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
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
|
subroutine dqc25s(f,a,b,bl,br,alfa,beta,ri,rj,rg,rh,result,
* abserr,resasc,integr,nev)
c***begin prologue dqc25s
c***date written 810101 (yymmdd)
c***revision date 830518 (yymmdd)
c***category no. h2a2a2
c***keywords 25-point clenshaw-curtis integration
c***author piessens,robert,appl. math. & progr. div. - k.u.leuven
c de doncker,elise,appl. math. & progr. div. - k.u.leuven
c***purpose to compute i = integral of f*w over (bl,br), with error
c estimate, where the weight function w has a singular
c behaviour of algebraico-logarithmic type at the points
c a and/or b. (bl,br) is a part of (a,b).
c***description
c
c integration rules for integrands having algebraico-logarithmic
c end point singularities
c standard fortran subroutine
c double precision version
c
c parameters
c f - double precision
c function subprogram defining the integrand
c f(x). the actual name for f needs to be declared
c e x t e r n a l in the driver program.
c
c a - double precision
c left end point of the original interval
c
c b - double precision
c right end point of the original interval, b.gt.a
c
c bl - double precision
c lower limit of integration, bl.ge.a
c
c br - double precision
c upper limit of integration, br.le.b
c
c alfa - double precision
c parameter in the weight function
c
c beta - double precision
c parameter in the weight function
c
c ri,rj,rg,rh - double precision
c modified chebyshev moments for the application
c of the generalized clenshaw-curtis
c method (computed in subroutine dqmomo)
c
c result - double precision
c approximation to the integral
c result is computed by using a generalized
c clenshaw-curtis method if b1 = a or br = b.
c in all other cases the 15-point kronrod
c rule is applied, obtained by optimal addition of
c abscissae to the 7-point gauss rule.
c
c abserr - double precision
c estimate of the modulus of the absolute error,
c which should equal or exceed abs(i-result)
c
c resasc - double precision
c approximation to the integral of abs(f*w-i/(b-a))
c
c integr - integer
c which determines the weight function
c = 1 w(x) = (x-a)**alfa*(b-x)**beta
c = 2 w(x) = (x-a)**alfa*(b-x)**beta*log(x-a)
c = 3 w(x) = (x-a)**alfa*(b-x)**beta*log(b-x)
c = 4 w(x) = (x-a)**alfa*(b-x)**beta*log(x-a)*
c log(b-x)
c
c nev - integer
c number of integrand evaluations
c***references (none)
c***routines called dqcheb,dqk15w
c***end prologue dqc25s
c
double precision a,abserr,alfa,b,beta,bl,br,centr,cheb12,cheb24,
* dabs,dc,dlog,f,factor,fix,fval,hlgth,resabs,resasc,result,res12,
* res24,rg,rh,ri,rj,u,dqwgts,x
integer i,integr,isym,nev
c
dimension cheb12(13),cheb24(25),fval(25),rg(25),rh(25),ri(25),
* rj(25),x(11)
c
external f,dqwgts
c
c the vector x contains the values cos(k*pi/24)
c k = 1, ..., 11, to be used for the computation of the
c chebyshev series expansion of f.
