MB03YA

Annihilation of one or two entries on the subdiagonal of a Hessenberg matrix corresponding to zero elements on the diagonal of a triangular matrix

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

To annihilate one or two entries on the subdiagonal of the
Hessenberg matrix A for dealing with zero elements on the diagonal
of the triangular matrix B.

MB03YA is an auxiliary routine called by SLICOT Library routines
MB03XP and MB03YD.

Specification
SUBROUTINE MB03YA( WANTT, WANTQ, WANTZ, N, ILO, IHI, ILOQ, IHIQ,
\$                   POS, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO )
C     .. Scalar Arguments ..
LOGICAL            WANTQ, WANTT, WANTZ
INTEGER            IHI, IHIQ, ILO, ILOQ, INFO, LDA, LDB, LDQ, LDZ,
\$                   N, POS
C     .. Array Arguments ..
DOUBLE PRECISION   A(LDA,*), B(LDB,*), Q(LDQ,*), Z(LDZ,*)

Arguments

Mode Parameters

WANTT   LOGICAL
Indicates whether the user wishes to compute the full
Schur form or the eigenvalues only, as follows:
= .TRUE. :  Compute the full Schur form;
= .FALSE.:  compute the eigenvalues only.

WANTQ   LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Q as follows:
= .TRUE. :  The matrix Q is updated;
= .FALSE.:  the matrix Q is not required.

WANTZ   LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Z as follows:
= .TRUE. :  The matrix Z is updated;
= .FALSE.:  the matrix Z is not required.

Input/Output Parameters
N       (input) INTEGER
The order of the matrices A and B. N >= 0.

ILO     (input) INTEGER
IHI     (input) INTEGER
It is assumed that the matrices A and B are already
(quasi) upper triangular in rows and columns 1:ILO-1 and
IHI+1:N. The routine works primarily with the submatrices
in rows and columns ILO to IHI, but applies the
transformations to all the rows and columns of the
matrices A and B, if WANTT = .TRUE..
1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.

ILOQ    (input) INTEGER
IHIQ    (input) INTEGER
Specify the rows of Q and Z to which transformations
must be applied if WANTQ = .TRUE. and WANTZ = .TRUE.,
respectively.
1 <= ILOQ <= ILO; IHI <= IHIQ <= N.

POS     (input) INTEGER
The position of the zero element on the diagonal of B.
ILO <= POS <= IHI.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper Hessenberg matrix A.
On exit, the leading N-by-N part of this array contains
the updated matrix A where A(POS,POS-1) = 0, if POS > ILO,
and A(POS+1,POS) = 0, if POS < IHI.

LDA     INTEGER
The leading dimension of the array A.  LDA >= MAX(1,N).

B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain an upper triangular matrix B with B(POS,POS) = 0.
On exit, the leading N-by-N part of this array contains
the updated upper triangular matrix B.

LDB     INTEGER
The leading dimension of the array B.  LDB >= MAX(1,N).

Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., then the leading N-by-N part
of this array must contain the current matrix Q of
transformations accumulated by MB03XP.
On exit, if WANTQ = .TRUE., then the leading N-by-N part
of this array contains the matrix Q updated in the
submatrix Q(ILOQ:IHIQ,ILO:IHI).
If WANTQ = .FALSE., Q is not referenced.

LDQ     INTEGER
The leading dimension of the array Q.  LDQ >= 1.
If WANTQ = .TRUE., LDQ >= MAX(1,N).

Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., then the leading N-by-N part
of this array must contain the current matrix Z of
transformations accumulated by MB03XP.
On exit, if WANTZ = .TRUE., then the leading N-by-N part
of this array contains the matrix Z updated in the
submatrix Z(ILOQ:IHIQ,ILO:IHI).
If WANTZ = .FALSE., Z is not referenced.

LDZ     INTEGER
The leading dimension of the array Z.  LDZ >= 1.
If WANTZ = .TRUE., LDZ >= MAX(1,N).

Error Indicator
INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

Method
The method is illustrated by Wilkinson diagrams for N = 5,
POS = 3:

[ x x x x x ]       [ x x x x x ]
[ x x x x x ]       [ o x x x x ]
A = [ o x x x x ],  B = [ o o o x x ].
[ o o x x x ]       [ o o o x x ]
[ o o o x x ]       [ o o o o x ]

First, a QR factorization is applied to A(1:3,1:3) and the
resulting nonzero in the updated matrix B is immediately
annihilated by a Givens rotation acting on columns 1 and 2:

[ x x x x x ]       [ x x x x x ]
[ x x x x x ]       [ o x x x x ]
A = [ o o x x x ],  B = [ o o o x x ].
[ o o x x x ]       [ o o o x x ]
[ o o o x x ]       [ o o o o x ]

Secondly, an RQ factorization is applied to A(4:5,4:5) and the
resulting nonzero in the updated matrix B is immediately
annihilated by a Givens rotation acting on rows 4 and 5:

[ x x x x x ]       [ x x x x x ]
[ x x x x x ]       [ o x x x x ]
A = [ o o x x x ],  B = [ o o o x x ].
[ o o o x x ]       [ o o o x x ]
[ o o o x x ]       [ o o o o x ]

References
[1] Bojanczyk, A.W., Golub, G.H., and Van Dooren, P.
The periodic Schur decomposition: Algorithms and applications.
Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
1992.

Numerical Aspects
The algorithm requires O(N**2) floating point operations and is
backward stable.