1 SUBROUTINE ZGEEQUB( 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 C( * ), R( * )
20 COMPLEX*16 A( LDA, * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * ZGEEQUB computes row and column scalings intended to equilibrate an
27 * M-by-N matrix A and reduce its condition number. R returns the row
28 * scale factors and C the column scale factors, chosen to try to make
29 * the largest element in each row and column of the matrix B with
30 * elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
31 * the radix.
32 *
33 * R(i) and C(j) are restricted to be a power of the radix between
34 * SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
35 * of these scaling factors is not guaranteed to reduce the condition
36 * number of A but works well in practice.
37 *
38 * This routine differs from ZGEEQU by restricting the scaling factors
39 * to a power of the radix. Baring over- and underflow, scaling by
40 * these factors introduces no additional rounding errors. However, the
41 * scaled entries' magnitured are no longer approximately 1 but lie
42 * between sqrt(radix) and 1/sqrt(radix).
43 *
44 * Arguments
45 * =========
46 *
47 * M (input) INTEGER
48 * The number of rows of the matrix A. M >= 0.
49 *
50 * N (input) INTEGER
51 * The number of columns of the matrix A. N >= 0.
52 *
53 * A (input) COMPLEX*16 array, dimension (LDA,N)
54 * The M-by-N matrix whose equilibration factors are
55 * to be computed.
56 *
57 * LDA (input) INTEGER
58 * The leading dimension of the array A. LDA >= max(1,M).
59 *
60 * R (output) DOUBLE PRECISION array, dimension (M)
61 * If INFO = 0 or INFO > M, R contains the row scale factors
62 * for A.
63 *
64 * C (output) DOUBLE PRECISION array, dimension (N)
65 * If INFO = 0, C contains the column scale factors for A.
66 *
67 * ROWCND (output) DOUBLE PRECISION
68 * If INFO = 0 or INFO > M, ROWCND contains the ratio of the
69 * smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
70 * AMAX is neither too large nor too small, it is not worth
71 * scaling by R.
72 *
73 * COLCND (output) DOUBLE PRECISION
74 * If INFO = 0, COLCND contains the ratio of the smallest
75 * C(i) to the largest C(i). If COLCND >= 0.1, it is not
76 * worth scaling by C.
77 *
78 * AMAX (output) DOUBLE PRECISION
79 * Absolute value of largest matrix element. If AMAX is very
80 * close to overflow or very close to underflow, the matrix
81 * should be scaled.
82 *
83 * INFO (output) INTEGER
84 * = 0: successful exit
85 * < 0: if INFO = -i, the i-th argument had an illegal value
86 * > 0: if INFO = i, and i is
87 * <= M: the i-th row of A is exactly zero
88 * > M: the (i-M)-th column of A is exactly zero
89 *
90 * =====================================================================
91 *
92 * .. Parameters ..
93 DOUBLE PRECISION ONE, ZERO
94 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
95 * ..
96 * .. Local Scalars ..
97 INTEGER I, J
98 DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
99 COMPLEX*16 ZDUM
100 * ..
101 * .. External Functions ..
102 DOUBLE PRECISION DLAMCH
103 EXTERNAL DLAMCH
104 * ..
105 * .. External Subroutines ..
106 EXTERNAL XERBLA
107 * ..
108 * .. Intrinsic Functions ..
109 INTRINSIC ABS, MAX, MIN, LOG, REAL, DIMAG
110 * ..
111 * .. Statement Functions ..
112 DOUBLE PRECISION CABS1
113 * ..
114 * .. Statement Function definitions ..
115 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
116 * ..
117 * .. Executable Statements ..
118 *
119 * Test the input parameters.
120 *
121 INFO = 0
122 IF( M.LT.0 ) THEN
123 INFO = -1
124 ELSE IF( N.LT.0 ) THEN
125 INFO = -2
126 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
127 INFO = -4
128 END IF
129 IF( INFO.NE.0 ) THEN
130 CALL XERBLA( 'ZGEEQUB', -INFO )
131 RETURN
132 END IF
133 *
134 * Quick return if possible.
135 *
136 IF( M.EQ.0 .OR. N.EQ.0 ) THEN
137 ROWCND = ONE
138 COLCND = ONE
139 AMAX = ZERO
140 RETURN
141 END IF
142 *
143 * Get machine constants. Assume SMLNUM is a power of the radix.
144 *
145 SMLNUM = DLAMCH( 'S' )
146 BIGNUM = ONE / SMLNUM
147 RADIX = DLAMCH( 'B' )
148 LOGRDX = LOG( RADIX )
149 *
150 * Compute row scale factors.
151 *
152 DO 10 I = 1, M
153 R( I ) = ZERO
154 10 CONTINUE
155 *
156 * Find the maximum element in each row.
157 *
158 DO 30 J = 1, N
159 DO 20 I = 1, M
160 R( I ) = MAX( R( I ), CABS1( A( I, J ) ) )
161 20 CONTINUE
162 30 CONTINUE
163 DO I = 1, M
164 IF( R( I ).GT.ZERO ) THEN
165 R( I ) = RADIX**INT( LOG(R( I ) ) / LOGRDX )
166 END IF
167 END DO
168 *
169 * Find the maximum and minimum scale factors.
