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
|
SUBROUTINE dzror(status,x,fx,xlo,xhi,qleft,qhi)
C**********************************************************************
C
C SUBROUTINE DZROR(STATUS, X, FX, XLO, XHI, QLEFT, QHI)
C Double precision ZeRo of a function -- Reverse Communication
C
C
C Function
C
C
C Performs the zero finding. STZROR must have been called before
C this routine in order to set its parameters.
C
C
C Arguments
C
C
C STATUS <--> At the beginning of a zero finding problem, STATUS
C should be set to 0 and ZROR invoked. (The value
C of other parameters will be ignored on this call.)
C
C When ZROR needs the function evaluated, it will set
C STATUS to 1 and return. The value of the function
C should be set in FX and ZROR again called without
C changing any of its other parameters.
C
C When ZROR has finished without error, it will return
C with STATUS 0. In that case (XLO,XHI) bound the answe
C
C If ZROR finds an error (which implies that F(XLO)-Y an
C F(XHI)-Y have the same sign, it returns STATUS -1. In
C this case, XLO and XHI are undefined.
C INTEGER STATUS
C
C X <-- The value of X at which F(X) is to be evaluated.
C DOUBLE PRECISION X
C
C FX --> The value of F(X) calculated when ZROR returns with
C STATUS = 1.
C DOUBLE PRECISION FX
C
C XLO <-- When ZROR returns with STATUS = 0, XLO bounds the
C inverval in X containing the solution below.
C DOUBLE PRECISION XLO
C
C XHI <-- When ZROR returns with STATUS = 0, XHI bounds the
C inverval in X containing the solution above.
C DOUBLE PRECISION XHI
C
C QLEFT <-- .TRUE. if the stepping search terminated unsucessfully
C at XLO. If it is .FALSE. the search terminated
C unsucessfully at XHI.
C QLEFT is LOGICAL
C
C QHI <-- .TRUE. if F(X) .GT. Y at the termination of the
C search and .FALSE. if F(X) .LT. Y at the
C termination of the search.
C QHI is LOGICAL
C
C**********************************************************************
C .. Scalar Arguments ..
DOUBLE PRECISION fx,x,xhi,xlo,zabstl,zreltl,zxhi,zxlo
INTEGER status
LOGICAL qhi,qleft
C ..
C .. Save statement ..
SAVE
C ..
C .. Local Scalars ..
DOUBLE PRECISION a,abstol,b,c,d,fa,fb,fc,fd,fda,fdb,m,mb,p,q,
+ reltol,tol,w,xxhi,xxlo,zx
INTEGER ext,i99999
LOGICAL first,qrzero
C ..
C .. Intrinsic Functions ..
INTRINSIC abs,max,sign
C ..
C .. Statement Functions ..
DOUBLE PRECISION ftol
C ..
C .. Statement Function definitions ..
ftol(zx) = 0.5D0*max(abstol,reltol*abs(zx))
C ..
C .. Executable Statements ..
IF (status.GT.0) GO TO 280
xlo = xxlo
xhi = xxhi
b = xlo
x = xlo
C GET-FUNCTION-VALUE
ASSIGN 10 TO i99999
GO TO 270
10 fb = fx
xlo = xhi
a = xlo
x = xlo
C GET-FUNCTION-VALUE
ASSIGN 20 TO i99999
GO TO 270
C
C Check that F(ZXLO) < 0 < F(ZXHI) or
C F(ZXLO) > 0 > F(ZXHI)
C
20 IF (.NOT. (fb.LT.0.0D0)) GO TO 40
IF (.NOT. (fx.LT.0.0D0)) GO TO 30
status = -1
qleft = fx .LT. fb
qhi = .FALSE.
RETURN
30 CONTINUE
40 IF (.NOT. (fb.GT.0.0D0)) GO TO 60
IF (.NOT. (fx.GT.0.0D0)) GO TO 50
status = -1
qleft = fx .GT. fb
qhi = .TRUE.
RETURN
50 CONTINUE
60 fa = fx
C
first = .TRUE.
