LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
dlaed4.f
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1 *> \brief \b DLAED4 used by DSTEDC. Finds a single root of the secular equation.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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7 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER I, INFO, N
25 * DOUBLE PRECISION DLAM, RHO
26 * ..
27 * .. Array Arguments ..
28 * DOUBLE PRECISION D( * ), DELTA( * ), Z( * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> This subroutine computes the I-th updated eigenvalue of a symmetric
38 *> rank-one modification to a diagonal matrix whose elements are
39 *> given in the array d, and that
40 *>
41 *> D(i) < D(j) for i < j
42 *>
43 *> and that RHO > 0. This is arranged by the calling routine, and is
44 *> no loss in generality. The rank-one modified system is thus
45 *>
46 *> diag( D ) + RHO * Z * Z_transpose.
47 *>
48 *> where we assume the Euclidean norm of Z is 1.
49 *>
50 *> The method consists of approximating the rational functions in the
51 *> secular equation by simpler interpolating rational functions.
52 *> \endverbatim
53 *
54 * Arguments:
55 * ==========
56 *
57 *> \param[in] N
58 *> \verbatim
59 *> N is INTEGER
60 *> The length of all arrays.
61 *> \endverbatim
62 *>
63 *> \param[in] I
64 *> \verbatim
65 *> I is INTEGER
66 *> The index of the eigenvalue to be computed. 1 <= I <= N.
67 *> \endverbatim
68 *>
69 *> \param[in] D
70 *> \verbatim
71 *> D is DOUBLE PRECISION array, dimension (N)
72 *> The original eigenvalues. It is assumed that they are in
73 *> order, D(I) < D(J) for I < J.
74 *> \endverbatim
75 *>
76 *> \param[in] Z
77 *> \verbatim
78 *> Z is DOUBLE PRECISION array, dimension (N)
79 *> The components of the updating vector.
80 *> \endverbatim
81 *>
82 *> \param[out] DELTA
83 *> \verbatim
84 *> DELTA is DOUBLE PRECISION array, dimension (N)
85 *> If N > 2, DELTA contains (D(j) - lambda_I) in its j-th
86 *> component. If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
87 *> for detail. The vector DELTA contains the information necessary
88 *> to construct the eigenvectors by DLAED3 and DLAED9.
89 *> \endverbatim
90 *>
91 *> \param[in] RHO
92 *> \verbatim
93 *> RHO is DOUBLE PRECISION
94 *> The scalar in the symmetric updating formula.
95 *> \endverbatim
96 *>
97 *> \param[out] DLAM
98 *> \verbatim
99 *> DLAM is DOUBLE PRECISION
100 *> The computed lambda_I, the I-th updated eigenvalue.
101 *> \endverbatim
102 *>
103 *> \param[out] INFO
104 *> \verbatim
105 *> INFO is INTEGER
106 *> = 0: successful exit
107 *> > 0: if INFO = 1, the updating process failed.
108 *> \endverbatim
109 *
110 *> \par Internal Parameters:
111 * =========================
112 *>
113 *> \verbatim
114 *> Logical variable ORGATI (origin-at-i?) is used for distinguishing
115 *> whether D(i) or D(i+1) is treated as the origin.
116 *>
117 *> ORGATI = .true. origin at i
118 *> ORGATI = .false. origin at i+1
119 *>
120 *> Logical variable SWTCH3 (switch-for-3-poles?) is for noting
121 *> if we are working with THREE poles!
122 *>
123 *> MAXIT is the maximum number of iterations allowed for each
124 *> eigenvalue.
125 *> \endverbatim
126 *
127 * Authors:
128 * ========
129 *
130 *> \author Univ. of Tennessee
131 *> \author Univ. of California Berkeley
132 *> \author Univ. of Colorado Denver
133 *> \author NAG Ltd.
134 *
135 *> \ingroup auxOTHERcomputational
136 *
137 *> \par Contributors:
138 * ==================
139 *>
140 *> Ren-Cang Li, Computer Science Division, University of California
141 *> at Berkeley, USA
142 *>
143 * =====================================================================
144  SUBROUTINE dlaed4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
145 *
146 * -- LAPACK computational routine --
147 * -- LAPACK is a software package provided by Univ. of Tennessee, --
148 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
149 *
150 * .. Scalar Arguments ..