c
data x(1) / 0.9914448613 7381041114 4557526928 563d0 /
data x(2) / 0.9659258262 8906828674 9743199728 897d0 /
data x(3) / 0.9238795325 1128675612 8183189396 788d0 /
data x(4) / 0.8660254037 8443864676 3723170752 936d0 /
data x(5) / 0.7933533402 9123516457 9776961501 299d0 /
data x(6) / 0.7071067811 8654752440 0844362104 849d0 /
data x(7) / 0.6087614290 0872063941 6097542898 164d0 /
data x(8) / 0.5000000000 0000000000 0000000000 000d0 /
data x(9) / 0.3826834323 6508977172 8459984030 399d0 /
data x(10) / 0.2588190451 0252076234 8898837624 048d0 /
data x(11) / 0.1305261922 2005159154 8406227895 489d0 /
c
c list of major variables
c -----------------------
c
c fval - value of the function f at the points
c (br-bl)*0.5*cos(k*pi/24)+(br+bl)*0.5
c k = 0, ..., 24
c cheb12 - coefficients of the chebyshev series expansion
c of degree 12, for the function f, in the
c interval (bl,br)
c cheb24 - coefficients of the chebyshev series expansion
c of degree 24, for the function f, in the
c interval (bl,br)
c res12 - approximation to the integral obtained from cheb12
c res24 - approximation to the integral obtained from cheb24
c dqwgts - external function subprogram defining
c the four possible weight functions
c hlgth - half-length of the interval (bl,br)
c centr - mid point of the interval (bl,br)
c
c***first executable statement dqc25s
nev = 25
if(bl.eq.a.and.(alfa.ne.0.0d+00.or.integr.eq.2.or.integr.eq.4))
* go to 10
if(br.eq.b.and.(beta.ne.0.0d+00.or.integr.eq.3.or.integr.eq.4))
* go to 140
c
c if a.gt.bl and b.lt.br, apply the 15-point gauss-kronrod
c scheme.
c
c
call dqk15w(f,dqwgts,a,b,alfa,beta,integr,bl,br,
* result,abserr,resabs,resasc)
nev = 15
go to 270
c
c this part of the program is executed only if a = bl.
c ----------------------------------------------------
c
c compute the chebyshev series expansion of the
c following function
c f1 = (0.5*(b+b-br-a)-0.5*(br-a)*x)**beta
c *f(0.5*(br-a)*x+0.5*(br+a))
c
10 hlgth = 0.5d+00*(br-bl)
centr = 0.5d+00*(br+bl)
fix = b-centr
fval(1) = 0.5d+00*f(hlgth+centr)*(fix-hlgth)**beta
fval(13) = f(centr)*(fix**beta)
fval(25) = 0.5d+00*f(centr-hlgth)*(fix+hlgth)**beta
do 20 i=2,12
u = hlgth*x(i-1)
isym = 26-i
fval(i) = f(u+centr)*(fix-u)**beta
fval(isym) = f(centr-u)*(fix+u)**beta
20 continue
factor = hlgth**(alfa+0.1d+01)
result = 0.0d+00
abserr = 0.0d+00
res12 = 0.0d+00
res24 = 0.0d+00
if(integr.gt.2) go to 70
call dqcheb(x,fval,cheb12,cheb24)
c
c integr = 1 (or 2)
c
do 30 i=1,13
res12 = res12+cheb12(i)*ri(i)
res24 = res24+cheb24(i)*ri(i)
30 continue
do 40 i=14,25
res24 = res24+cheb24(i)*ri(i)
40 continue
if(integr.eq.1) go to 130
c
c integr = 2
c
dc = dlog(br-bl)
result = res24*dc
abserr = dabs((res24-res12)*dc)
res12 = 0.0d+00
res24 = 0.0d+00
do 50 i=1,13
res12 = res12+cheb12(i)*rg(i)
res24 = res12+cheb24(i)*rg(i)
50 continue
do 60 i=14,25
res24 = res24+cheb24(i)*rg(i)
60 continue
go to 130
c
c compute the chebyshev series expansion of the
c following function
c f4 = f1*log(0.5*(b+b-br-a)-0.5*(br-a)*x)
c
70 fval(1) = fval(1)*dlog(fix-hlgth)
fval(13) = fval(13)*dlog(fix)
fval(25) = fval(25)*dlog(fix+hlgth)
do 80 i=2,12
u = hlgth*x(i-1)
isym = 26-i
fval(i) = fval(i)*dlog(fix-u)
fval(isym) = fval(isym)*dlog(fix+u)
80 continue
call dqcheb(x,fval,cheb12,cheb24)
c
c integr = 3 (or 4)
c
do 90 i=1,13
res12 = res12+cheb12(i)*ri(i)
res24 = res24+cheb24(i)*ri(i)
90 continue
do 100 i=14,25
res24 = res24+cheb24(i)*ri(i)
100 continue
if(integr.eq.3) go to 130
c
c integr = 4
c
dc = dlog(br-bl)
result = res24*dc
abserr = dabs((res24-res12)*dc)
res12 = 0.0d+00
res24 = 0.0d+00
do 110 i=1,13
res12 = res12+cheb12(i)*rg(i)
res24 = res24+cheb24(i)*rg(i)
110 continue
do 120 i=14,25
res24 = res24+cheb24(i)*rg(i)
120 continue
130 result = (result+res24)*factor
abserr = (abserr+dabs(res24-res12))*factor
go to 270
c
c this part of the program is executed only if b = br.
