nag_complex_cholesky (f01bnc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_complex_cholesky (f01bnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_complex_cholesky (f01bnc) computes a Cholesky factorization of a complex positive definite Hermitian matrix.

2  Specification

#include <nag.h>
#include <nagf01.h>
void  nag_complex_cholesky (Integer n, Complex a[], Integer tda, double p[], NagError *fail)

3  Description

nag_complex_cholesky (f01bnc) computes the Cholesky factorization of a complex positive definite Hermitian matrix A = U H U , where U  is a complex upper triangular matrix with real diagonal elements.

4  References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5  Arguments

1:     nIntegerInput
On entry: n , the order of the matrix A .
Constraint: n1 .
2:     a[n×tda]ComplexInput/Output
On entry: the lower triangle of the n  by n  positive definite Hermitian matrix A . The elements of the array above the diagonal need not be set.
On exit: the off-diagonal elements of the upper triangular matrix U . The lower triangle of A  is unchanged.
3:     tdaIntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: tdan .
4:     p[n]doubleOutput
On exit: the reciprocals of the real diagonal elements of U .
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, tda=value  while n=value . These arguments must satisfy tdan .
NE_DIAG_IMAG_NON_ZERO
Matrix diagonal element a[value×tda+value] has nonzero imaginary part.
NE_INT_ARG_LT
On entry, n=value.
Constraint: n1.
NE_NOT_POS_DEF
The matrix is not positive definite, possibly due to rounding errors.

7  Accuracy

The Cholesky factorization of a positive definite matrix is known for its remarkable numerical stability. The computed matrix U  satisfies the relation U H U = A + E  where the 2-norms of A  and E  are related by
E c ε A ,
c  is a modest function of n , and ε  is the machine precision.

8  Further Comments

The time taken by nag_complex_cholesky (f01bnc) is approximately proportional to n 3 .

9  Example

To compute the Cholesky factorization of the well-conditioned positive definite Hermitian matrix
- 15 -2 1 - 2 i -1 2 1 - 4 + 3 i -1 1 + 2 i - 20 1 - 2 + 2 i -2 3 - 3 i -1 2 2 - 2 - 1 i - 18 1 - 1 + 2 i 1 - 4 - 3 i -2 3 + 3 i 1 - 1 - 2 i - 26 .

9.1  Program Text

Program Text (f01bnce.c)

9.2  Program Data

Program Data (f01bnce.d)

9.3  Program Results

Program Results (f01bnce.r)


nag_complex_cholesky (f01bnc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG C Library Manual

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