1      SUBROUTINE SLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
2     $                   DN1, DN2, TAU, TTYPE )
3*
4*  -- LAPACK auxiliary routine (version 3.0) --
5*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
6*     Courant Institute, Argonne National Lab, and Rice University
7*     October 31, 1999
8*
9*     .. Scalar Arguments ..
10      INTEGER            I0, N0, N0IN, PP, TTYPE
11      REAL               DMIN, DMIN1, DMIN2, DN, DN1, DN2, TAU
12*     ..
13*     .. Array Arguments ..
14      REAL               Z( * )
15*     ..
16*
17*  Purpose
18*  =======
19*
20*  SLASQ4 computes an approximation TAU to the smallest eigenvalue
21*  using values of d from the previous transform.
22*
23*  I0    (input) INTEGER
24*        First index.
25*
26*  N0    (input) INTEGER
27*        Last index.
28*
29*  Z     (input) REAL array, dimension ( 4*N )
30*        Z holds the qd array.
31*
32*  PP    (input) INTEGER
33*        PP=0 for ping, PP=1 for pong.
34*
35*  NOIN  (input) INTEGER
36*        The value of N0 at start of EIGTEST.
37*
38*  DMIN  (input) REAL
39*        Minimum value of d.
40*
41*  DMIN1 (input) REAL
42*        Minimum value of d, excluding D( N0 ).
43*
44*  DMIN2 (input) REAL
45*        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
46*
47*  DN    (input) REAL
48*        d(N)
49*
50*  DN1   (input) REAL
51*        d(N-1)
52*
53*  DN2   (input) REAL
54*        d(N-2)
55*
56*  TAU   (output) REAL
57*        This is the shift.
58*
59*  TTYPE (output) INTEGER
60*        Shift type.
61*
62*  Further Details
63*  ===============
64*  CNST1 = 9/16
65*
66*  =====================================================================
67*
68*     .. Parameters ..
69      REAL               CNST1, CNST2, CNST3
70      PARAMETER          ( CNST1 = 0.5630E0, CNST2 = 1.010E0,
71     $                   CNST3 = 1.050E0 )
72      REAL               QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
73      PARAMETER          ( QURTR = 0.250E0, THIRD = 0.3330E0,
74     $                   HALF = 0.50E0, ZERO = 0.0E0, ONE = 1.0E0,
75     $                   TWO = 2.0E0, HUNDRD = 100.0E0 )
76*     ..
77*     .. Local Scalars ..
78      INTEGER            I4, NN, NP
79      REAL               A2, B1, B2, G, GAM, GAP1, GAP2, S
80*     ..
81*     .. Intrinsic Functions ..
82      INTRINSIC          MAX, MIN, SQRT
83*     ..
84*     .. Save statement ..
85      SAVE               G
86*     ..
87*     .. Data statement ..
88      DATA               G / ZERO /
89*     ..
90*     .. Executable Statements ..
91*
92*     A negative DMIN forces the shift to take that absolute value
93*     TTYPE records the type of shift.
94*
95      IF( DMIN.LE.ZERO ) THEN
96         TAU = -DMIN
97         TTYPE = -1
98         RETURN
99      END IF
100*
101      NN = 4*N0 + PP
102      IF( N0IN.EQ.N0 ) THEN
103*
104*        No eigenvalues deflated.
105*
106         IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
107*
108            B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
109            B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
110            A2 = Z( NN-7 ) + Z( NN-5 )
111*
112*           Cases 2 and 3.
113*
114            IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
115               GAP2 = DMIN2 - A2 - DMIN2*QURTR
116               IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
117                  GAP1 = A2 - DN - ( B2 / GAP2 )*B2
118               ELSE
119                  GAP1 = A2 - DN - ( B1+B2 )
120               END IF
121               IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
122                  S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
123                  TTYPE = -2
124               ELSE
125                  S = ZERO
126                  IF( DN.GT.B1 )
127     $               S = DN - B1
128                  IF( A2.GT.( B1+B2 ) )
129     $               S = MIN( S, A2-( B1+B2 ) )
130                  S = MAX( S, THIRD*DMIN )
131                  TTYPE = -3
132               END IF
133            ELSE
134*
135*              Case 4.
136*
137               TTYPE = -4
138               S = QURTR*DMIN
139               IF( DMIN.EQ.DN ) THEN
140                  GAM = DN
141                  A2 = ZERO
142                  IF( Z( NN-5 ) .GT. Z( NN-7 ) )
143     $               RETURN
144                  B2 = Z( NN-5 ) / Z( NN-7 )
145                  NP = NN - 9
146               ELSE
147                  NP = NN - 2*PP
148                  B2 = Z( NP-2 )
149                  GAM = DN1
150                  IF( Z( NP-4 ) .GT. Z( NP-2 ) )
151     $               RETURN
152                  A2 = Z( NP-4 ) / Z( NP-2 )
153                  IF( Z( NN-9 ) .GT. Z( NN-11 ) )
154     $               RETURN
155                  B2 = Z( NN-9 ) / Z( NN-11 )
156                  NP = NN - 13
157               END IF
158*
159*              Approximate contribution to norm squared from I < NN-1.
