1CTGSY2(1) LAPACK auxiliary routine (version 3.1) CTGSY2(1)
2
6 CTGSY2 - the generalized Sylvester equation A * R - L * B = scale D
7 * R - L * E = scale * F using Level 1 and 2 BLAS, where R and L are
8 unknown M-by-N matrices,
9
11 SUBROUTINE CTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD,
12 E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO )
13 CHARACTER TRANS
15 INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
17 REAL RDSCAL, RDSUM, SCALE
19 COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ),
21 E( LDE, * ), F( LDF, * )
22
24 CTGSY2 solves the generalized Sylvester equation (A, D), (B, E) and (C,
25 F) are given matrix pairs of size M-by-M, N-by-N and M-by-N, respec‐
26 tively. A, B, D and E are upper triangular (i.e., (A,D) and (B,E) in
27 generalized Schur form).
28 The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
30 scaling factor chosen to avoid overflow.
31 In matrix notation solving equation (1) corresponds to solve Zx = scale
33 * b, where Z is defined as
34 Z = [ kron(In, A) -kron(B', Im) ] (2)
36 [ kron(In, D) -kron(E', Im) ],
37 Ik is the identity matrix of size k and X' is the transpose of X.
39 kron(X, Y) is the Kronecker product between the matrices X and Y.
40 If TRANS = 'C', y in the conjugate transposed system Z'y = scale*b is
42 solved for, which is equivalent to solve for R and L in
43 A' * R + D' * L = scale * C (3)
45 R * B' + L * E' = scale * -F
46 This case is used to compute an estimate of Dif[(A, D), (B, E)] = =
48 sigma_min(Z) using reverse communicaton with CLACON.
49 CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL of an
51 upper bound on the separation between to matrix pairs. Then the input
52 (A, D), (B, E) are sub-pencils of two matrix pairs in CTGSYL.
53
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: A contribution from this subsystem to a Frobenius norm-
64 based estimate of the separation between two matrix pairs is
65 computed. (look ahead strategy is used). =2: A contribution
66 from this subsystem to a Frobenius norm-based estimate of the
67 separation between two matrix pairs is computed. (SGECON on
68 sub-systems is used.) Not referenced if TRANS = 'T'.
69 M (input) INTEGER
71 On entry, M specifies the order of A and D, and the row dimen‐
72 sion of C, F, R and L.
73 N (input) INTEGER
75 On entry, N specifies the order of B and E, and the column
76 dimension of C, F, R and L.
77 A (input) COMPLEX array, dimension (LDA, M)
79 On entry, A contains an upper triangular matrix.
80 LDA (input) INTEGER
82 The leading dimension of the matrix A. LDA >= max(1, M).
83 B (input) COMPLEX array, dimension (LDB, N)
85 On entry, B contains an upper triangular matrix.
86 LDB (input) INTEGER
88 The leading dimension of the matrix B. LDB >= max(1, N).
89 C (input/output) COMPLEX array, dimension (LDC, N)
91 On entry, C contains the right-hand-side of the first matrix
92 equation in (1). On exit, if IJOB = 0, C has been overwritten
93 by the solution R.
94 LDC (input) INTEGER
96 The leading dimension of the matrix C. LDC >= max(1, M).
97 D (input) COMPLEX array, dimension (LDD, M)
99 On entry, D contains an upper triangular matrix.
100 LDD (input) INTEGER
102 The leading dimension of the matrix D. LDD >= max(1, M).
103 E (input) COMPLEX array, dimension (LDE, N)
105 On entry, E contains an upper triangular matrix.
106 LDE (input) INTEGER
108 The leading dimension of the matrix E. LDE >= max(1, N).
109 F (input/output) COMPLEX array, dimension (LDF, N)
111 On entry, F contains the right-hand-side of the second matrix
112 equation in (1). On exit, if IJOB = 0, F has been overwritten
113 by the solution L.
114 LDF (input) INTEGER
116 The leading dimension of the matrix F. LDF >= max(1, M).
117 SCALE (output) REAL
119 On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions R and
120 L (C and F on entry) will hold the solutions to a slightly per‐
121 turbed system but the input matrices A, B, D and E have not
122 been changed. If SCALE = 0, R and L will hold the solutions to
123 the homogeneous system with C = F = 0. Normally, SCALE = 1.
124 RDSUM (input/output) REAL
126 On entry, the sum of squares of computed contributions to the
127 Dif-estimate under computation by CTGSYL, where the scaling
128 factor RDSCAL (see below) has been factored out. On exit, the
129 corresponding sum of squares updated with the contributions
130 from the current sub-system. If TRANS = 'T' RDSUM is not
131 touched. NOTE: RDSUM only makes sense when CTGSY2 is called by
132 CTGSYL.
133 RDSCAL (input/output) REAL
135 On entry, scaling factor used to prevent overflow in RDSUM. On
136 exit, RDSCAL is updated w.r.t. the current contributions in
137 RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL
138 only makes sense when CTGSY2 is called by CTGSYL.
139 INFO (output) INTEGER
141 On exit, if INFO is set to =0: Successful exit
142 <0: If INFO = -i, input argument number i is illegal.
143 >0: The matrix pairs (A, D) and (B, E) have common or very
144 close eigenvalues.
145
147 Based on contributions by
148 Bo Kagstrom and Peter Poromaa, Department of Computing Science,
149 Umea University, S-901 87 Umea, Sweden.
150 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 CTGSY2(1)