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1.1 bertrand 1: SUBROUTINE DSYEQUB( 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: DOUBLE PRECISION A( LDA, * ), S( * ), WORK( * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * DSYEQUB computes row and column scalings intended to equilibrate a
26: * symmetric matrix A and reduce its condition number
27: * (with respect to the two-norm). S contains the scale factors,
28: * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
29: * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
30: * choice of S puts the condition number of B within a factor N of the
31: * smallest possible condition number over all possible diagonal
32: * scalings.
33: *
34: * Arguments
35: * =========
36: *
37: * UPLO (input) CHARACTER*1
38: * Specifies whether the details of the factorization are stored
39: * as an upper or lower triangular matrix.
40: * = 'U': Upper triangular, form is A = U*D*U**T;
41: * = 'L': Lower triangular, form is A = L*D*L**T.
42: *
43: * N (input) INTEGER
44: * The order of the matrix A. N >= 0.
45: *
46: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
47: * The N-by-N symmetric matrix whose scaling
48: * factors are to be computed. Only the diagonal elements of A
49: * are referenced.
50: *
51: * LDA (input) INTEGER
52: * The leading dimension of the array A. LDA >= max(1,N).
53: *
54: * S (output) DOUBLE PRECISION array, dimension (N)
55: * If INFO = 0, S contains the scale factors for A.
56: *
57: * SCOND (output) DOUBLE PRECISION
58: * If INFO = 0, S contains the ratio of the smallest S(i) to
59: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
60: * large nor too small, it is not worth scaling by S.
61: *
62: * AMAX (output) DOUBLE PRECISION
63: * Absolute value of largest matrix element. If AMAX is very
64: * close to overflow or very close to underflow, the matrix
65: * should be scaled.
66: *
67: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
68: *
69: * INFO (output) INTEGER
70: * = 0: successful exit
71: * < 0: if INFO = -i, the i-th argument had an illegal value
72: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
73: *
74: * Further Details
75: * ======= =======
76: *
77: * Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization",
78: * Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
79: * DOI 10.1023/B:NUMA.0000016606.32820.69
80: * Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf
81: *
82: * =====================================================================
83: *
84: * .. Parameters ..
85: DOUBLE PRECISION ONE, ZERO
86: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
87: INTEGER MAX_ITER
88: PARAMETER ( MAX_ITER = 100 )
89: * ..
90: * .. Local Scalars ..
91: INTEGER I, J, ITER
92: DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
93: $ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
94: LOGICAL UP
95: * ..
96: * .. External Functions ..
97: DOUBLE PRECISION DLAMCH
98: LOGICAL LSAME
99: EXTERNAL DLAMCH, LSAME
100: * ..
101: * .. External Subroutines ..
102: EXTERNAL DLASSQ
103: * ..
104: * .. Intrinsic Functions ..
105: INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT
106: * ..
107: * .. Executable Statements ..
108: *
109: * Test input parameters.
110: *
111: INFO = 0
112: IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
113: INFO = -1
114: ELSE IF ( N .LT. 0 ) THEN
115: INFO = -2
116: ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
117: INFO = -4
118: END IF
119: IF ( INFO .NE. 0 ) THEN
120: CALL XERBLA( 'DSYEQUB', -INFO )
121: RETURN
122: END IF
123:
124: UP = LSAME( UPLO, 'U' )
125: AMAX = ZERO
126: *
127: * Quick return if possible.
128: *
129: IF ( N .EQ. 0 ) THEN
130: SCOND = ONE
131: RETURN
132: END IF
133:
134: DO I = 1, N
135: S( I ) = ZERO
136: END DO
137:
138: AMAX = ZERO
139: IF ( UP ) THEN
140: DO J = 1, N
141: DO I = 1, J-1
142: S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
143: S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
144: AMAX = MAX( AMAX, ABS( A(I, J) ) )
145: END DO
146: S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
147: AMAX = MAX( AMAX, ABS( A( J, J ) ) )
148: END DO
149: ELSE
150: DO J = 1, N
151: S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
152: AMAX = MAX( AMAX, ABS( A( J, J ) ) )
153: DO I = J+1, N
154: S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
155: S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
156: AMAX = MAX( AMAX, ABS( A( I, J ) ) )
157: END DO
158: END DO
159: END IF
160: DO J = 1, N
161: S( J ) = 1.0D+0 / S( J )
162: END DO
163:
164: TOL = ONE / SQRT(2.0D0 * N)
165:
166: DO ITER = 1, MAX_ITER
167: SCALE = 0.0D+0
168: SUMSQ = 0.0D+0
169: * BETA = |A|S
170: DO I = 1, N
171: WORK(I) = ZERO
172: END DO
173: IF ( UP ) THEN
174: DO J = 1, N
175: DO I = 1, J-1
176: T = ABS( A( I, J ) )
177: WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
178: WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
179: END DO
180: WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
181: END DO
182: ELSE
183: DO J = 1, N
184: WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
185: DO I = J+1, N
186: T = ABS( A( I, J ) )
187: WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
188: WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
189: END DO
190: END DO
191: END IF
192:
193: * avg = s^T beta / n
194: AVG = 0.0D+0
195: DO I = 1, N
196: AVG = AVG + S( I )*WORK( I )
197: END DO
198: AVG = AVG / N
199:
200: STD = 0.0D+0
201: DO I = 2*N+1, 3*N
202: WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG
203: END DO
204: CALL DLASSQ( N, WORK( 2*N+1 ), 1, SCALE, SUMSQ )
205: STD = SCALE * SQRT( SUMSQ / N )
206:
207: IF ( STD .LT. TOL * AVG ) GOTO 999
208:
209: DO I = 1, N
210: T = ABS( A( I, I ) )
211: SI = S( I )
212: C2 = ( N-1 ) * T
213: C1 = ( N-2 ) * ( WORK( I ) - T*SI )
214: C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
215: D = C1*C1 - 4*C0*C2
216:
217: IF ( D .LE. 0 ) THEN
218: INFO = -1
219: RETURN
220: END IF
221: SI = -2*C0 / ( C1 + SQRT( D ) )
222:
223: D = SI - S( I )
224: U = ZERO
225: IF ( UP ) THEN
226: DO J = 1, I
227: T = ABS( A( J, I ) )
228: U = U + S( J )*T
229: WORK( J ) = WORK( J ) + D*T
230: END DO
231: DO J = I+1,N
232: T = ABS( A( I, J ) )
233: U = U + S( J )*T
234: WORK( J ) = WORK( J ) + D*T
235: END DO
236: ELSE
237: DO J = 1, I
238: T = ABS( A( I, J ) )
239: U = U + S( J )*T
240: WORK( J ) = WORK( J ) + D*T
241: END DO
242: DO J = I+1,N
243: T = ABS( A( J, I ) )
244: U = U + S( J )*T
245: WORK( J ) = WORK( J ) + D*T
246: END DO
247: END IF
248:
249: AVG = AVG + ( U + WORK( I ) ) * D / N
250: S( I ) = SI
251:
252: END DO
253:
254: END DO
255:
256: 999 CONTINUE
257:
258: SMLNUM = DLAMCH( 'SAFEMIN' )
259: BIGNUM = ONE / SMLNUM
260: SMIN = BIGNUM
261: SMAX = ZERO
262: T = ONE / SQRT(AVG)
263: BASE = DLAMCH( 'B' )
264: U = ONE / LOG( BASE )
265: DO I = 1, N
266: S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
267: SMIN = MIN( SMIN, S( I ) )
268: SMAX = MAX( SMAX, S( I ) )
269: END DO
270: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
271: *
272: END