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<HTML>
<HEAD><TITLE>MB02DD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB02DD">MB02DD</A></H2>
<H3>
Updating Cholesky factorization of a positive definite block Toeplitz 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 update the Cholesky factor and the generator and/or the
Cholesky factor of the inverse of a symmetric positive definite
(s.p.d.) block Toeplitz matrix T, given the information from
a previous factorization and additional blocks in TA of its first
block row, or its first block column, depending on the routine
parameter TYPET. Transformation information is stored.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB02DD( JOB, TYPET, K, M, N, TA, LDTA, T, LDT, G,
$ LDG, R, LDR, L, LDL, CS, LCS, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOB, TYPET
INTEGER INFO, K, LCS, LDG, LDL, LDR, LDT, LDTA, LDWORK,
$ M, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), G(LDG, *), L(LDL,*), R(LDR,*),
$ T(LDT,*), TA(LDTA,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
JOB CHARACTER*1
Specifies the output of the routine, as follows:
= 'R': updates the generator G of the inverse and
computes the new columns / rows for the Cholesky
factor R of T;
= 'A': updates the generator G, computes the new
columns / rows for the Cholesky factor R of T and
the new rows / columns for the Cholesky factor L
of the inverse;
= 'O': only computes the new columns / rows for the
Cholesky factor R of T.
TYPET CHARACTER*1
Specifies the type of T, as follows:
= 'R': the first block row of an s.p.d. block Toeplitz
matrix was/is defined; if demanded, the Cholesky
factors R and L are upper and lower triangular,
respectively, and G contains the transposed
generator of the inverse;
= 'C': the first block column of an s.p.d. block Toeplitz
matrix was/is defined; if demanded, the Cholesky
factors R and L are lower and upper triangular,
respectively, and G contains the generator of the
inverse. This choice results in a column oriented
algorithm which is usually faster.
Note: in this routine, the notation x / y means that
x corresponds to TYPET = 'R' and y corresponds to
TYPET = 'C'.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
K (input) INTEGER
The number of rows / columns in T, which should be equal
to the blocksize. K >= 0.
M (input) INTEGER
The number of blocks in TA. M >= 0.
N (input) INTEGER
The number of blocks in T. N >= 0.
TA (input/output) DOUBLE PRECISION array, dimension
(LDTA,M*K) / (LDTA,K)
On entry, the leading K-by-M*K / M*K-by-K part of this
array must contain the (N+1)-th to (N+M)-th blocks in the
first block row / column of an s.p.d. block Toeplitz
matrix.
On exit, if INFO = 0, the leading K-by-M*K / M*K-by-K part
of this array contains information on the Householder
transformations used, such that the array
[ T TA ] / [ T ]
[ TA ]
serves as the new transformation matrix T for further
applications of this routine.
LDTA INTEGER
The leading dimension of the array TA.
LDTA >= MAX(1,K), if TYPET = 'R';
LDTA >= MAX(1,M*K), if TYPET = 'C'.
T (input) DOUBLE PRECISION array, dimension (LDT,N*K) /
(LDT,K)
The leading K-by-N*K / N*K-by-K part of this array must
contain transformation information generated by the SLICOT
Library routine MB02CD, i.e., in the first K-by-K block,
the upper / lower Cholesky factor of T(1:K,1:K), and in
the remaining part, the Householder transformations
applied during the initial factorization process.
LDT INTEGER
The leading dimension of the array T.
LDT >= MAX(1,K), if TYPET = 'R';
LDT >= MAX(1,N*K), if TYPET = 'C'.
G (input/output) DOUBLE PRECISION array, dimension
(LDG,( N + M )*K) / (LDG,2*K)
On entry, if JOB = 'R', or 'A', then the leading
2*K-by-N*K / N*K-by-2*K part of this array must contain,
in the first K-by-K block of the second block row /
column, the lower right block of the Cholesky factor of
the inverse of T, and in the remaining part, the generator
of the inverse of T.
On exit, if INFO = 0 and JOB = 'R', or 'A', then the
leading 2*K-by-( N + M )*K / ( N + M )*K-by-2*K part of
this array contains the same information as on entry, now
for the updated Toeplitz matrix. Actually, to obtain a
generator of the inverse one has to set
G(K+1:2*K, 1:K) = 0, if TYPET = 'R';
G(1:K, K+1:2*K) = 0, if TYPET = 'C'.
