1*> \brief \b DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download DLASQ2 + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq2.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq2.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq2.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DLASQ2( N, Z, INFO )
22*
23*       .. Scalar Arguments ..
24*       INTEGER            INFO, N
25*       ..
26*       .. Array Arguments ..
27*       DOUBLE PRECISION   Z( * )
28*       ..
29*
30*
31*> \par Purpose:
32*  =============
33*>
34*> \verbatim
35*>
36*> DLASQ2 computes all the eigenvalues of the symmetric positive
37*> definite tridiagonal matrix associated with the qd array Z to high
38*> relative accuracy are computed to high relative accuracy, in the
39*> absence of denormalization, underflow and overflow.
40*>
41*> To see the relation of Z to the tridiagonal matrix, let L be a
42*> unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
43*> let U be an upper bidiagonal matrix with 1's above and diagonal
44*> Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
45*> symmetric tridiagonal to which it is similar.
46*>
47*> Note : DLASQ2 defines a logical variable, IEEE, which is true
48*> on machines which follow ieee-754 floating-point standard in their
49*> handling of infinities and NaNs, and false otherwise. This variable
50*> is passed to DLASQ3.
51*> \endverbatim
52*
53*  Arguments:
54*  ==========
55*
56*> \param[in] N
57*> \verbatim
58*>          N is INTEGER
59*>        The number of rows and columns in the matrix. N >= 0.
60*> \endverbatim
61*>
62*> \param[in,out] Z
63*> \verbatim
64*>          Z is DOUBLE PRECISION array, dimension ( 4*N )
65*>        On entry Z holds the qd array. On exit, entries 1 to N hold
66*>        the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
67*>        trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
68*>        N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
69*>        holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
70*>        shifts that failed.
71*> \endverbatim
72*>
73*> \param[out] INFO
74*> \verbatim
75*>          INFO is INTEGER
76*>        = 0: successful exit
77*>        < 0: if the i-th argument is a scalar and had an illegal
78*>             value, then INFO = -i, if the i-th argument is an
79*>             array and the j-entry had an illegal value, then
80*>             INFO = -(i*100+j)
81*>        > 0: the algorithm failed
82*>              = 1, a split was marked by a positive value in E
83*>              = 2, current block of Z not diagonalized after 100*N
84*>                   iterations (in inner while loop).  On exit Z holds
85*>                   a qd array with the same eigenvalues as the given Z.
86*>              = 3, termination criterion of outer while loop not met
87*>                   (program created more than N unreduced blocks)
88*> \endverbatim
89*
90*  Authors:
91*  ========
92*
93*> \author Univ. of Tennessee
94*> \author Univ. of California Berkeley
95*> \author Univ. of Colorado Denver
96*> \author NAG Ltd.
97*
98*> \date December 2016
99*
100*> \ingroup auxOTHERcomputational
101*
102*> \par Further Details:
103*  =====================
104*>
105*> \verbatim
106*>
107*>  Local Variables: I0:N0 defines a current unreduced segment of Z.
108*>  The shifts are accumulated in SIGMA. Iteration count is in ITER.
109*>  Ping-pong is controlled by PP (alternates between 0 and 1).
110*> \endverbatim
111*>
112*  =====================================================================
113      SUBROUTINE DLASQ2( N, Z, INFO )
114*
115*  -- LAPACK computational routine (version 3.7.0) --
116*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
117*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
118*     December 2016
119*
120*     .. Scalar Arguments ..
121      INTEGER            INFO, N
122*     ..
123*     .. Array Arguments ..
124      DOUBLE PRECISION   Z( * )
125*     ..
126*
127*  =====================================================================
128*
129*     .. Parameters ..
130      DOUBLE PRECISION   CBIAS
131      PARAMETER          ( CBIAS = 1.50D0 )
132      DOUBLE PRECISION   ZERO, HALF, ONE, TWO, FOUR, HUNDRD
133      PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
134     $                     TWO = 2.0D0, FOUR = 4.0D0, HUNDRD = 100.0D0 )
135*     ..
136*     .. Local Scalars ..