151  INTEGER I, INFO, N
152  DOUBLE PRECISION DLAM, RHO
153 * ..
154 * .. Array Arguments ..
155  DOUBLE PRECISION D( * ), DELTA( * ), Z( * )
156 * ..
157 *
158 * =====================================================================
159 *
160 * .. Parameters ..
161  INTEGER MAXIT
162  parameter( maxit = 30 )
163  DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN
164  parameter( zero = 0.0d0, one = 1.0d0, two = 2.0d0,
165  \$ three = 3.0d0, four = 4.0d0, eight = 8.0d0,
166  \$ ten = 10.0d0 )
167 * ..
168 * .. Local Scalars ..
169  LOGICAL ORGATI, SWTCH, SWTCH3
170  INTEGER II, IIM1, IIP1, IP1, ITER, J, NITER
171  DOUBLE PRECISION A, B, C, DEL, DLTLB, DLTUB, DPHI, DPSI, DW,
172  \$ EPS, ERRETM, ETA, MIDPT, PHI, PREW, PSI,
173  \$ RHOINV, TAU, TEMP, TEMP1, W
174 * ..
175 * .. Local Arrays ..
176  DOUBLE PRECISION ZZ( 3 )
177 * ..
178 * .. External Functions ..
179  DOUBLE PRECISION DLAMCH
180  EXTERNAL dlamch
181 * ..
182 * .. External Subroutines ..
183  EXTERNAL dlaed5, dlaed6
184 * ..
185 * .. Intrinsic Functions ..
186  INTRINSIC abs, max, min, sqrt
187 * ..
188 * .. Executable Statements ..
189 *
190 * Since this routine is called in an inner loop, we do no argument
191 * checking.
192 *
193 * Quick return for N=1 and 2.
194 *
195  info = 0
196  IF( n.EQ.1 ) THEN
197 *
198 * Presumably, I=1 upon entry
199 *
200  dlam = d( 1 ) + rho*z( 1 )*z( 1 )
201  delta( 1 ) = one
202  RETURN
203  END IF
204  IF( n.EQ.2 ) THEN
205  CALL dlaed5( i, d, z, delta, rho, dlam )
206  RETURN
207  END IF
208 *
209 * Compute machine epsilon
210 *
211  eps = dlamch( 'Epsilon' )
212  rhoinv = one / rho
213 *
214 * The case I = N
215 *
216  IF( i.EQ.n ) THEN
217 *
218 * Initialize some basic variables
219 *
220  ii = n - 1
221  niter = 1
222 *
223 * Calculate initial guess
224 *
225  midpt = rho / two
226 *
227 * If ||Z||_2 is not one, then TEMP should be set to
228 * RHO * ||Z||_2^2 / TWO
229 *
230  DO 10 j = 1, n
231  delta( j ) = ( d( j )-d( i ) ) - midpt
232  10 CONTINUE
233 *
234  psi = zero
235  DO 20 j = 1, n - 2
236  psi = psi + z( j )*z( j ) / delta( j )
237  20 CONTINUE
238 *
239  c = rhoinv + psi
240  w = c + z( ii )*z( ii ) / delta( ii ) +
241  \$ z( n )*z( n ) / delta( n )
242 *
243  IF( w.LE.zero ) THEN
244  temp = z( n-1 )*z( n-1 ) / ( d( n )-d( n-1 )+rho ) +
245  \$ z( n )*z( n ) / rho
246  IF( c.LE.temp ) THEN
247  tau = rho
248  ELSE
249  del = d( n ) - d( n-1 )
