nag_zheev (f08fnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zheev (f08fnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zheev (f08fnc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex n by n Hermitian matrix A.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zheev (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, Complex a[], Integer pda, double w[], NagError *fail)

3  Description

The Hermitian matrix A is first reduced to real tridiagonal form, using unitary similarity transformations, and then the QR algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
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:     jobNag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
job=Nag_EigVals
Only eigenvalues are computed.
job=Nag_DoBoth
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_EigVals or Nag_DoBoth.
3:     uploNag_UploTypeInput
On entry: if uplo=Nag_Upper, the upper triangular part of A is stored.
If uplo=Nag_Lower, the lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     a[dim]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the n by n Hermitian matrix A.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if job=Nag_DoBoth, then a contains the orthonormal eigenvectors of the matrix A.
If job=Nag_EigVals, then on exit the lower triangle (if uplo=Nag_Lower) or the upper triangle (if uplo=Nag_Upper) of a, including the diagonal, is overwritten.
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
7:     w[n]doubleOutput
On exit: the eigenvalues in ascending order.
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_CONVERGENCE
The algorithm failed to converge; value off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,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 eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2 = Oε A2 ,
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8  Further Comments

Each eigenvector is normalized so that the element of largest absolute value is real and positive.
The total number of floating point operations is proportional to n3.
The real analogue of this function is nag_dsyev (f08fac).

9  Example

This example finds all the eigenvalues and eigenvectors of the Hermitian matrix
A = 1 2-i 3-i 4-i 2+i 2 3-2i 4-2i 3+i 3+2i 3 4-3i 4+i 4+2i 4+3i 4 ,
together with approximate error bounds for the computed eigenvalues and eigenvectors.

9.1  Program Text

Program Text (f08fnce.c)

9.2  Program Data

Program Data (f08fnce.d)

9.3  Program Results

Program Results (f08fnce.r)


nag_zheev (f08fnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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