LDG INTEGER
The leading dimension of the array G.
LDG >= MAX(1,2*K), if TYPET = 'R' and JOB = 'R', or 'A';
LDG >= MAX(1,( N + M )*K),
if TYPET = 'C' and JOB = 'R', or 'A';
LDG >= 1, if JOB = 'O'.
R (input/output) DOUBLE PRECISION array, dimension
(LDR,M*K) / (LDR,( N + M )*K)
On input, the leading N*K-by-K part of R(K+1,1) /
K-by-N*K part of R(1,K+1) contains the last block column /
row of the previous Cholesky factor R.
On exit, if INFO = 0, then the leading
( N + M )*K-by-M*K / M*K-by-( N + M )*K part of this
array contains the last M*K columns / rows of the upper /
lower Cholesky factor of T. The elements in the strictly
lower / upper triangular part are not referenced.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX(1, ( N + M )*K), if TYPET = 'R';
LDR >= MAX(1, M*K), if TYPET = 'C'.
L (output) DOUBLE PRECISION array, dimension
(LDL,( N + M )*K) / (LDL,M*K)
If INFO = 0 and JOB = 'A', then the leading
M*K-by-( N + M )*K / ( N + M )*K-by-M*K part of this
array contains the last M*K rows / columns of the lower /
upper Cholesky factor of the inverse of T. The elements
in the strictly upper / lower triangular part are not
referenced.
LDL INTEGER
The leading dimension of the array L.
LDL >= MAX(1, M*K), if TYPET = 'R' and JOB = 'A';
LDL >= MAX(1, ( N + M )*K), if TYPET = 'C' and JOB = 'A';
LDL >= 1, if JOB = 'R', or 'O'.
CS (input/output) DOUBLE PRECISION array, dimension (LCS)
On input, the leading 3*(N-1)*K part of this array must
contain the necessary information about the hyperbolic
rotations and Householder transformations applied
previously by SLICOT Library routine MB02CD.
On exit, if INFO = 0, then the leading 3*(N+M-1)*K part of
this array contains information about all the hyperbolic
rotations and Householder transformations applied during
the whole process.
LCS INTEGER
The length of the array CS. LCS >= 3*(N+M-1)*K.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -19, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,(N+M-1)*K).
For optimum performance LDWORK should be larger.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction algorithm failed. The block Toeplitz
matrix associated with [ T TA ] / [ T' TA' ]' is
not (numerically) positive definite.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
Householder transformations and modified hyperbolic rotations
are used in the Schur algorithm [1], [2].
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The implemented method is numerically stable.
3 2
The algorithm requires 0(K ( N M + M ) ) floating point
operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
For min(K,N,M) = 0, the routine sets DWORK(1) = 1 and returns.
Although the calculations could still be performed when N = 0,
but min(K,M) > 0, this case is not considered as an "update".
SLICOT Library routine MB02CD should be called with the argument
M instead of N.
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* MB02DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, MMAX, NMAX
PARAMETER ( KMAX = 20, MMAX = 20, NMAX = 20 )
INTEGER LCS, LDG, LDL, LDR, LDT, LDWORK
PARAMETER ( LDG = KMAX*( MMAX + NMAX ),
$ LDL = KMAX*( MMAX + NMAX ),
$ LDR = KMAX*( MMAX + NMAX ),
$ LDT = KMAX*( MMAX + NMAX ),
$ LDWORK = ( MMAX + NMAX - 1 )*KMAX )
PARAMETER ( LCS = 3*LDWORK )
* .. Local Scalars ..
INTEGER I, INFO, J, K, M, N, S
CHARACTER JOB, TYPET
* .. Local Arrays ..
* The arrays are dimensioned for both TYPET = 'R' and TYPET = 'C'.
* Arrays G and T could be smaller.
* For array G, it is assumed that MMAX + NMAX >= 2.
* The matrix TA is also stored in the array T.