137      LOGICAL            IEEE
138      INTEGER            I0, I1, I4, IINFO, IPN4, ITER, IWHILA, IWHILB,
139     $                   K, KMIN, N0, N1, NBIG, NDIV, NFAIL, PP, SPLT,
140     $                   TTYPE
141      DOUBLE PRECISION   D, DEE, DEEMIN, DESIG, DMIN, DMIN1, DMIN2, DN,
142     $                   DN1, DN2, E, EMAX, EMIN, EPS, G, OLDEMN, QMAX,
143     $                   QMIN, S, SAFMIN, SIGMA, T, TAU, TEMP, TOL,
144     $                   TOL2, TRACE, ZMAX, TEMPE, TEMPQ
145*     ..
146*     .. External Subroutines ..
147      EXTERNAL           DLASQ3, DLASRT, XERBLA
148*     ..
149*     .. External Functions ..
150      INTEGER            ILAENV
151      DOUBLE PRECISION   DLAMCH
152      EXTERNAL           DLAMCH, ILAENV
153*     ..
154*     .. Intrinsic Functions ..
155      INTRINSIC          ABS, DBLE, MAX, MIN, SQRT
156*     ..
157*     .. Executable Statements ..
158*
159*     Test the input arguments.
160*     (in case DLASQ2 is not called by DLASQ1)
161*
162      INFO = 0
163      EPS = DLAMCH( 'Precision' )
164      SAFMIN = DLAMCH( 'Safe minimum' )
165      TOL = EPS*HUNDRD
166      TOL2 = TOL**2
167*
168      IF( N.LT.0 ) THEN
169         INFO = -1
170         CALL XERBLA( 'DLASQ2', 1 )
171         RETURN
172      ELSE IF( N.EQ.0 ) THEN
173         RETURN
174      ELSE IF( N.EQ.1 ) THEN
175*
176*        1-by-1 case.
177*
178         IF( Z( 1 ).LT.ZERO ) THEN
179            INFO = -201
180            CALL XERBLA( 'DLASQ2', 2 )
181         END IF
182         RETURN
183      ELSE IF( N.EQ.2 ) THEN
184*
185*        2-by-2 case.
186*
187         IF( Z( 2 ).LT.ZERO .OR. Z( 3 ).LT.ZERO ) THEN
188            INFO = -2
189            CALL XERBLA( 'DLASQ2', 2 )
190            RETURN
191         ELSE IF( Z( 3 ).GT.Z( 1 ) ) THEN
192            D = Z( 3 )
193            Z( 3 ) = Z( 1 )
194            Z( 1 ) = D
195         END IF
196         Z( 5 ) = Z( 1 ) + Z( 2 ) + Z( 3 )
197         IF( Z( 2 ).GT.Z( 3 )*TOL2 ) THEN
198            T = HALF*( ( Z( 1 )-Z( 3 ) )+Z( 2 ) )
199            S = Z( 3 )*( Z( 2 ) / T )
200            IF( S.LE.T ) THEN
201               S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) )
202            ELSE
203               S = Z( 3 )*( Z( 2 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
204            END IF
205            T = Z( 1 ) + ( S+Z( 2 ) )
206            Z( 3 ) = Z( 3 )*( Z( 1 ) / T )
207            Z( 1 ) = T
208         END IF
209         Z( 2 ) = Z( 3 )
210         Z( 6 ) = Z( 2 ) + Z( 1 )
211         RETURN
212      END IF
213*
214*     Check for negative data and compute sums of q's and e's.
215*
216      Z( 2*N ) = ZERO
217      EMIN = Z( 2 )
218      QMAX = ZERO
219      ZMAX = ZERO
220      D = ZERO
221      E = ZERO
222*
223      DO 10 K = 1, 2*( N-1 ), 2
224         IF( Z( K ).LT.ZERO ) THEN
225            INFO = -( 200+K )
226            CALL XERBLA( 'DLASQ2', 2 )
227            RETURN
228         ELSE IF( Z( K+1 ).LT.ZERO ) THEN
229            INFO = -( 200+K+1 )
230            CALL XERBLA( 'DLASQ2', 2 )
231            RETURN
232         END IF
233         D = D + Z( K )
234         E = E + Z( K+1 )
235         QMAX = MAX( QMAX, Z( K ) )
236         EMIN = MIN( EMIN, Z( K+1 ) )
237         ZMAX = MAX( QMAX, ZMAX, Z( K+1 ) )
238   10 CONTINUE
239      IF( Z( 2*N-1 ).LT.ZERO ) THEN
240         INFO = -( 200+2*N-1 )
241         CALL XERBLA( 'DLASQ2', 2 )
242         RETURN
243      END IF
244      D = D + Z( 2*N-1 )
245      QMAX = MAX( QMAX, Z( 2*N-1 ) )
246      ZMAX = MAX( QMAX, ZMAX )
247*
248*     Check for diagonality.