70 c = a
fc = fa
ext = 0
80 IF (.NOT. (abs(fc).LT.abs(fb))) GO TO 100
IF (.NOT. (c.NE.a)) GO TO 90
d = a
fd = fa
90 a = b
fa = fb
xlo = c
b = xlo
fb = fc
c = a
fc = fa
100 tol = ftol(xlo)
m = (c+b)*.5D0
mb = m - b
IF (.NOT. (abs(mb).GT.tol)) GO TO 240
IF (.NOT. (ext.GT.3)) GO TO 110
w = mb
GO TO 190
110 tol = sign(tol,mb)
p = (b-a)*fb
IF (.NOT. (first)) GO TO 120
q = fa - fb
first = .FALSE.
GO TO 130
120 fdb = (fd-fb)/ (d-b)
fda = (fd-fa)/ (d-a)
p = fda*p
q = fdb*fa - fda*fb
130 IF (.NOT. (p.LT.0.0D0)) GO TO 140
p = -p
q = -q
140 IF (ext.EQ.3) p = p*2.0D0
IF (.NOT. ((p*1.0D0).EQ.0.0D0.OR.p.LE. (q*tol))) GO TO 150
w = tol
GO TO 180
150 IF (.NOT. (p.LT. (mb*q))) GO TO 160
w = p/q
GO TO 170
160 w = mb
170 CONTINUE
180 CONTINUE
190 d = a
fd = fa
a = b
fa = fb
b = b + w
xlo = b
x = xlo
C GET-FUNCTION-VALUE
ASSIGN 200 TO i99999
GO TO 270
200 fb = fx
IF (.NOT. ((fc*fb).GE.0.0D0)) GO TO 210
GO TO 70
210 IF (.NOT. (w.EQ.mb)) GO TO 220
ext = 0
GO TO 230
220 ext = ext + 1
230 GO TO 80
240 xhi = c
qrzero = (fc.GE.0.0D0 .AND. fb.LE.0.0D0) .OR.
+ (fc.LT.0.0D0 .AND. fb.GE.0.0D0)
IF (.NOT. (qrzero)) GO TO 250
status = 0
GO TO 260
250 status = -1
260 RETURN
ENTRY dstzr(zxlo,zxhi,zabstl,zreltl)
C**********************************************************************
C
C SUBROUTINE DSTZR( XLO, XHI, ABSTOL, RELTOL )
C Double precision SeT ZeRo finder - Reverse communication version
C
C
C Function
C
C
C
C Sets quantities needed by ZROR. The function of ZROR
C and the quantities set is given here.
C
C Concise Description - Given a function F
C find XLO such that F(XLO) = 0.
C
C More Precise Description -
C
C Input condition. F is a double precision function of a single
C double precision argument and XLO and XHI are such that
C F(XLO)*F(XHI) .LE. 0.0
C
C If the input condition is met, QRZERO returns .TRUE.
C and output values of XLO and XHI satisfy the following
C F(XLO)*F(XHI) .LE. 0.
C ABS(F(XLO) .LE. ABS(F(XHI)
C ABS(XLO-XHI) .LE. TOL(X)
C where
C TOL(X) = MAX(ABSTOL,RELTOL*ABS(X))
C
C If this algorithm does not find XLO and XHI satisfying
C these conditions then QRZERO returns .FALSE. This
C implies that the input condition was not met.
C
C
C Arguments
C
C
C XLO --> The left endpoint of the interval to be
C searched for a solution.
C XLO is DOUBLE PRECISION
C
C XHI --> The right endpoint of the interval to be
C for a solution.
C XHI is DOUBLE PRECISION
C
C ABSTOL, RELTOL --> Two numbers that determine the accuracy
C of the solution. See function for a
C precise definition.
C ABSTOL is DOUBLE PRECISION
C RELTOL is DOUBLE PRECISION
C
C
C Method
C
C
C Algorithm R of the paper 'Two Efficient Algorithms with
C Guaranteed Convergence for Finding a Zero of a Function'
C by J. C. P. Bus and T. J. Dekker in ACM Transactions on
C Mathematical Software, Volume 1, no. 4 page 330
C (Dec. '75) is employed to find the zero of F(X)-Y.
C
C**********************************************************************
xxlo = zxlo
xxhi = zxhi
abstol = zabstl
reltol = zreltl
RETURN
STOP '*** EXECUTION FLOWING INTO FLECS PROCEDURES ***'
C TO GET-FUNCTION-VALUE
270 status = 1
RETURN
280 CONTINUE
GO TO i99999
END
|
