nag_zhbevd (f08hqc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zhbevd (f08hqc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zhbevd (f08hqc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian band matrix. 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_zhbevd (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, Integer kd, Complex ab[], Integer pdab, double w[], Complex z[], Integer pdz, NagError *fail)

3  Description

nag_zhbevd (f08hqc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian band matrix A. In other words, it can compute the spectral factorization of A as
A=ZΛZH,
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 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_DoNothing
Only eigenvalues are computed.
job=Nag_EigVecs
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.
uplo=Nag_Upper
The upper triangular part of A is stored.
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:     kdIntegerInput
On entry: if uplo=Nag_Upper, the number of superdiagonals, kd, of the matrix A.
If uplo=Nag_Lower, the number of subdiagonals, kd, of the matrix A.
Constraint: kd0.
6:     ab[dim]ComplexInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the upper or lower triangle of the n by n Hermitian band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[kd+i-j+j-1×pdab], for j=1,,n and i=max1,j-kd,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+kd;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+kd;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[kd+j-i+i-1×pdab], for i=1,,n and j=max1,i-kd,,i.
On exit: ab is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix T are returned in ab using the same storage format as described above.
7:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkd+1.
8:     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.
9:     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. The ith column of Z contains the eigenvector which corresponds to the eigenvalue w[i-1].
If job=Nag_DoNothing, 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_EigVecs, pdz max1,n ;
  • if job=Nag_DoNothing, 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
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.
NE_ENUM_INT_2
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_EigVecs, pdz max1,n ;
if job=Nag_DoNothing, pdz1.
NE_INT
On entry, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdz=value.
Constraint: pdz>0.
NE_INT_2
On entry, pdab=value and kd=value.
Constraint: pdabkd+1.
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

The real analogue of this function is nag_dsbevd (f08hcc).

9  Example

This example computes all the eigenvalues and eigenvectors of the Hermitian band matrix A, where
A = 1+0i 2-1i 3-1i 0+0i 0+0i 2+1i 2+0i 3-2i 4-2i 0+0i 3+1i 3+2i 3+0i 4-3i 5-3i 0+0i 4+2i 4+3i 4+0i 5-4i 0+0i 0+0i 5+3i 5+4i 5+0i .

9.1  Program Text

Program Text (f08hqce.c)

9.2  Program Data

Program Data (f08hqce.d)

9.3  Program Results

Program Results (f08hqce.r)


nag_zhbevd (f08hqc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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