nag_zhpevd (f08gqc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zhpevd (f08gqc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zhpevd (f08gqc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix held in packed storage. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the QL or QR algorithm.

2  Specification

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

3  Description

nag_zhpevd (f08gqc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix A (held in packed storage). In other words, it can compute the spectral factorization of A as
where Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the (complex) unitary matrix whose columns are the eigenvectors zi. Thus
Azi=λizi,  i=1,2,,n.

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
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 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.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_DoNothing or Nag_EigVecs.
3:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of A is stored.
The upper triangular part of A is stored.
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:     ap[dim]ComplexInput/Output
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the upper or lower triangle of the n by n Hermitian matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ap[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ap[i-1×i/2+j-1], for ij.
On exit: ap is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A.
6:     w[dim]doubleOutput
Note: the dimension, dim, of the array w must be at least max1,n.
On exit: the eigenvalues of the matrix A in ascending order.
7:     z[dim]ComplexOutput
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when job=Nag_EigVecs;
  • 1 when job=Nag_DoNothing.
The i,jth element of the matrix Z is stored in
  • z[j-1×pdz+i-1] when order=Nag_ColMajor;
  • z[i-1×pdz+j-1] when order=Nag_RowMajor.
On exit: if job=Nag_EigVecs, z is overwritten by the unitary matrix Z which contains the eigenvectors of A.
If job=Nag_DoNothing, z is not referenced.
8:     pdzIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
  • if job=Nag_EigVecs, pdz max1,n ;
  • if job=Nag_DoNothing, pdz1.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
If fail.errnum=value and job=Nag_DoNothing, the algorithm failed to converge; value elements of an intermediate tridiagonal form did not converge to zero; if fail.errnum=value and job=Nag_EigVecs, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column value/n+1 through value mod n+1.
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_EigVecs, pdz max1,n ;
if job=Nag_DoNothing, pdz1.
On entry, n=value.
Constraint: n0.
On entry, pdz=value.
Constraint: pdz>0.
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

The real analogue of this function is nag_dspevd (f08gcc).

9  Example

This example computes all the eigenvalues and eigenvectors of the Hermitian matrix A, where
A = 1.0+0.0i 2.0-1.0i 3.0-1.0i 4.0-1.0i 2.0+1.0i 2.0+0.0i 3.0-2.0i 4.0-2.0i 3.0+1.0i 3.0+2.0i 3.0+0.0i 4.0-3.0i 4.0+1.0i 4.0+2.0i 4.0+3.0i 4.0+0.0i .

9.1  Program Text

Program Text (f08gqce.c)

9.2  Program Data

Program Data (f08gqce.d)

9.3  Program Results

Program Results (f08gqce.r)

nag_zhpevd (f08gqc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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