1 SUBROUTINE ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
2 *
3 * -- LAPACK routine (version 3.2.2) --
4 * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
5 * -- Jason Riedy of Univ. of California Berkeley. --
6 * -- June 2010 --
7 *
8 * -- LAPACK is a software package provided by Univ. of Tennessee, --
9 * -- Univ. of California Berkeley and NAG Ltd. --
10 *
11 IMPLICIT NONE
12 * ..
13 * .. Scalar Arguments ..
14 INTEGER INFO, LDA, N
15 DOUBLE PRECISION AMAX, SCOND
16 CHARACTER UPLO
17 * ..
18 * .. Array Arguments ..
19 COMPLEX*16 A( LDA, * ), WORK( * )
20 DOUBLE PRECISION S( * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * ZSYEQUB computes row and column scalings intended to equilibrate a
27 * symmetric matrix A and reduce its condition number
28 * (with respect to the two-norm). S contains the scale factors,
29 * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
30 * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
31 * choice of S puts the condition number of B within a factor N of the
32 * smallest possible condition number over all possible diagonal
33 * scalings.
34 *
35 * Arguments
36 * =========
37 *
38 * N (input) INTEGER
39 * The order of the matrix A. N >= 0.
40 *
41 * A (input) COMPLEX*16 array, dimension (LDA,N)
42 * The N-by-N symmetric matrix whose scaling
43 * factors are to be computed. Only the diagonal elements of A
44 * are referenced.
45 *
46 * LDA (input) INTEGER
47 * The leading dimension of the array A. LDA >= max(1,N).
48 *
49 * S (output) DOUBLE PRECISION array, dimension (N)
50 * If INFO = 0, S contains the scale factors for A.
51 *
52 * SCOND (output) DOUBLE PRECISION
53 * If INFO = 0, S contains the ratio of the smallest S(i) to
54 * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
55 * large nor too small, it is not worth scaling by S.
56 *
57 * AMAX (output) DOUBLE PRECISION
58 * Absolute value of largest matrix element. If AMAX is very
59 * close to overflow or very close to underflow, the matrix
60 * should be scaled.
61 * INFO (output) INTEGER
62 * = 0: successful exit
63 * < 0: if INFO = -i, the i-th argument had an illegal value
64 * > 0: if INFO = i, the i-th diagonal element is nonpositive.
65 *
66 * =====================================================================
67 *
68 * .. Parameters ..
69 DOUBLE PRECISION ONE, ZERO
70 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
71 INTEGER MAX_ITER
72 PARAMETER ( MAX_ITER = 100 )
73 * ..
74 * .. Local Scalars ..
75 INTEGER I, J, ITER
76 DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D,
77 $ BASE, SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
78 LOGICAL UP
79 COMPLEX*16 ZDUM
80 * ..
81 * .. External Functions ..
82 DOUBLE PRECISION DLAMCH
83 LOGICAL LSAME
84 EXTERNAL DLAMCH, LSAME
85 * ..
86 * .. External Subroutines ..
87 EXTERNAL ZLASSQ
88 * ..
89 * .. Intrinsic Functions ..
90 INTRINSIC ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT
91 * ..
92 * .. Statement Functions ..
93 DOUBLE PRECISION CABS1
94 * ..
95 * .. Statement Function Definitions ..
96 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
97 *
98 * Test input parameters.
99 *
100 INFO = 0
101 IF (.NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
102 INFO = -1
103 ELSE IF ( N .LT. 0 ) THEN
104 INFO = -2
105 ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
106 INFO = -4
107 END IF
108 IF ( INFO .NE. 0 ) THEN
109 CALL XERBLA( 'ZHEEQUB', -INFO )
110 RETURN
111 END IF
112
113 UP = LSAME( UPLO, 'U' )
114 AMAX = ZERO
115 *
116 * Quick return if possible.
