1ZGEBAL(1) LAPACK routine (version 3.1) ZGEBAL(1)
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6 ZGEBAL - a general complex matrix A
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9 SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
10 CHARACTER JOB
12 INTEGER IHI, ILO, INFO, LDA, N
14 DOUBLE PRECISION SCALE( * )
16 COMPLEX*16 A( LDA, * )
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20 ZGEBAL balances a general complex matrix A. This involves, first, per‐
21 muting A by a similarity transformation to isolate eigenvalues in the
22 first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and sec‐
23 ond, applying a diagonal similarity transformation to rows and columns
24 ILO to IHI to make the rows and columns as close in norm as possible.
25 Both steps are optional.
26 Balancing may reduce the 1-norm of the matrix, and improve the accuracy
28 of the computed eigenvalues and/or eigenvectors.
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32 JOB (input) CHARACTER*1
33 Specifies the operations to be performed on A:
34 = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0 for
35 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 matrix A. 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
44 the balanced matrix. If JOB = 'N', A is not referenced. See
45 Further Details. LDA (input) INTEGER The leading dimension
46 of the array A. LDA >= max(1,N).
47 ILO (output) INTEGER
49 IHI (output) INTEGER ILO and IHI are set to integers such
50 that on exit A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I =
51 IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N.
52 SCALE (output) DOUBLE PRECISION array, dimension (N)
54 Details of the permutations and scaling factors applied to A.
55 If P(j) is the index of the row and column interchanged with
56 row and column j and D(j) is the scaling factor applied to row
57 and column j, then SCALE(j) = P(j) for j = 1,...,ILO-1 =
58 D(j) for j = ILO,...,IHI = P(j) for j = IHI+1,...,N. The
59 order in which the interchanges are made is N to IHI+1, then 1
60 to ILO-1.
61 INFO (output) INTEGER
63 = 0: successful exit.
64 < 0: if INFO = -i, the i-th argument had an illegal value.
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67 The permutations consist of row and column interchanges which put the
68 matrix in the form
69 ( T1 X Y )
71 P A P = ( 0 B Z )
72 ( 0 0 T2 )
73 where T1 and T2 are upper triangular matrices whose eigenvalues lie
75 along the diagonal. The column indices ILO and IHI mark the starting
76 and ending columns of the submatrix B. Balancing consists of applying a
77 diagonal similarity transformation inv(D) * B * D to make the 1-norms
78 of each row of B and its corresponding column nearly equal. The output
79 matrix is
80 ( T1 X*D Y )
82 ( 0 inv(D)*B*D inv(D)*Z ).
83 ( 0 0 T2 )
84 Information about the permutations P and the diagonal matrix D is
86 returned in the vector SCALE.
87 This subroutine is based on the EISPACK routine CBAL.
89 Modified by Tzu-Yi Chen, Computer Science Division, University of
91 California at Berkeley, USA
92 LAPACK routine (version 3.1) November 2006 ZGEBAL(1)