Questions and comments should be directed to B. S. Garbow,
APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
------------------------------------------------------------------
***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
1976.
***ROUTINES CALLED  BALANC,BALBAK,ELMHES,ELTRAN,HQR,HQR2
***END PROLOGUE  RG

***BEGIN PROLOGUE  RG
***DATE WRITTEN   760101   (YYMMDD)
***REVISION DATE  830518   (YYMMDD)
***CATEGORY NO.  D4A2
***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
***AUTHOR  SMITH, B. T., ET AL.
***PURPOSE  Computes eigenvalues and, optionally, eigenvectors of a
real general matrix.
***DESCRIPTION

This subroutine calls the recommended sequence of
subroutines from the eigensystem subroutine package (EISPACK)
To find the eigenvalues and eigenvectors (if desired)
of a REAL GENERAL matrix.

WR  and  WI  contain the real and imaginary parts,
respectively, of the eigenvalues.  Complex conjugate
pairs of eigenvalues appear consecutively with the
eigenvalue having the positive imaginary part first.

Z  contains the real and imaginary parts of the eigenvectors
if MATZ is not zero.  If the J-th eigenvalue is real, the
J-th column of  Z  contains its eigenvector.  If the J-TH
eigenvalue is complex with positive imaginary part, the
J-th and (J+1)-th columns of  Z  contain the real and
imaginary parts of its eigenvector.  The conjugate of this
vector is the eigenvector for the conjugate eigenvalue.

***BEGIN PROLOGUE  BALANC
***DATE WRITTEN   760101   (YYMMDD)
***REVISION DATE  830518   (YYMMDD)
***CATEGORY NO.  D4C1A
***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
***AUTHOR  SMITH, B. T., ET AL.
***PURPOSE  Balances a general REAL matrix and isolates eigenvalue
whenever possible.
***DESCRIPTION

LOW and IGH are two integers such that A(I,J)
is equal to zero if
(1) I is greater than J and
(2) J=1,...,LOW-1 or I=IGH+1,...,N.

Suppose that the principal submatrix in rows LOW through IGH
has been balanced, that P(J) denotes the index interchanged
with J during the permutation step, and that the elements
of the diagonal matrix used are denoted by D(I,J).  Then
SCALE(J) = P(J),    for J = 1,...,LOW-1
= D(J,J),      J = LOW,...,IGH
= P(J)         J = IGH+1,...,N.
The order in which the interchanges are made is N to IGH+1,
then 1 TO LOW-1.

Questions and comments should be directed to B. S. Garbow,
Applied Mathematics Division, ARGONNE NATIONAL LABORATORY
------------------------------------------------------------------
***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
1976.
***ROUTINES CALLED  (NONE)
***END PROLOGUE  BALANC

***BEGIN PROLOGUE  BALBAK
***DATE WRITTEN   760101   (YYMMDD)
***REVISION DATE  830518   (YYMMDD)
***CATEGORY NO.  D4C4
***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
***AUTHOR  SMITH, B. T., ET AL.
***PURPOSE  Forms eigenvectors of REAL general matrix from
eigenvectors of matrix output from BALANC.
***DESCRIPTION

Questions and comments should be directed to B. S. Garbow,
Applied Mathematics Division, ARGONNE NATIONAL LABORATORY
------------------------------------------------------------------
***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
1976.
***ROUTINES CALLED  (NONE)
***END PROLOGUE  BALBAK

***BEGIN PROLOGUE  ELMHES
***DATE WRITTEN   760101   (YYMMDD)
***REVISION DATE  830518   (YYMMDD)
***CATEGORY NO.  D4C1B2
***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
***AUTHOR  SMITH, B. T., ET AL.
***PURPOSE  Reduces REAL general matrix to upper Hessenberg form
stabilized elementary similarity transformations.
***DESCRIPTION

Given a REAL GENERAL matrix, this subroutine
reduces a submatrix situated in rows and columns
LOW through IGH to upper Hessenberg form by
stabilized elementary similarity transformations.