117 *
118 IF ( N .EQ. 0 ) THEN
119 SCOND = ONE
120 RETURN
121 END IF
122
123 DO I = 1, N
124 S( I ) = ZERO
125 END DO
126
127 AMAX = ZERO
128 IF ( UP ) THEN
129 DO J = 1, N
130 DO I = 1, J-1
131 S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
132 S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
133 AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
134 END DO
135 S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
136 AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
137 END DO
138 ELSE
139 DO J = 1, N
140 S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
141 AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
142 DO I = J+1, N
143 S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
144 S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
145 AMAX = MAX( AMAX, CABS1( A(I, J ) ) )
146 END DO
147 END DO
148 END IF
149 DO J = 1, N
150 S( J ) = 1.0D+0 / S( J )
151 END DO
152
153 TOL = ONE / SQRT( 2.0D0 * N )
154
155 DO ITER = 1, MAX_ITER
156 SCALE = 0.0D+0
157 SUMSQ = 0.0D+0
158 * beta = |A|s
159 DO I = 1, N
160 WORK( I ) = ZERO
161 END DO
162 IF ( UP ) THEN
163 DO J = 1, N
164 DO I = 1, J-1
165 T = CABS1( A( I, J ) )
166 WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
167 WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
168 END DO
169 WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
170 END DO
171 ELSE
172 DO J = 1, N
173 WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
174 DO I = J+1, N
175 T = CABS1( A( I, J ) )
176 WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
177 WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
178 END DO
179 END DO
180 END IF
181
182 * avg = s^T beta / n
183 AVG = 0.0D+0
184 DO I = 1, N
185 AVG = AVG + S( I )*WORK( I )
186 END DO
187 AVG = AVG / N
188
189 STD = 0.0D+0
190 DO I = 2*N+1, 3*N
191 WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG
192 END DO
193 CALL ZLASSQ( N, WORK( 2*N+1 ), 1, SCALE, SUMSQ )
194 STD = SCALE * SQRT( SUMSQ / N )
195
196 IF ( STD .LT. TOL * AVG ) GOTO 999
197
198 DO I = 1, N
199 T = CABS1( A( I, I ) )
200 SI = S( I )
201 C2 = ( N-1 ) * T
202 C1 = ( N-2 ) * ( WORK( I ) - T*SI )
203 C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
204
205 D = C1*C1 - 4*C0*C2
206 IF ( D .LE. 0 ) THEN
207 INFO = -1
208 RETURN
209 END IF
210 SI = -2*C0 / ( C1 + SQRT( D ) )
211
212 D = SI - S(I)
213 U = ZERO
214 IF ( UP ) THEN
215 DO J = 1, I
216 T = CABS1( A( J, I ) )
217 U = U + S( J )*T
218 WORK( J ) = WORK( J ) + D*T
219 END DO
220 DO J = I+1,N
221 T = CABS1( A( I, J ) )
222 U = U + S( J )*T
223 WORK( J ) = WORK( J ) + D*T
224 END DO
225 ELSE
226 DO J = 1, I
227 T = CABS1( A( I, J ) )
228 U = U + S( J )*T
229 WORK( J ) = WORK( J ) + D*T
230 END DO
231 DO J = I+1,N
232 T = CABS1( A( J, I ) )
233 U = U + S( J )*T
234 WORK( J ) = WORK( J ) + D*T
235 END DO
236 END IF
237 AVG = AVG + ( U + WORK( I ) ) * D / N
238 S( I ) = SI
239 END DO
240
241 END DO
242
243 999 CONTINUE
244
245 SMLNUM = DLAMCH( 'SAFEMIN' )
246 BIGNUM = ONE / SMLNUM
247 SMIN = BIGNUM
248 SMAX = ZERO
249 T = ONE / SQRT( AVG )
250 BASE = DLAMCH( 'B' )
251 U = ONE / LOG( BASE )
252 DO I = 1, N
253 S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
254 SMIN = MIN( SMIN, S( I ) )
255 SMAX = MAX( SMAX, S( I ) )
256 END DO
257 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
258
259 END
2 *
3 * -- LAPACK routine (version 3.2.2) --
4 * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
5 * -- Jason Riedy of Univ. of California Berkeley. --
6 * -- June 2010 --
7 *
8 * -- LAPACK is a software package provided by Univ. of Tennessee, --
9 * -- Univ. of California Berkeley and NAG Ltd. --
10 *
11 IMPLICIT NONE
12 * ..
