NAG FL Interface
f08jsf (zsteqr)

1 Purpose

f08jsf computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix which has been reduced to tridiagonal form.

2 Specification

Fortran Interface
Subroutine f08jsf ( compz, n, d, e, z, ldz, work, info)
Integer, Intent (In) :: n, ldz
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), work(*)
Complex (Kind=nag_wp), Intent (Inout) :: z(ldz,*)
Character (1), Intent (In) :: compz
C Header Interface
#include <nag.h>
void  f08jsf_ (const char *compz, const Integer *n, double d[], double e[], Complex z[], const Integer *ldz, double work[], Integer *info, const Charlen length_compz)
The routine may be called by the names f08jsf, nagf_lapackeig_zsteqr or its LAPACK name zsteqr.

3 Description

f08jsf 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
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 routine 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 f08jsf, which must be called with compz='V'. The routines which must be called to perform the reduction to tridiagonal form and form Q are:
full matrix f08fsf and f08ftf
full matrix, packed storage f08gsf and f08gtf
band matrix f08hsf with vect='V'.
f08jsf 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 f08jff instead. If T is positive definite, small eigenvalues can be computed more accurately by f08juf.

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: compz Character(1) Input
On entry: indicates whether the eigenvectors are to be computed.
Only the eigenvalues are computed (and the array z is not referenced).
The eigenvalues and eigenvectors of A are computed (and the array z must contain the matrix Q on entry).
The eigenvalues and eigenvectors of T are computed (and the array z is initialized by the routine).
Constraint: compz='N', 'V' or 'I'.
2: n Integer Input
On entry: n, the order of the matrix T.
Constraint: n0.
3: d* Real (Kind=nag_wp) array Input/Output
Note: the dimension 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 info>0 (in which case see Section 6).
4: e* Real (Kind=nag_wp) array Input/Output
Note: the dimension 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.
5: zldz* Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array z must be at least max1,n if compz='V' or 'I' and at least 1 if compz='N'.
On entry: if compz='V', z must contain the unitary matrix Q from the reduction to tridiagonal form.
If compz='I', z need not be set.
On exit: if compz='V' or 'I', the n required orthonormal eigenvectors stored as columns of Z; the ith column corresponds to the ith eigenvalue, where i=1,2,,n, unless info>0.
If compz='N', z is not referenced.
6: ldz Integer Input
On entry: the first dimension of the array z as declared in the (sub)program from which f08jsf is called.
  • if compz='V' or 'I', ldz max1,n ;
  • if compz='N', ldz1.
7: work* Real (Kind=nag_wp) array Workspace
Note: the dimension of the array work must be at least max1,2×n-1 if compz='V' or 'I' and at least 1 if compz='N'.
If compz='N', work is not referenced.
8: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
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 unitarily similar to T. value off-diagonal elements have not converged to zero.

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 Parallelism and Performance

f08jsf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jsf 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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of real floating-point operations is typically about 24n2 if compz='N' and about 14n3 if compz='V' or 'I', but depends on how rapidly the algorithm converges. When compz='N', the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when compz='V' or 'I' can be vectorized and on some machines may be performed much faster.
The real analogue of this routine is f08jef.

10 Example

See Section 10 in f08ftf, f08gtf or f08hsf, which illustrate the use of this routine to compute the eigenvalues and eigenvectors of a full or band Hermitian matrix.