250  a = -c*del + z( n-1 )*z( n-1 ) + z( n )*z( n )
251  b = z( n )*z( n )*del
252  IF( a.LT.zero ) THEN
253  tau = two*b / ( sqrt( a*a+four*b*c )-a )
254  ELSE
255  tau = ( a+sqrt( a*a+four*b*c ) ) / ( two*c )
256  END IF
257  END IF
258 *
259 * It can be proved that
260 * D(N)+RHO/2 <= LAMBDA(N) < D(N)+TAU <= D(N)+RHO
261 *
262  dltlb = midpt
263  dltub = rho
264  ELSE
265  del = d( n ) - d( n-1 )
266  a = -c*del + z( n-1 )*z( n-1 ) + z( n )*z( n )
267  b = z( n )*z( n )*del
268  IF( a.LT.zero ) THEN
269  tau = two*b / ( sqrt( a*a+four*b*c )-a )
270  ELSE
271  tau = ( a+sqrt( a*a+four*b*c ) ) / ( two*c )
272  END IF
273 *
274 * It can be proved that
275 * D(N) < D(N)+TAU < LAMBDA(N) < D(N)+RHO/2
276 *
277  dltlb = zero
278  dltub = midpt
279  END IF
280 *
281  DO 30 j = 1, n
282  delta( j ) = ( d( j )-d( i ) ) - tau
283  30 CONTINUE
284 *
285 * Evaluate PSI and the derivative DPSI
286 *
287  dpsi = zero
288  psi = zero
289  erretm = zero
290  DO 40 j = 1, ii
291  temp = z( j ) / delta( j )
292  psi = psi + z( j )*temp
293  dpsi = dpsi + temp*temp
294  erretm = erretm + psi
295  40 CONTINUE
296  erretm = abs( erretm )
297 *
298 * Evaluate PHI and the derivative DPHI
299 *
300  temp = z( n ) / delta( n )
301  phi = z( n )*temp
302  dphi = temp*temp
303  erretm = eight*( -phi-psi ) + erretm - phi + rhoinv +
304  \$ abs( tau )*( dpsi+dphi )
305 *
306  w = rhoinv + phi + psi
307 *
308 * Test for convergence
309 *
310  IF( abs( w ).LE.eps*erretm ) THEN
311  dlam = d( i ) + tau
312  GO TO 250
313  END IF
314 *
315  IF( w.LE.zero ) THEN
316  dltlb = max( dltlb, tau )
317  ELSE
318  dltub = min( dltub, tau )
319  END IF
320 *
321 * Calculate the new step
322 *
323  niter = niter + 1
324  c = w - delta( n-1 )*dpsi - delta( n )*dphi
325  a = ( delta( n-1 )+delta( n ) )*w -
326  \$ delta( n-1 )*delta( n )*( dpsi+dphi )
327  b = delta( n-1 )*delta( n )*w
328  IF( c.LT.zero )
329  \$ c = abs( c )
330  IF( c.EQ.zero ) THEN
331 * ETA = B/A
332 * ETA = RHO - TAU
333  eta = dltub - tau
334  ELSE IF( a.GE.zero ) THEN
335  eta = ( a+sqrt( abs( a*a-four*b*c ) ) ) / ( two*c )
336  ELSE
337  eta = two*b / ( a-sqrt( abs( a*a-four*b*c ) ) )
338  END IF
339 *
340 * Note, eta should be positive if w is negative, and
341 * eta should be negative otherwise. However,
342 * if for some reason caused by roundoff, eta*w > 0,
343 * we simply use one Newton step instead. This way
344 * will guarantee eta*w < 0.
