1 SUBROUTINE ZSYEQUB( 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 * UPLO (input) CHARACTER*1
39 * Specifies whether the details of the factorization are stored
40 * as an upper or lower triangular matrix.
41 * = 'U': Upper triangular, form is A = U*D*U**T;
42 * = 'L': Lower triangular, form is A = L*D*L**T.
43 *
44 * N (input) INTEGER
45 * The order of the matrix A. N >= 0.
46 *
47 * A (input) COMPLEX*16 array, dimension (LDA,N)
48 * The N-by-N symmetric matrix whose scaling
49 * factors are to be computed. Only the diagonal elements of A
50 * are referenced.
51 *
52 * LDA (input) INTEGER
53 * The leading dimension of the array A. LDA >= max(1,N).
54 *
55 * S (output) DOUBLE PRECISION array, dimension (N)
56 * If INFO = 0, S contains the scale factors for A.
57 *
58 * SCOND (output) DOUBLE PRECISION
59 * If INFO = 0, S contains the ratio of the smallest S(i) to
60 * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
61 * large nor too small, it is not worth scaling by S.
62 *
63 * AMAX (output) DOUBLE PRECISION
64 * Absolute value of largest matrix element. If AMAX is very
65 * close to overflow or very close to underflow, the matrix
66 * should be scaled.
67 *
68 * WORK (workspace) COMPLEX*16 array, dimension (3*N)
69 *
70 * INFO (output) INTEGER
71 * = 0: successful exit
72 * < 0: if INFO = -i, the i-th argument had an illegal value
73 * > 0: if INFO = i, the i-th diagonal element is nonpositive.
74 *
75 * Further Details
76 * ======= =======
77 *
78 * Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization",
79 * Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
80 * DOI 10.1023/B:NUMA.0000016606.32820.69
81 * Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf
82 *
83 * =====================================================================
84 *
85 * .. Parameters ..
86 DOUBLE PRECISION ONE, ZERO
87 PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
88 INTEGER MAX_ITER
89 PARAMETER ( MAX_ITER = 100 )
90 * ..
91 * .. Local Scalars ..
92 INTEGER I, J, ITER
93 DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
94 $ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
95 LOGICAL UP
96 COMPLEX*16 ZDUM
97 * ..
98 * .. External Functions ..
99 DOUBLE PRECISION DLAMCH
100 LOGICAL LSAME
101 EXTERNAL DLAMCH, LSAME
102 * ..
103 * .. External Subroutines ..
104 EXTERNAL ZLASSQ
105 * ..
106 * .. Intrinsic Functions ..
107 INTRINSIC ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT
108 * ..
109 * .. Statement Functions ..
110 DOUBLE PRECISION CABS1
111 * ..
112 * Statement Function Definitions
113 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
114 * ..
115 * .. Executable Statements ..
116 *
117 * Test the input parameters.
118 *
119 INFO = 0
120 IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
121 INFO = -1
122 ELSE IF ( N .LT. 0 ) THEN
123 INFO = -2
124 ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
125 INFO = -4
126 END IF
127 IF ( INFO .NE. 0 ) THEN
128 CALL XERBLA( 'ZSYEQUB', -INFO )
129 RETURN
130 END IF
131
132 UP = LSAME( UPLO, 'U' )
133 AMAX = ZERO
134 *
135 * Quick return if possible.
136 *
137 IF ( N .EQ. 0 ) THEN
138 SCOND = ONE
139 RETURN
140 END IF
141
142 DO I = 1, N
143 S( I ) = ZERO
144 END DO
145
146 AMAX = ZERO
147 IF ( UP ) THEN
148 DO J = 1, N
149 DO I = 1, J-1
150 S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
151 S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
152 AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
153 END DO
154 S( J ) = MAX( S( J ), CABS1( A( J, J) ) )
155 AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
156 END DO
157 ELSE
158 DO J = 1, N
159 S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
160 AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
161 DO I = J+1, N
162 S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
163 S( J ) = MAX( S( J ), CABS1 (A( I, J ) ) )
164 AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
165 END DO
166 END DO
167 END IF
168 DO J = 1, N
169 S( J ) = 1.0D+0 / S( J )
170 END DO
171
172 TOL = ONE / SQRT( 2.0D0 * N )
173
174 DO ITER = 1, MAX_ITER
175 SCALE = 0.0D+0
176 SUMSQ = 0.0D+0
177 * beta = |A|s
178 DO I = 1, N
179 WORK( I ) = ZERO
180 END DO
181 IF ( UP ) THEN
182 DO J = 1, N
183 DO I = 1, J-1
184 T = CABS1( A( I, J ) )
185 WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
186 WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
