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# zstein

```
NAME
ZSTEIN - compute the eigenvectors of a real symmetric tridi-
agonal matrix T corresponding to specified eigenvalues,
using inverse iteration

SYNOPSIS
SUBROUTINE ZSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ,
WORK, IWORK, IFAIL, INFO )

INTEGER        INFO, LDZ, M, N

INTEGER        IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
IWORK( * )

DOUBLE         PRECISION D( * ), E( * ), W( * ), WORK( *
)

COMPLEX*16     Z( LDZ, * )

PURPOSE
ZSTEIN computes the eigenvectors of a real symmetric tridi-
agonal matrix T corresponding to specified eigenvalues,
using inverse iteration.

The maximum number of iterations allowed for each eigenvec-
tor is specified by an internal parameter MAXITS (currently
set to 5).

Although the eigenvectors are real, they are stored in a
complex array, which may be passed to ZUNMTR or ZUPMTR for
back
transformation to the eigenvectors of a complex Hermitian
matrix which was reduced to tridiagonal form.

ARGUMENTS
N       (input) INTEGER
The order of the matrix.  N >= 0.

D       (input) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.

E       (input) DOUBLE PRECISION array, dimension (N)
The (n-1) subdiagonal elements of the tridiagonal
matrix T, stored in elements 1 to N-1; E(N) need not
be set.

M       (input) INTEGER
The number of eigenvectors to be found.  0 <= M <=
N.

W       (input) DOUBLE PRECISION array, dimension (N)
The first M elements of W contain the eigenvalues
for which eigenvectors are to be computed.  The
eigenvalues should be grouped by split-off block and
ordered from smallest to largest within the block.
( The output array W from DSTEBZ with ORDER = 'B' is
expected here. )

IBLOCK  (input) INTEGER array, dimension (N)
The submatrix indices associated with the
corresponding eigenvalues in W; IBLOCK(i)=1 if
eigenvalue W(i) belongs to the first submatrix from
the top, =2 if W(i) belongs to the second submatrix,
etc.  ( The output array IBLOCK from DSTEBZ is
expected here. )

ISPLIT  (input) INTEGER array, dimension (N)
The splitting points, at which T breaks up into sub-
matrices.  The first submatrix consists of
rows/columns 1 to ISPLIT( 1 ), the second of
rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc.
( The output array ISPLIT from DSTEBZ is expected
here. )

Z       (output) COMPLEX*16 array, dimension (LDZ, M)
The computed eigenvectors.  The eigenvector associ-
ated with the eigenvalue W(i) is stored in the i-th
column of Z.  Any vector which fails to converge is
set to its current iterate after MAXITS iterations.
The imaginary parts of the eigenvectors are set to
zero.

LDZ     (input) INTEGER
The leading dimension of the array Z.  LDZ >=
max(1,N).

WORK    (workspace) DOUBLE PRECISION array, dimension (5*N)

IWORK   (workspace) INTEGER array, dimension (N)

IFAIL   (output) INTEGER array, dimension (M)
On normal exit, all elements of IFAIL are zero.  If
one or more eigenvectors fail to converge after MAX-
ITS iterations, then their indices are stored in
array IFAIL.

INFO    (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
value
> 0: if INFO = i, then i eigenvectors failed to con-
verge in MAXITS iterations.  Their indices are

stored in array IFAIL.

PARAMETERS
MAXITS  INTEGER, default = 5
The maximum number of iterations performed.

EXTRA   INTEGER, default = 2
The number of iterations performed after norm growth
criterion is satisfied, should be at least 1.
```