcon (f07hgc) estimates the condition number of a real symmetric positive definite band matrix $A$, where $A$ has been factorized by nag_dpbtrf (f07hdc).

## 2  Specification

 #include #include
 void nag_dpbcon (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, const double ab[], Integer pdab, double anorm, double *rcond, NagError *fail)

## 3  Description

nag_dpbcon (f07hgc) estimates the condition number (in the $1$-norm) of a real symmetric positive definite band 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_dsb_norm (f16rec) to compute ${‖A‖}_{1}$ and a call to nag_dpbtrf (f07hdc) 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:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     uploNag_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:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     kdIntegerInput
On entry: ${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
5:     ab[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$.
On entry: the Cholesky factor of $A$, as returned by nag_dpbtrf (f07hdc).
6:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kd}}+1$.
7:     anormdoubleInput
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling nag_dsb_norm (f16rec) with its argument ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$. anorm must be computed either before calling nag_dpbtrf (f07hdc) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.
8:     rconddouble *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.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{kd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kd}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}>0$.
NE_INT_2
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$ and ${\mathbf{kd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kd}}+1$.
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.
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.

A call to nag_dpbcon (f07hgc) 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 $4nk$ floating point operations (assuming $n\gg k$) but takes considerably longer than a call to nag_dpbtrs (f07hec) 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_zpbcon (f07huc).

## 9  Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 .$
Here $A$ is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_dpbtrf (f07hdc). The true condition number in the $1$-norm is $74.15$.

### 9.1  Program Text

Program Text (f07hgce.c)

### 9.2  Program Data

Program Data (f07hgce.d)

### 9.3  Program Results

Program Results (f07hgce.r)