187 END DO
188 WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
189 END DO
190 ELSE
191 DO J = 1, N
192 WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
193 DO I = J+1, N
194 T = CABS1( A( I, J ) )
195 WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
196 WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
197 END DO
198 END DO
199 END IF
200
201 * avg = s^T beta / n
202 AVG = 0.0D+0
203 DO I = 1, N
204 AVG = AVG + S( I )*WORK( I )
205 END DO
206 AVG = AVG / N
207
208 STD = 0.0D+0
209 DO I = N+1, 2*N
210 WORK( I ) = S( I-N ) * WORK( I-N ) - AVG
211 END DO
212 CALL ZLASSQ( N, WORK( N+1 ), 1, SCALE, SUMSQ )
213 STD = SCALE * SQRT( SUMSQ / N )
214
215 IF ( STD .LT. TOL * AVG ) GOTO 999
216
217 DO I = 1, N
218 T = CABS1( A( I, I ) )
219 SI = S( I )
220 C2 = ( N-1 ) * T
221 C1 = ( N-2 ) * ( WORK( I ) - T*SI )
222 C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
223 D = C1*C1 - 4*C0*C2
224
225 IF ( D .LE. 0 ) THEN
226 INFO = -1
227 RETURN
228 END IF
229 SI = -2*C0 / ( C1 + SQRT( D ) )
230
231 D = SI - S( I )
232 U = ZERO
233 IF ( UP ) THEN
234 DO J = 1, I
235 T = CABS1( A( J, I ) )
236 U = U + S( J )*T
237 WORK( J ) = WORK( J ) + D*T
238 END DO
239 DO J = I+1,N
240 T = CABS1( A( I, J ) )
241 U = U + S( J )*T
242 WORK( J ) = WORK( J ) + D*T
243 END DO
244 ELSE
245 DO J = 1, I
246 T = CABS1( A( I, J ) )
247 U = U + S( J )*T
248 WORK( J ) = WORK( J ) + D*T
249 END DO
250 DO J = I+1,N
251 T = CABS1( A( J, I ) )
252 U = U + S( J )*T
253 WORK( J ) = WORK( J ) + D*T
254 END DO
255 END IF
256 AVG = AVG + ( U + WORK( I ) ) * D / N
257 S( I ) = SI
258 END DO
259 END DO
260
261 999 CONTINUE
262
263 SMLNUM = DLAMCH( 'SAFEMIN' )
264 BIGNUM = ONE / SMLNUM
265 SMIN = BIGNUM
266 SMAX = ZERO
267 T = ONE / SQRT( AVG )
268 BASE = DLAMCH( 'B' )
269 U = ONE / LOG( BASE )
270 DO I = 1, N
271 S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
272 SMIN = MIN( SMIN, S( I ) )
273 SMAX = MAX( SMAX, S( I ) )
274 END DO
275 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
276 *
277 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 * UPLO (input) CHARACTER*1
39 * Specifies whether the details of the factorization are stored
40 * as an upper or lower triangular matrix.
41 * = 'U': Upper triangular, form is A = U*D*U**T;
42 * = 'L': Lower triangular, form is A = L*D*L**T.
43 *
44 * N (input) INTEGER
45 * The order of the matrix A. N >= 0.
46 *
47 * A (input) COMPLEX*16 array, dimension (LDA,N)
48 * The N-by-N symmetric matrix whose scaling
49 * factors are to be computed. Only the diagonal elements of A
50 * are referenced.
51 *
52 * LDA (input) INTEGER
53 * The leading dimension of the array A. LDA >= max(1,N).
54 *
55 * S (output) DOUBLE PRECISION array, dimension (N)
56 * If INFO = 0, S contains the scale factors for A.
57 *
58 * SCOND (output) DOUBLE PRECISION
59 * If INFO = 0, S contains the ratio of the smallest S(i) to
60 * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
61 * large nor too small, it is not worth scaling by S.
62 *
63 * AMAX (output) DOUBLE PRECISION
64 * Absolute value of largest matrix element. If AMAX is very
65 * close to overflow or very close to underflow, the matrix
66 * should be scaled.
67 *
68 * WORK (workspace) COMPLEX*16 array, dimension (3*N)
69 *
70 * INFO (output) INTEGER
71 * = 0: successful exit
72 * < 0: if INFO = -i, the i-th argument had an illegal value
73 * > 0: if INFO = i, the i-th diagonal element is nonpositive.
74 *
75 * Further Details
76 * ======= =======
77 *
78 * Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization",
79 * Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
80 * DOI 10.1023/B:NUMA.0000016606.32820.69
81 * Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf
82 *
83 * =====================================================================
84 *
85 * .. Parameters ..
86 DOUBLE PRECISION ONE, ZERO
87 PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
88 INTEGER MAX_ITER
89 PARAMETER ( MAX_ITER = 100 )
90 * ..
91 * .. Local Scalars ..
92 INTEGER I, J, ITER
93 DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
94 $ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
95 LOGICAL UP
96 COMPLEX*16 ZDUM
97 * ..
98 * .. External Functions ..
99 DOUBLE PRECISION DLAMCH
100 LOGICAL LSAME
101 EXTERNAL DLAMCH, LSAME
102 * ..
103 * .. External Subroutines ..
104 EXTERNAL ZLASSQ
105 * ..
