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
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) 

C++ Header Interface
#include <nag.h> extern "C" {
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
$\Lambda $ is a diagonal matrix whose diagonal elements are the eigenvalues
${\lambda}_{i}$, and
$Z$ is the orthogonal matrix whose columns are the eigenvectors
${z}_{i}$. Thus
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
$T$:
In this case, the matrix
$Q$ must be formed explicitly and passed to
f08jsf, which must be called with
${\mathbf{compz}}=\text{'V'}$. The routines which must be called to perform the reduction to tridiagonal form and form
$Q$ are:
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
${\Vert {z}_{i}\Vert}_{2}=1$, 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 CS8992) University of Tennessee, Knoxville
https://www.netlib.org/lapack/lawnspdf/lawn17.pdf
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia
5
Arguments

1:
$\mathbf{compz}$ – Character(1)
Input

On entry: indicates whether the eigenvectors are to be computed.
 ${\mathbf{compz}}=\text{'N'}$
 Only the eigenvalues are computed (and the array z is not referenced).
 ${\mathbf{compz}}=\text{'V'}$
 The eigenvalues and eigenvectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
 ${\mathbf{compz}}=\text{'I'}$
 The eigenvalues and eigenvectors of $T$ are computed (and the array z is initialized by the routine).
Constraint:
${\mathbf{compz}}=\text{'N'}$, $\text{'V'}$ or $\text{'I'}$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $T$.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the dimension of the array
d
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the diagonal elements of the tridiagonal matrix $T$.
On exit: the
$n$ eigenvalues in ascending order, unless
${\mathbf{info}}>{\mathbf{0}}$ (in which case see
Section 6).

4:
$\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the dimension of the array
e
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}1\right)$.
On entry: the offdiagonal elements of the tridiagonal matrix $T$.
On exit:
e is overwritten.

5:
$\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Complex (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
z
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if
${\mathbf{compz}}=\text{'V'}$ or
$\text{'I'}$ and at least
$1$ if
${\mathbf{compz}}=\text{'N'}$.
On entry: if
${\mathbf{compz}}=\text{'V'}$,
z must contain the unitary matrix
$Q$ from the reduction to tridiagonal form.
If
${\mathbf{compz}}=\text{'I'}$,
z need not be set.
On exit: if
${\mathbf{compz}}=\text{'V'}$ or
$\text{'I'}$, the
$n$ required orthonormal eigenvectors stored as columns of
$Z$; the
$i$th column corresponds to the
$i$th eigenvalue, where
$i=1,2,\dots ,n$, unless
${\mathbf{info}}>{\mathbf{0}}$.
If
${\mathbf{compz}}=\text{'N'}$,
z is not referenced.

6:
$\mathbf{ldz}$ – Integer
Input

On entry: the first dimension of the array
z as declared in the (sub)program from which
f08jsf is called.
Constraints:
 if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
 if ${\mathbf{compz}}=\text{'N'}$, ${\mathbf{ldz}}\ge 1$.

7:
$\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) array
Workspace

Note: the dimension of the array
work
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\times \left({\mathbf{n}}1\right)\right)$ if
${\mathbf{compz}}=\text{'V'}$ or
$\text{'I'}$ and at least
$1$ if
${\mathbf{compz}}=\text{'N'}$.
If
${\mathbf{compz}}=\text{'N'}$,
work is not referenced.

8:
$\mathbf{info}$ – Integer
Output
On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
 ${\mathbf{info}}<0$
If ${\mathbf{info}}=i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
 ${\mathbf{info}}>0$

The algorithm has failed to find all the eigenvalues after a total of
$30\times {\mathbf{n}}$ iterations. In this case,
d and
e contain on exit the diagonal and offdiagonal elements, respectively, of a tridiagonal matrix unitarily similar to
$T$.
$\u2329\mathit{\text{value}}\u232a$ offdiagonal elements have not converged to zero.
7
Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrix
$\left(T+E\right)$, where
and
$\epsilon $ is the
machine precision.
If
${\lambda}_{i}$ is an exact eigenvalue and
${\stackrel{~}{\lambda}}_{i}$ is the corresponding computed value, then
where
$c\left(n\right)$ is a modestly increasing function of
$n$.
If
${z}_{i}$ is the corresponding exact eigenvector, and
${\stackrel{~}{z}}_{i}$ is the corresponding computed eigenvector, then the angle
$\theta \left({\stackrel{~}{z}}_{i},{z}_{i}\right)$ between them is bounded as follows:
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 implementationspecific information.
The total number of real floatingpoint operations is typically about $24{n}^{2}$ if ${\mathbf{compz}}=\text{'N'}$ and about $14{n}^{3}$ if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$, but depends on how rapidly the algorithm converges. When ${\mathbf{compz}}=\text{'N'}$, the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$ 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.