Purpose
To compute the following matrices
-1
G = B*R *B',
- -1
A = A - B*R *L',
- -1
Q = Q - L*R *L',
where A, B, Q, R, L, and G are N-by-N, N-by-M, N-by-N, M-by-M,
N-by-M, and N-by-N matrices, respectively, with Q, R and G
symmetric matrices.
When R is well-conditioned with respect to inversion, standard
algorithms for solving linear-quadratic optimization problems will
then also solve optimization problems with coupling weighting
matrix L. Moreover, a gain in efficiency is possible using matrix
G in the deflating subspace algorithms (see SLICOT Library routine
SB02OD).
Specification
SUBROUTINE SB02MT( JOBG, JOBL, FACT, UPLO, N, M, A, LDA, B, LDB,
$ Q, LDQ, R, LDR, L, LDL, IPIV, OUFACT, G, LDG,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOBG, JOBL, UPLO
INTEGER INFO, LDA, LDB, LDG, LDL, LDQ, LDR, LDWORK, M,
$ N, OUFACT
C .. Array Arguments ..
INTEGER IPIV(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), G(LDG,*),
$ L(LDL,*), Q(LDQ,*), R(LDR,*)
Arguments
Mode Parameters
JOBG CHARACTER*1
Specifies whether or not the matrix G is to be computed,
as follows:
= 'G': Compute G;
= 'N': Do not compute G.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
FACT CHARACTER*1
Specifies how the matrix R is given (factored or not), as
follows:
= 'N': Array R contains the matrix R;
= 'C': Array R contains the Cholesky factor of R;
= 'U': Array R contains the symmetric indefinite UdU' or
LdL' factorization of R.
UPLO CHARACTER*1
Specifies which triangle of the matrices R and Q (if
JOBL = 'N') is stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, Q, and G, and the number of
rows of the matrices B and L. N >= 0.
M (input) INTEGER
The order of the matrix R, and the number of columns of
the matrices B and L. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if JOBL = 'N', the leading N-by-N part of this
array must contain the matrix A.
On exit, if JOBL = 'N', and INFO = 0, the leading N-by-N
- -1
part of this array contains the matrix A = A - B*R L'.
If JOBL = 'Z', this array is not referenced.
LDA INTEGER
The leading dimension of array A.
LDA >= MAX(1,N) if JOBL = 'N';
LDA >= 1 if JOBL = 'Z'.
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the matrix B.
On exit, if OUFACT = 1, and INFO = 0, the leading N-by-M
-1
part of this array contains the matrix B*chol(R) .
On exit, B is unchanged if OUFACT = 2 (hence also when
FACT = 'U').
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBL = 'N', the leading N-by-N upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively, of
the symmetric matrix Q. The stricly lower triangular part
(if UPLO = 'U') or stricly upper triangular part (if
UPLO = 'L') is not referenced.
On exit, if JOBL = 'N' and INFO = 0, the leading N-by-N
upper triangular part (if UPLO = 'U') or lower triangular
part (if UPLO = 'L') of this array contains the upper
triangular part or lower triangular part, respectively, of
- -1
the symmetric matrix Q = Q - L*R *L'.
If JOBL = 'Z', this array is not referenced.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,N) if JOBL = 'N';
LDQ >= 1 if JOBL = 'Z'.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry, if FACT = 'N', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively,
of the symmetric input weighting matrix R.
On entry, if FACT = 'C', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the Cholesky
factor of the positive definite input weighting matrix R
(as produced by LAPACK routine DPOTRF).
On entry, if FACT = 'U', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the factors of
the UdU' or LdL' factorization, respectively, of the
symmetric indefinite input weighting matrix R (as produced
by LAPACK routine DSYTRF).
If FACT = 'N', the stricly lower triangular part (if UPLO
= 'U') or stricly upper triangular part (if UPLO = 'L') of
this array is used as workspace.
On exit, if OUFACT = 1, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the Cholesky factor of the given input weighting
matrix.
On exit, if OUFACT = 2, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the factors of the UdU' or LdL' factorization,
respectively, of the given input weighting matrix.
On exit R is unchanged if FACT = 'C' or 'U'.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,M).
L (input/output) DOUBLE PRECISION array, dimension (LDL,M)
On entry, if JOBL = 'N', the leading N-by-M part of this
array must contain the matrix L.
On exit, if JOBL = 'N', OUFACT = 1, and INFO = 0, the
leading N-by-M part of this array contains the matrix
-1
L*chol(R) .
On exit, L is unchanged if OUFACT = 2 (hence also when
FACT = 'U').
L is not referenced if JOBL = 'Z'.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N';
LDL >= 1 if JOBL = 'Z'.
IPIV (input/output) INTEGER array, dimension (M)
On entry, if FACT = 'U', this array must contain details
of the interchanges performed and the block structure of
the d factor in the UdU' or LdL' factorization of matrix R
(as produced by LAPACK routine DSYTRF).
On exit, if OUFACT = 2, this array contains details of
the interchanges performed and the block structure of the
d factor in the UdU' or LdL' factorization of matrix R,
as produced by LAPACK routine DSYTRF.
This array is not referenced if FACT = 'C'.
OUFACT (output) INTEGER
Information about the factorization finally used.
OUFACT = 1: Cholesky factorization of R has been used;
OUFACT = 2: UdU' (if UPLO = 'U') or LdL' (if UPLO = 'L')
factorization of R has been used.
G (output) DOUBLE PRECISION array, dimension (LDG,N)
If JOBG = 'G', and INFO = 0, the leading N-by-N upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array contains the upper
triangular part (if UPLO = 'U') or lower triangular part
-1
(if UPLO = 'L'), respectively, of the matrix G = B*R B'.
If JOBG = 'N', this array is not referenced.
LDG INTEGER
The leading dimension of array G.
LDG >= MAX(1,N) if JOBG = 'G',
LDG >= 1 if JOBG = 'N'.
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK; if FACT = 'N', DWORK(2) contains the reciprocal
condition number of the given matrix R.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1 if FACT = 'C';
LDWORK >= MAX(2,3*M,N*M) if FACT = 'N';
LDWORK >= MAX(1,N*M) if FACT = 'U'.
For optimum performance LDWORK should be larger than 3*M,
if FACT = 'N'.
The N*M workspace is not needed for FACT = 'N', if matrix
R is positive definite.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if the i-th element (1 <= i <= M) of the d factor is
exactly zero; the UdU' (or LdL') factorization has
been completed, but the block diagonal matrix d is
exactly singular;
= M+1: if the matrix R is numerically singular.
Method
- - The matrices G, and/or A and Q are evaluated using the given or computed symmetric factorization of R.Numerical Aspects
The routine should not be used when R is ill-conditioned.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
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