345 *
346  IF( w*eta.GT.zero )
347  \$ eta = -w / ( dpsi+dphi )
348  temp = tau + eta
349  IF( temp.GT.dltub .OR. temp.LT.dltlb ) THEN
350  IF( w.LT.zero ) THEN
351  eta = ( dltub-tau ) / two
352  ELSE
353  eta = ( dltlb-tau ) / two
354  END IF
355  END IF
356  DO 50 j = 1, n
357  delta( j ) = delta( j ) - eta
358  50 CONTINUE
359 *
360  tau = tau + eta
361 *
362 * Evaluate PSI and the derivative DPSI
363 *
364  dpsi = zero
365  psi = zero
366  erretm = zero
367  DO 60 j = 1, ii
368  temp = z( j ) / delta( j )
369  psi = psi + z( j )*temp
370  dpsi = dpsi + temp*temp
371  erretm = erretm + psi
372  60 CONTINUE
373  erretm = abs( erretm )
374 *
375 * Evaluate PHI and the derivative DPHI
376 *
377  temp = z( n ) / delta( n )
378  phi = z( n )*temp
379  dphi = temp*temp
380  erretm = eight*( -phi-psi ) + erretm - phi + rhoinv +
381  \$ abs( tau )*( dpsi+dphi )
382 *
383  w = rhoinv + phi + psi
384 *
385 * Main loop to update the values of the array DELTA
386 *
387  iter = niter + 1
388 *
389  DO 90 niter = iter, maxit
390 *
391 * Test for convergence
392 *
393  IF( abs( w ).LE.eps*erretm ) THEN
394  dlam = d( i ) + tau
395  GO TO 250
396  END IF
397 *
398  IF( w.LE.zero ) THEN
399  dltlb = max( dltlb, tau )
400  ELSE
401  dltub = min( dltub, tau )
402  END IF
403 *
404 * Calculate the new step
405 *
406  c = w - delta( n-1 )*dpsi - delta( n )*dphi
407  a = ( delta( n-1 )+delta( n ) )*w -
408  \$ delta( n-1 )*delta( n )*( dpsi+dphi )
409  b = delta( n-1 )*delta( n )*w
410  IF( a.GE.zero ) THEN
411  eta = ( a+sqrt( abs( a*a-four*b*c ) ) ) / ( two*c )
412  ELSE
413  eta = two*b / ( a-sqrt( abs( a*a-four*b*c ) ) )
414  END IF
415 *
416 * Note, eta should be positive if w is negative, and
417 * eta should be negative otherwise. However,
418 * if for some reason caused by roundoff, eta*w > 0,
419 * we simply use one Newton step instead. This way
420 * will guarantee eta*w < 0.
421 *
422  IF( w*eta.GT.zero )
423  \$ eta = -w / ( dpsi+dphi )
424  temp = tau + eta
425  IF( temp.GT.dltub .OR. temp.LT.dltlb ) THEN
426  IF( w.LT.zero ) THEN
427  eta = ( dltub-tau ) / two
428  ELSE
429  eta = ( dltlb-tau ) / two
430  END IF
431  END IF
432  DO 70 j = 1, n
433  delta( j ) = delta( j ) - eta
434  70 CONTINUE
435 *
436  tau = tau + eta
437 *
438 * Evaluate PSI and the derivative DPSI
439 *
440  dpsi = zero
441  psi = zero
442  erretm = zero
443  DO 80 j = 1, ii
444  temp = z( j ) / delta( j )
445  psi = psi + z( j )*temp
446  dpsi = dpsi + temp*temp
447  erretm = erretm + psi
448  80 CONTINUE
449  erretm = abs( erretm )
450 *
451 * Evaluate PHI and the derivative DPHI
452 *
453  temp = z( n ) / delta( n )
454  phi = z( n )*temp
455  dphi = temp*temp
456  erretm = eight*( -phi-psi ) + erretm - phi + rhoinv +
457  \$ abs( tau )*( dpsi+dphi )
458 *
459  w = rhoinv + phi + psi
460  90 CONTINUE
461 *
462 * Return with INFO = 1, NITER = MAXIT and not converged
463 *
464  info = 1
465  dlam = d( i ) + tau
466  GO TO 250
467 *
468 * End for the case I = N
469 *
470  ELSE
471 *
472 * The case for I < N
473 *
474  niter = 1
475  ip1 = i + 1
476 *
477 * Calculate initial guess
478 *
479  del = d( ip1 ) - d( i )
480  midpt = del / two
481  DO 100 j = 1, n
482  delta( j ) = ( d( j )-d( i ) ) - midpt
483  100 CONTINUE
484 *
485  psi = zero
486  DO 110 j = 1, i - 1
487  psi = psi + z( j )*z( j ) / delta( j )
488  110 CONTINUE
489 *
490  phi = zero
491  DO 120 j = n, i + 2, -1
492  phi = phi + z( j )*z( j ) / delta( j )
493  120 CONTINUE
494  c = rhoinv + psi + phi
495  w = c + z( i )*z( i ) / delta( i ) +
496  \$ z( ip1 )*z( ip1 ) / delta( ip1 )
497 *
498  IF( w.GT.zero ) THEN
499 *
500 * d(i)< the ith eigenvalue < (d(i)+d(i+1))/2
501 *
502 * We choose d(i) as origin.
