1STGSYL(1) LAPACK routine (version 3.2) STGSYL(1)
2
6 STGSYL - solves the generalized Sylvester equation
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9 SUBROUTINE STGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD,
10 E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO
11 )
12 CHARACTER TRANS
14 INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, M,
16 N
17 REAL DIF, SCALE
19 INTEGER IWORK( * )
21 REAL A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ),
23 E( LDE, * ), F( LDF, * ), WORK( * )
24
26 STGSYL solves the generalized Sylvester equation:
27 A * R - L * B = scale * C (1)
28 D * R - L * E = scale * F
29 where R and L are unknown m-by-n matrices, (A, D), (B, E) and (C, F)
30 are given matrix pairs of size m-by-m, n-by-n and m-by-n, respectively,
31 with real entries. (A, D) and (B, E) must be in generalized (real)
32 Schur canonical form, i.e. A, B are upper quasi triangular and D, E are
33 upper triangular.
34 The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
35 scaling factor chosen to avoid overflow.
36 In matrix notation (1) is equivalent to solve Zx = scale b, where Z is
37 defined as
38 Z = [ kron(In, A) -kron(B', Im) ] (2)
39 [ kron(In, D) -kron(E', Im) ].
40 Here Ik is the identity matrix of size k and X' is the transpose of X.
41 kron(X, Y) is the Kronecker product between the matrices X and Y. If
42 TRANS = 'T', STGSYL solves the transposed system Z'*y = scale*b, which
43 is equivalent to solve for R and L in
44 A' * R + D' * L = scale * C (3)
45 R * B' + L * E' = scale * (-F)
46 This case (TRANS = 'T') is used to compute an one-norm-based estimate
47 of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D) and
48 (B,E), using SLACON.
49 If IJOB >= 1, STGSYL computes a Frobenius norm-based estimate of
50 Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
51 reciprocal of the smallest singular value of Z. See [1-2] for more
52 information.
53 This is a level 3 BLAS algorithm.
54
56 TRANS (input) CHARACTER*1
57 = 'N', solve the generalized Sylvester equation (1). = 'T',
58 solve the 'transposed' system (3).
59 IJOB (input) INTEGER
61 Specifies what kind of functionality to be performed. =0:
62 solve (1) only.
63 =1: The functionality of 0 and 3.
64 =2: The functionality of 0 and 4.
65 =3: Only an estimate of Dif[(A,D), (B,E)] is computed. (look
66 ahead strategy IJOB = 1 is used). =4: Only an estimate of
67 Dif[(A,D), (B,E)] is computed. ( SGECON on sub-systems is used
68 ). Not referenced if TRANS = 'T'.
69 M (input) INTEGER
71 The order of the matrices A and D, and the row dimension of the
72 matrices C, F, R and L.
73 N (input) INTEGER
75 The order of the matrices B and E, and the column dimension of
76 the matrices C, F, R and L.
77 A (input) REAL array, dimension (LDA, M)
79 The upper quasi triangular matrix A.
80 LDA (input) INTEGER
82 The leading dimension of the array A. LDA >= max(1, M).
83 B (input) REAL array, dimension (LDB, N)
85 The upper quasi triangular matrix B.
86 LDB (input) INTEGER
88 The leading dimension of the array B. LDB >= max(1, N).
89 C (input/output) REAL array, dimension (LDC, N)
91 On entry, C contains the right-hand-side of the first matrix
92 equation in (1) or (3). On exit, if IJOB = 0, 1 or 2, C has
93 been overwritten by the solution R. If IJOB = 3 or 4 and TRANS
94 = 'N', C holds R, the solution achieved during the computation
95 of the Dif-estimate.
96 LDC (input) INTEGER
98 The leading dimension of the array C. LDC >= max(1, M).
99 D (input) REAL array, dimension (LDD, M)
101 The upper triangular matrix D.
102 LDD (input) INTEGER
104 The leading dimension of the array D. LDD >= max(1, M).
105 E (input) REAL array, dimension (LDE, N)
107 The upper triangular matrix E.
108 LDE (input) INTEGER
110 The leading dimension of the array E. LDE >= max(1, N).
111 F (input/output) REAL array, dimension (LDF, N)
113 On entry, F contains the right-hand-side of the second matrix
114 equation in (1) or (3). On exit, if IJOB = 0, 1 or 2, F has
115 been overwritten by the solution L. If IJOB = 3 or 4 and TRANS
116 = 'N', F holds L, the solution achieved during the computation
117 of the Dif-estimate.
118 LDF (input) INTEGER
120 The leading dimension of the array F. LDF >= max(1, M).
121 DIF (output) REAL
123 On exit DIF is the reciprocal of a lower bound of the recipro‐
124 cal of the Dif-function, i.e. DIF is an upper bound of
125 Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2). IF IJOB =
126 0 or TRANS = 'T', DIF is not touched.
127 SCALE (output) REAL
129 On exit SCALE is the scaling factor in (1) or (3). If 0 <
130 SCALE < 1, C and F hold the solutions R and L, resp., to a
131 slightly perturbed system but the input matrices A, B, D and E
132 have not been changed. If SCALE = 0, C and F hold the solutions
133 R and L, respectively, to the homogeneous system with C = F =
134 0. Normally, SCALE = 1.
135 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
137 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
138 LWORK (input) INTEGER
140 The dimension of the array WORK. LWORK > = 1. If IJOB = 1 or 2
141 and TRANS = 'N', LWORK >= max(1,2*M*N). If LWORK = -1, then a
142 workspace query is assumed; the routine only calculates the
143 optimal size of the WORK array, returns this value as the first
144 entry of the WORK array, and no error message related to LWORK
145 is issued by XERBLA.
146 IWORK (workspace) INTEGER array, dimension (M+N+6)
148 INFO (output) INTEGER
150 =0: successful exit
151 <0: If INFO = -i, the i-th argument had an illegal value.
152 >0: (A, D) and (B, E) have common or close eigenvalues.
153
155 Based on contributions by
156 Bo Kagstrom and Peter Poromaa, Department of Computing Science,
157 Umea University, S-901 87 Umea, Sweden.
158 [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
159 for Solving the Generalized Sylvester Equation and Estimating the
160 Separation between Regular Matrix Pairs, Report UMINF - 93.23,
161 Department of Computing Science, Umea University, S-901 87 Umea,
162 Sweden, December 1993, Revised April 1994, Also as LAPACK Working
163 Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
164 No 1, 1996.
165 [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
166 Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
167 Appl., 15(4):1045-1060, 1994
168 [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
169 Condition Estimators for Solving the Generalized Sylvester
170 Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
171 July 1989, pp 745-751.
172 LAPACK routine (version 3.2) November 2008 STGSYL(1)