DOUBLE PRECISION CS(LCS), DWORK(LDWORK),
$ G(LDG, KMAX*( MMAX + NMAX )),
$ L(LDL, KMAX*( MMAX + NMAX )),
$ R(LDR, KMAX*( MMAX + NMAX )),
$ T(LDT, KMAX*( MMAX + NMAX ))
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DLACPY, MB02CD, MB02DD
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, K, M, JOB, TYPET
S = ( N + M )*K
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N
ELSE
IF ( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) K
ELSE
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
IF ( LSAME( TYPET, 'R' ) ) THEN
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,S ), I = 1,K )
ELSE
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,K ), I = 1,S )
END IF
* Compute the Cholesky factors.
CALL MB02CD( JOB, TYPET, K, N, T, LDT, G, LDG, R, LDR, L,
$ LDL, CS, LCS, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N*K
WRITE ( NOUT, FMT = 99990 ) ( R(I,J), J = 1, N*K )
10 CONTINUE
IF ( LSAME( JOB, 'R' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
IF ( LSAME( TYPET, 'R' ) ) THEN
DO 20 I = 1, 2*K
WRITE ( NOUT, FMT = 99990 )
$ ( G(I,J), J = 1, N*K )
20 CONTINUE
ELSE
DO 30 I = 1, N*K
WRITE ( NOUT, FMT = 99990 )
$ ( G(I,J), J = 1, 2*K )
30 CONTINUE
END IF
END IF
IF ( LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N*K
WRITE ( NOUT, FMT = 99990 )
$ ( L(I,J), J = 1, N*K )
40 CONTINUE
END IF
* Update the Cholesky factors.
IF ( LSAME( TYPET, 'R' ) ) THEN
* Copy the last block column of R.
CALL DLACPY( 'All', N*K, K, R(1,(N-1)*K+1), LDR,
$ R(K+1,N*K+1), LDR )
CALL MB02DD( JOB, TYPET, K, M, N, T(1,N*K+1), LDT,
$ T, LDT, G, LDG, R(1,N*K+1), LDR,
$ L(N*K+1,1), LDL, CS, LCS, DWORK,
$ LDWORK, INFO )
ELSE
* Copy the last block row of R.
CALL DLACPY( 'All', K, N*K, R((N-1)*K+1,1), LDR,
$ R(N*K+1,K+1), LDR )
CALL MB02DD( JOB, TYPET, K, M, N, T(N*K+1,1), LDT,
$ T, LDT, G, LDG, R(N*K+1,1), LDR,
$ L(1,N*K+1), LDL, CS, LCS, DWORK,
$ LDWORK, INFO )
END IF
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99993 )
DO 50 I = 1, S
WRITE ( NOUT, FMT = 99990 ) ( R(I,J), J = 1, S )
50 CONTINUE
IF ( LSAME( JOB, 'R' ) .OR. LSAME( JOB, 'A' ) )
$ THEN
WRITE ( NOUT, FMT = 99992 )
IF ( LSAME( TYPET, 'R' ) ) THEN
DO 60 I = 1, 2*K
WRITE ( NOUT, FMT = 99990 )
$ ( G(I,J), J = 1, S )
60 CONTINUE
ELSE
DO 70 I = 1, S
WRITE ( NOUT, FMT = 99990 )
$ ( G(I,J), J = 1, 2*K )
70 CONTINUE
END IF
END IF
IF ( LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99991 )
DO 80 I = 1, S
WRITE ( NOUT, FMT = 99990 )
$ ( L(I,J), J = 1, S )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT ( ' MB02DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT ( ' INFO on exit from MB02CD = ',I2)
99997 FORMAT ( ' INFO on exit from MB02DD = ',I2)
99996 FORMAT ( ' The Cholesky factor is ')
99995 FORMAT (/' The inverse generator is ')
99994 FORMAT (/' The inverse Cholesky factor is ')
99993 FORMAT (/' The updated Cholesky factor is ')
99992 FORMAT (/' The updated inverse generator is ')
99991 FORMAT (/' The updated inverse Cholesky factor is ')
99990 FORMAT (20(1X,F8.4))
