1SGGES(1) LAPACK driver routine (version 3.2) SGGES(1)
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6 SGGES - computes for a pair of N-by-N real nonsymmetric matrices (A,B),
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9 SUBROUTINE SGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
10 SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
11 WORK, LWORK, BWORK, INFO )
12 CHARACTER JOBVSL, JOBVSR, SORT
14 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
16 LOGICAL BWORK( * )
18 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
20 BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( *
21 )
22 LOGICAL SELCTG
24 EXTERNAL SELCTG
26
28 SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
29 the generalized eigenvalues, the generalized real Schur form (S,T),
30 optionally, the left and/or right matrices of Schur vectors (VSL and
31 VSR). This gives the generalized Schur factorization
32 (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
33 Optionally, it also orders the eigenvalues so that a selected cluster
34 of eigenvalues appears in the leading diagonal blocks of the upper
35 quasi-triangular matrix S and the upper triangular matrix T.The leading
36 columns of VSL and VSR then form an orthonormal basis for the corre‐
37 sponding left and right eigenspaces (deflating subspaces). (If only
38 the generalized eigenvalues are needed, use the driver SGGEV instead,
39 which is faster.)
40 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or
41 a ratio alpha/beta = w, such that A - w*B is singular. It is usually
42 represented as the pair (alpha,beta), as there is a reasonable inter‐
43 pretation for beta=0 or both being zero.
44 A pair of matrices (S,T) is in generalized real Schur form if T is
45 upper triangular with non-negative diagonal and S is block upper trian‐
46 gular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real
47 generalized eigenvalues, while 2-by-2 blocks of S will be "standard‐
48 ized" by making the corresponding elements of T have the form:
49 [ a 0 ]
50 [ 0 b ]
51 and the pair of corresponding 2-by-2 blocks in S and T will have a com‐
52 plex conjugate pair of generalized eigenvalues.
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55 JOBVSL (input) CHARACTER*1
56 = 'N': do not compute the left Schur vectors;
57 = 'V': compute the left Schur vectors.
58 JOBVSR (input) CHARACTER*1
60 = 'N': do not compute the right Schur vectors;
61 = 'V': compute the right Schur vectors.
62 SORT (input) CHARACTER*1
64 Specifies whether or not to order the eigenvalues on the diago‐
65 nal of the generalized Schur form. = 'N': Eigenvalues are not
66 ordered;
67 = 'S': Eigenvalues are ordered (see SELCTG);
68 SELCTG (external procedure) LOGICAL FUNCTION of three REAL arguments
70 SELCTG must be declared EXTERNAL in the calling subroutine. If
71 SORT = 'N', SELCTG is not referenced. If SORT = 'S', SELCTG is
72 used to select eigenvalues to sort to the top left of the Schur
73 form. An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected
74 if SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
75 one of a complex conjugate pair of eigenvalues is selected,
76 then both complex eigenvalues are selected. Note that in the
77 ill-conditioned case, a selected complex eigenvalue may no
78 longer satisfy SELCTG(ALPHAR(j),ALPHAI(j), BETA(j)) = .TRUE.
79 after ordering. INFO is to be set to N+2 in this case.
80 N (input) INTEGER
82 The order of the matrices A, B, VSL, and VSR. N >= 0.
83 A (input/output) REAL array, dimension (LDA, N)
85 On entry, the first of the pair of matrices. On exit, A has
86 been overwritten by its generalized Schur form S.
87 LDA (input) INTEGER
89 The leading dimension of A. LDA >= max(1,N).
90 B (input/output) REAL array, dimension (LDB, N)
92 On entry, the second of the pair of matrices. On exit, B has
93 been overwritten by its generalized Schur form T.
94 LDB (input) INTEGER
96 The leading dimension of B. LDB >= max(1,N).
97 SDIM (output) INTEGER
99 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
100 genvalues (after sorting) for which SELCTG is true. (Complex
101 conjugate pairs for which SELCTG is true for either eigenvalue
102 count as 2.)
103 ALPHAR (output) REAL array, dimension (N)
105 ALPHAI (output) REAL array, dimension (N) BETA (output)
106 REAL array, dimension (N) On exit, (ALPHAR(j) +
107 ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigen‐
108 values. ALPHAR(j) + ALPHAI(j)*i, and BETA(j),j=1,...,N are
109 the diagonals of the complex Schur form (S,T) that would result
110 if the 2-by-2 diagonal blocks of the real Schur form of (A,B)
111 were further reduced to triangular form using 2-by-2 complex
112 unitary transformations. If ALPHAI(j) is zero, then the j-th
113 eigenvalue is real; if positive, then the j-th and (j+1)-st ei‐
114 genvalues are a complex conjugate pair, with ALPHAI(j+1) nega‐
115 tive. Note: the quotients ALPHAR(j)/BETA(j) and
116 ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j)
117 may even be zero. Thus, the user should avoid naively comput‐
118 ing the ratio. However, ALPHAR and ALPHAI will be always less
119 than and usually comparable with norm(A) in magnitude, and BETA
120 always less than and usually comparable with norm(B).
121 VSL (output) REAL array, dimension (LDVSL,N)
123 If JOBVSL = 'V', VSL will contain the left Schur vectors. Not
124 referenced if JOBVSL = 'N'.
125 LDVSL (input) INTEGER
127 The leading dimension of the matrix VSL. LDVSL >=1, and if JOB‐
128 VSL = 'V', LDVSL >= N.
129 VSR (output) REAL array, dimension (LDVSR,N)
131 If JOBVSR = 'V', VSR will contain the right Schur vectors. Not
132 referenced if JOBVSR = 'N'.
133 LDVSR (input) INTEGER
135 The leading dimension of the matrix VSR. LDVSR >= 1, and if
136 JOBVSR = 'V', LDVSR >= N.
137 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
139 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
140 LWORK (input) INTEGER
142 The dimension of the array WORK. If N = 0, LWORK >= 1, else
143 LWORK >= max(8*N,6*N+16). For good performance , LWORK must
144 generally be larger. If LWORK = -1, then a workspace query is
145 assumed; the routine only calculates the optimal size of the
146 WORK array, returns this value as the first entry of the WORK
147 array, and no error message related to LWORK is issued by
148 XERBLA.
149 BWORK (workspace) LOGICAL array, dimension (N)
151 Not referenced if SORT = 'N'.
152 INFO (output) INTEGER
154 = 0: successful exit
155 < 0: if INFO = -i, the i-th argument had an illegal value.
156 = 1,...,N: The QZ iteration failed. (A,B) are not in Schur
157 form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct
158 for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed
159 in SHGEQZ.
160 =N+2: after reordering, roundoff changed values of some complex
161 eigenvalues so that leading eigenvalues in the Generalized
162 Schur form no longer satisfy SELCTG=.TRUE. This could also be
163 caused due to scaling. =N+3: reordering failed in STGSEN.
164 LAPACK driver routine (version 3.N2o)vember 2008 SGGES(1)