503 *
504  orgati = .true.
505  a = c*del + z( i )*z( i ) + z( ip1 )*z( ip1 )
506  b = z( i )*z( i )*del
507  IF( a.GT.zero ) THEN
508  tau = two*b / ( a+sqrt( abs( a*a-four*b*c ) ) )
509  ELSE
510  tau = ( a-sqrt( abs( a*a-four*b*c ) ) ) / ( two*c )
511  END IF
512  dltlb = zero
513  dltub = midpt
514  ELSE
515 *
516 * (d(i)+d(i+1))/2 <= the ith eigenvalue < d(i+1)
517 *
518 * We choose d(i+1) as origin.
519 *
520  orgati = .false.
521  a = c*del - z( i )*z( i ) - z( ip1 )*z( ip1 )
522  b = z( ip1 )*z( ip1 )*del
523  IF( a.LT.zero ) THEN
524  tau = two*b / ( a-sqrt( abs( a*a+four*b*c ) ) )
525  ELSE
526  tau = -( a+sqrt( abs( a*a+four*b*c ) ) ) / ( two*c )
527  END IF
528  dltlb = -midpt
529  dltub = zero
530  END IF
531 *
532  IF( orgati ) THEN
533  DO 130 j = 1, n
534  delta( j ) = ( d( j )-d( i ) ) - tau
535  130 CONTINUE
536  ELSE
537  DO 140 j = 1, n
538  delta( j ) = ( d( j )-d( ip1 ) ) - tau
539  140 CONTINUE
540  END IF
541  IF( orgati ) THEN
542  ii = i
543  ELSE
544  ii = i + 1
545  END IF
546  iim1 = ii - 1
547  iip1 = ii + 1
548 *
549 * Evaluate PSI and the derivative DPSI
550 *
551  dpsi = zero
552  psi = zero
553  erretm = zero
554  DO 150 j = 1, iim1
555  temp = z( j ) / delta( j )
556  psi = psi + z( j )*temp
557  dpsi = dpsi + temp*temp
558  erretm = erretm + psi
559  150 CONTINUE
560  erretm = abs( erretm )
561 *
562 * Evaluate PHI and the derivative DPHI
563 *
564  dphi = zero
565  phi = zero
566  DO 160 j = n, iip1, -1
567  temp = z( j ) / delta( j )
568  phi = phi + z( j )*temp
569  dphi = dphi + temp*temp
570  erretm = erretm + phi
571  160 CONTINUE
572 *
573  w = rhoinv + phi + psi
574 *
575 * W is the value of the secular function with
576 * its ii-th element removed.
577 *
578  swtch3 = .false.
579  IF( orgati ) THEN
580  IF( w.LT.zero )
581  \$ swtch3 = .true.
582  ELSE
583  IF( w.GT.zero )
584  \$ swtch3 = .true.
585  END IF
586  IF( ii.EQ.1 .OR. ii.EQ.n )
587  \$ swtch3 = .false.
