nag_zhpgv (f08tnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zhpgv (f08tnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zhpgv (f08tnc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form
Az=λBz ,   ABz=λz   or   BAz=λz ,
where A and B are Hermitian, stored in packed format, and B is also positive definite.

2  Specification

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

3  Description

nag_zhpgv (f08tnc) first performs a Cholesky factorization of the matrix B as B=UHU , when uplo=Nag_Upper or B=LLH , when uplo=Nag_Lower. The generalized problem is then reduced to a standard symmetric eigenvalue problem
Cx=λx ,
which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem Az=λBz , the eigenvectors are normalized so that the matrix of eigenvectors, Z, satisfies
ZH A Z = Λ   and   ZH B Z = I ,
where Λ  is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem A B z = λ z  we correspondingly have
Z-1 A Z-H = Λ   and   ZH B Z = I ,
and for B A z = λ z  we have
ZH A Z = Λ   and   ZH B-1 Z = I .

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:     itypeIntegerInput
On entry: specifies the problem type to be solved.
itype=1
Az=λBz.
itype=2
ABz=λz.
itype=3
BAz=λz.
Constraint: itype=1, 2 or 3.
3:     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.
4:     uploNag_UploTypeInput
On entry: if uplo=Nag_Upper, the upper triangles of A and B are stored.
If uplo=Nag_Lower, the lower triangles of A and B are stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
5:     nIntegerInput
On entry: n, the order of the matrices A and B.
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: the contents of ap are destroyed.
7:     bp[dim]ComplexInput/Output
Note: the dimension, dim, of the array bp must be at least max1,n×n+1/2.
On entry: the upper or lower triangle of the n by n Hermitian matrix B, packed by rows or columns.
The storage of elements Bij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Bij is stored in bp[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Bij is stored in bp[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Bij is stored in bp[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Bij is stored in bp[i-1×i/2+j-1], for ij.
On exit: the triangular factor U or L from the Cholesky factorization B=UHU or B=LLH, in the same storage format as B.
8:     w[n]doubleOutput
On exit: the eigenvalues in ascending order.
9:     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, z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:
  • if itype=1 or 2, ZHBZ=I;
  • if itype=3, ZHB-1Z=I.
If job=Nag_EigVals, z is not referenced.
10:   pdzIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if job=Nag_DoBoth, pdz max1,n ;
  • otherwise pdz1.
11:   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_ENUM_INT_2
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_DoBoth, pdz max1,n ;
otherwise pdz1.
NE_INT
On entry, itype=value.
Constraint: itype=1, 2 or 3.
On entry, n=value.
Constraint: n0.
On entry, pdz=value.
Constraint: pdz>0.
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.
NE_MAT_NOT_POS_DEF
If fail.errnum=n+value, for 1valuen, then the leading minor of order value of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

7  Accuracy

If B is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of B differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of B would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.
The example program below illustrates the computation of approximate error bounds.

8  Further Comments

The total number of floating point operations is proportional to n3 .
The real analogue of this function is nag_dspgv (f08tac).

9  Example

This example finds all the eigenvalues and eigenvectors of the generalized Hermitian eigenproblem Az = λ Bz , where
A = -7.36i+0.00 0.77-0.43i -0.64-0.92i 3.01-6.97i 0.77+0.43i 3.49i+0.00 2.19+4.45i 1.90+3.73i -0.64+0.92i 2.19-4.45i 0.12i+0.00 2.88-3.17i 3.01+6.97i 1.90-3.73i 2.88+3.17i -2.54i+0.00
and
B = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00 ,
together with an estimate of the condition number of B, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for nag_zhpgvd (f08tqc) illustrates solving a generalized symmetric eigenproblem of the form ABz=λz .

9.1  Program Text

Program Text (f08tnce.c)

9.2  Program Data

Program Data (f08tnce.d)

9.3  Program Results

Program Results (f08tnce.r)


nag_zhpgv (f08tnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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