249*
250      IF( E.EQ.ZERO ) THEN
251         DO 20 K = 2, N
252            Z( K ) = Z( 2*K-1 )
253   20    CONTINUE
254         CALL DLASRT( 'D', N, Z, IINFO )
255         Z( 2*N-1 ) = D
256         RETURN
257      END IF
258*
259      TRACE = D + E
260*
261*     Check for zero data.
262*
263      IF( TRACE.EQ.ZERO ) THEN
264         Z( 2*N-1 ) = ZERO
265         RETURN
266      END IF
267*
268*     Check whether the machine is IEEE conformable.
269*
270      IEEE = ILAENV( 10, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 .AND.
271     $       ILAENV( 11, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1
272*
273*     Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...).
274*
275      DO 30 K = 2*N, 2, -2
276         Z( 2*K ) = ZERO
277         Z( 2*K-1 ) = Z( K )
278         Z( 2*K-2 ) = ZERO
279         Z( 2*K-3 ) = Z( K-1 )
280   30 CONTINUE
281*
282      I0 = 1
283      N0 = N
284*
285*     Reverse the qd-array, if warranted.
286*
287      IF( CBIAS*Z( 4*I0-3 ).LT.Z( 4*N0-3 ) ) THEN
288         IPN4 = 4*( I0+N0 )
289         DO 40 I4 = 4*I0, 2*( I0+N0-1 ), 4
290            TEMP = Z( I4-3 )
291            Z( I4-3 ) = Z( IPN4-I4-3 )
292            Z( IPN4-I4-3 ) = TEMP
293            TEMP = Z( I4-1 )
294            Z( I4-1 ) = Z( IPN4-I4-5 )
295            Z( IPN4-I4-5 ) = TEMP
296   40    CONTINUE
297      END IF
298*
299*     Initial split checking via dqd and Li's test.
300*
301      PP = 0
302*
303      DO 80 K = 1, 2
304*
305         D = Z( 4*N0+PP-3 )
306         DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4
307            IF( Z( I4-1 ).LE.TOL2*D ) THEN
308               Z( I4-1 ) = -ZERO
309               D = Z( I4-3 )
310            ELSE
311               D = Z( I4-3 )*( D / ( D+Z( I4-1 ) ) )
312            END IF
313   50    CONTINUE
314*
315*        dqd maps Z to ZZ plus Li's test.
316*
317         EMIN = Z( 4*I0+PP+1 )
318         D = Z( 4*I0+PP-3 )
319         DO 60 I4 = 4*I0 + PP, 4*( N0-1 ) + PP, 4
320            Z( I4-2*PP-2 ) = D + Z( I4-1 )
321            IF( Z( I4-1 ).LE.TOL2*D ) THEN
322               Z( I4-1 ) = -ZERO
323               Z( I4-2*PP-2 ) = D
324               Z( I4-2*PP ) = ZERO
325               D = Z( I4+1 )
326            ELSE IF( SAFMIN*Z( I4+1 ).LT.Z( I4-2*PP-2 ) .AND.
327     $               SAFMIN*Z( I4-2*PP-2 ).LT.Z( I4+1 ) ) THEN
328               TEMP = Z( I4+1 ) / Z( I4-2*PP-2 )
329               Z( I4-2*PP ) = Z( I4-1 )*TEMP
330               D = D*TEMP
331            ELSE
332               Z( I4-2*PP ) = Z( I4+1 )*( Z( I4-1 ) / Z( I4-2*PP-2 ) )
333               D = Z( I4+1 )*( D / Z( I4-2*PP-2 ) )
334            END IF
335            EMIN = MIN( EMIN, Z( I4-2*PP ) )
336   60    CONTINUE
337         Z( 4*N0-PP-2 ) = D
338*
339*        Now find qmax.
340*
341         QMAX = Z( 4*I0-PP-2 )
342         DO 70 I4 = 4*I0 - PP + 2, 4*N0 - PP - 2, 4
343            QMAX = MAX( QMAX, Z( I4 ) )
344   70    CONTINUE
345*
346*        Prepare for the next iteration on K.
347*
348         PP = 1 - PP
349   80 CONTINUE
350*
351*     Initialise variables to pass to DLASQ3.