588 *
589  temp = z( ii ) / delta( ii )
590  dw = dpsi + dphi + temp*temp
591  temp = z( ii )*temp
592  w = w + temp
593  erretm = eight*( phi-psi ) + erretm + two*rhoinv +
594  \$ three*abs( temp ) + abs( tau )*dw
595 *
596 * Test for convergence
597 *
598  IF( abs( w ).LE.eps*erretm ) THEN
599  IF( orgati ) THEN
600  dlam = d( i ) + tau
601  ELSE
602  dlam = d( ip1 ) + tau
603  END IF
604  GO TO 250
605  END IF
606 *
607  IF( w.LE.zero ) THEN
608  dltlb = max( dltlb, tau )
609  ELSE
610  dltub = min( dltub, tau )
611  END IF
612 *
613 * Calculate the new step
614 *
615  niter = niter + 1
616  IF( .NOT.swtch3 ) THEN
617  IF( orgati ) THEN
618  c = w - delta( ip1 )*dw - ( d( i )-d( ip1 ) )*
619  \$ ( z( i ) / delta( i ) )**2
620  ELSE
621  c = w - delta( i )*dw - ( d( ip1 )-d( i ) )*
622  \$ ( z( ip1 ) / delta( ip1 ) )**2
623  END IF
624  a = ( delta( i )+delta( ip1 ) )*w -
625  \$ delta( i )*delta( ip1 )*dw
626  b = delta( i )*delta( ip1 )*w
627  IF( c.EQ.zero ) THEN
628  IF( a.EQ.zero ) THEN
629  IF( orgati ) THEN
630  a = z( i )*z( i ) + delta( ip1 )*delta( ip1 )*
631  \$ ( dpsi+dphi )
632  ELSE
633  a = z( ip1 )*z( ip1 ) + delta( i )*delta( i )*
634  \$ ( dpsi+dphi )
635  END IF
636  END IF
637  eta = b / a
638  ELSE IF( a.LE.zero ) THEN
639  eta = ( a-sqrt( abs( a*a-four*b*c ) ) ) / ( two*c )
640  ELSE
641  eta = two*b / ( a+sqrt( abs( a*a-four*b*c ) ) )
642  END IF
643  ELSE
644 *
645 * Interpolation using THREE most relevant poles
646 *
647  temp = rhoinv + psi + phi
648  IF( orgati ) THEN
649  temp1 = z( iim1 ) / delta( iim1 )
650  temp1 = temp1*temp1
651  c = temp - delta( iip1 )*( dpsi+dphi ) -
652  \$ ( d( iim1 )-d( iip1 ) )*temp1
653  zz( 1 ) = z( iim1 )*z( iim1 )
654  zz( 3 ) = delta( iip1 )*delta( iip1 )*
655  \$ ( ( dpsi-temp1 )+dphi )
656  ELSE
657  temp1 = z( iip1 ) / delta( iip1 )
658  temp1 = temp1*temp1
659  c = temp - delta( iim1 )*( dpsi+dphi ) -
660  \$ ( d( iip1 )-d( iim1 ) )*temp1
661  zz( 1 ) = delta( iim1 )*delta( iim1 )*
662  \$ ( dpsi+( dphi-temp1 ) )
663  zz( 3 ) = z( iip1 )*z( iip1 )
664  END IF
665  zz( 2 ) = z( ii )*z( ii )
666  CALL dlaed6( niter, orgati, c, delta( iim1 ), zz, w, eta,
667  \$ info )
668  IF( info.NE.0 )
669  \$ GO TO 250
670  END IF
671 *
672 * Note, eta should be positive if w is negative, and
673 * eta should be negative otherwise. However,
674 * if for some reason caused by roundoff, eta*w > 0,
675 * we simply use one Newton step instead. This way
676 * will guarantee eta*w < 0.
677 *
678  IF( w*eta.GE.zero )
679  \$ eta = -w / dw
680  temp = tau + eta
681  IF( temp.GT.dltub .OR. temp.LT.dltlb ) THEN
682  IF( w.LT.zero ) THEN
683  eta = ( dltub-tau ) / two
684  ELSE
685  eta = ( dltlb-tau ) / two
686  END IF
687  END IF
688 *
689  prew = w
690 *
691  DO 180 j = 1, n
692  delta( j ) = delta( j ) - eta
693  180 CONTINUE
694 *
695 * Evaluate PSI and the derivative DPSI
696 *
697  dpsi = zero
698  psi = zero
699  erretm = zero
700  DO 190 j = 1, iim1
701  temp = z( j ) / delta( j )
702  psi = psi + z( j )*temp
703  dpsi = dpsi + temp*temp
704  erretm = erretm + psi
705  190 CONTINUE
706  erretm = abs( erretm )
707 *
708 * Evaluate PHI and the derivative DPHI
709 *
710  dphi = zero
711  phi = zero
712  DO 200 j = n, iip1, -1
713  temp = z( j ) / delta( j )
714  phi = phi + z( j )*temp
715  dphi = dphi + temp*temp
716  erretm = erretm + phi
717  200 CONTINUE
718 *
719  temp = z( ii ) / delta( ii )
720  dw = dpsi + dphi + temp*temp
721  temp = z( ii )*temp
722  w = rhoinv + phi + psi + temp
723  erretm = eight*( phi-psi ) + erretm + two*rhoinv +
724  \$ three*abs( temp ) + abs( tau+eta )*dw
725 *
726  swtch = .false.
