nag_zstegr (f08jyc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_zstegr (f08jyc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zstegr (f08jyc) computes all the eigenvalues and, optionally, all the eigenvectors of a real n by n symmetric tridiagonal matrix.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zstegr (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Integer n, double d[], double e[], double vl, double vu, Integer il, Integer iu, Integer *m, double w[], Complex z[], Integer pdz, Integer isuppz[], NagError *fail)

3  Description

nag_zstegr (f08jyc) computes all the eigenvalues and, optionally, the eigenvectors, of a real symmetric tridiagonal matrix T. That is, the function computes the spectral factorization of T given by
T = ZΛZT ,
where Λ is a diagonal matrix whose diagonal elements are the eigenvalues, λi, of T and Z is an orthogonal matrix whose columns are the eigenvectors, zi, of T. Thus
Tzi= λi zi ,   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 explicitly applied to the output matrix Z. The functions which must be called to perform the reduction to tridiagonal form and apply Q are:
full matrix nag_zhetrd (f08fsc) and nag_zunmtr (f08fuc)
full matrix, packed storage nag_zhptrd (f08gsc) and nag_zupmtr (f08guc)
band matrix nag_zhbtrd (f08hsc) with vect=Nag_FormQ and nag_zgemm (f16zac).
This function uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example Parlett and Dhillon (2000) and Dhillon and Parlett (2004) for further details. nag_zstegr (f08jyc) can usually compute all the eigenvalues and eigenvectors in On2 floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal functions in this chapter when all the eigenvectors are required, particularly so compared to those functions that are based on the QR algorithm.

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
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps. SIAM J. Appl. Math. 25 858–899
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

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: indicates which eigenvalues should be returned.
All eigenvalues will be found.
All eigenvalues in the half-open interval vl,vu will be found.
The ilth through iuth eigenvectors will be found.
Constraint: range=Nag_AllValues, Nag_Interval or Nag_Indices.
4:     nIntegerInput
On entry: n, the order of the matrix T.
Constraint: n0.
5:     d[dim]doubleInput/Output
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: the n diagonal elements of the tridiagonal matrix T.
On exit: d is overwritten.
6:     e[dim]doubleInput/Output
Note: the dimension, dim, of the array e must be at least max1,n.
On entry: e[0] to e[n-2] are the subdiagonal elements of the tridiagonal matrix T. e[n-1] need not be set.
On exit: e is overwritten.
7:     vldoubleInput
8:     vudoubleInput
On entry: if range=Nag_Interval, vl and vu contain the lower and upper bounds respectively 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, il and iu contains the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If range=Nag_AllValues or Nag_Interval, il and iu are not referenced.
  • if range=Nag_Indices and n>0, 1 il iu n ;
  • if range=Nag_Indices and n=0, il=1 and iu=0.
11:   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.
12:   w[dim]doubleOutput
Note: the dimension, dim, of the array w must be at least max1,n.
On exit: the eigenvalues in ascending order.
13:   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 columns of z contain the orthonormal eigenvectors of the matrix T, with the ith column of Z holding the eigenvector associated with w[i-1].
If job=Nag_EigVals, z is not referenced.
14:   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.
15:   isuppz[dim]IntegerOutput
Note: the dimension, dim, of the array isuppz must be at least max1,2×m.
On exit: the support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The ith eigenvector is nonzero only in elements isuppz[2×i-2] through isuppz[2×i-1].
16:   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.
Inverse iteration failed to converge.
The dqds algorithm failed to 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, 1 il iu n ;
if range=Nag_Indices and n=0, il=1 and iu=0.
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

See Section 4.7 of Anderson et al. (1999) and Barlow and Demmel (1990) for further details.

8  Parallelism and Performance

nag_zstegr (f08jyc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zstegr (f08jyc) 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 total number of floating-point operations required to compute all the eigenvalues and eigenvectors is approximately proportional to n2.
The real analogue of this function is nag_dstegr (f08jlc).

10  Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
T = 1.0 1.0 0.0 0.0 1.0 4.0 2.0 0.0 0.0 2.0 9.0 3.0 0.0 0.0 3.0 16.0 .

10.1  Program Text

Program Text (f08jyce.c)

10.2  Program Data

Program Data (f08jyce.d)

10.3  Program Results

Program Results (f08jyce.r)

nag_zstegr (f08jyc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

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