160*
161               A2 = A2 + B2
162               DO 10 I4 = NP, 4*I0 - 1 + PP, -4
163                  IF( B2.EQ.ZERO )
164     $               GO TO 20
165                  B1 = B2
166                  IF( Z( I4 ) .GT. Z( I4-2 ) )
167     $               RETURN
168                  B2 = B2*( Z( I4 ) / Z( I4-2 ) )
169                  A2 = A2 + B2
170                  IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
171     $               GO TO 20
172   10          CONTINUE
173   20          CONTINUE
174               A2 = CNST3*A2
175*
176*              Rayleigh quotient residual bound.
177*
178               IF( A2.LT.CNST1 )
179     $            S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
180            END IF
181         ELSE IF( DMIN.EQ.DN2 ) THEN
182*
183*           Case 5.
184*
185            TTYPE = -5
186            S = QURTR*DMIN
187*
188*           Compute contribution to norm squared from I > NN-2.
189*
190            NP = NN - 2*PP
191            B1 = Z( NP-2 )
192            B2 = Z( NP-6 )
193            GAM = DN2
194            IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
195     $         RETURN
196            A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
197*
198*           Approximate contribution to norm squared from I < NN-2.
199*
200            IF( N0-I0.GT.2 ) THEN
201               B2 = Z( NN-13 ) / Z( NN-15 )
202               A2 = A2 + B2
203               DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
204                  IF( B2.EQ.ZERO )
205     $               GO TO 40
206                  B1 = B2
207                  IF( Z( I4 ) .GT. Z( I4-2 ) )
208     $               RETURN
209                  B2 = B2*( Z( I4 ) / Z( I4-2 ) )
210                  A2 = A2 + B2
211                  IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
212     $               GO TO 40
213   30          CONTINUE
214   40          CONTINUE
215               A2 = CNST3*A2
216            END IF
217*
218            IF( A2.LT.CNST1 )
219     $         S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
220         ELSE
221*
222*           Case 6, no information to guide us.
223*
224            IF( TTYPE.EQ.-6 ) THEN
225               G = G + THIRD*( ONE-G )
226            ELSE IF( TTYPE.EQ.-18 ) THEN
227               G = QURTR*THIRD
228            ELSE
229               G = QURTR
230            END IF
231            S = G*DMIN
232            TTYPE = -6
233         END IF
234*
235      ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
236*
237*        One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
238*
239         IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN
240*
241*           Cases 7 and 8.
242*
243            TTYPE = -7
244            S = THIRD*DMIN1
245            IF( Z( NN-5 ).GT.Z( NN-7 ) )
246     $         RETURN
247            B1 = Z( NN-5 ) / Z( NN-7 )
248            B2 = B1
249            IF( B2.EQ.ZERO )
250     $         GO TO 60
251            DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
252               A2 = B1
253               IF( Z( I4 ).GT.Z( I4-2 ) )
254     $            RETURN
255               B1 = B1*( Z( I4 ) / Z( I4-2 ) )
256               B2 = B2 + B1
257               IF( HUNDRD*MAX( B1, A2 ).LT.B2 )
258     $            GO TO 60
259   50       CONTINUE
260   60       CONTINUE
261            B2 = SQRT( CNST3*B2 )
262            A2 = DMIN1 / ( ONE+B2**2 )
263            GAP2 = HALF*DMIN2 - A2
264            IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
265               S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
266            ELSE
267               S = MAX( S, A2*( ONE-CNST2*B2 ) )
268               TTYPE = -8
269            END IF
270         ELSE
271*
272*           Case 9.
273*
274            S = QURTR*DMIN1
275            IF( DMIN1.EQ.DN1 )
276     $         S = HALF*DMIN1
277            TTYPE = -9
278         END IF
279*
280      ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
281*
282*        Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
283*
284*        Cases 10 and 11.
285*
286         IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN
287            TTYPE = -10
288            S = THIRD*DMIN2
289            IF( Z( NN-5 ).GT.Z( NN-7 ) )
290     $         RETURN
291            B1 = Z( NN-5 ) / Z( NN-7 )
292            B2 = B1
293            IF( B2.EQ.ZERO )
294     $         GO TO 80
295            DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
296               IF( Z( I4 ).GT.Z( I4-2 ) )
297     $            RETURN
298               B1 = B1*( Z( I4 ) / Z( I4-2 ) )
299               B2 = B2 + B1
300               IF( HUNDRD*B1.LT.B2 )
301     $            GO TO 80
302   70       CONTINUE
303   80       CONTINUE
304            B2 = SQRT( CNST3*B2 )
305            A2 = DMIN2 / ( ONE+B2**2 )
306            GAP2 = Z( NN-7 ) + Z( NN-9 ) -
307     $             SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
308            IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
309               S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
310            ELSE
311               S = MAX( S, A2*( ONE-CNST2*B2 ) )
312            END IF
313         ELSE
314            S = QURTR*DMIN2
315            TTYPE = -11
316         END IF
317      ELSE IF( N0IN.GT.( N0+2 ) ) THEN
318*
319*        Case 12, more than two eigenvalues deflated. No information.
320*
321         S = ZERO
322         TTYPE = -12
323      END IF
324*
325      TAU = S
326      RETURN
327*
328*     End of SLASQ4
329*
330      END
331