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HQR3 and EXCHNG: FORTRAN Subroutines for Calculating the Eigenvalues of a Real Upper Hessenberg Matrix in a Prescribed Order

IP.com Disclosure Number: IPCOM000147997D
Original Publication Date: 1975-Aug-31
Included in the Prior Art Database: 2007-Mar-28

Publishing Venue

Software Patent Institute

Related People

Stewart, G.W.: AUTHOR [+2]

Abstract

Technical Report TR-403 N00014-67-A-0218-0018 August 1975 FQR3 and EXCHNG: FORTRAN Subroutines for Calculating the Eigenvalues of a Real Upper Hessenberg Matrix ina Prescribed Order

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Page 1 of 19

Technical Report TR-403
N00014-67-A-0218-0018 August 1975

FQR3 and EXCHNG: FORTRAN Subroutines
for Calculating the Eigenvalues of
a Real Upper Hessenberg Matrix in
a Prescribed Order

by
G. W. Stewart

Abstract

HQR.3 and EXCHNG are FORTRAN subroutines for computing the eigenvalues .

of a real upper Hessenberg matrix A in a prescribed order.

I

                                            HQR3 uses
orthogonal similarity transformations to reduce A to a block triangular
form in which the diagonal blocks are 1 x 1 or 2 x 2. The 1 x 1 blocks
contain real eigenvalues, and the 2 x 2 blocks contain conjugate pairs
of complex eigenvalues. The eigenvalues are ordered in decreasing order
of absolute value along the diagonal.. EXWNG is a FORTRAN,subrouti.e
to

interchange two consecutive diagonal blocks of a real upper Hessenberg
matrix. The blocks are of order at most two. By repeated use of EXCHNG,
the eigenvalues of the matrix produced by HQEW can be reordered in any

I

desired manner.

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HQR3 and EXCHNG: FORTRAN Subroutines
for Calculating the Eigenvalues of

a Real Upper Hessenberg Matrix in

*

a Prescribed Order

1. Usage
HQEW is a FORTRAN subroutine to reduce a real upper Hessenberg matrix

A to quasi-triangular form B by a unitary similarity transformation U:

The diagonal of B consists of 1 x 1 and 2 x 2 blocks as illustrated below:

The 1 x 1 blocks contain the real eigenvalues of A, and the 2 x 2 blocks
contain the complex eigenvalues, a conjugate pair to each bloqk. The
blocks are ordered so that the eigenvalues appear in descendjg order of
absolute value along the diagonal. The transformation U is postmultiplied
into an array V, which presumably contains earlier transformations performed

  The calling sequence for HQR3 is
with (starred parameters are altered by the subroutine)

h his work was supported in part by the Office of Naval Research under
Contract No. N0014-67-A-0218-0018.

l

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*A A doubly subscripted real array containing the matrix
to be reduced. On return A. contains the final quasi-
triangular matrix. The elements of the array below the
third subdiagonal are unaltered by the subroutine.

W A doubly subscripted real array into which the reducing
transformation is postmu1tip:lied.

N An integer containing the order of A and V.'

       Integers prescribing what part of A is to be reduced.
NUP Specifically A(NL@W- 1,

               NLflW) md A (NUP , NUP+l) are assumed
zero and only the block from NLOW through NUP is reduced.
However, the transformation is performed on all of the matrix

A so that the final result is similar to A.

EPS A convergence criterion. Maximal accuracy will be attained
if EPS is set to B - ~ ,
where f3 is the base of the floating

       point word and t is the lmgth of its fraction. Smaller
values of EPS will increase the amount of work without signifi-
cantly impro...