13 * .. Scalar Arguments ..
14 INTEGER INFO, LDA, N
15 DOUBLE PRECISION AMAX, SCOND
16 CHARACTER UPLO
17 * ..
18 * .. Array Arguments ..
19 COMPLEX*16 A( LDA, * ), WORK( * )
20 DOUBLE PRECISION S( * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * ZSYEQUB computes row and column scalings intended to equilibrate a
27 * symmetric matrix A and reduce its condition number
28 * (with respect to the two-norm). S contains the scale factors,
29 * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
30 * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
31 * choice of S puts the condition number of B within a factor N of the
32 * smallest possible condition number over all possible diagonal
33 * scalings.
34 *
35 * Arguments
36 * =========
37 *
38 * N (input) INTEGER
39 * The order of the matrix A. N >= 0.
40 *
41 * A (input) COMPLEX*16 array, dimension (LDA,N)
42 * The N-by-N symmetric matrix whose scaling
43 * factors are to be computed. Only the diagonal elements of A
44 * are referenced.
45 *
46 * LDA (input) INTEGER
47 * The leading dimension of the array A. LDA >= max(1,N).
48 *
49 * S (output) DOUBLE PRECISION array, dimension (N)
50 * If INFO = 0, S contains the scale factors for A.
51 *
52 * SCOND (output) DOUBLE PRECISION
53 * If INFO = 0, S contains the ratio of the smallest S(i) to
54 * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
55 * large nor too small, it is not worth scaling by S.
56 *
57 * AMAX (output) DOUBLE PRECISION
58 * Absolute value of largest matrix element. If AMAX is very
59 * close to overflow or very close to underflow, the matrix
60 * should be scaled.
61 * INFO (output) INTEGER
62 * = 0: successful exit
63 * < 0: if INFO = -i, the i-th argument had an illegal value
64 * > 0: if INFO = i, the i-th diagonal element is nonpositive.
65 *
66 * =====================================================================
67 *
68 * .. Parameters ..
69 DOUBLE PRECISION ONE, ZERO
70 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
71 INTEGER MAX_ITER
72 PARAMETER ( MAX_ITER = 100 )
73 * ..
74 * .. Local Scalars ..
75 INTEGER I, J, ITER
76 DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D,
77 $ BASE, SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
78 LOGICAL UP
79 COMPLEX*16 ZDUM
80 * ..
81 * .. External Functions ..
82 DOUBLE PRECISION DLAMCH
83 LOGICAL LSAME
84 EXTERNAL DLAMCH, LSAME
85 * ..
86 * .. External Subroutines ..
87 EXTERNAL ZLASSQ
88 * ..
89 * .. Intrinsic Functions ..
90 INTRINSIC ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT
91 * ..
92 * .. Statement Functions ..
93 DOUBLE PRECISION CABS1
94 * ..
95 * .. Statement Function Definitions ..
96 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
97 *
98 * Test input parameters.
99 *
100 INFO = 0
101 IF (.NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
102 INFO = -1
103 ELSE IF ( N .LT. 0 ) THEN
104 INFO = -2
105 ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
106 INFO = -4
107 END IF
108 IF ( INFO .NE. 0 ) THEN
109 CALL XERBLA( 'ZHEEQUB', -INFO )
110 RETURN
111 END IF
112
113 UP = LSAME( UPLO, 'U' )
114 AMAX = ZERO
115 *
116 * Quick return if possible.
