Purpose
To compute the Cholesky factor U of the matrix X,
H
X = op(U) * op(U),
which is the solution of either the generalized c-stable
continuous-time Lyapunov equation
H H
op(A) * X * op(E) + op(E) * X * op(A)
2 H
= - SCALE * op(B) * op(B), (1)
or the generalized d-stable discrete-time Lyapunov equation
H H
op(A) * X * op(A) - op(E) * X * op(E)
2 H
= - SCALE * op(B) * op(B), (2)
without first finding X and without the need to form the matrix
op(B)**H * op(B).
op(K) is either K or K**H for K = A, B, E, U. A and E are N-by-N
matrices, op(B) is an M-by-N matrix. The resulting matrix U is an
N-by-N upper triangular matrix with non-negative entries on its
main diagonal. SCALE is an output scale factor set to avoid
overflow in U.
In the continuous-time case (1) the pencil A - lambda * E must be
c-stable (that is, all eigenvalues must have negative real parts).
In the discrete-time case (2) the pencil A - lambda * E must be
d-stable (that is, the moduli of all eigenvalues must be smaller
than one).
Specification
SUBROUTINE SG03BZ( DICO, FACT, TRANS, N, M, A, LDA, E, LDE, Q,
$ LDQ, Z, LDZ, B, LDB, SCALE, ALPHA, BETA, DWORK,
$ ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDB, LDE, LDQ, LDZ, LZWORK, M, N
CHARACTER DICO, FACT, TRANS
C .. Array Arguments ..
COMPLEX*16 A(LDA,*), ALPHA(*), B(LDB,*), BETA(*), E(LDE,*),
$ Q(LDQ,*), Z(LDZ,*), ZWORK(*)
DOUBLE PRECISION DWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies which type of the equation is considered:
= 'C': Continuous-time equation (1);
= 'D': Discrete-time equation (2).
FACT CHARACTER*1
Specifies whether the generalized (complex) Schur
factorization of the pencil A - lambda * E is supplied on
entry or not:
= 'N': Factorization is not supplied;
= 'F': Factorization is supplied.
TRANS CHARACTER*1
Specifies whether the conjugate transposed equation is to
be solved or not:
= 'N': op(A) = A, op(E) = E;
= 'C': op(A) = A**H, op(E) = E**H.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The number of rows in the matrix op(B). M >= 0.
If M = 0, A and E are unchanged on exit, and Q, Z, ALPHA
and BETA are not set.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, if FACT = 'F', then the leading N-by-N upper
triangular part of this array must contain the generalized
Schur factor A_s of the matrix A (see definition (3) in
section METHOD). A_s must be an upper triangular matrix.
The elements below the upper triangular part of the array
A are used as workspace.
If FACT = 'N', then the leading N-by-N part of this array
must contain the matrix A.
On exit, if FACT = 'N', the leading N-by-N upper
triangular part of this array contains the generalized
Schur factor A_s of the matrix A. (A_s is an upper
triangular matrix.) If FACT = 'F', the leading N-by-N
upper triangular part of this array is unchanged.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) COMPLEX*16 array, dimension (LDE,N)
On entry, if FACT = 'F', then the leading N-by-N upper
triangular part of this array must contain the generalized
Schur factor E_s of the matrix E (see definition (4) in
section METHOD). E_s must be an upper triangular matrix.
The elements below the upper triangular part of the array
E are used as workspace.
If FACT = 'N', then the leading N-by-N part of this array
must contain the coefficient matrix E of the equation.
On exit, if FACT = 'N', the leading N-by-N upper
triangular part of this array contains the generalized
Schur factor E_s of the matrix E. (E_s is an upper
triangular matrix.) If FACT = 'F', the leading N-by-N
upper triangular part of this array is unchanged.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the unitary matrix Q from the
generalized Schur factorization (see definitions (3) and
(4) in section METHOD), or an identity matrix (if the
original equation has upper triangular matrices A and E).
If FACT = 'N', Q need not be set on entry.
On exit, if FACT = 'N', the leading N-by-N part of this
array contains the unitary matrix Q from the generalized
Schur factorization. If FACT = 'F', this array is
unchanged.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the unitary matrix Z from the
generalized Schur factorization (see definitions (3) and
(4) in section METHOD), or an identity matrix (if the
original equation has upper triangular matrices A and E).