106 * .. Intrinsic Functions ..
107 INTRINSIC ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT
108 * ..
109 * .. Statement Functions ..
110 DOUBLE PRECISION CABS1
111 * ..
112 * Statement Function Definitions
113 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
114 * ..
115 * .. Executable Statements ..
116 *
117 * Test the input parameters.
118 *
119 INFO = 0
120 IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
121 INFO = -1
122 ELSE IF ( N .LT. 0 ) THEN
123 INFO = -2
124 ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
125 INFO = -4
126 END IF
127 IF ( INFO .NE. 0 ) THEN
128 CALL XERBLA( 'ZSYEQUB', -INFO )
129 RETURN
130 END IF
131
132 UP = LSAME( UPLO, 'U' )
133 AMAX = ZERO
134 *
135 * Quick return if possible.
136 *
137 IF ( N .EQ. 0 ) THEN
138 SCOND = ONE
139 RETURN
140 END IF
141
142 DO I = 1, N
143 S( I ) = ZERO
144 END DO
145
146 AMAX = ZERO
147 IF ( UP ) THEN
148 DO J = 1, N
149 DO I = 1, J-1
150 S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
151 S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
152 AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
153 END DO
154 S( J ) = MAX( S( J ), CABS1( A( J, J) ) )
155 AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
156 END DO
157 ELSE
158 DO J = 1, N
159 S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
160 AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
161 DO I = J+1, N
162 S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
163 S( J ) = MAX( S( J ), CABS1 (A( I, J ) ) )
164 AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
165 END DO
166 END DO
167 END IF
168 DO J = 1, N
169 S( J ) = 1.0D+0 / S( J )
170 END DO
171
172 TOL = ONE / SQRT( 2.0D0 * N )
173
174 DO ITER = 1, MAX_ITER
175 SCALE = 0.0D+0
176 SUMSQ = 0.0D+0
177 * beta = |A|s
178 DO I = 1, N
179 WORK( I ) = ZERO
180 END DO
181 IF ( UP ) THEN
182 DO J = 1, N
183 DO I = 1, J-1
184 T = CABS1( A( I, J ) )
185 WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
186 WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
187 END DO
188 WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
189 END DO
190 ELSE
191 DO J = 1, N
192 WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
193 DO I = J+1, N
194 T = CABS1( A( I, J ) )
195 WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
196 WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
197 END DO
198 END DO
199 END IF
200
201 * avg = s^T beta / n
202 AVG = 0.0D+0
203 DO I = 1, N
204 AVG = AVG + S( I )*WORK( I )
205 END DO
206 AVG = AVG / N
207
208 STD = 0.0D+0
209 DO I = N+1, 2*N
210 WORK( I ) = S( I-N ) * WORK( I-N ) - AVG
211 END DO
212 CALL ZLASSQ( N, WORK( N+1 ), 1, SCALE, SUMSQ )
213 STD = SCALE * SQRT( SUMSQ / N )
214
215 IF ( STD .LT. TOL * AVG ) GOTO 999
216
217 DO I = 1, N
218 T = CABS1( A( I, I ) )
219 SI = S( I )
220 C2 = ( N-1 ) * T
221 C1 = ( N-2 ) * ( WORK( I ) - T*SI )
222 C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
223 D = C1*C1 - 4*C0*C2
224
225 IF ( D .LE. 0 ) THEN
226 INFO = -1
227 RETURN
228 END IF
229 SI = -2*C0 / ( C1 + SQRT( D ) )
230
231 D = SI - S( I )
232 U = ZERO
233 IF ( UP ) THEN
234 DO J = 1, I
235 T = CABS1( A( J, I ) )
236 U = U + S( J )*T
237 WORK( J ) = WORK( J ) + D*T
238 END DO
239 DO J = I+1,N
240 T = CABS1( A( I, J ) )
241 U = U + S( J )*T
242 WORK( J ) = WORK( J ) + D*T
243 END DO
244 ELSE
245 DO J = 1, I
246 T = CABS1( A( I, J ) )
247 U = U + S( J )*T
248 WORK( J ) = WORK( J ) + D*T
249 END DO
250 DO J = I+1,N
251 T = CABS1( A( J, I ) )
252 U = U + S( J )*T
253 WORK( J ) = WORK( J ) + D*T
254 END DO
255 END IF
256 AVG = AVG + ( U + WORK( I ) ) * D / N
257 S( I ) = SI
258 END DO
259 END DO
260
261 999 CONTINUE
262
263 SMLNUM = DLAMCH( 'SAFEMIN' )
264 BIGNUM = ONE / SMLNUM
265 SMIN = BIGNUM
266 SMAX = ZERO
267 T = ONE / SQRT( AVG )
268 BASE = DLAMCH( 'B' )
269 U = ONE / LOG( BASE )
270 DO I = 1, N
271 S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
272 SMIN = MIN( SMIN, S( I ) )
273 SMAX = MAX( SMAX, S( I ) )
274 END DO
275 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
276 *
277 END