1DTGSY2(1) LAPACK auxiliary routine (version 3.2) DTGSY2(1)
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6 DTGSY2 - solves the generalized Sylvester equation
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9 SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD,
10 E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ,
11 INFO )
12 CHARACTER TRANS
14 INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N, PQ
16 DOUBLE PRECISION RDSCAL, RDSUM, SCALE
18 INTEGER IWORK( * )
20 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ), D(
22 LDD, * ), E( LDE, * ), F( LDF, * )
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25 DTGSY2 solves the generalized Sylvester equation:
26 A * R - L * B = scale * C (1)
27 D * R - L * E = scale * F,
28 using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,
29 (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M, N-by-N
30 and M-by-N, respectively, with real entries. (A, D) and (B, E) must be
31 in generalized Schur canonical form, i.e. A, B are upper quasi triangu‐
32 lar and D, E are upper triangular. The solution (R, L) overwrites (C,
33 F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid over‐
34 flow.
35 In matrix notation solving equation (1) corresponds to solve Z*x =
36 scale*b, where Z is defined as
37 Z = [ kron(In, A) -kron(B', Im) ] (2)
38 [ kron(In, D) -kron(E', Im) ],
39 Ik is the identity matrix of size k and X' is the transpose of X.
40 kron(X, Y) is the Kronecker product between the matrices X and Y. In
41 the process of solving (1), we solve a number of such systems where
42 Dim(In), Dim(In) = 1 or 2.
43 If TRANS = 'T', solve the transposed system Z'*y = scale*b for y, which
44 is equivalent to solve for R and L in
45 A' * R + D' * L = scale * C (3)
46 R * B' + L * E' = scale * -F
47 This case is used to compute an estimate of Dif[(A, D), (B, E)] =
48 sigma_min(Z) using reverse communicaton with DLACON.
49 DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL of an
50 upper bound on the separation between to matrix pairs. Then the input
51 (A, D), (B, E) are sub-pencils of the matrix pair in DTGSYL. See DTGSYL
52 for details.
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55 TRANS (input) CHARACTER*1
56 = 'N', solve the generalized Sylvester equation (1). = 'T':
57 solve the 'transposed' system (3).
58 IJOB (input) INTEGER
60 Specifies what kind of functionality to be performed. = 0:
61 solve (1) only.
62 = 1: A contribution from this subsystem to a Frobenius norm-
63 based estimate of the separation between two matrix pairs is
64 computed. (look ahead strategy is used). = 2: A contribution
65 from this subsystem to a Frobenius norm-based estimate of the
66 separation between two matrix pairs is computed. (DGECON on
67 sub-systems is used.) Not referenced if TRANS = 'T'.
68 M (input) INTEGER
70 On entry, M specifies the order of A and D, and the row dimen‐
71 sion of C, F, R and L.
72 N (input) INTEGER
74 On entry, N specifies the order of B and E, and the column
75 dimension of C, F, R and L.
76 A (input) DOUBLE PRECISION array, dimension (LDA, M)
78 On entry, A contains an upper quasi triangular matrix.
79 LDA (input) INTEGER
81 The leading dimension of the matrix A. LDA >= max(1, M).
82 B (input) DOUBLE PRECISION array, dimension (LDB, N)
84 On entry, B contains an upper quasi triangular matrix.
85 LDB (input) INTEGER
87 The leading dimension of the matrix B. LDB >= max(1, N).
88 C (input/output) DOUBLE PRECISION array, dimension (LDC, N)
90 On entry, C contains the right-hand-side of the first matrix
91 equation in (1). On exit, if IJOB = 0, C has been overwritten
92 by the solution R.
93 LDC (input) INTEGER
95 The leading dimension of the matrix C. LDC >= max(1, M).
96 D (input) DOUBLE PRECISION array, dimension (LDD, M)
98 On entry, D contains an upper triangular matrix.
99 LDD (input) INTEGER
101 The leading dimension of the matrix D. LDD >= max(1, M).
102 E (input) DOUBLE PRECISION array, dimension (LDE, N)
104 On entry, E contains an upper triangular matrix.
105 LDE (input) INTEGER
107 The leading dimension of the matrix E. LDE >= max(1, N).
108 F (input/output) DOUBLE PRECISION array, dimension (LDF, N)
110 On entry, F contains the right-hand-side of the second matrix
111 equation in (1). On exit, if IJOB = 0, F has been overwritten
112 by the solution L.
113 LDF (input) INTEGER
115 The leading dimension of the matrix F. LDF >= max(1, M).
116 SCALE (output) DOUBLE PRECISION
118 On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions R and
119 L (C and F on entry) will hold the solutions to a slightly per‐
120 turbed system but the input matrices A, B, D and E have not
121 been changed. If SCALE = 0, R and L will hold the solutions to
122 the homogeneous system with C = F = 0. Normally, SCALE = 1.
123 RDSUM (input/output) DOUBLE PRECISION
125 On entry, the sum of squares of computed contributions to the
126 Dif-estimate under computation by DTGSYL, where the scaling
127 factor RDSCAL (see below) has been factored out. On exit, the
128 corresponding sum of squares updated with the contributions
129 from the current sub-system. If TRANS = 'T' RDSUM is not
130 touched. NOTE: RDSUM only makes sense when DTGSY2 is called by
131 DTGSYL.
132 RDSCAL (input/output) DOUBLE PRECISION
134 On entry, scaling factor used to prevent overflow in RDSUM. On
135 exit, RDSCAL is updated w.r.t. the current contributions in
136 RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL
137 only makes sense when DTGSY2 is called by DTGSYL.
138 IWORK (workspace) INTEGER array, dimension (M+N+2)
140 PQ (output) INTEGER
142 On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
143 8-by-8) solved by this routine.
144 INFO (output) INTEGER
146 On exit, if INFO is set to =0: Successful exit
147 <0: If INFO = -i, the i-th argument had an illegal value.
148 >0: The matrix pairs (A, D) and (B, E) have common or very
149 close eigenvalues.
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152 Based on contributions by
153 Bo Kagstrom and Peter Poromaa, Department of Computing Science,
154 Umea University, S-901 87 Umea, Sweden.
155 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DTGSY2(1)