# NAG Library Routine Document

## 1Purpose

f07vuf (ztbcon) estimates the condition number of a complex triangular band matrix.

## 2Specification

Fortran Interface
 Subroutine f07vuf ( norm, uplo, diag, n, kd, ab, ldab, work, info)
 Integer, Intent (In) :: n, kd, ldab Integer, Intent (Out) :: info 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) :: norm, uplo, diag
#include nagmk26.h
 void f07vuf_ (const char *norm, const char *uplo, const char *diag, const Integer *n, const Integer *kd, const Complex ab[], const Integer *ldab, double *rcond, Complex work[], double rwork[], Integer *info, const Charlen length_norm, const Charlen length_uplo, const Charlen length_diag)
The routine may be called by its LAPACK name ztbcon.

## 3Description

f07vuf (ztbcon) estimates the condition number of a complex triangular band matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1A=A1A-11 or κ∞A=A∞A-1∞ .$
Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{T}}\right)$.
Because the condition number is infinite if $A$ is singular, the routine actually returns an estimate of the reciprocal of the condition number.
The routine computes ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$ exactly, and uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$ or ${‖{A}^{-1}‖}_{\infty }$.

## 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:     $\mathbf{norm}$ – Character(1)Input
On entry: indicates whether ${\kappa }_{1}\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.
${\mathbf{norm}}=\text{'1'}$ or $\text{'O'}$
${\kappa }_{1}\left(A\right)$ is estimated.
${\mathbf{norm}}=\text{'I'}$
${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: ${\mathbf{norm}}=\text{'1'}$, $\text{'O'}$ or $\text{'I'}$.
2:     $\mathbf{uplo}$ – Character(1)Input
On entry: specifies whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathbf{diag}$ – Character(1)Input
On entry: indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{diag}}=\text{'N'}$
$A$ is a nonunit triangular matrix.
${\mathbf{diag}}=\text{'U'}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{diag}}=\text{'N'}$ or $\text{'U'}$.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{kd}$ – IntegerInput
On entry: ${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{uplo}}=\text{'L'}$.
Constraint: ${\mathbf{kd}}\ge 0$.
6:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – 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 $n$ by $n$ triangular band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
7:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07vuf (ztbcon) is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
8:     $\mathbf{rcond}$ – 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.
9:     $\mathbf{work}\left(2×{\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayWorkspace
10:   $\mathbf{rwork}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
11:   $\mathbf{info}$ – 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

f07vuf (ztbcon) 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 f07vuf (ztbcon) involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{H}}x=b$; the number is usually $5$ and never more than $11$. Each solution involves approximately $8nk$ real floating-point operations (assuming $n\gg k$) but takes considerably longer than a call to f07vsf (ztbtrs) 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 f07vgf (dtbcon).

## 10Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= -1.94+4.43i 0.00+0.00i 0.00+0.00i 0.00+0.00i -3.39+3.44i 4.12-4.27i 0.00+0.00i 0.00+0.00i 1.62+3.68i -1.84+5.53i 0.43-2.66i 0.00+0.00i 0.00+0.00i -2.77-1.93i 1.74-0.04i 0.44+0.10i .$
Here $A$ is treated as a lower triangular band matrix with two subdiagonals. The true condition number in the $1$-norm is $71.51$.

### 10.1Program Text

Program Text (f07vufe.f90)

### 10.2Program Data

Program Data (f07vufe.d)

### 10.3Program Results

Program Results (f07vufe.r)

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