nag_zupgtr (f08gtc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zupgtr (f08gtc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zupgtr (f08gtc) generates the complex unitary matrix Q, which was determined by nag_zhptrd (f08gsc) when reducing a Hermitian matrix to tridiagonal form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zupgtr (Nag_OrderType order, Nag_UploType uplo, Integer n, const Complex ap[], const Complex tau[], Complex q[], Integer pdq, NagError *fail)

3  Description

nag_zupgtr (f08gtc) is intended to be used after a call to nag_zhptrd (f08gsc), which reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: A=QTQH. nag_zhptrd (f08gsc) represents the unitary matrix Q as a product of n-1 elementary reflectors.
This function may be used to generate Q explicitly as a square matrix.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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:     uploNag_UploTypeInput
On entry: this must be the same argument uplo as supplied to nag_zhptrd (f08gsc).
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix Q.
Constraint: n0.
4:     ap[dim]const ComplexInput
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_zhptrd (f08gsc).
5:     tau[dim]const ComplexInput
Note: the dimension, dim, of the array tau must be at least max1,n-1.
On entry: further details of the elementary reflectors, as returned by nag_zhptrd (f08gsc).
6:     q[dim]ComplexOutput
Note: the dimension, dim, of the array q must be at least max1,pdq×n.
The i,jth element of the matrix Q is stored in
  • q[j-1×pdq+i-1] when order=Nag_ColMajor;
  • q[i-1×pdq+j-1] when order=Nag_RowMajor.
On exit: the n by n unitary matrix Q.
7:     pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraint: pdqmax1,n.
8:     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, n=value.
Constraint: n0.
On entry, pdq=value.
Constraint: pdq>0.
NE_INT_2
On entry, pdq=value and n=value.
Constraint: pdqmax1,n.
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 matrix Q differs from an exactly unitary matrix by a matrix E such that
E2 = Oε ,
where ε is the machine precision.

8  Further Comments

The total number of real floating point operations is approximately 163n3.
The real analogue of this function is nag_dopgtr (f08gfc).

9  Example

This example computes all the eigenvalues and eigenvectors of the matrix A, where
A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ,
using packed storage. Here A is Hermitian and must first be reduced to tridiagonal form by nag_zhptrd (f08gsc). The program then calls nag_zupgtr (f08gtc) to form Q, and passes this matrix to nag_zsteqr (f08jsc) which computes the eigenvalues and eigenvectors of A.

9.1  Program Text

Program Text (f08gtce.c)

9.2  Program Data

Program Data (f08gtce.d)

9.3  Program Results

Program Results (f08gtce.r)


nag_zupgtr (f08gtc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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