727  IF( orgati ) THEN
728  IF( -w.GT.abs( prew ) / ten )
729  \$ swtch = .true.
730  ELSE
731  IF( w.GT.abs( prew ) / ten )
732  \$ swtch = .true.
733  END IF
734 *
735  tau = tau + eta
736 *
737 * Main loop to update the values of the array DELTA
738 *
739  iter = niter + 1
740 *
741  DO 240 niter = iter, maxit
742 *
743 * Test for convergence
744 *
745  IF( abs( w ).LE.eps*erretm ) THEN
746  IF( orgati ) THEN
747  dlam = d( i ) + tau
748  ELSE
749  dlam = d( ip1 ) + tau
750  END IF
751  GO TO 250
752  END IF
753 *
754  IF( w.LE.zero ) THEN
755  dltlb = max( dltlb, tau )
756  ELSE
757  dltub = min( dltub, tau )
758  END IF
759 *
760 * Calculate the new step
761 *
762  IF( .NOT.swtch3 ) THEN
763  IF( .NOT.swtch ) THEN
764  IF( orgati ) THEN
765  c = w - delta( ip1 )*dw -
766  \$ ( d( i )-d( ip1 ) )*( z( i ) / delta( i ) )**2
767  ELSE
768  c = w - delta( i )*dw - ( d( ip1 )-d( i ) )*
769  \$ ( z( ip1 ) / delta( ip1 ) )**2
770  END IF
771  ELSE
772  temp = z( ii ) / delta( ii )
773  IF( orgati ) THEN
774  dpsi = dpsi + temp*temp
775  ELSE
776  dphi = dphi + temp*temp
777  END IF
778  c = w - delta( i )*dpsi - delta( ip1 )*dphi
779  END IF
780  a = ( delta( i )+delta( ip1 ) )*w -
781  \$ delta( i )*delta( ip1 )*dw
782  b = delta( i )*delta( ip1 )*w
783  IF( c.EQ.zero ) THEN
784  IF( a.EQ.zero ) THEN
785  IF( .NOT.swtch ) THEN
786  IF( orgati ) THEN
787  a = z( i )*z( i ) + delta( ip1 )*
788  \$ delta( ip1 )*( dpsi+dphi )
789  ELSE
790  a = z( ip1 )*z( ip1 ) +
791  \$ delta( i )*delta( i )*( dpsi+dphi )
792  END IF
793  ELSE
794  a = delta( i )*delta( i )*dpsi +
795  \$ delta( ip1 )*delta( ip1 )*dphi
796  END IF
797  END IF
798  eta = b / a
799  ELSE IF( a.LE.zero ) THEN
800  eta = ( a-sqrt( abs( a*a-four*b*c ) ) ) / ( two*c )
801  ELSE
802  eta = two*b / ( a+sqrt( abs( a*a-four*b*c ) ) )
803  END IF
804  ELSE
805 *
806 * Interpolation using THREE most relevant poles
807 *
808  temp = rhoinv + psi + phi
809  IF( swtch ) THEN
810  c = temp - delta( iim1 )*dpsi - delta( iip1 )*dphi
811  zz( 1 ) = delta( iim1 )*delta( iim1 )*dpsi
812  zz( 3 ) = delta( iip1 )*delta( iip1 )*dphi
813  ELSE
814  IF( orgati ) THEN
815  temp1 = z( iim1 ) / delta( iim1 )
816  temp1 = temp1*temp1
817  c = temp - delta( iip1 )*( dpsi+dphi ) -
818  \$ ( d( iim1 )-d( iip1 ) )*temp1
819  zz( 1 ) = z( iim1 )*z( iim1 )
820  zz( 3 ) = delta( iip1 )*delta( iip1 )*
821  \$ ( ( dpsi-temp1 )+dphi )
822  ELSE
823  temp1 = z( iip1 ) / delta( iip1 )
824  temp1 = temp1*temp1
825  c = temp - delta( iim1 )*( dpsi+dphi ) -
