summaryrefslogtreecommitdiff
path: root/SRC/dlaed6.f
blob: daa8db39e4ce4a5f6a26c70017b3a3224b4027ab (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
*> \brief \b DLAED6 used by sstedc. Computes one Newton step in solution of the secular equation.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed6.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            ORGATI
*       INTEGER            INFO, KNITER
*       DOUBLE PRECISION   FINIT, RHO, TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( 3 ), Z( 3 )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED6 computes the positive or negative root (closest to the origin)
*> of
*>                  z(1)        z(2)        z(3)
*> f(x) =   rho + --------- + ---------- + ---------
*>                 d(1)-x      d(2)-x      d(3)-x
*>
*> It is assumed that
*>
*>       if ORGATI = .true. the root is between d(2) and d(3);
*>       otherwise it is between d(1) and d(2)
*>
*> This routine will be called by DLAED4 when necessary. In most cases,
*> the root sought is the smallest in magnitude, though it might not be
*> in some extremely rare situations.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] KNITER
*> \verbatim
*>          KNITER is INTEGER
*>               Refer to DLAED4 for its significance.
*> \endverbatim
*>
*> \param[in] ORGATI
*> \verbatim
*>          ORGATI is LOGICAL
*>               If ORGATI is true, the needed root is between d(2) and
*>               d(3); otherwise it is between d(1) and d(2).  See
*>               DLAED4 for further details.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>               Refer to the equation f(x) above.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (3)
*>               D satisfies d(1) < d(2) < d(3).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (3)
*>               Each of the elements in z must be positive.
*> \endverbatim
*>
*> \param[in] FINIT
*> \verbatim
*>          FINIT is DOUBLE PRECISION
*>               The value of f at 0. It is more accurate than the one
*>               evaluated inside this routine (if someone wants to do
*>               so).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>               The root of the equation f(x).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>               = 0: successful exit
*>               > 0: if INFO = 1, failure to converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup auxOTHERcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  10/02/03: This version has a few statements commented out for thread
*>  safety (machine parameters are computed on each entry). SJH.
*>
*>  05/10/06: Modified from a new version of Ren-Cang Li, use
*>     Gragg-Thornton-Warner cubic convergent scheme for better stability.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Ren-Cang Li, Computer Science Division, University of California
*>     at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
*  -- LAPACK computational routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      LOGICAL            ORGATI
      INTEGER            INFO, KNITER
      DOUBLE PRECISION   FINIT, RHO, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( 3 ), Z( 3 )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 40 )
      DOUBLE PRECISION   ZERO, ONE, TWO, THREE, FOUR, EIGHT
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0 )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DSCALE( 3 ), ZSCALE( 3 )
*     ..
*     .. Local Scalars ..
      LOGICAL            SCALE
      INTEGER            I, ITER, NITER
      DOUBLE PRECISION   A, B, BASE, C, DDF, DF, EPS, ERRETM, ETA, F,
     $                   FC, SCLFAC, SCLINV, SMALL1, SMALL2, SMINV1,
     $                   SMINV2, TEMP, TEMP1, TEMP2, TEMP3, TEMP4,
     $                   LBD, UBD
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, LOG, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
      IF( ORGATI ) THEN
         LBD = D(2)
         UBD = D(3)
      ELSE
         LBD = D(1)
         UBD = D(2)
      END IF
      IF( FINIT .LT. ZERO )THEN
         LBD = ZERO
      ELSE
         UBD = ZERO
      END IF
*
      NITER = 1
      TAU = ZERO
      IF( KNITER.EQ.2 ) THEN
         IF( ORGATI ) THEN
            TEMP = ( D( 3 )-D( 2 ) ) / TWO
            C = RHO + Z( 1 ) / ( ( D( 1 )-D( 2 ) )-TEMP )
            A = C*( D( 2 )+D( 3 ) ) + Z( 2 ) + Z( 3 )
            B = C*D( 2 )*D( 3 ) + Z( 2 )*D( 3 ) + Z( 3 )*D( 2 )
         ELSE
            TEMP = ( D( 1 )-D( 2 ) ) / TWO
            C = RHO + Z( 3 ) / ( ( D( 3 )-D( 2 ) )-TEMP )
            A = C*( D( 1 )+D( 2 ) ) + Z( 1 ) + Z( 2 )
            B = C*D( 1 )*D( 2 ) + Z( 1 )*D( 2 ) + Z( 2 )*D( 1 )
         END IF
         TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
         A = A / TEMP
         B = B / TEMP
         C = C / TEMP
         IF( C.EQ.ZERO ) THEN
            TAU = B / A
         ELSE IF( A.LE.ZERO ) THEN
            TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
         ELSE
            TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
         END IF
         IF( TAU .LT. LBD .OR. TAU .GT. UBD )
     $      TAU = ( LBD+UBD )/TWO
         IF( D(1).EQ.TAU .OR. D(2).EQ.TAU .OR. D(3).EQ.TAU ) THEN
            TAU = ZERO
         ELSE
            TEMP = FINIT + TAU*Z(1)/( D(1)*( D( 1 )-TAU ) ) +
     $                     TAU*Z(2)/( D(2)*( D( 2 )-TAU ) ) +
     $                     TAU*Z(3)/( D(3)*( D( 3 )-TAU ) )
            IF( TEMP .LE. ZERO )THEN
               LBD = TAU
            ELSE
               UBD = TAU
            END IF
            IF( ABS( FINIT ).LE.ABS( TEMP ) )
     $         TAU = ZERO
         END IF
      END IF
*
*     get machine parameters for possible scaling to avoid overflow
*
*     modified by Sven: parameters SMALL1, SMINV1, SMALL2,
*     SMINV2, EPS are not SAVEd anymore between one call to the
*     others but recomputed at each call
*
      EPS = DLAMCH( 'Epsilon' )
      BASE = DLAMCH( 'Base' )
      SMALL1 = BASE**( INT( LOG( DLAMCH( 'SafMin' ) ) / LOG( BASE ) /
     $         THREE ) )
      SMINV1 = ONE / SMALL1
      SMALL2 = SMALL1*SMALL1
      SMINV2 = SMINV1*SMINV1
*
*     Determine if scaling of inputs necessary to avoid overflow
*     when computing 1/TEMP**3
*
      IF( ORGATI ) THEN
         TEMP = MIN( ABS( D( 2 )-TAU ), ABS( D( 3 )-TAU ) )
      ELSE
         TEMP = MIN( ABS( D( 1 )-TAU ), ABS( D( 2 )-TAU ) )
      END IF
      SCALE = .FALSE.
