nag_zgbcon (f07buc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgbcon (f07buc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgbcon (f07buc) estimates the condition number of a complex band matrix A, where A has been factorized by nag_zgbtrf (f07brc).

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zgbcon (Nag_OrderType order, Nag_NormType norm, Integer n, Integer kl, Integer ku, const Complex ab[], Integer pdab, const Integer ipiv[], double anorm, double *rcond, NagError *fail)

3  Description

nag_zgbcon (f07buc) estimates the condition number of a complex band matrix A, in either the 1-norm or the -norm:
κ1A=A1A-11   or   κA=AA-1 .
Note that κA=κ1AH.
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 should be preceded by a call to nag_zgb_norm (f16ubc) to compute A1 or A, and a call to nag_zgbtrf (f07brc) to compute the LU factorization of A. The function then 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:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     klIntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
5:     kuIntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
6:     ab[dim]const ComplexInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the LU factorization of A, as returned by nag_zgbtrf (f07brc).
7:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.
Constraint: pdab2×kl+ku+1.
8:     ipiv[dim]const IntegerInput
Note: the dimension, dim, of the array ipiv must be at least max1,n.
On entry: the pivot indices, as returned by nag_zgbtrf (f07brc).
9:     anormdoubleInput
On entry: if norm=Nag_OneNorm, the 1-norm of the original matrix A.
If norm=Nag_InfNorm, the -norm of the original matrix A.
anorm may be computed by calling nag_zgb_norm (f16ubc) with the same value for the argument norm.
anorm must be computed either before calling nag_zgbtrf (f07brc) or else from a copy of the original matrix A (see Section 9).
Constraint: anorm0.0.
10:   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.
11:   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, kl=value.
Constraint: kl0.
On entry, ku=value.
Constraint: ku0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
NE_INT_3
On entry, pdab=value, kl=value and ku=value.
Constraint: pdab2×kl+ku+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, anorm=value.
Constraint: anorm0.0.

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_zgbcon (f07buc) 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 8n2kl+ku real floating point operations (assuming nkl and nku) but takes considerably longer than a call to nag_zgbtrs (f07bsc) 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_dgbcon (f07bgc).

9  Example

This example estimates the condition number in the 1-norm of the matrix A, where
A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i .

9.1  Program Text

Program Text (f07buce.c)

9.2  Program Data

Program Data (f07buce.d)

9.3  Program Results

Program Results (f07buce.r)


nag_zgbcon (f07buc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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