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
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