If FACT = 'N', Z need not be set on entry.
On exit, if FACT = 'N', the leading N-by-N part of this
array contains the unitary matrix Z from the generalized
Schur factorization. If FACT = 'F', this array is
unchanged.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,N1)
On entry, if TRANS = 'C', the leading N-by-M part of this
array must contain the matrix B and N1 >= MAX(M,N).
If TRANS = 'N', the leading M-by-N part of this array
must contain the matrix B and N1 >= N.
On exit, if INFO = 0, the leading N-by-N part of this
array contains the Cholesky factor U of the solution
matrix X of the problem, X = op(U)**H * op(U).
If M = 0 and N > 0, then U is set to zero.
LDB INTEGER
The leading dimension of the array B.
If TRANS = 'C', LDB >= MAX(1,N).
If TRANS = 'N', LDB >= MAX(1,M,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in U.
0 < SCALE <= 1.
ALPHA (output) COMPLEX*16 arrays, dimension (N)
BETA If INFO = 0, 5, 6, or 7, then ALPHA(j)/BETA(j),
j = 1, ... , N, are the eigenvalues of the matrix pencil
A - lambda * E (the diagonals of the complex Schur form).
All BETA(j) are non-negative real numbers.
ALPHA will be always less than and usually comparable with
norm(A) in magnitude, and BETA always less than and
usually comparable with norm(B).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK), where
LDWORK = 0, if MIN(M,N) = 0 or
FACT = 'F' and N <= 1; else,
LDWORK = N-1, if FACT = 'F' and DICO = 'C';
LDWORK = MAX(N-1,10), if FACT = 'F' and DICO = 'D';
LDWORK = 8*N, if FACT = 'N'.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
On exit, if INFO = -21, ZWORK(1) returns the minimum value
of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= MAX(1,3*N-3,2*N).
For good performance, LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the ZWORK
array, returns this value as the first entry of the ZWORK
array, and no error message related to LZWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 4: FACT = 'N' and the pencil A - lambda * E cannot be
reduced to generalized Schur form: LAPACK routine
ZGGES has failed to converge;
= 5: DICO = 'C' and the pencil A - lambda * E is not
c-stable;
= 6: DICO = 'D' and the pencil A - lambda * E is not
d-stable;
= 7: the LAPACK routine ZSTEIN utilized to factorize M3
failed to converge in the discrete-time case (see
section METHOD for SLICOT Library routine SG03BS).
This error is unlikely to occur.
Method
An extension [2] of Hammarling's method [1] to generalized
Lyapunov equations is utilized to solve (1) or (2).
First the pencil A - lambda * E is reduced to complex generalized
Schur form A_s - lambda * E_s by means of unitary transformations
(QZ-algorithm):
A_s = Q**H * A * Z (upper triangular), (3)
E_s = Q**H * E * Z (upper triangular). (4)
If the pencil A - lambda * E has already been factorized prior to
calling the routine, however, then the factors A_s, E_s, Q and Z
may be supplied and the initial factorization omitted.
Depending on the parameters TRANS and M, the N-by-N upper
triangular matrix B_s is defined as follows. In any case Q_B is
an M-by-M unitary matrix, which need not be accumulated.
1. If TRANS = 'N' and M < N, B_s is the upper triangular matrix
from the QR-factorization
( Q_B O ) ( B * Z )
( ) * B_s = ( ),
( O I ) ( O )
where the O's are zero matrices of proper size and I is the
identity matrix of order N-M.
2. If TRANS = 'N' and M >= N, B_s is the upper triangular matrix
from the (rectangular) QR-factorization
( B_s )
Q_B * ( ) = B * Z,
( O )
where O is the (M-N)-by-N zero matrix.
3. If TRANS = 'C' and M < N, B_s is the upper triangular matrix
from the RQ-factorization
( Q_B O )
(B_s O ) * ( ) = ( Q**H * B O ).
( O I )
4. If TRANS = 'C' and M >= N, B_s is the upper triangular matrix
from the (rectangular) RQ-factorization
( B_s O ) * Q_B = Q**H * B,
where O is the N-by-(M-N) zero matrix.
