nag_zsteqr (f08jsc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zsteqr (f08jsc)

+ Contents

    1  Purpose
    7  Accuracy
    9  Example

1  Purpose

nag_zsteqr (f08jsc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix which has been reduced to tridiagonal form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zsteqr (Nag_OrderType order, Nag_ComputeZType compz, Integer n, double d[], double e[], Complex z[], Integer pdz, NagError *fail)

3  Description

nag_zsteqr (f08jsc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix T. In other words, it can compute the spectral factorization of T as
T=ZΛZT,
where Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix whose columns are the eigenvectors zi. Thus
Tzi=λizi,  i=1,2,,n.
The function stores the real orthogonal matrix Z in a complex array, so that it may also be used to compute all the eigenvalues and eigenvectors of a complex Hermitian matrix A which has been reduced to tridiagonal form T:
A =QTQH, where ​Q​ is unitary =QZΛQZH.
In this case, the matrix Q must be formed explicitly and passed to nag_zsteqr (f08jsc), which must be called with compz=Nag_UpdateZ. The functions which must be called to perform the reduction to tridiagonal form and form Q are:
full matrix nag_zhetrd (f08fsc) and nag_zungtr (f08ftc)
full matrix, packed storage nag_zhptrd (f08gsc) and nag_zupgtr (f08gtc)
band matrix nag_zhbtrd (f08hsc) with vect=Nag_FormQ.
nag_zsteqr (f08jsc) uses the implicitly shifted QR algorithm, switching between the QR and QL variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that zi2=1, but are determined only to within a complex factor of absolute value 1.
If only the eigenvalues of T are required, it is more efficient to call nag_dsterf (f08jfc) instead. If T is positive definite, small eigenvalues can be computed more accurately by nag_zpteqr (f08juc).

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

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:     compzNag_ComputeZTypeInput
On entry: indicates whether the eigenvectors are to be computed.
compz=Nag_NotZ
Only the eigenvalues are computed (and the array z is not referenced).
compz=Nag_InitZ
The eigenvalues and eigenvectors of T are computed (and the array z is initialized by the function).
compz=Nag_UpdateZ
The eigenvalues and eigenvectors of A are computed (and the array z must contain the matrix Q on entry).
Constraint: compz=Nag_NotZ, Nag_UpdateZ or Nag_InitZ.
3:     nIntegerInput
On entry: n, the order of the matrix T.
Constraint: n0.
4:     d[dim]doubleInput/Output
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: the diagonal elements of the tridiagonal matrix T.
On exit: the n eigenvalues in ascending order, unless fail.code= NE_CONVERGENCE (in which case see Section 6).
5:     e[dim]doubleInput/Output
Note: the dimension, dim, of the array e must be at least max1,n-1.
On entry: the off-diagonal elements of the tridiagonal matrix T.
On exit: e is overwritten.
6:     z[dim]ComplexInput/Output
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when compz=Nag_UpdateZ or Nag_InitZ and order=Nag_ColMajor;
  • max1,×pdz when compz=Nag_UpdateZ or Nag_InitZ and order=Nag_RowMajor;
  • 1 when compz=Nag_NotZ.
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 entry: if compz=Nag_UpdateZ, z must contain the unitary matrix Q from the reduction to tridiagonal form.
If compz=Nag_InitZ, z need not be set.
On exit: if compz=Nag_InitZ or Nag_UpdateZ, the n required orthonormal eigenvectors stored as columns of Z; the ith column corresponds to the ith eigenvalue, where i=1,2,,n, unless fail.code= NE_CONVERGENCE.
If compz=Nag_NotZ, z is not referenced.
7:     pdzIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if order=Nag_ColMajor,
    • if compz=Nag_InitZ or Nag_UpdateZ, pdz max1,n ;
    • if compz=Nag_NotZ, pdz1;
  • if order=Nag_RowMajor,
    • if compz=Nag_UpdateZ or Nag_InitZ, pdzmax1,n;
    • if compz=Nag_NotZ, pdz1.
8:     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 has failed to find all the eigenvalues after a total of 30×n iterations. In this case, d and e contain on exit the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix similar to T. value off-diagonal elements have not converged to zero.
NE_ENUM_INT_2
On entry, compz=value, pdz=value and n=value.
Constraint: if compz=Nag_InitZ or Nag_UpdateZ, pdz max1,n ;
if compz=Nag_NotZ, pdz1.
On entry, compz=value, pdz=value, n=value.
Constraint: if compz=Nag_UpdateZ or Nag_InitZ, pdzmax1,n;
if compz=Nag_NotZ, pdz1.
NE_INT
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.

7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix T+E, where
E2 = Oε T2 ,
and ε is the machine precision.
If λi is an exact eigenvalue and λ~i is the corresponding computed value, then
λ~i - λi c n ε T2 ,
where cn is a modestly increasing function of n.
If zi is the corresponding exact eigenvector, and z~i is the corresponding computed eigenvector, then the angle θz~i,zi between them is bounded as follows:
θ z~i,zi cnεT2 minijλi-λj .
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

8  Further Comments

The total number of real floating point operations is typically about 24n2 if compz=Nag_NotZ and about 14n3 if compz=Nag_UpdateZ or Nag_InitZ, but depends on how rapidly the algorithm converges. When compz=Nag_NotZ, the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when compz=Nag_UpdateZ or Nag_InitZ can be vectorized and on some machines may be performed much faster.
The real analogue of this function is nag_dsteqr (f08jec).

9  Example

See Section 9 in nag_zungtr (f08ftc), nag_zupgtr (f08gtc) or nag_zhbtrd (f08hsc), which illustrate the use of this function to compute the eigenvalues and eigenvectors of a full or band Hermitian matrix.

nag_zsteqr (f08jsc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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