      IF( TEMP.LE.SMALL1 ) THEN
         SCALE = .TRUE.
         IF( TEMP.LE.SMALL2 ) THEN
*
*        Scale up by power of radix nearest 1/SAFMIN**(2/3)
*
            SCLFAC = SMINV2
            SCLINV = SMALL2
         ELSE
*
*        Scale up by power of radix nearest 1/SAFMIN**(1/3)
*
            SCLFAC = SMINV1
            SCLINV = SMALL1
         END IF
*
*        Scaling up safe because D, Z, TAU scaled elsewhere to be O(1)
*
         DO 10 I = 1, 3
            DSCALE( I ) = D( I )*SCLFAC
            ZSCALE( I ) = Z( I )*SCLFAC
   10    CONTINUE
         TAU = TAU*SCLFAC
         LBD = LBD*SCLFAC
         UBD = UBD*SCLFAC
      ELSE
*
*        Copy D and Z to DSCALE and ZSCALE
*
         DO 20 I = 1, 3
            DSCALE( I ) = D( I )
            ZSCALE( I ) = Z( I )
   20    CONTINUE
      END IF
*
      FC = ZERO
      DF = ZERO
      DDF = ZERO
      DO 30 I = 1, 3
         TEMP = ONE / ( DSCALE( I )-TAU )
         TEMP1 = ZSCALE( I )*TEMP
         TEMP2 = TEMP1*TEMP
         TEMP3 = TEMP2*TEMP
         FC = FC + TEMP1 / DSCALE( I )
         DF = DF + TEMP2
         DDF = DDF + TEMP3
   30 CONTINUE
      F = FINIT + TAU*FC
*
      IF( ABS( F ).LE.ZERO )
     $   GO TO 60
      IF( F .LE. ZERO )THEN
         LBD = TAU
      ELSE
         UBD = TAU
      END IF
*
*        Iteration begins -- Use Gragg-Thornton-Warner cubic convergent
*                            scheme
*
*     It is not hard to see that
*
*           1) Iterations will go up monotonically
*              if FINIT < 0;
*
*           2) Iterations will go down monotonically
*              if FINIT > 0.
*
      ITER = NITER + 1
*
      DO 50 NITER = ITER, MAXIT
*
         IF( ORGATI ) THEN
            TEMP1 = DSCALE( 2 ) - TAU
            TEMP2 = DSCALE( 3 ) - TAU
         ELSE
            TEMP1 = DSCALE( 1 ) - TAU
            TEMP2 = DSCALE( 2 ) - TAU
         END IF
         A = ( TEMP1+TEMP2 )*F - TEMP1*TEMP2*DF
         B = TEMP1*TEMP2*F
         C = F - ( TEMP1+TEMP2 )*DF + TEMP1*TEMP2*DDF
         TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
         A = A / TEMP
         B = B / TEMP
         C = C / TEMP
         IF( C.EQ.ZERO ) THEN
            ETA = B / A
         ELSE IF( A.LE.ZERO ) THEN
            ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
         ELSE
            ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
         END IF
         IF( F*ETA.GE.ZERO ) THEN
            ETA = -F / DF
         END IF
*
         TAU = TAU + ETA
         IF( TAU .LT. LBD .OR. TAU .GT. UBD )
     $      TAU = ( LBD + UBD )/TWO
*
         FC = ZERO
         ERRETM = ZERO
         DF = ZERO
         DDF = ZERO
         DO 40 I = 1, 3
            IF ( ( DSCALE( I )-TAU ).NE.ZERO ) THEN
               TEMP = ONE / ( DSCALE( I )-TAU )
               TEMP1 = ZSCALE( I )*TEMP
               TEMP2 = TEMP1*TEMP
               TEMP3 = TEMP2*TEMP
               TEMP4 = TEMP1 / DSCALE( I )
               FC = FC + TEMP4
               ERRETM = ERRETM + ABS( TEMP4 )
               DF = DF + TEMP2
               DDF = DDF + TEMP3
            ELSE
               GO TO 60
            END IF
   40    CONTINUE
         F = FINIT + TAU*FC
         ERRETM = EIGHT*( ABS( FINIT )+ABS( TAU )*ERRETM ) +
     $            ABS( TAU )*DF
         IF( ( ABS( F ).LE.FOUR*EPS*ERRETM ) .OR.
     $      ( (UBD-LBD).LE.FOUR*EPS*ABS(TAU) )  )
     $      GO TO 60
         IF( F .LE. ZERO )THEN
            LBD = TAU
         ELSE
            UBD = TAU
         END IF
   50 CONTINUE
      INFO = 1
   60 CONTINUE
*
*     Undo scaling
*
      IF( SCALE )
     $   TAU = TAU*SCLINV
      RETURN
*
*     End of DLAED6
*
      END