1: SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
2: $ INFO )
3: *
4: * -- LAPACK routine (version 3.2) --
5: * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
6: * -- Jason Riedy of Univ. of California Berkeley. --
7: * -- November 2008 --
8: *
9: * -- LAPACK is a software package provided by Univ. of Tennessee, --
10: * -- Univ. of California Berkeley and NAG Ltd. --
11: *
12: IMPLICIT NONE
13: * ..
14: * .. Scalar Arguments ..
15: INTEGER INFO, LDA, M, N
16: DOUBLE PRECISION AMAX, COLCND, ROWCND
17: * ..
18: * .. Array Arguments ..
19: DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * DGEEQUB computes row and column scalings intended to equilibrate an
26: * M-by-N matrix A and reduce its condition number. R returns the row
27: * scale factors and C the column scale factors, chosen to try to make
28: * the largest element in each row and column of the matrix B with
29: * elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
30: * the radix.
31: *
32: * R(i) and C(j) are restricted to be a power of the radix between
33: * SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
34: * of these scaling factors is not guaranteed to reduce the condition
35: * number of A but works well in practice.
36: *
37: * This routine differs from DGEEQU by restricting the scaling factors
38: * to a power of the radix. Baring over- and underflow, scaling by
39: * these factors introduces no additional rounding errors. However, the
40: * scaled entries' magnitured are no longer approximately 1 but lie
41: * between sqrt(radix) and 1/sqrt(radix).
42: *
43: * Arguments
44: * =========
45: *
46: * M (input) INTEGER
47: * The number of rows of the matrix A. M >= 0.
48: *
49: * N (input) INTEGER
50: * The number of columns of the matrix A. N >= 0.
51: *
52: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
53: * The M-by-N matrix whose equilibration factors are
54: * to be computed.
55: *
56: * LDA (input) INTEGER
57: * The leading dimension of the array A. LDA >= max(1,M).
58: *
59: * R (output) DOUBLE PRECISION array, dimension (M)
60: * If INFO = 0 or INFO > M, R contains the row scale factors
61: * for A.
62: *
63: * C (output) DOUBLE PRECISION array, dimension (N)
64: * If INFO = 0, C contains the column scale factors for A.
65: *
66: * ROWCND (output) DOUBLE PRECISION
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
69: * AMAX is neither too large nor too small, it is not worth
70: * scaling by R.
71: *
72: * COLCND (output) DOUBLE PRECISION
73: * If INFO = 0, COLCND contains the ratio of the smallest
74: * C(i) to the largest C(i). If COLCND >= 0.1, it is not
75: * worth scaling by C.
76: *
77: * AMAX (output) DOUBLE PRECISION
78: * Absolute value of largest matrix element. If AMAX is very
79: * close to overflow or very close to underflow, the matrix
80: * should be scaled.
81: *
82: * 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: *
89: * =====================================================================
90: *
91: * .. Parameters ..
92: DOUBLE PRECISION ONE, ZERO
93: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
94: * ..
95: * .. Local Scalars ..
96: INTEGER I, J
97: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
98: * ..
99: * .. External Functions ..
100: DOUBLE PRECISION DLAMCH
101: EXTERNAL DLAMCH
102: * ..
103: * .. External Subroutines ..
104: EXTERNAL XERBLA
105: * ..
106: * .. Intrinsic Functions ..
107: INTRINSIC ABS, MAX, MIN, LOG
108: * ..
109: * .. Executable Statements ..
110: *
111: * Test the input parameters.
112: *
113: INFO = 0
114: IF( M.LT.0 ) THEN
115: INFO = -1
116: ELSE IF( N.LT.0 ) THEN
117: INFO = -2
118: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
119: INFO = -4
120: END IF
121: IF( INFO.NE.0 ) THEN
122: CALL XERBLA( 'DGEEQUB', -INFO )
123: RETURN
124: END IF
125: *
126: * Quick return if possible.
127: *
128: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
129: ROWCND = ONE
130: COLCND = ONE
131: AMAX = ZERO
132: RETURN
133: END IF
134: *
135: * Get machine constants. Assume SMLNUM is a power of the radix.
136: *
137: SMLNUM = DLAMCH( 'S' )
138: BIGNUM = ONE / SMLNUM
139: RADIX = DLAMCH( 'B' )
140: LOGRDX = LOG( RADIX )
141: *
142: * Compute row scale factors.
143: *
144: DO 10 I = 1, M
145: R( I ) = ZERO
146: 10 CONTINUE
147: *
148: * Find the maximum element in each row.
149: *
150: DO 30 J = 1, N
151: DO 20 I = 1, M
152: R( I ) = MAX( R( I ), ABS( A( I, J ) ) )
153: 20 CONTINUE
154: 30 CONTINUE
155: DO I = 1, M
156: IF( R( I ).GT.ZERO ) THEN
157: R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
158: END IF
159: END DO
160: *
161: * Find the maximum and minimum scale factors.
162: *
163: RCMIN = BIGNUM
164: RCMAX = ZERO
165: DO 40 I = 1, M
166: RCMAX = MAX( RCMAX, R( I ) )
167: RCMIN = MIN( RCMIN, R( I ) )
168: 40 CONTINUE
169: AMAX = RCMAX
170: *
171: IF( RCMIN.EQ.ZERO ) THEN
172: *
173: * Find the first zero scale factor and return an error code.
174: *
175: DO 50 I = 1, M
176: IF( R( I ).EQ.ZERO ) THEN
177: INFO = I
178: RETURN
179: END IF
180: 50 CONTINUE
181: ELSE
182: *
183: * Invert the scale factors.
184: *
185: DO 60 I = 1, M
186: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
187: 60 CONTINUE
188: *
189: * Compute ROWCND = min(R(I)) / max(R(I)).
190: *
191: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
192: END IF
193: *
194: * Compute column scale factors
195: *
196: DO 70 J = 1, N
197: C( J ) = ZERO
198: 70 CONTINUE
199: *
200: * Find the maximum element in each column,
201: * assuming the row scaling computed above.
202: *
203: DO 90 J = 1, N
204: DO 80 I = 1, M
205: C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) )
206: 80 CONTINUE
207: IF( C( J ).GT.ZERO ) THEN
208: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
209: END IF
210: 90 CONTINUE
211: *
212: * Find the maximum and minimum scale factors.
213: *
214: RCMIN = BIGNUM
215: RCMAX = ZERO
216: DO 100 J = 1, N
217: RCMIN = MIN( RCMIN, C( J ) )
218: RCMAX = MAX( RCMAX, C( J ) )
219: 100 CONTINUE
220: *
221: IF( RCMIN.EQ.ZERO ) THEN
222: *
223: * Find the first zero scale factor and return an error code.
224: *
225: DO 110 J = 1, N
226: IF( C( J ).EQ.ZERO ) THEN
227: INFO = M + J
228: RETURN
229: END IF
230: 110 CONTINUE
231: ELSE
232: *
233: * Invert the scale factors.
234: *
235: DO 120 J = 1, N
236: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
237: 120 CONTINUE
238: *
239: * Compute COLCND = min(C(J)) / max(C(J)).
240: *
241: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
242: END IF
243: *
244: RETURN
245: *
246: * End of DGEEQUB
247: *
248: END
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