170 *
171 RCMIN = BIGNUM
172 RCMAX = ZERO
173 DO 40 I = 1, M
174 RCMAX = MAX( RCMAX, R( I ) )
175 RCMIN = MIN( RCMIN, R( I ) )
176 40 CONTINUE
177 AMAX = RCMAX
178 *
179 IF( RCMIN.EQ.ZERO ) THEN
180 *
181 * Find the first zero scale factor and return an error code.
182 *
183 DO 50 I = 1, M
184 IF( R( I ).EQ.ZERO ) THEN
185 INFO = I
186 RETURN
187 END IF
188 50 CONTINUE
189 ELSE
190 *
191 * Invert the scale factors.
192 *
193 DO 60 I = 1, M
194 R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
195 60 CONTINUE
196 *
197 * Compute ROWCND = min(R(I)) / max(R(I)).
198 *
199 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
200 END IF
201 *
202 * Compute column scale factors.
203 *
204 DO 70 J = 1, N
205 C( J ) = ZERO
206 70 CONTINUE
207 *
208 * Find the maximum element in each column,
209 * assuming the row scaling computed above.
210 *
211 DO 90 J = 1, N
212 DO 80 I = 1, M
213 C( J ) = MAX( C( J ), CABS1( A( I, J ) )*R( I ) )
214 80 CONTINUE
215 IF( C( J ).GT.ZERO ) THEN
216 C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
217 END IF
218 90 CONTINUE
219 *
220 * Find the maximum and minimum scale factors.
221 *
222 RCMIN = BIGNUM
223 RCMAX = ZERO
224 DO 100 J = 1, N
225 RCMIN = MIN( RCMIN, C( J ) )
226 RCMAX = MAX( RCMAX, C( J ) )
227 100 CONTINUE
228 *
229 IF( RCMIN.EQ.ZERO ) THEN
230 *
231 * Find the first zero scale factor and return an error code.
232 *
233 DO 110 J = 1, N
234 IF( C( J ).EQ.ZERO ) THEN
235 INFO = M + J
236 RETURN
237 END IF
238 110 CONTINUE
239 ELSE
240 *
241 * Invert the scale factors.
242 *
243 DO 120 J = 1, N
244 C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
245 120 CONTINUE
246 *
247 * Compute COLCND = min(C(J)) / max(C(J)).
248 *
249 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
250 END IF
251 *
252 RETURN
253 *
254 * End of ZGEEQUB
255 *
256 END
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 C( * ), R( * )
20 COMPLEX*16 A( LDA, * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * ZGEEQUB computes row and column scalings intended to equilibrate an
27 * M-by-N matrix A and reduce its condition number. R returns the row
28 * scale factors and C the column scale factors, chosen to try to make
29 * the largest element in each row and column of the matrix B with
30 * elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
31 * the radix.
32 *
33 * R(i) and C(j) are restricted to be a power of the radix between
34 * SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
35 * of these scaling factors is not guaranteed to reduce the condition
36 * number of A but works well in practice.
37 *
38 * This routine differs from ZGEEQU by restricting the scaling factors
39 * to a power of the radix. Baring over- and underflow, scaling by
40 * these factors introduces no additional rounding errors. However, the
41 * scaled entries' magnitured are no longer approximately 1 but lie
42 * between sqrt(radix) and 1/sqrt(radix).
43 *
44 * Arguments
45 * =========
46 *
47 * M (input) INTEGER
48 * The number of rows of the matrix A. M >= 0.
49 *
50 * N (input) INTEGER
51 * The number of columns of the matrix A. N >= 0.
52 *
53 * A (input) COMPLEX*16 array, dimension (LDA,N)
54 * The M-by-N matrix whose equilibration factors are
55 * to be computed.
56 *
57 * LDA (input) INTEGER
58 * The leading dimension of the array A. LDA >= max(1,M).
59 *
60 * R (output) DOUBLE PRECISION array, dimension (M)
61 * If INFO = 0 or INFO > M, R contains the row scale factors
62 * for A.
63 *
64 * C (output) DOUBLE PRECISION array, dimension (N)
65 * If INFO = 0, C contains the column scale factors for A.
66 *
67 * ROWCND (output) DOUBLE PRECISION
68 * If INFO = 0 or INFO > M, ROWCND contains the ratio of the
69 * smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
70 * AMAX is neither too large nor too small, it is not worth
71 * scaling by R.
72 *
73 * COLCND (output) DOUBLE PRECISION
74 * If INFO = 0, COLCND contains the ratio of the smallest
75 * C(i) to the largest C(i). If COLCND >= 0.1, it is not
76 * worth scaling by C.
77 *
78 * AMAX (output) DOUBLE PRECISION
79 * Absolute value of largest matrix element. If AMAX is very
80 * close to overflow or very close to underflow, the matrix
81 * should be scaled.
