1ZGGBAL(1) LAPACK routine (version 3.2) ZGGBAL(1)
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6 ZGGBAL - balances 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. Balancing
27 may reduce the 1-norm of the matrices, and improve the accuracy of the
28 computed eigenvalues and/or eigenvectors in the generalized eigenvalue
29 problem A*x = lambda*B*x.
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32 JOB (input) CHARACTER*1
33 Specifies the operations to be performed on A and B:
34 = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 and
35 RSCALE(I) = 1.0 for i=1,...,N; = 'P': permute only;
36 = 'S': scale only;
37 = 'B': both permute and scale.
38 N (input) INTEGER
40 The order of the matrices A and B. N >= 0.
41 A (input/output) COMPLEX*16 array, dimension (LDA,N)
43 On entry, the input matrix A. On exit, A is overwritten by the
44 balanced matrix. If JOB = 'N', A is not referenced.
45 LDA (input) INTEGER
47 The leading dimension of the array A. LDA >= max(1,N).
48 B (input/output) COMPLEX*16 array, dimension (LDB,N)
50 On entry, the input matrix B. On exit, B is overwritten by the
51 balanced matrix. If JOB = 'N', B is not referenced.
52 LDB (input) INTEGER
54 The leading dimension of the array B. LDB >= max(1,N).
55 ILO (output) INTEGER
57 IHI (output) INTEGER ILO and IHI are set to integers such
58 that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j =
59 1,...,ILO-1 or i = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1
60 and IHI = N.
61 LSCALE (output) DOUBLE PRECISION array, dimension (N)
63 Details of the permutations and scaling factors applied to the
64 left side of A and B. If P(j) is the index of the row inter‐
65 changed with row j, and D(j) is the scaling factor applied to
66 row j, then LSCALE(j) = P(j) for J = 1,...,ILO-1 = D(j)
67 for J = ILO,...,IHI = P(j) for J = IHI+1,...,N. The order
68 in which the interchanges are made is N to IHI+1, then 1 to
69 ILO-1.
70 RSCALE (output) DOUBLE PRECISION array, dimension (N)
72 Details of the permutations and scaling factors applied to the
73 right side of A and B. If P(j) is the index of the column
74 interchanged with column j, and D(j) is the scaling factor
75 applied to column j, then RSCALE(j) = P(j) for J =
76 1,...,ILO-1 = D(j) for J = ILO,...,IHI = P(j) for J =
77 IHI+1,...,N. The order in which the interchanges are made is N
78 to IHI+1, then 1 to ILO-1.
79 WORK (workspace) REAL array, dimension (lwork)
81 lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at
82 least 1 when JOB = 'N' or 'P'.
83 INFO (output) INTEGER
85 = 0: successful exit
86 < 0: if INFO = -i, the i-th argument had an illegal value.
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89 See R.C. WARD, Balancing the generalized eigenvalue problem,
90 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
91 LAPACK routine (version 3.2) November 2008 ZGGBAL(1)