# NAG Library Routine Document

## 1Purpose

f07huf (zpbcon) estimates the condition number of a complex Hermitian positive definite band matrix $A$, where $A$ has been factorized by f07hrf (zpbtrf).

## 2Specification

Fortran Interface
 Subroutine f07huf ( uplo, n, kd, ab, ldab, work, info)
 Integer, Intent (In) :: n, kd, ldab Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: anorm Real (Kind=nag_wp), Intent (Out) :: rcond, rwork(n) Complex (Kind=nag_wp), Intent (In) :: ab(ldab,*) Complex (Kind=nag_wp), Intent (Out) :: work(2*n) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f07huf_ ( const char *uplo, const Integer *n, const Integer *kd, const Complex ab[], const Integer *ldab, const double *anorm, double *rcond, Complex work[], double rwork[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zpbcon.

## 3Description

f07huf (zpbcon) estimates the condition number (in the $1$-norm) of a complex Hermitian positive definite band matrix $A$:
 $κ1A=A1A-11 .$
Since $A$ is Hermitian, ${\kappa }_{1}\left(A\right)={\kappa }_{\infty }\left(A\right)={‖A‖}_{\infty }{‖{A}^{-1}‖}_{\infty }$.
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 f06uef to compute ${‖A‖}_{1}$ and a call to f07hrf (zpbtrf) 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.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{H}}$, 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:     $\mathbf{kd}$ – IntegerInput
On entry: ${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
4:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab 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 f07hrf (zpbtrf).
5:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07huf (zpbcon) is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
6:     $\mathbf{anorm}$ – Real (Kind=nag_wp)Input
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling f06uef with its argument ${\mathbf{norm}}=\text{'1'}$. anorm must be computed either before calling f07hrf (zpbtrf) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.
7:     $\mathbf{rcond}$ – 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.
8:     $\mathbf{work}\left(2×{\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayWorkspace
9:     $\mathbf{rwork}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
10:   $\mathbf{info}$ – 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

f07huf (zpbcon) 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 f07huf (zpbcon) involves solving a number of systems of linear equations of the form $Ax=b$; the number is usually $5$ and never more than $11$. Each solution involves approximately $16nk$ real floating-point operations (assuming $n\gg k$) but takes considerably longer than a call to f07hsf (zpbtrs) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The real analogue of this routine is f07hgf (dpbcon).

## 10Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 9.39+0.00i 1.08-1.73i 0.00+0.00i 0.00+0.00i 1.08+1.73i 1.69+0.00i -0.04+0.29i 0.00+0.00i 0.00+0.00i -0.04-0.29i 2.65+0.00i -0.33+2.24i 0.00+0.00i 0.00+0.00i -0.33-2.24i 2.17+0.00i .$
Here $A$ is Hermitian positive definite, and is treated as a band matrix, which must first be factorized by f07hrf (zpbtrf). The true condition number in the $1$-norm is $153.45$.

### 10.1Program Text

Program Text (f07hufe.f90)

### 10.2Program Data

Program Data (f07hufe.d)

### 10.3Program Results

Program Results (f07hufe.r)

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