1ZGBEQUB(1) LAPACK routine (version 3.2) ZGBEQUB(1)
2
6 ZGBEQUB - computes row and column scalings intended to equilibrate an
7 M-by-N matrix A and reduce its condition number
8
10 SUBROUTINE ZGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX,
11 INFO )
12 IMPLICIT NONE
14 INTEGER INFO, KL, KU, LDAB, M, N
16 DOUBLE PRECISION AMAX, COLCND, ROWCND
18 DOUBLE PRECISION C( * ), R( * )
20 COMPLEX*16 AB( LDAB, * )
22
24 ZGBEQUB computes row and column scalings intended to equilibrate an M-
25 by-N matrix A and reduce its condition number. R returns the row scale
26 factors and C the column scale factors, chosen to try to make the
27 largest element in each row and column of the matrix B with elements
28 B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most the radix.
29 R(i) and C(j) are restricted to be a power of the radix between SMLNUM
30 = smallest safe number and BIGNUM = largest safe number. Use of these
31 scaling factors is not guaranteed to reduce the condition number of A
32 but works well in practice.
33 This routine differs from ZGEEQU by restricting the scaling factors to
34 a power of the radix. Baring over- and underflow, scaling by these
35 factors introduces no additional rounding errors. However, the scaled
36 entries' magnitured are no longer approximately 1 but lie between
37 sqrt(radix) and 1/sqrt(radix).
38
40 M (input) INTEGER
41 The number of rows of the matrix A. M >= 0.
42 N (input) INTEGER
44 The number of columns of the matrix A. N >= 0.
45 KL (input) INTEGER
47 The number of subdiagonals within the band of A. KL >= 0.
48 KU (input) INTEGER
50 The number of superdiagonals within the band of A. KU >= 0.
51 AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
53 On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
54 The j-th column of A is stored in the j-th column of the array
55 AB as follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j-
56 KU)<=i<=min(N,j+kl)
57 LDAB (input) INTEGER
59 The leading dimension of the array A. LDAB >= max(1,M).
60 R (output) DOUBLE PRECISION array, dimension (M)
62 If INFO = 0 or INFO > M, R contains the row scale factors for
63 A.
64 C (output) DOUBLE PRECISION array, dimension (N)
66 If INFO = 0, C contains the column scale factors for A.
67 ROWCND (output) DOUBLE PRECISION
69 If INFO = 0 or INFO > M, ROWCND contains the ratio of the
70 smallest R(i) to the largest R(i). If ROWCND >= 0.1 and AMAX
71 is neither too large nor too small, it is not worth scaling by
72 R.
73 COLCND (output) DOUBLE PRECISION
75 If INFO = 0, COLCND contains the ratio of the smallest C(i) to
76 the largest C(i). If COLCND >= 0.1, it is not worth scaling by
77 C.
78 AMAX (output) DOUBLE PRECISION
80 Absolute value of largest matrix element. If AMAX is very
81 close to overflow or very close to underflow, the matrix should
82 be scaled.
83 INFO (output) INTEGER
85 = 0: successful exit
86 < 0: if INFO = -i, the i-th argument had an illegal value
87 > 0: if INFO = i, and i is
88 <= M: the i-th row of A is exactly zero
89 > M: the (i-M)-th column of A is exactly zero
90 LAPACK routine (version 3.2) November 2008 ZGBEQUB(1)