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
|
subroutine dqags(f,a,b,epsabs,epsrel,result,abserr,neval,ier,
* limit,lenw,last,iwork,work)
c***begin prologue dqags
c***date written 800101 (yymmdd)
c***revision date 830518 (yymmdd)
c***category no. h2a1a1
c***keywords automatic integrator, general-purpose,
c (end-point) singularities, extrapolation,
c globally adaptive
c***author piessens,robert,appl. math. & progr. div. - k.u.leuven
c de doncker,elise,appl. math. & prog. div. - k.u.leuven
c***purpose the routine calculates an approximation result to a given
c definite integral i = integral of f over (a,b),
c hopefully satisfying following claim for accuracy
c abs(i-result).le.max(epsabs,epsrel*abs(i)).
c***description
c
c computation of a definite integral
c standard fortran subroutine
c double precision version
c
c
c parameters
c on entry
c f - double precision
c function subprogram defining the integrand
c function f(x). the actual name for f needs to be
c declared e x t e r n a l in the driver program.
c
c a - double precision
c lower limit of integration
c
c b - double precision
c upper limit of integration
c
c epsabs - double precision
c absolute accuracy requested
c epsrel - double precision
c relative accuracy requested
c if epsabs.le.0
c and epsrel.lt.max(50*rel.mach.acc.,0.5d-28),
c the routine will end with ier = 6.
c
c on return
c result - double precision
c approximation to the integral
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 neval - integer
c number of integrand evaluations
c
c ier - integer
c ier = 0 normal and reliable termination of the
c routine. it is assumed that the requested
c accuracy has been achieved.
c ier.gt.0 abnormal termination of the routine
c the estimates for integral and error are
c less reliable. it is assumed that the
c requested accuracy has not been achieved.
c error messages
c ier = 1 maximum number of subdivisions allowed
c has been achieved. one can allow more sub-
c divisions by increasing the value of limit
c (and taking the according dimension
c adjustments into account. however, if
c this yields no improvement it is advised
c to analyze the integrand in order to
c determine the integration difficulties. if
c the position of a local difficulty can be
c determined (e.g. singularity,
c discontinuity within the interval) one
c will probably gain from splitting up the
c interval at this point and calling the
c integrator on the subranges. if possible,
c an appropriate special-purpose integrator
c should be used, which is designed for
c handling the type of difficulty involved.
c = 2 the occurrence of roundoff error is detec-
c ted, which prevents the requested
c tolerance from being achieved.
c the error may be under-estimated.
c = 3 extremely bad integrand behaviour
c occurs at some points of the integration
c interval.
c = 4 the algorithm does not converge.
c roundoff error is detected in the
c extrapolation table. it is presumed that
c the requested tolerance cannot be
c achieved, and that the returned result is
c the best which can be obtained.
c = 5 the integral is probably divergent, or
c slowly convergent. it must be noted that
c divergence can occur with any other value
c of ier.
c = 6 the input is invalid, because
c (epsabs.le.0 and
c epsrel.lt.max(50*rel.mach.acc.,0.5d-28)
c or limit.lt.1 or lenw.lt.limit*4.
c result, abserr, neval, last are set to
c zero.except when limit or lenw is invalid,
c iwork(1), work(limit*2+1) and
c work(limit*3+1) are set to zero, work(1)
c is set to a and work(limit+1) to b.
c
c dimensioning parameters
c limit - integer
c dimensioning parameter for iwork
c limit determines the maximum number of subintervals
c in the partition of the given integration interval
c (a,b), limit.ge.1.
c if limit.lt.1, the routine will end with ier = 6.
c
c lenw - integer
c dimensioning parameter for work
c lenw must be at least limit*4.
c if lenw.lt.limit*4, the routine will end
c with ier = 6.
c
c last - integer
c on return, last equals the number of subintervals
c produced in the subdivision process, detemines the
c number of significant elements actually in the work
c arrays.
c
c work arrays
c iwork - integer
c vector of dimension at least limit, the first k
c elements of which contain pointers
c to the error estimates over the subintervals
c such that work(limit*3+iwork(1)),... ,
c work(limit*3+iwork(k)) form a decreasing
c sequence, with k = last if last.le.(limit/2+2),
c and k = limit+1-last otherwise
c
c work - double precision
c vector of dimension at least lenw
c on return
c work(1), ..., work(last) contain the left
c end-points of the subintervals in the
c partition of (a,b),
c work(limit+1), ..., work(limit+last) contain
c the right end-points,
c work(limit*2+1), ..., work(limit*2+last) contain
c the integral approximations over the subintervals,
c work(limit*3+1), ..., work(limit*3+last)
c contain the error estimates.
c
c***references (none)
c***routines called dqagse,xerror
c***end prologue dqags
c
c
double precision a,abserr,b,epsabs,epsrel,f,result,work
integer ier,iwork,last,lenw,limit,lvl,l1,l2,l3,neval
c
dimension iwork(limit),work(lenw)
c
external f
c
c check validity of limit and lenw.
c
c***first executable statement dqags
ier = 6
neval = 0
last = 0
result = 0.0d+00
abserr = 0.0d+00
if(limit.lt.1.or.lenw.lt.limit*4) go to 10
c
c prepare call for dqagse.
c
l1 = limit+1
l2 = limit+l1
l3 = limit+l2
c
call dqagse(f,a,b,epsabs,epsrel,limit,result,abserr,neval,
* ier,work(1),work(l1),work(l2),work(l3),iwork,last)
c
c call error handler if necessary.
c
lvl = 0
10 if(ier.eq.6) lvl = 1
if(ier.ne.0) call xerror('abnormal return from dqags',26,ier,lvl)
return
end
|
