nag_ztbcon (f07vuc) (PDF version)
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f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_ztbcon (f07vuc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_ztbcon (f07vuc) estimates the condition number of a complex triangular band matrix.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_ztbcon (Nag_OrderType order, Nag_NormType norm, Nag_UploType uplo, Nag_DiagType diag, Integer n, Integer kd, const Complex ab[], Integer pdab, double *rcond, NagError *fail)

3  Description

nag_ztbcon (f07vuc) estimates the condition number of a complex triangular band matrix A, in either the 1-norm or the -norm:
κ1A=A1A-11   or   κA=AA-1 .
Note that κA=κ1AT.
Because the condition number is infinite if A is singular, the function actually returns an estimate of the reciprocal of the condition number.
The function computes A1 or A exactly, and uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11 or A-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 order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     normNag_NormTypeInput
On entry: indicates whether κ1A or κA is estimated.
norm=Nag_OneNorm
κ1A is estimated.
norm=Nag_InfNorm
κA is estimated.
Constraint: norm=Nag_OneNorm or Nag_InfNorm.
3:     uploNag_UploTypeInput
On entry: specifies whether A is upper or lower triangular.
uplo=Nag_Upper
A is upper triangular.
uplo=Nag_Lower
A is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     diagNag_DiagTypeInput
On entry: indicates whether A is a nonunit or unit triangular matrix.
diag=Nag_NonUnitDiag
A is a nonunit triangular matrix.
diag=Nag_UnitDiag
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag=Nag_NonUnitDiag or Nag_UnitDiag.
5:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
6:     kdIntegerInput
On entry: kd, the number of superdiagonals of the matrix A if uplo=Nag_Upper, or the number of subdiagonals if uplo=Nag_Lower.
Constraint: kd0.
7:     ab[dim]const ComplexInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the n by n triangular band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[kd+i-j+j-1×pdab], for j=1,,n and i=max1,j-kd,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+kd;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+kd;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[kd+j-i+i-1×pdab], for i=1,,n and j=max1,i-kd,,i.
If diag=Nag_UnitDiag, the diagonal elements of AB are assumed to be 1, and are not referenced.
8:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkd+1.
9:     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.
10:   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.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
NE_INT_2
On entry, pdab=value and kd=value.
Constraint: pdabkd+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.

7  Accuracy

The computed estimate rcond is never less than the true value ρ, and in practice is nearly always less than 10ρ, although examples can be constructed where rcond is much larger.

8  Further Comments

A call to nag_ztbcon (f07vuc) involves solving a number of systems of linear equations of the form Ax=b or AHx=b; the number is usually 5 and never more than 11. Each solution involves approximately 8nk real floating point operations (assuming nk) but takes considerably longer than a call to nag_ztbtrs (f07vsc) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The real analogue of this function is nag_dtbcon (f07vgc).

9  Example

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.

9.1  Program Text

Program Text (f07vuce.c)

9.2  Program Data

Program Data (f07vuce.d)

9.3  Program Results

Program Results (f07vuce.r)


nag_ztbcon (f07vuc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012