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<HTML>
<HEAD><TITLE>MB03YA - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03YA">MB03YA</A></H2>
<H3>
Annihilation of one or two entries on the subdiagonal of a Hessenberg matrix corresponding to zero elements on the diagonal of a triangular matrix
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
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.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
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,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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).
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
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 ]
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm requires O(N**2) floating point operations and is
backward stable.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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