nag_zhpevx (f08gpc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zhpevx (f08gpc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zhpevx (f08gpc) computes selected eigenvalues and, optionally, eigenvectors of a complex n by n Hermitian matrix A in packed storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zhpevx (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Nag_UploType uplo, Integer n, Complex ap[], double vl, double vu, Integer il, Integer iu, double abstol, Integer *m, double w[], Complex z[], Integer pdz, Integer jfail[], NagError *fail)

3  Description

The Hermitian matrix A is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
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_EigVals or Nag_DoBoth.
3:     rangeNag_RangeTypeInput
On entry: if range=Nag_AllValues, all eigenvalues will be found.
If range=Nag_Interval, all eigenvalues in the half-open interval vl,vu will be found.
If range=Nag_Indices, the ilth to iuth eigenvalues will be found.
Constraint: range=Nag_AllValues, Nag_Interval or Nag_Indices.
4:     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.
5:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
6:     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.
7:     vldoubleInput
8:     vudoubleInput
On entry: if range=Nag_Interval, the lower and upper bounds of the interval to be searched for eigenvalues.
If range=Nag_AllValues or Nag_Indices, vl and vu are not referenced.
Constraint: if range=Nag_Interval, vl<vu.
9:     ilIntegerInput
10:   iuIntegerInput
On entry: if range=Nag_Indices, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range=Nag_AllValues or Nag_Interval, il and iu are not referenced.
  • if range=Nag_Indices and n=0, il=1 and iu=0;
  • if range=Nag_Indices and n>0, 1 il iu n .
11:   abstoldoubleInput
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval a,b  of width less than or equal to
abstol+ε maxa,b ,
where ε  is the machine precision. If abstol is less than or equal to zero, then ε T1  will be used in its place, where T is the tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2 × nag_real_safe_small_number , not zero. If this function returns with fail.code= NE_CONVERGENCE, indicating that some eigenvectors did not converge, try setting abstol to 2 × nag_real_safe_small_number . See Demmel and Kahan (1990).
12:   mInteger *Output
On exit: the total number of eigenvalues found. 0mn.
If range=Nag_AllValues, m=n.
If range=Nag_Indices, m=iu-il+1.
13:   w[n]doubleOutput
On exit: the selected eigenvalues in ascending order.
14:   z[dim]ComplexOutput
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when job=Nag_DoBoth;
  • 1 otherwise.
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_DoBoth, then
  • if fail.code= NE_NOERROR, the first m columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with w[i-1];
  • if an eigenvector fails to converge (fail.code= NE_CONVERGENCE), then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If job=Nag_EigVals, z is not referenced.
15:   pdzIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
  • if job=Nag_DoBoth, pdz max1,n ;
  • otherwise pdz1.
16:   jfail[dim]IntegerOutput
Note: the dimension, dim, of the array jfail must be at least max1,n.
On exit: if job=Nag_DoBoth, then
  • if fail.code= NE_NOERROR, the first m elements of jfail are zero;
  • if fail.code= NE_CONVERGENCE, jfail contains the indices of the eigenvectors that failed to converge.
If job=Nag_EigVals, jfail is not referenced.
17:   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.
The algorithm failed to converge; value eigenvectors did not converge.
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_DoBoth, pdz max1,n ;
otherwise pdz1.
On entry, range=value, il=value, iu=value and n=value.
Constraint: if range=Nag_Indices and n=0, il=1 and iu=0;
if range=Nag_Indices and n>0, 1 il iu n .
On entry, range=value, vl=value and vu=value.
Constraint: if range=Nag_Interval, vl<vu.
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 total number of floating point operations is proportional to n3.
The real analogue of this function is nag_dspevx (f08gbc).

9  Example

This example finds the eigenvalues in the half-open interval -2,2 , and the corresponding 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 .

9.1  Program Text

Program Text (f08gpce.c)

9.2  Program Data

Program Data (f08gpce.d)

9.3  Program Results

Program Results (f08gpce.r)

nag_zhpevx (f08gpc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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