352*
353      TTYPE = 0
354      DMIN1 = ZERO
355      DMIN2 = ZERO
356      DN    = ZERO
357      DN1   = ZERO
358      DN2   = ZERO
359      G     = ZERO
360      TAU   = ZERO
361*
362      ITER = 2
363      NFAIL = 0
364      NDIV = 2*( N0-I0 )
365*
366      DO 160 IWHILA = 1, N + 1
367         IF( N0.LT.1 )
368     $      GO TO 170
369*
370*        While array unfinished do
371*
372*        E(N0) holds the value of SIGMA when submatrix in I0:N0
373*        splits from the rest of the array, but is negated.
374*
375         DESIG = ZERO
376         IF( N0.EQ.N ) THEN
377            SIGMA = ZERO
378         ELSE
379            SIGMA = -Z( 4*N0-1 )
380         END IF
381         IF( SIGMA.LT.ZERO ) THEN
382            INFO = 1
383            RETURN
384         END IF
385*
386*        Find last unreduced submatrix's top index I0, find QMAX and
387*        EMIN. Find Gershgorin-type bound if Q's much greater than E's.
388*
389         EMAX = ZERO
390         IF( N0.GT.I0 ) THEN
391            EMIN = ABS( Z( 4*N0-5 ) )
392         ELSE
393            EMIN = ZERO
394         END IF
395         QMIN = Z( 4*N0-3 )
396         QMAX = QMIN
397         DO 90 I4 = 4*N0, 8, -4
398            IF( Z( I4-5 ).LE.ZERO )
399     $         GO TO 100
400            IF( QMIN.GE.FOUR*EMAX ) THEN
401               QMIN = MIN( QMIN, Z( I4-3 ) )
402               EMAX = MAX( EMAX, Z( I4-5 ) )
403            END IF
404            QMAX = MAX( QMAX, Z( I4-7 )+Z( I4-5 ) )
405            EMIN = MIN( EMIN, Z( I4-5 ) )
406   90    CONTINUE
407         I4 = 4
408*
409  100    CONTINUE
410         I0 = I4 / 4
411         PP = 0
412*
413         IF( N0-I0.GT.1 ) THEN
414            DEE = Z( 4*I0-3 )
415            DEEMIN = DEE
416            KMIN = I0
417            DO 110 I4 = 4*I0+1, 4*N0-3, 4
418               DEE = Z( I4 )*( DEE /( DEE+Z( I4-2 ) ) )
419               IF( DEE.LE.DEEMIN ) THEN
420                  DEEMIN = DEE
421                  KMIN = ( I4+3 )/4
422               END IF
423  110       CONTINUE
424            IF( (KMIN-I0)*2.LT.N0-KMIN .AND.
425     $         DEEMIN.LE.HALF*Z(4*N0-3) ) THEN
426               IPN4 = 4*( I0+N0 )
427               PP = 2
428               DO 120 I4 = 4*I0, 2*( I0+N0-1 ), 4
429                  TEMP = Z( I4-3 )
430                  Z( I4-3 ) = Z( IPN4-I4-3 )
431                  Z( IPN4-I4-3 ) = TEMP
432                  TEMP = Z( I4-2 )
433                  Z( I4-2 ) = Z( IPN4-I4-2 )
434                  Z( IPN4-I4-2 ) = TEMP
435                  TEMP = Z( I4-1 )
436                  Z( I4-1 ) = Z( IPN4-I4-5 )
437                  Z( IPN4-I4-5 ) = TEMP
438                  TEMP = Z( I4 )
439                  Z( I4 ) = Z( IPN4-I4-4 )
440                  Z( IPN4-I4-4 ) = TEMP
441  120          CONTINUE
442            END IF
443         END IF
444*
445*        Put -(initial shift) into DMIN.
446*
447         DMIN = -MAX( ZERO, QMIN-TWO*SQRT( QMIN )*SQRT( EMAX ) )
448*
449*        Now I0:N0 is unreduced.
450*        PP = 0 for ping, PP = 1 for pong.
451*        PP = 2 indicates that flipping was applied to the Z array and
452*               and that the tests for deflation upon entry in DLASQ3
453*               should not be performed.
454*
455         NBIG = 100*( N0-I0+1 )
456         DO 140 IWHILB = 1, NBIG
457            IF( I0.GT.N0 )
458     $         GO TO 150
459*
460*           While submatrix unfinished take a good dqds step.
