1CTGSY2(1) LAPACK auxiliary routine (version 3.2) CTGSY2(1)
2
6 CTGSY2 - solves the generalized Sylvester equation A * R - L * B =
7 scale D * R - L * E = scale * F using Level 1 and 2 BLAS, where R and
8 L are 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
29 scaling factor chosen to avoid overflow.
30 In matrix notation solving equation (1) corresponds to solve Zx = scale
31 * b, where Z is defined as
32 Z = [ kron(In, A) -kron(B', Im) ] (2)
33 [ kron(In, D) -kron(E', Im) ],
34 Ik is the identity matrix of size k and X' is the transpose of X.
35 kron(X, Y) is the Kronecker product between the matrices X and Y. If
36 TRANS = 'C', y in the conjugate transposed system Z'y = scale*b is
37 solved for, which is equivalent to solve for R and L in
38 A' * R + D' * L = scale * C (3)
39 R * B' + L * E' = scale * -F
40 This case is used to compute an estimate of Dif[(A, D), (B, E)] = =
41 sigma_min(Z) using reverse communicaton with CLACON.
42 CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL of an
43 upper bound on the separation between to matrix pairs. Then the input
44 (A, D), (B, E) are sub-pencils of two matrix pairs in CTGSYL.
45
47 TRANS (input) CHARACTER*1
48 = 'N', solve the generalized Sylvester equation (1). = 'T':
49 solve the 'transposed' system (3).
50 IJOB (input) INTEGER
52 Specifies what kind of functionality to be performed. =0:
53 solve (1) only.
54 =1: A contribution from this subsystem to a Frobenius norm-
55 based estimate of the separation between two matrix pairs is
56 computed. (look ahead strategy is used). =2: A contribution
57 from this subsystem to a Frobenius norm-based estimate of the
58 separation between two matrix pairs is computed. (SGECON on
59 sub-systems is used.) Not referenced if TRANS = 'T'.
60 M (input) INTEGER
62 On entry, M specifies the order of A and D, and the row dimen‐
63 sion of C, F, R and L.
64 N (input) INTEGER
66 On entry, N specifies the order of B and E, and the column
67 dimension of C, F, R and L.
68 A (input) COMPLEX array, dimension (LDA, M)
70 On entry, A contains an upper triangular matrix.
71 LDA (input) INTEGER
73 The leading dimension of the matrix A. LDA >= max(1, M).
74 B (input) COMPLEX array, dimension (LDB, N)
76 On entry, B contains an upper triangular matrix.
77 LDB (input) INTEGER
79 The leading dimension of the matrix B. LDB >= max(1, N).
80 C (input/output) COMPLEX array, dimension (LDC, N)
82 On entry, C contains the right-hand-side of the first matrix
83 equation in (1). On exit, if IJOB = 0, C has been overwritten
84 by the solution R.
85 LDC (input) INTEGER
87 The leading dimension of the matrix C. LDC >= max(1, M).
88 D (input) COMPLEX array, dimension (LDD, M)
90 On entry, D contains an upper triangular matrix.
91 LDD (input) INTEGER
93 The leading dimension of the matrix D. LDD >= max(1, M).
94 E (input) COMPLEX array, dimension (LDE, N)
96 On entry, E contains an upper triangular matrix.
97 LDE (input) INTEGER
99 The leading dimension of the matrix E. LDE >= max(1, N).
100 F (input/output) COMPLEX array, dimension (LDF, N)
102 On entry, F contains the right-hand-side of the second matrix
103 equation in (1). On exit, if IJOB = 0, F has been overwritten
104 by the solution L.
105 LDF (input) INTEGER
107 The leading dimension of the matrix F. LDF >= max(1, M).
108 SCALE (output) REAL
110 On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions R and
111 L (C and F on entry) will hold the solutions to a slightly per‐
112 turbed system but the input matrices A, B, D and E have not
113 been changed. If SCALE = 0, R and L will hold the solutions to
114 the homogeneous system with C = F = 0. Normally, SCALE = 1.
115 RDSUM (input/output) REAL
117 On entry, the sum of squares of computed contributions to the
118 Dif-estimate under computation by CTGSYL, where the scaling
119 factor RDSCAL (see below) has been factored out. On exit, the
120 corresponding sum of squares updated with the contributions
121 from the current sub-system. If TRANS = 'T' RDSUM is not
122 touched. NOTE: RDSUM only makes sense when CTGSY2 is called by
123 CTGSYL.
124 RDSCAL (input/output) REAL
126 On entry, scaling factor used to prevent overflow in RDSUM. On
127 exit, RDSCAL is updated w.r.t. the current contributions in
128 RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL
129 only makes sense when CTGSY2 is called by CTGSYL.
130 INFO (output) INTEGER
132 On exit, if INFO is set to =0: Successful exit
133 <0: If INFO = -i, input argument number i is illegal.
134 >0: The matrix pairs (A, D) and (B, E) have common or very
135 close eigenvalues.
136
138 Based on contributions by
139 Bo Kagstrom and Peter Poromaa, Department of Computing Science,
140 Umea University, S-901 87 Umea, Sweden.
141 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 CTGSY2(1)