117 *
118 IF ( N .EQ. 0 ) THEN
119 SCOND = ONE
120 RETURN
121 END IF
122
123 DO I = 1, N
124 S( I ) = ZERO
125 END DO
126
127 AMAX = ZERO
128 IF ( UP ) THEN
129 DO J = 1, N
130 DO I = 1, J-1
131 S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
132 S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
133 AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
134 END DO
135 S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
136 AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
137 END DO
138 ELSE
139 DO J = 1, N
140 S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
141 AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
142 DO I = J+1, N
143 S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
144 S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
145 AMAX = MAX( AMAX, CABS1( A(I, J ) ) )
146 END DO
147 END DO
148 END IF
149 DO J = 1, N
150 S( J ) = 1.0D+0 / S( J )
151 END DO
152
153 TOL = ONE / SQRT( 2.0D0 * N )
154
155 DO ITER = 1, MAX_ITER
156 SCALE = 0.0D+0
157 SUMSQ = 0.0D+0
158 * beta = |A|s
159 DO I = 1, N
160 WORK( I ) = ZERO
161 END DO
162 IF ( UP ) THEN
163 DO J = 1, N
164 DO I = 1, J-1
165 T = CABS1( A( I, J ) )
166 WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
167 WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
168 END DO
169 WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
170 END DO
171 ELSE
172 DO J = 1, N
173 WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
174 DO I = J+1, N
175 T = CABS1( A( I, J ) )
176 WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
177 WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
178 END DO
179 END DO
180 END IF
181
182 * avg = s^T beta / n
183 AVG = 0.0D+0
184 DO I = 1, N
185 AVG = AVG + S( I )*WORK( I )
186 END DO
187 AVG = AVG / N
188
189 STD = 0.0D+0
190 DO I = 2*N+1, 3*N
191 WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG
192 END DO
193 CALL ZLASSQ( N, WORK( 2*N+1 ), 1, SCALE, SUMSQ )
194 STD = SCALE * SQRT( SUMSQ / N )
195
196 IF ( STD .LT. TOL * AVG ) GOTO 999
197
198 DO I = 1, N
199 T = CABS1( A( I, I ) )
200 SI = S( I )
201 C2 = ( N-1 ) * T
202 C1 = ( N-2 ) * ( WORK( I ) - T*SI )
203 C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
204
205 D = C1*C1 - 4*C0*C2
206 IF ( D .LE. 0 ) THEN
207 INFO = -1
208 RETURN
209 END IF
210 SI = -2*C0 / ( C1 + SQRT( D ) )
211
212 D = SI - S(I)
213 U = ZERO
214 IF ( UP ) THEN
215 DO J = 1, I
216 T = CABS1( A( J, I ) )
217 U = U + S( J )*T
218 WORK( J ) = WORK( J ) + D*T
219 END DO
220 DO J = I+1,N
221 T = CABS1( A( I, J ) )
222 U = U + S( J )*T
223 WORK( J ) = WORK( J ) + D*T
224 END DO
225 ELSE
226 DO J = 1, I
227 T = CABS1( A( I, J ) )
228 U = U + S( J )*T
229 WORK( J ) = WORK( J ) + D*T
230 END DO
231 DO J = I+1,N
232 T = CABS1( A( J, I ) )
233 U = U + S( J )*T
234 WORK( J ) = WORK( J ) + D*T
235 END DO
236 END IF
237 AVG = AVG + ( U + WORK( I ) ) * D / N
238 S( I ) = SI
239 END DO
240
241 END DO
242
243 999 CONTINUE
244
245 SMLNUM = DLAMCH( 'SAFEMIN' )
246 BIGNUM = ONE / SMLNUM
247 SMIN = BIGNUM
248 SMAX = ZERO
249 T = ONE / SQRT( AVG )
250 BASE = DLAMCH( 'B' )
251 U = ONE / LOG( BASE )
252 DO I = 1, N
253 S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
254 SMIN = MIN( SMIN, S( I ) )
255 SMAX = MAX( SMAX, S( I ) )
256 END DO
257 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
258
259 END