1: SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
2: $ DN1, DN2, TAU, TTYPE, G )
3: *
4: * -- LAPACK routine (version 3.2) --
5: *
6: * -- Contributed by Osni Marques of the Lawrence Berkeley National --
7: * -- Laboratory and Beresford Parlett of the Univ. of California at --
8: * -- Berkeley --
9: * -- November 2008 --
10: *
11: * -- LAPACK is a software package provided by Univ. of Tennessee, --
12: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
13: *
14: * .. Scalar Arguments ..
15: INTEGER I0, N0, N0IN, PP, TTYPE
16: DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
17: * ..
18: * .. Array Arguments ..
19: DOUBLE PRECISION Z( * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * DLASQ4 computes an approximation TAU to the smallest eigenvalue
26: * using values of d from the previous transform.
27: *
28: * I0 (input) INTEGER
29: * First index.
30: *
31: * N0 (input) INTEGER
32: * Last index.
33: *
34: * Z (input) DOUBLE PRECISION array, dimension ( 4*N )
35: * Z holds the qd array.
36: *
37: * PP (input) INTEGER
38: * PP=0 for ping, PP=1 for pong.
39: *
40: * NOIN (input) INTEGER
41: * The value of N0 at start of EIGTEST.
42: *
43: * DMIN (input) DOUBLE PRECISION
44: * Minimum value of d.
45: *
46: * DMIN1 (input) DOUBLE PRECISION
47: * Minimum value of d, excluding D( N0 ).
48: *
49: * DMIN2 (input) DOUBLE PRECISION
50: * Minimum value of d, excluding D( N0 ) and D( N0-1 ).
51: *
52: * DN (input) DOUBLE PRECISION
53: * d(N)
54: *
55: * DN1 (input) DOUBLE PRECISION
56: * d(N-1)
57: *
58: * DN2 (input) DOUBLE PRECISION
59: * d(N-2)
60: *
61: * TAU (output) DOUBLE PRECISION
62: * This is the shift.
63: *
64: * TTYPE (output) INTEGER
65: * Shift type.
66: *
67: * G (input/output) REAL
68: * G is passed as an argument in order to save its value between
69: * calls to DLASQ4.
70: *
71: * Further Details
72: * ===============
73: * CNST1 = 9/16
74: *
75: * =====================================================================
76: *
77: * .. Parameters ..
78: DOUBLE PRECISION CNST1, CNST2, CNST3
79: PARAMETER ( CNST1 = 0.5630D0, CNST2 = 1.010D0,
80: $ CNST3 = 1.050D0 )
81: DOUBLE PRECISION QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
82: PARAMETER ( QURTR = 0.250D0, THIRD = 0.3330D0,
83: $ HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0,
84: $ TWO = 2.0D0, HUNDRD = 100.0D0 )
85: * ..
86: * .. Local Scalars ..
87: INTEGER I4, NN, NP
88: DOUBLE PRECISION A2, B1, B2, GAM, GAP1, GAP2, S
89: * ..
90: * .. Intrinsic Functions ..
91: INTRINSIC MAX, MIN, SQRT
92: * ..
93: * .. Executable Statements ..
94: *
95: * A negative DMIN forces the shift to take that absolute value
96: * TTYPE records the type of shift.
97: *
98: IF( DMIN.LE.ZERO ) THEN
99: TAU = -DMIN
100: TTYPE = -1
101: RETURN
102: END IF
103: *
104: NN = 4*N0 + PP
105: IF( N0IN.EQ.N0 ) THEN
106: *
107: * No eigenvalues deflated.
108: *
109: IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
110: *
111: B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
112: B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
113: A2 = Z( NN-7 ) + Z( NN-5 )
114: *
115: * Cases 2 and 3.
116: *
117: IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
118: GAP2 = DMIN2 - A2 - DMIN2*QURTR
119: IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
120: GAP1 = A2 - DN - ( B2 / GAP2 )*B2
121: ELSE
122: GAP1 = A2 - DN - ( B1+B2 )
123: END IF
124: IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
125: S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
126: TTYPE = -2
127: ELSE
128: S = ZERO
129: IF( DN.GT.B1 )
130: $ S = DN - B1
131: IF( A2.GT.( B1+B2 ) )
132: $ S = MIN( S, A2-( B1+B2 ) )
133: S = MAX( S, THIRD*DMIN )
134: TTYPE = -3
135: END IF
136: ELSE
137: *
138: * Case 4.
139: *
140: TTYPE = -4
141: S = QURTR*DMIN
142: IF( DMIN.EQ.DN ) THEN
143: GAM = DN
144: A2 = ZERO
145: IF( Z( NN-5 ) .GT. Z( NN-7 ) )
146: $ RETURN
147: B2 = Z( NN-5 ) / Z( NN-7 )
148: NP = NN - 9
149: ELSE
150: NP = NN - 2*PP
151: B2 = Z( NP-2 )
152: GAM = DN1
153: IF( Z( NP-4 ) .GT. Z( NP-2 ) )
154: $ RETURN
155: A2 = Z( NP-4 ) / Z( NP-2 )
156: IF( Z( NN-9 ) .GT. Z( NN-11 ) )
157: $ RETURN
158: B2 = Z( NN-9 ) / Z( NN-11 )
159: NP = NN - 13
160: END IF
161: *
162: * Approximate contribution to norm squared from I < NN-1.
