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1.1 bertrand 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