1: *> \brief \b DSYEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSYEQUB + dependencies
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, N
25: * DOUBLE PRECISION AMAX, SCOND
26: * CHARACTER UPLO
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), S( * ), WORK( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DSYEQUB computes row and column scalings intended to equilibrate a
39: *> symmetric matrix A (with respect to the Euclidean norm) and reduce
40: *> its condition number. The scale factors S are computed by the BIN
41: *> algorithm (see references) so that the scaled matrix B with elements
42: *> B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
43: *> the smallest possible condition number over all possible diagonal
44: *> scalings.
45: *> \endverbatim
46: *
47: * Arguments:
48: * ==========
49: *
50: *> \param[in] UPLO
51: *> \verbatim
52: *> UPLO is CHARACTER*1
53: *> = 'U': Upper triangle of A is stored;
54: *> = 'L': Lower triangle of A is stored.
55: *> \endverbatim
56: *>
57: *> \param[in] N
58: *> \verbatim
59: *> N is INTEGER
60: *> The order of the matrix A. N >= 0.
61: *> \endverbatim
62: *>
63: *> \param[in] A
64: *> \verbatim
65: *> A is DOUBLE PRECISION array, dimension (LDA,N)
66: *> The N-by-N symmetric matrix whose scaling factors are to be
67: *> computed.
68: *> \endverbatim
69: *>
70: *> \param[in] LDA
71: *> \verbatim
72: *> LDA is INTEGER
73: *> The leading dimension of the array A. LDA >= max(1,N).
74: *> \endverbatim
75: *>
76: *> \param[out] S
77: *> \verbatim
78: *> S is DOUBLE PRECISION array, dimension (N)
79: *> If INFO = 0, S contains the scale factors for A.
80: *> \endverbatim
81: *>
82: *> \param[out] SCOND
83: *> \verbatim
84: *> SCOND is DOUBLE PRECISION
85: *> If INFO = 0, S contains the ratio of the smallest S(i) to
86: *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
87: *> large nor too small, it is not worth scaling by S.
88: *> \endverbatim
89: *>
90: *> \param[out] AMAX
91: *> \verbatim
92: *> AMAX is DOUBLE PRECISION
93: *> Largest absolute value of any matrix element. If AMAX is
94: *> very close to overflow or very close to underflow, the
95: *> matrix should be scaled.
96: *> \endverbatim
97: *>
98: *> \param[out] WORK
99: *> \verbatim
100: *> WORK is DOUBLE PRECISION array, dimension (2*N)
101: *> \endverbatim
102: *>
103: *> \param[out] INFO
104: *> \verbatim
105: *> INFO is INTEGER
106: *> = 0: successful exit
107: *> < 0: if INFO = -i, the i-th argument had an illegal value
108: *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
109: *> \endverbatim
110: *
111: * Authors:
112: * ========
113: *
114: *> \author Univ. of Tennessee
115: *> \author Univ. of California Berkeley
116: *> \author Univ. of Colorado Denver
117: *> \author NAG Ltd.
118: *
119: *> \date December 2016
120: *
121: *> \ingroup doubleSYcomputational
122: *
123: *> \par References:
124: * ================
125: *>
126: *> Livne, O.E. and Golub, G.H., "Scaling by Binormalization", \n
127: *> Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. \n
128: *> DOI 10.1023/B:NUMA.0000016606.32820.69 \n
129: *> Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679
130: *>
131: * =====================================================================
132: SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
133: *
134: * -- LAPACK computational routine (version 3.7.0) --
135: * -- LAPACK is a software package provided by Univ. of Tennessee, --
136: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
137: * December 2016
138: *
139: * .. Scalar Arguments ..
140: INTEGER INFO, LDA, N
141: DOUBLE PRECISION AMAX, SCOND
142: CHARACTER UPLO
143: * ..
144: * .. Array Arguments ..
145: DOUBLE PRECISION A( LDA, * ), S( * ), WORK( * )
146: * ..
147: *
148: * =====================================================================
149: *
150: * .. Parameters ..
151: DOUBLE PRECISION ONE, ZERO
152: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
153: INTEGER MAX_ITER
154: PARAMETER ( MAX_ITER = 100 )
155: * ..
156: * .. Local Scalars ..
157: INTEGER I, J, ITER
158: DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
159: $ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
160: LOGICAL UP
161: * ..
162: * .. External Functions ..
163: DOUBLE PRECISION DLAMCH
164: LOGICAL LSAME
165: EXTERNAL DLAMCH, LSAME
166: * ..
167: * .. External Subroutines ..
168: EXTERNAL DLASSQ
169: * ..
170: * .. Intrinsic Functions ..
171: INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT
172: * ..
173: * .. Executable Statements ..
174: *
175: * Test the input parameters.