163: *
164: A2 = A2 + B2
165: DO 10 I4 = NP, 4*I0 - 1 + PP, -4
166: IF( B2.EQ.ZERO )
167: $ GO TO 20
168: B1 = B2
169: IF( Z( I4 ) .GT. Z( I4-2 ) )
170: $ RETURN
171: B2 = B2*( Z( I4 ) / Z( I4-2 ) )
172: A2 = A2 + B2
173: IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
174: $ GO TO 20
175: 10 CONTINUE
176: 20 CONTINUE
177: A2 = CNST3*A2
178: *
179: * Rayleigh quotient residual bound.
180: *
181: IF( A2.LT.CNST1 )
182: $ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
183: END IF
184: ELSE IF( DMIN.EQ.DN2 ) THEN
185: *
186: * Case 5.
187: *
188: TTYPE = -5
189: S = QURTR*DMIN
190: *
191: * Compute contribution to norm squared from I > NN-2.
192: *
193: NP = NN - 2*PP
194: B1 = Z( NP-2 )
195: B2 = Z( NP-6 )
196: GAM = DN2
197: IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
198: $ RETURN
199: A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
200: *
201: * Approximate contribution to norm squared from I < NN-2.
202: *
203: IF( N0-I0.GT.2 ) THEN
204: B2 = Z( NN-13 ) / Z( NN-15 )
205: A2 = A2 + B2
206: DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
207: IF( B2.EQ.ZERO )
208: $ GO TO 40
209: B1 = B2
210: IF( Z( I4 ) .GT. Z( I4-2 ) )
211: $ RETURN
212: B2 = B2*( Z( I4 ) / Z( I4-2 ) )
213: A2 = A2 + B2
214: IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
215: $ GO TO 40
216: 30 CONTINUE
217: 40 CONTINUE
218: A2 = CNST3*A2
219: END IF
220: *
221: IF( A2.LT.CNST1 )
222: $ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
223: ELSE
224: *
225: * Case 6, no information to guide us.
226: *
227: IF( TTYPE.EQ.-6 ) THEN
228: G = G + THIRD*( ONE-G )
229: ELSE IF( TTYPE.EQ.-18 ) THEN
230: G = QURTR*THIRD
231: ELSE
232: G = QURTR
233: END IF
234: S = G*DMIN
235: TTYPE = -6
236: END IF
237: *
238: ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
239: *
240: * One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
241: *
242: IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN
243: *
244: * Cases 7 and 8.
245: *
246: TTYPE = -7
247: S = THIRD*DMIN1
248: IF( Z( NN-5 ).GT.Z( NN-7 ) )
249: $ RETURN
250: B1 = Z( NN-5 ) / Z( NN-7 )
251: B2 = B1
252: IF( B2.EQ.ZERO )
253: $ GO TO 60
254: DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
255: A2 = B1
256: IF( Z( I4 ).GT.Z( I4-2 ) )
257: $ RETURN
258: B1 = B1*( Z( I4 ) / Z( I4-2 ) )
259: B2 = B2 + B1
260: IF( HUNDRD*MAX( B1, A2 ).LT.B2 )
261: $ GO TO 60
262: 50 CONTINUE
263: 60 CONTINUE
264: B2 = SQRT( CNST3*B2 )
265: A2 = DMIN1 / ( ONE+B2**2 )
266: GAP2 = HALF*DMIN2 - A2
267: IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
268: S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
269: ELSE
270: S = MAX( S, A2*( ONE-CNST2*B2 ) )
271: TTYPE = -8
272: END IF
273: ELSE
274: *
275: * Case 9.
276: *
277: S = QURTR*DMIN1
278: IF( DMIN1.EQ.DN1 )
279: $ S = HALF*DMIN1
280: TTYPE = -9
281: END IF
282: *
283: ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
284: *
285: * Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
286: *
287: * Cases 10 and 11.
288: *
289: IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN
290: TTYPE = -10
291: S = THIRD*DMIN2
292: IF( Z( NN-5 ).GT.Z( NN-7 ) )
293: $ RETURN
294: B1 = Z( NN-5 ) / Z( NN-7 )
295: B2 = B1
296: IF( B2.EQ.ZERO )
297: $ GO TO 80
298: DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
299: IF( Z( I4 ).GT.Z( I4-2 ) )
300: $ RETURN
301: B1 = B1*( Z( I4 ) / Z( I4-2 ) )
302: B2 = B2 + B1
303: IF( HUNDRD*B1.LT.B2 )
304: $ GO TO 80
305: 70 CONTINUE
306: 80 CONTINUE
307: B2 = SQRT( CNST3*B2 )
308: A2 = DMIN2 / ( ONE+B2**2 )
309: GAP2 = Z( NN-7 ) + Z( NN-9 ) -
310: $ SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
311: IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
312: S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
313: ELSE
314: S = MAX( S, A2*( ONE-CNST2*B2 ) )
315: END IF
316: ELSE
317: S = QURTR*DMIN2
318: TTYPE = -11
319: END IF
320: ELSE IF( N0IN.GT.( N0+2 ) ) THEN
321: *
322: * Case 12, more than two eigenvalues deflated. No information.
323: *
324: S = ZERO
325: TTYPE = -12
326: END IF
327: *
328: TAU = S
329: RETURN
330: *
331: * End of DLASQ4
332: *
333: END
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