Questions and comments should be directed to B. S. Garbow,
APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
------------------------------------------------------------------
***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
1976.
***ROUTINES CALLED  (NONE)
***END PROLOGUE  ELMHES

***BEGIN PROLOGUE  ELTRAN
***DATE WRITTEN   760101   (YYMMDD)
***REVISION DATE  830518   (YYMMDD)
***CATEGORY NO.  D4C4
***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
***AUTHOR  SMITH, B. T., ET AL.
***PURPOSE  Accumulates the stabilized elementary similarity
transformations used in the reduction of a REAL general
matrix to upper Hessenberg form by ELMHES.
***DESCRIPTION

Questions and comments should be directed to B. S. Garbow,
APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
------------------------------------------------------------------
***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
1976.
***ROUTINES CALLED  (NONE)
***END PROLOGUE  ELTRAN

***BEGIN PROLOGUE  HQR
***DATE WRITTEN   760101   (YYMMDD)
***REVISION DATE  830518   (YYMMDD)
***CATEGORY NO.  D4C2B
***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
***AUTHOR  SMITH, B. T., ET AL.
***PURPOSE  Computes eigenvalues of a REAL upper Hessenberg matrix
using the QR method.
***DESCRIPTION

H contains the upper Hessenberg matrix.  Information about
the transformations used in the reduction to Hessenberg
form by  ELMHES  or  ORTHES, if performed, is stored
in the remaining triangle under the Hessenberg matrix.

WR and WI contain the real and imaginary parts,
respectively, of the eigenvalues.  The eigenvalues
are unordered except that complex conjugate pairs
of values appear consecutively with the eigenvalue
having the positive imaginary part first.  If an
error exit is made, the eigenvalues should be correct
for indices IERR+1,...,N.

IERR is set to
Zero       for normal return,
J          if the J-th eigenvalue has not been
determined after a total of 30*N iterations.

Questions and comments should be directed to B. S. Garbow,
APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
------------------------------------------------------------------
***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
1976.
***ROUTINES CALLED  (NONE)
***END PROLOGUE  HQR

***BEGIN PROLOGUE  HQR2
***DATE WRITTEN   760101   (YYMMDD)
***REVISION DATE  830518   (YYMMDD)
***CATEGORY NO.  D4C2B
***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
***AUTHOR  SMITH, B. T., ET AL.
***PURPOSE  Computes eigenvalues and eigenvectors of real upper
Hessenberg matrix using QR method.
***DESCRIPTION

This subroutine finds the eigenvalues and eigenvectors
of a REAL UPPER Hessenberg matrix by the QR method.  The
eigenvectors of a REAL GENERAL matrix can also be found
if  ELMHES  and  ELTRAN  or  ORTHES  and  ORTRAN  have
been used to reduce this general matrix to Hessenberg form
and to accumulate the similarity transformations.

Z contains the transformation matrix produced by  ELTRAN
after the reduction by  ELMHES, or by  ORTRAN  after the
reduction by  ORTHES, if performed.  If the eigenvectors
of the Hessenberg matrix are desired, Z must contain the
identity matrix.

WR and WI contain the real and imaginary parts,
respectively, of the eigenvalues.  The eigenvalues
are unordered except that complex conjugate pairs
of values appear consecutively with the eigenvalue
having the positive imaginary part first.  If an
error exit is made, the eigenvalues should be correct
for indices IERR+1,...,N.

Z contains the real and imaginary parts of the eigenvectors.
If the I-th eigenvalue is real, the I-th column of Z
contains its eigenvector.  If the I-th eigenvalue is complex
with positive imaginary part, the I-th and (I+1)-th
columns of Z contain the real and imaginary parts of its
eigenvector.  The eigenvectors are unnormalized.  If an
error exit is made, none of the eigenvectors has been found.

IERR is set to
Zero       for normal return,
J          if the J-th eigenvalue has not been
determined after a total of 30*N iterations.

Questions and comments should be directed to B. S. Garbow,
APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
------------------------------------------------------------------
***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
1976.
***ROUTINES CALLED  CDIV
***END PROLOGUE  HQR2