1ZGGBAL(1) LAPACK routine (version 3.1) ZGGBAL(1)
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6 ZGGBAL - a pair of general complex matrices (A,B)
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9 SUBROUTINE ZGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
10 WORK, INFO )
11 CHARACTER JOB
13 INTEGER IHI, ILO, INFO, LDA, LDB, N
15 DOUBLE PRECISION LSCALE( * ), RSCALE( * ), WORK( * )
17 COMPLEX*16 A( LDA, * ), B( LDB, * )
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21 ZGGBAL balances a pair of general complex matrices (A,B). This
22 involves, first, permuting A and B by similarity transformations to
23 isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N ele‐
24 ments on the diagonal; and second, applying a diagonal similarity
25 transformation to rows and columns ILO to IHI to make the rows and col‐
26 umns as close in norm as possible. Both steps are optional.
27 Balancing may reduce the 1-norm of the matrices, and improve the accu‐
29 racy of the computed eigenvalues and/or eigenvectors in the generalized
30 eigenvalue problem A*x = lambda*B*x.
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34 JOB (input) CHARACTER*1
35 Specifies the operations to be performed on A and B:
36 = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 and
37 RSCALE(I) = 1.0 for i=1,...,N; = 'P': permute only;
38 = 'S': scale only;
39 = 'B': both permute and scale.
40 N (input) INTEGER
42 The order of the matrices A and B. N >= 0.
43 A (input/output) COMPLEX*16 array, dimension (LDA,N)
45 On entry, the input matrix A. On exit, A is overwritten by the
46 balanced matrix. If JOB = 'N', A is not referenced.
47 LDA (input) INTEGER
49 The leading dimension of the array A. LDA >= max(1,N).
50 B (input/output) COMPLEX*16 array, dimension (LDB,N)
52 On entry, the input matrix B. On exit, B is overwritten by the
53 balanced matrix. If JOB = 'N', B is not referenced.
54 LDB (input) INTEGER
56 The leading dimension of the array B. LDB >= max(1,N).
57 ILO (output) INTEGER
59 IHI (output) INTEGER ILO and IHI are set to integers such
60 that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j =
61 1,...,ILO-1 or i = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1
62 and IHI = N.
63 LSCALE (output) DOUBLE PRECISION array, dimension (N)
65 Details of the permutations and scaling factors applied to the
66 left side of A and B. If P(j) is the index of the row inter‐
67 changed with row j, and D(j) is the scaling factor applied to
68 row j, then LSCALE(j) = P(j) for J = 1,...,ILO-1 = D(j)
69 for J = ILO,...,IHI = P(j) for J = IHI+1,...,N. The order
70 in which the interchanges are made is N to IHI+1, then 1 to
71 ILO-1.
72 RSCALE (output) DOUBLE PRECISION array, dimension (N)
74 Details of the permutations and scaling factors applied to the
75 right side of A and B. If P(j) is the index of the column
76 interchanged with column j, and D(j) is the scaling factor
77 applied to column j, then RSCALE(j) = P(j) for J =
78 1,...,ILO-1 = D(j) for J = ILO,...,IHI = P(j) for J =
79 IHI+1,...,N. The order in which the interchanges are made is N
80 to IHI+1, then 1 to ILO-1.
81 WORK (workspace) REAL array, dimension (lwork)
83 lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at
84 least 1 when JOB = 'N' or 'P'.
85 INFO (output) INTEGER
87 = 0: successful exit
88 < 0: if INFO = -i, the i-th argument had an illegal value.
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91 See R.C. WARD, Balancing the generalized eigenvalue problem,
92 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
93 LAPACK routine (version 3.1) November 2006 ZGGBAL(1)