99989 FORMAT (/' N is out of range.',/' N = ',I5)
99988 FORMAT (/' K is out of range.',/' K = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
MB02DD EXAMPLE PROGRAM DATA
3 2 2 A R
3.0000 1.0000 0.1000 0.1000 0.2000 0.0500 0.1000 0.0400 0.01 0.02
1.0000 4.0000 0.4000 0.1000 0.0400 0.2000 0.0300 0.0200 0.03 0.01
</PRE>
<B>Program Results</B>
<PRE>
MB02DD EXAMPLE PROGRAM RESULTS
The Cholesky factor is
1.7321 0.5774 0.0577 0.0577 0.1155 0.0289
0.0000 1.9149 0.1915 0.0348 -0.0139 0.0957
0.0000 0.0000 1.7205 0.5754 0.0558 0.0465
0.0000 0.0000 0.0000 1.9142 0.1890 0.0357
0.0000 0.0000 0.0000 0.0000 1.7169 0.5759
0.0000 0.0000 0.0000 0.0000 0.0000 1.9118
The inverse generator is
-0.2355 0.5231 -0.0642 0.0077 0.0187 -0.0265
-0.5568 -0.0568 0.0229 0.0060 0.0363 0.0000
0.5825 0.0000 -0.0387 0.0052 0.0003 -0.0575
-0.1754 0.5231 0.0119 -0.0265 -0.0110 0.0076
The inverse Cholesky factor is
0.5774 0.0000 0.0000 0.0000 0.0000 0.0000
-0.1741 0.5222 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0581 0.5812 0.0000 0.0000 0.0000
-0.0142 0.0080 -0.1747 0.5224 0.0000 0.0000
-0.0387 0.0052 0.0003 -0.0575 0.5825 0.0000
0.0119 -0.0265 -0.0110 0.0076 -0.1754 0.5231
The updated Cholesky factor is
1.7321 0.5774 0.0577 0.0577 0.1155 0.0289 0.0577 0.0231 0.0058 0.0115
0.0000 1.9149 0.1915 0.0348 -0.0139 0.0957 -0.0017 0.0035 0.0139 0.0017
0.0000 0.0000 1.7205 0.5754 0.0558 0.0465 0.1145 0.0279 0.0564 0.0227
0.0000 0.0000 0.0000 1.9142 0.1890 0.0357 -0.0152 0.0953 -0.0017 0.0033
0.0000 0.0000 0.0000 0.0000 1.7169 0.5759 0.0523 0.0453 0.1146 0.0273
0.0000 0.0000 0.0000 0.0000 0.0000 1.9118 0.1902 0.0357 -0.0157 0.0955
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.7159 0.5757 0.0526 0.0450
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.9118 0.1901 0.0357
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.7159 0.5757
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.9117
The updated inverse generator is
-0.5599 0.3310 -0.0305 0.0098 0.0392 -0.0209 0.0191 -0.0010 -0.0045 0.0035
-0.2289 -0.4091 0.0612 -0.0012 0.0125 0.0182 0.0042 0.0017 0.0014 0.0000
0.5828 0.0000 0.0027 -0.0029 -0.0195 0.0072 -0.0393 0.0057 0.0016 -0.0580
-0.1755 0.5231 -0.0037 0.0022 0.0005 -0.0022 0.0125 -0.0266 -0.0109 0.0077
The updated inverse Cholesky factor is
0.5774 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-0.1741 0.5222 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0581 0.5812 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-0.0142 0.0080 -0.1747 0.5224 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-0.0387 0.0052 0.0003 -0.0575 0.5825 0.0000 0.0000 0.0000 0.0000 0.0000
0.0119 -0.0265 -0.0110 0.0076 -0.1754 0.5231 0.0000 0.0000 0.0000 0.0000
-0.0199 0.0073 -0.0391 0.0056 0.0017 -0.0580 0.5828 0.0000 0.0000 0.0000
0.0007 -0.0023 0.0122 -0.0265 -0.0110 0.0077 -0.1755 0.5231 0.0000 0.0000
0.0027 -0.0029 -0.0195 0.0072 -0.0393 0.0057 0.0016 -0.0580 0.5828 0.0000
-0.0037 0.0022 0.0005 -0.0022 0.0125 -0.0266 -0.0109 0.0077 -0.1755 0.5231
</PRE>
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