826  \$ ( d( iip1 )-d( iim1 ) )*temp1
827  zz( 1 ) = delta( iim1 )*delta( iim1 )*
828  \$ ( dpsi+( dphi-temp1 ) )
829  zz( 3 ) = z( iip1 )*z( iip1 )
830  END IF
831  END IF
832  CALL dlaed6( niter, orgati, c, delta( iim1 ), zz, w, eta,
833  \$ info )
834  IF( info.NE.0 )
835  \$ GO TO 250
836  END IF
837 *
838 * Note, eta should be positive if w is negative, and
839 * eta should be negative otherwise. However,
840 * if for some reason caused by roundoff, eta*w > 0,
841 * we simply use one Newton step instead. This way
842 * will guarantee eta*w < 0.
843 *
844  IF( w*eta.GE.zero )
845  \$ eta = -w / dw
846  temp = tau + eta
847  IF( temp.GT.dltub .OR. temp.LT.dltlb ) THEN
848  IF( w.LT.zero ) THEN
849  eta = ( dltub-tau ) / two
850  ELSE
851  eta = ( dltlb-tau ) / two
852  END IF
853  END IF
854 *
855  DO 210 j = 1, n
856  delta( j ) = delta( j ) - eta
857  210 CONTINUE
858 *
859  tau = tau + eta
860  prew = w
861 *
862 * Evaluate PSI and the derivative DPSI
863 *
864  dpsi = zero
865  psi = zero
866  erretm = zero
867  DO 220 j = 1, iim1
868  temp = z( j ) / delta( j )
869  psi = psi + z( j )*temp
870  dpsi = dpsi + temp*temp
871  erretm = erretm + psi
872  220 CONTINUE
873  erretm = abs( erretm )
874 *
875 * Evaluate PHI and the derivative DPHI
876 *
877  dphi = zero
878  phi = zero
879  DO 230 j = n, iip1, -1
880  temp = z( j ) / delta( j )
881  phi = phi + z( j )*temp
882  dphi = dphi + temp*temp
883  erretm = erretm + phi
884  230 CONTINUE
885 *
886  temp = z( ii ) / delta( ii )
887  dw = dpsi + dphi + temp*temp
888  temp = z( ii )*temp
889  w = rhoinv + phi + psi + temp
890  erretm = eight*( phi-psi ) + erretm + two*rhoinv +
891  \$ three*abs( temp ) + abs( tau )*dw
892  IF( w*prew.GT.zero .AND. abs( w ).GT.abs( prew ) / ten )
893  \$ swtch = .NOT.swtch
894 *
895  240 CONTINUE
896 *
897 * Return with INFO = 1, NITER = MAXIT and not converged
898 *
899  info = 1
900  IF( orgati ) THEN
901  dlam = d( i ) + tau
902  ELSE
903  dlam = d( ip1 ) + tau
904  END IF
905 *
906  END IF
907 *
908  250 CONTINUE
909 *
910  RETURN
911 *
912 * End of DLAED4
913 *
914  END
subroutine dlaed4(N, I, D, Z, DELTA, RHO, DLAM, INFO)
DLAED4 used by DSTEDC. Finds a single root of the secular equation.
Definition: dlaed4.f:145
subroutine dlaed6(KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO)
DLAED6 used by DSTEDC. Computes one Newton step in solution of the secular equation.
Definition: dlaed6.f:140
subroutine dlaed5(I, D, Z, DELTA, RHO, DLAM)
DLAED5 used by DSTEDC. Solves the 2-by-2 secular equation.
Definition: dlaed5.f:108