c ----------------------------------------------------
c
c compute the chebyshev series expansion of the
c following function
c f2 = (0.5*(b+bl-a-a)+0.5*(b-bl)*x)**alfa
c *f(0.5*(b-bl)*x+0.5*(b+bl))
c
140 hlgth = 0.5d+00*(br-bl)
centr = 0.5d+00*(br+bl)
fix = centr-a
fval(1) = 0.5d+00*f(hlgth+centr)*(fix+hlgth)**alfa
fval(13) = f(centr)*(fix**alfa)
fval(25) = 0.5d+00*f(centr-hlgth)*(fix-hlgth)**alfa
do 150 i=2,12
u = hlgth*x(i-1)
isym = 26-i
fval(i) = f(u+centr)*(fix+u)**alfa
fval(isym) = f(centr-u)*(fix-u)**alfa
150 continue
factor = hlgth**(beta+0.1d+01)
result = 0.0d+00
abserr = 0.0d+00
res12 = 0.0d+00
res24 = 0.0d+00
if(integr.eq.2.or.integr.eq.4) go to 200
c
c integr = 1 (or 3)
c
call dqcheb(x,fval,cheb12,cheb24)
do 160 i=1,13
res12 = res12+cheb12(i)*rj(i)
res24 = res24+cheb24(i)*rj(i)
160 continue
do 170 i=14,25
res24 = res24+cheb24(i)*rj(i)
170 continue
if(integr.eq.1) go to 260
c
c integr = 3
c
dc = dlog(br-bl)
result = res24*dc
abserr = dabs((res24-res12)*dc)
res12 = 0.0d+00
res24 = 0.0d+00
do 180 i=1,13
res12 = res12+cheb12(i)*rh(i)
res24 = res24+cheb24(i)*rh(i)
180 continue
do 190 i=14,25
res24 = res24+cheb24(i)*rh(i)
190 continue
go to 260
c
c compute the chebyshev series expansion of the
c following function
c f3 = f2*log(0.5*(b-bl)*x+0.5*(b+bl-a-a))
c
200 fval(1) = fval(1)*dlog(hlgth+fix)
fval(13) = fval(13)*dlog(fix)
fval(25) = fval(25)*dlog(fix-hlgth)
do 210 i=2,12
u = hlgth*x(i-1)
isym = 26-i
fval(i) = fval(i)*dlog(u+fix)
fval(isym) = fval(isym)*dlog(fix-u)
210 continue
call dqcheb(x,fval,cheb12,cheb24)
c
c integr = 2 (or 4)
c
do 220 i=1,13
res12 = res12+cheb12(i)*rj(i)
res24 = res24+cheb24(i)*rj(i)
220 continue
do 230 i=14,25
res24 = res24+cheb24(i)*rj(i)
230 continue
if(integr.eq.2) go to 260
dc = dlog(br-bl)
result = res24*dc
abserr = dabs((res24-res12)*dc)
res12 = 0.0d+00
res24 = 0.0d+00
c
c integr = 4
c
do 240 i=1,13
res12 = res12+cheb12(i)*rh(i)
res24 = res24+cheb24(i)*rh(i)
240 continue
do 250 i=14,25
res24 = res24+cheb24(i)*rh(i)
250 continue
260 result = (result+res24)*factor
abserr = (abserr+dabs(res24-res12))*factor
270 return
end
|