176: *
177: INFO = 0
178: IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
179: INFO = -1
180: ELSE IF ( N .LT. 0 ) THEN
181: INFO = -2
182: ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
183: INFO = -4
184: END IF
185: IF ( INFO .NE. 0 ) THEN
186: CALL XERBLA( 'DSYEQUB', -INFO )
187: RETURN
188: END IF
189:
190: UP = LSAME( UPLO, 'U' )
191: AMAX = ZERO
192: *
193: * Quick return if possible.
194: *
195: IF ( N .EQ. 0 ) THEN
196: SCOND = ONE
197: RETURN
198: END IF
199:
200: DO I = 1, N
201: S( I ) = ZERO
202: END DO
203:
204: AMAX = ZERO
205: IF ( UP ) THEN
206: DO J = 1, N
207: DO I = 1, J-1
208: S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
209: S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
210: AMAX = MAX( AMAX, ABS( A( I, J ) ) )
211: END DO
212: S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
213: AMAX = MAX( AMAX, ABS( A( J, J ) ) )
214: END DO
215: ELSE
216: DO J = 1, N
217: S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
218: AMAX = MAX( AMAX, ABS( A( J, J ) ) )
219: DO I = J+1, N
220: S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
221: S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
222: AMAX = MAX( AMAX, ABS( A( I, J ) ) )
223: END DO
224: END DO
225: END IF
226: DO J = 1, N
227: S( J ) = 1.0D0 / S( J )
228: END DO
229:
230: TOL = ONE / SQRT( 2.0D0 * N )
231:
232: DO ITER = 1, MAX_ITER
233: SCALE = 0.0D0
234: SUMSQ = 0.0D0
235: * beta = |A|s
236: DO I = 1, N
237: WORK( I ) = ZERO
238: END DO
239: IF ( UP ) THEN
240: DO J = 1, N
241: DO I = 1, J-1
242: WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
243: WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
244: END DO
245: WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
246: END DO
247: ELSE
248: DO J = 1, N
249: WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
250: DO I = J+1, N
251: WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
252: WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
253: END DO
254: END DO
255: END IF
256:
257: * avg = s^T beta / n
258: AVG = 0.0D0
259: DO I = 1, N
260: AVG = AVG + S( I )*WORK( I )
261: END DO
262: AVG = AVG / N
263:
264: STD = 0.0D0
265: DO I = N+1, 2*N
266: WORK( I ) = S( I-N ) * WORK( I-N ) - AVG
267: END DO
268: CALL DLASSQ( N, WORK( N+1 ), 1, SCALE, SUMSQ )
269: STD = SCALE * SQRT( SUMSQ / N )
270:
271: IF ( STD .LT. TOL * AVG ) GOTO 999
272:
273: DO I = 1, N
274: T = ABS( A( I, I ) )
275: SI = S( I )
276: C2 = ( N-1 ) * T
277: C1 = ( N-2 ) * ( WORK( I ) - T*SI )
278: C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
279: D = C1*C1 - 4*C0*C2
280:
281: IF ( D .LE. 0 ) THEN
282: INFO = -1
283: RETURN
284: END IF
285: SI = -2*C0 / ( C1 + SQRT( D ) )
286:
287: D = SI - S( I )
288: U = ZERO
289: IF ( UP ) THEN
290: DO J = 1, I
291: T = ABS( A( J, I ) )
292: U = U + S( J )*T
293: WORK( J ) = WORK( J ) + D*T
294: END DO
295: DO J = I+1,N
296: T = ABS( A( I, J ) )
297: U = U + S( J )*T
298: WORK( J ) = WORK( J ) + D*T
299: END DO
300: ELSE
301: DO J = 1, I
302: T = ABS( A( I, J ) )
303: U = U + S( J )*T
304: WORK( J ) = WORK( J ) + D*T
305: END DO
306: DO J = I+1,N
307: T = ABS( A( J, I ) )
308: U = U + S( J )*T
309: WORK( J ) = WORK( J ) + D*T
310: END DO
311: END IF
312:
313: AVG = AVG + ( U + WORK( I ) ) * D / N
314: S( I ) = SI
315: END DO
316: END DO
317:
318: 999 CONTINUE
319:
320: SMLNUM = DLAMCH( 'SAFEMIN' )
321: BIGNUM = ONE / SMLNUM
322: SMIN = BIGNUM
323: SMAX = ZERO
324: T = ONE / SQRT( AVG )
325: BASE = DLAMCH( 'B' )
326: U = ONE / LOG( BASE )
327: DO I = 1, N
328: S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
329: SMIN = MIN( SMIN, S( I ) )
330: SMAX = MAX( SMAX, S( I ) )
331: END DO
332: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
333: *
334: END
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