1SGBEQUB(1) LAPACK routine (version 3.2) SGBEQUB(1)
2
6 SGBEQUB - computes row and column scalings intended to equilibrate an
7 M-by-N matrix A and reduce its condition number
8
10 SUBROUTINE SGBEQUB( 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 REAL AMAX, COLCND, ROWCND
18 REAL AB( LDAB, * ), C( * ), R( * )
20
22 SGBEQUB computes row and column scalings intended to equilibrate an M-
23 by-N matrix A and reduce its condition number. R returns the row scale
24 factors and C the column scale factors, chosen to try to make the
25 largest element in each row and column of the matrix B with elements
26 B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most the radix.
27 R(i) and C(j) are restricted to be a power of the radix between SMLNUM
28 = smallest safe number and BIGNUM = largest safe number. Use of these
29 scaling factors is not guaranteed to reduce the condition number of A
30 but works well in practice.
31 This routine differs from SGEEQU by restricting the scaling factors to
32 a power of the radix. Baring over- and underflow, scaling by these
33 factors introduces no additional rounding errors. However, the scaled
34 entries' magnitured are no longer approximately 1 but lie between
35 sqrt(radix) and 1/sqrt(radix).
36
38 M (input) INTEGER
39 The number of rows of the matrix A. M >= 0.
40 N (input) INTEGER
42 The number of columns of the matrix A. N >= 0.
43 KL (input) INTEGER
45 The number of subdiagonals within the band of A. KL >= 0.
46 KU (input) INTEGER
48 The number of superdiagonals within the band of A. KU >= 0.
49 AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
51 On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
52 The j-th column of A is stored in the j-th column of the array
53 AB as follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j-
54 KU)<=i<=min(N,j+kl)
55 LDAB (input) INTEGER
57 The leading dimension of the array A. LDAB >= max(1,M).
58 R (output) REAL array, dimension (M)
60 If INFO = 0 or INFO > M, R contains the row scale factors for
61 A.
62 C (output) REAL array, dimension (N)
64 If INFO = 0, C contains the column scale factors for A.
65 ROWCND (output) REAL
67 If INFO = 0 or INFO > M, ROWCND contains the ratio of the
68 smallest R(i) to the largest R(i). If ROWCND >= 0.1 and AMAX
69 is neither too large nor too small, it is not worth scaling by
70 R.
71 COLCND (output) REAL
73 If INFO = 0, COLCND contains the ratio of the smallest C(i) to
74 the largest C(i). If COLCND >= 0.1, it is not worth scaling by
75 C.
76 AMAX (output) REAL
78 Absolute value of largest matrix element. If AMAX is very
79 close to overflow or very close to underflow, the matrix should
80 be scaled.
81 INFO (output) INTEGER
83 = 0: successful exit
84 < 0: if INFO = -i, the i-th argument had an illegal value
85 > 0: if INFO = i, and i is
86 <= M: the i-th row of A is exactly zero
87 > M: the (i-M)-th column of A is exactly zero
88 LAPACK routine (version 3.2) November 2008 SGBEQUB(1)