nag_zhpevd (f08gqc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG 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  Parallelism and Performance

nag_zhpevd (f08gqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhpevd (f08gqc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

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

10  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 .

10.1  Program Text

Program Text (f08gqce.c)

10.2  Program Data

Program Data (f08gqce.d)

10.3  Program Results

Program Results (f08gqce.r)

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014