461*
462            CALL DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
463     $                   ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
464     $                   DN2, G, TAU )
465*
466            PP = 1 - PP
467*
468*           When EMIN is very small check for splits.
469*
470            IF( PP.EQ.0 .AND. N0-I0.GE.3 ) THEN
471               IF( Z( 4*N0 ).LE.TOL2*QMAX .OR.
472     $             Z( 4*N0-1 ).LE.TOL2*SIGMA ) THEN
473                  SPLT = I0 - 1
474                  QMAX = Z( 4*I0-3 )
475                  EMIN = Z( 4*I0-1 )
476                  OLDEMN = Z( 4*I0 )
477                  DO 130 I4 = 4*I0, 4*( N0-3 ), 4
478                     IF( Z( I4 ).LE.TOL2*Z( I4-3 ) .OR.
479     $                   Z( I4-1 ).LE.TOL2*SIGMA ) THEN
480                        Z( I4-1 ) = -SIGMA
481                        SPLT = I4 / 4
482                        QMAX = ZERO
483                        EMIN = Z( I4+3 )
484                        OLDEMN = Z( I4+4 )
485                     ELSE
486                        QMAX = MAX( QMAX, Z( I4+1 ) )
487                        EMIN = MIN( EMIN, Z( I4-1 ) )
488                        OLDEMN = MIN( OLDEMN, Z( I4 ) )
489                     END IF
490  130             CONTINUE
491                  Z( 4*N0-1 ) = EMIN
492                  Z( 4*N0 ) = OLDEMN
493                  I0 = SPLT + 1
494               END IF
495            END IF
496*
497  140    CONTINUE
498*
499         INFO = 2
500*
501*        Maximum number of iterations exceeded, restore the shift
502*        SIGMA and place the new d's and e's in a qd array.
503*        This might need to be done for several blocks
504*
505         I1 = I0
506         N1 = N0
507 145     CONTINUE
508         TEMPQ = Z( 4*I0-3 )
509         Z( 4*I0-3 ) = Z( 4*I0-3 ) + SIGMA
510         DO K = I0+1, N0
511            TEMPE = Z( 4*K-5 )
512            Z( 4*K-5 ) = Z( 4*K-5 ) * (TEMPQ / Z( 4*K-7 ))
513            TEMPQ = Z( 4*K-3 )
514            Z( 4*K-3 ) = Z( 4*K-3 ) + SIGMA + TEMPE - Z( 4*K-5 )
515         END DO
516*
517*        Prepare to do this on the previous block if there is one
518*
519         IF( I1.GT.1 ) THEN
520            N1 = I1-1
521            DO WHILE( ( I1.GE.2 ) .AND. ( Z(4*I1-5).GE.ZERO ) )
522               I1 = I1 - 1
523            END DO
524            SIGMA = -Z(4*N1-1)
525            GO TO 145
526         END IF
527
528         DO K = 1, N
529            Z( 2*K-1 ) = Z( 4*K-3 )
530*
531*        Only the block 1..N0 is unfinished.  The rest of the e's
532*        must be essentially zero, although sometimes other data
533*        has been stored in them.
534*
535            IF( K.LT.N0 ) THEN
536               Z( 2*K ) = Z( 4*K-1 )
537            ELSE
538               Z( 2*K ) = 0
539            END IF
540         END DO
541         RETURN
542*
543*        end IWHILB
544*
545  150    CONTINUE
546*
547  160 CONTINUE
548*
549      INFO = 3
550      RETURN
551*
552*     end IWHILA
553*
554  170 CONTINUE
555*
556*     Move q's to the front.
557*
558      DO 180 K = 2, N
559         Z( K ) = Z( 4*K-3 )
560  180 CONTINUE
561*
562*     Sort and compute sum of eigenvalues.
563*
564      CALL DLASRT( 'D', N, Z, IINFO )
565*
566      E = ZERO
567      DO 190 K = N, 1, -1
568         E = E + Z( K )
569  190 CONTINUE
570*
571*     Store trace, sum(eigenvalues) and information on performance.
572*
573      Z( 2*N+1 ) = TRACE
574      Z( 2*N+2 ) = E
575      Z( 2*N+3 ) = DBLE( ITER )
576      Z( 2*N+4 ) = DBLE( NDIV ) / DBLE( N**2 )
577      Z( 2*N+5 ) = HUNDRD*NFAIL / DBLE( ITER )
578      RETURN
579*
580*     End of DLASQ2
581*
582      END
583