82 *
83 * INFO (output) INTEGER
84 * = 0: successful exit
85 * < 0: if INFO = -i, the i-th argument had an illegal value
86 * > 0: if INFO = i, and i is
87 * <= M: the i-th row of A is exactly zero
88 * > M: the (i-M)-th column of A is exactly zero
89 *
90 * =====================================================================
91 *
92 * .. Parameters ..
93 DOUBLE PRECISION ONE, ZERO
94 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
95 * ..
96 * .. Local Scalars ..
97 INTEGER I, J
98 DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
99 COMPLEX*16 ZDUM
100 * ..
101 * .. External Functions ..
102 DOUBLE PRECISION DLAMCH
103 EXTERNAL DLAMCH
104 * ..
105 * .. External Subroutines ..
106 EXTERNAL XERBLA
107 * ..
108 * .. Intrinsic Functions ..
109 INTRINSIC ABS, MAX, MIN, LOG, REAL, DIMAG
110 * ..
111 * .. Statement Functions ..
112 DOUBLE PRECISION CABS1
113 * ..
114 * .. Statement Function definitions ..
115 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
116 * ..
117 * .. Executable Statements ..
118 *
119 * Test the input parameters.
120 *
121 INFO = 0
122 IF( M.LT.0 ) THEN
123 INFO = -1
124 ELSE IF( N.LT.0 ) THEN
125 INFO = -2
126 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
127 INFO = -4
128 END IF
129 IF( INFO.NE.0 ) THEN
130 CALL XERBLA( 'ZGEEQUB', -INFO )
131 RETURN
132 END IF
133 *
134 * Quick return if possible.
135 *
136 IF( M.EQ.0 .OR. N.EQ.0 ) THEN
137 ROWCND = ONE
138 COLCND = ONE
139 AMAX = ZERO
140 RETURN
141 END IF
142 *
143 * Get machine constants. Assume SMLNUM is a power of the radix.
144 *
145 SMLNUM = DLAMCH( 'S' )
146 BIGNUM = ONE / SMLNUM
147 RADIX = DLAMCH( 'B' )
148 LOGRDX = LOG( RADIX )
149 *
150 * Compute row scale factors.
151 *
152 DO 10 I = 1, M
153 R( I ) = ZERO
154 10 CONTINUE
155 *
156 * Find the maximum element in each row.
157 *
158 DO 30 J = 1, N
159 DO 20 I = 1, M
160 R( I ) = MAX( R( I ), CABS1( A( I, J ) ) )
161 20 CONTINUE
162 30 CONTINUE
163 DO I = 1, M
164 IF( R( I ).GT.ZERO ) THEN
165 R( I ) = RADIX**INT( LOG(R( I ) ) / LOGRDX )
166 END IF
167 END DO
168 *
169 * Find the maximum and minimum scale factors.
170 *
171 RCMIN = BIGNUM
172 RCMAX = ZERO
173 DO 40 I = 1, M
174 RCMAX = MAX( RCMAX, R( I ) )
175 RCMIN = MIN( RCMIN, R( I ) )
176 40 CONTINUE
177 AMAX = RCMAX
178 *
179 IF( RCMIN.EQ.ZERO ) THEN
180 *
181 * Find the first zero scale factor and return an error code.
182 *
183 DO 50 I = 1, M
184 IF( R( I ).EQ.ZERO ) THEN
185 INFO = I
186 RETURN
187 END IF
188 50 CONTINUE
189 ELSE
190 *
191 * Invert the scale factors.
192 *
193 DO 60 I = 1, M
194 R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
195 60 CONTINUE
196 *
197 * Compute ROWCND = min(R(I)) / max(R(I)).
198 *
199 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
200 END IF
201 *
202 * Compute column scale factors.
203 *
204 DO 70 J = 1, N
205 C( J ) = ZERO
206 70 CONTINUE
207 *
208 * Find the maximum element in each column,
209 * assuming the row scaling computed above.
210 *
211 DO 90 J = 1, N
212 DO 80 I = 1, M
213 C( J ) = MAX( C( J ), CABS1( A( I, J ) )*R( I ) )
214 80 CONTINUE
215 IF( C( J ).GT.ZERO ) THEN
216 C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
217 END IF
218 90 CONTINUE
219 *
220 * Find the maximum and minimum scale factors.
221 *
222 RCMIN = BIGNUM
223 RCMAX = ZERO
224 DO 100 J = 1, N
225 RCMIN = MIN( RCMIN, C( J ) )
226 RCMAX = MAX( RCMAX, C( J ) )
227 100 CONTINUE
228 *
229 IF( RCMIN.EQ.ZERO ) THEN
230 *
231 * Find the first zero scale factor and return an error code.
232 *
233 DO 110 J = 1, N
234 IF( C( J ).EQ.ZERO ) THEN
235 INFO = M + J
236 RETURN
237 END IF
238 110 CONTINUE
239 ELSE
240 *
241 * Invert the scale factors.
242 *
243 DO 120 J = 1, N
244 C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
245 120 CONTINUE
246 *
247 * Compute COLCND = min(C(J)) / max(C(J)).
248 *
249 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
250 END IF
251 *
252 RETURN
253 *
254 * End of ZGEEQUB
255 *
256 END