Assuming SCALE = 1, the transformation of A, E and B described
above leads to the reduced continuous-time equation
H H
op(A_s) op(U_s) op(U_s) op(E_s)
H H
+ op(E_s) op(U_s) op(U_s) op(A_s)
H
= - op(B_s) op(B_s) (5)
or to the reduced discrete-time equation
H H
op(A_s) op(U_s) op(U_s) op(A_s)
H H
- op(E_s) op(U_s) op(U_s) op(E_s)
H
= - op(B_s) op(B_s). (6)
For brevity we restrict ourself to equation (5) and the case
TRANS = 'N'. The other three cases can be treated in a similar
fashion.
We use the following partitioning for the matrices A_s, E_s, B_s,
and U_s
( A11 A12 ) ( E11 E12 )
A_s = ( ), E_s = ( ),
( 0 A22 ) ( 0 E22 )
( B11 B12 ) ( U11 U12 )
B_s = ( ), U_s = ( ). (7)
( 0 B22 ) ( 0 U22 )
The size of the (1,1)-blocks is 1-by-1.
We compute U11, U12**H, and U22 in three steps.
Step I:
From (5) and (7) we get the 1-by-1 equation
H H H H
A11 * U11 * U11 * E11 + E11 * U11 * U11 * A11
H
= - B11 * B11.
For brevity, details are omitted here. See [2]. The technique
for computing U11 is similar to those applied to standard
Lyapunov equations in Hammarling's algorithm ([1], section 5).
Furthermore, the auxiliary scalars M1 and M2 defined as follows
M1 = A11 / E11 ,
M2 = B11 / E11 / U11 ,
are computed in a numerically reliable way.
Step II:
The generalized Sylvester equation
H H H H
A22 * U12 + E22 * U12 * M1 =
H H H H H
- B12 * M2 - A12 * U11 - E12 * U11 * M1
is solved for U12**H, as a linear system of order N-1.
Step III:
It can be shown that
H H H H
A22 * U22 * U22 * E22 + E22 * U22 * U22 * A22 =
H H
- B22 * B22 - y * y (8)
holds, where y is defined as
H H H H H
y = B12 - ( E12 * U11 + E22 * U12 ) * M2 .
If B22_tilde is the square triangular matrix arising from the
(rectangular) QR-factorization
( B22_tilde ) ( B22 )
Q_B_tilde * ( ) = ( ),
( O ) ( y**H )
where Q_B_tilde is a unitary matrix of order N, then
H H H
- B22 * B22 - y * y = - B22_tilde * B22_tilde.
Replacing the right hand side in (8) by the term
- B22_tilde**H * B22_tilde leads to a reduced generalized
Lyapunov equation like (5), but of dimension N-1.
The recursive application of the steps I to III yields the
solution U_s of the equation (5).
It remains to compute the solution matrix U of the original
problem (1) or (2) from the matrix U_s. To this end we transform
the solution back (with respect to the transformation that led
from (1) to (5) (from (2) to (6)) and apply the QR-factorization
(RQ-factorization). The upper triangular solution matrix U is
obtained by
Q_U * U = U_s * Q**H (if TRANS = 'N'),
or
U * Q_U = Z * U_s (if TRANS = 'C'),
where Q_U is an N-by-N unitary matrix. Again, the unitary matrix
Q_U need not be accumulated.
References
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-323, 1982.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
The number of flops required by the routine is given by the
following table. Note that we count a single floating point
arithmetic operation as one flop.
| FACT = 'F' FACT = 'N'
---------+--------------------------------------------------
M <= N | (13*N**3+6*M*N**2 (211*N**3+6*M*N**2
| +6*M**2*N-2*M**3)/3 +6*M**2*N-2*M**3)/3
|
M > N | (11*N**3+12*M*N**2)/3 (209*N**3+12*M*N**2)/3
Further Comments
The Lyapunov equation may be very ill-conditioned. In particular, if DICO = 'D' and the pencil A - lambda * E has a pair of almost reciprocal eigenvalues, or DICO = 'C' and the pencil has an almost degenerate pair of eigenvalues, then the Lyapunov equation will be ill-conditioned. Perturbed values were used to solve the equation. A condition estimate can be obtained from the routine SG03AD.Example
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