00001 subroutine qagi(f,bound,inf,epsabs,epsrel,result,abserr,neval, 00002 * ier,limit,lenw,last,iwork,work) 00003 c***begin prologue qagi 00004 c***date written 800101 (yymmdd) 00005 c***revision date 830518 (yymmdd) 00006 c***category no. h2a3a1,h2a4a1 00007 c***keywords automatic integrator, infinite intervals, 00008 c general-purpose, transformation, extrapolation, 00009 c globally adaptive 00010 c***author piessens,robert,appl. math. & progr. div. - k.u.leuven 00011 c de doncker,elise,appl. math. & progr. div. -k.u.leuven 00012 c***purpose the routine calculates an approximation result to a given 00013 c integral i = integral of f over (bound,+infinity) 00014 c or i = integral of f over (-infinity,bound) 00015 c or i = integral of f over (-infinity,+infinity) 00016 c hopefully satisfying following claim for accuracy 00017 c abs(i-result).le.max(epsabs,epsrel*abs(i)). 00018 c***description 00019 c 00020 c integration over infinite intervals 00021 c standard fortran subroutine 00022 c 00023 c parameters 00024 c on entry 00025 c f - subroutine f(x,result) defining the integrand 00026 c function f(x). the actual name for f needs to be 00027 c declared e x t e r n a l in the driver program. 00028 c 00029 c bound - real 00030 c finite bound of integration range 00031 c (has no meaning if interval is doubly-infinite) 00032 c 00033 c inf - integer 00034 c indicating the kind of integration range involved 00035 c inf = 1 corresponds to (bound,+infinity), 00036 c inf = -1 to (-infinity,bound), 00037 c inf = 2 to (-infinity,+infinity). 00038 c 00039 c epsabs - real 00040 c absolute accuracy requested 00041 c epsrel - real 00042 c relative accuracy requested 00043 c if epsabs.le.0 00044 c and epsrel.lt.max(50*rel.mach.acc.,0.5d-28), 00045 c the routine will end with ier = 6. 00046 c 00047 c 00048 c on return 00049 c result - real 00050 c approximation to the integral 00051 c 00052 c abserr - real 00053 c estimate of the modulus of the absolute error, 00054 c which should equal or exceed abs(i-result) 00055 c 00056 c neval - integer 00057 c number of integrand evaluations 00058 c 00059 c ier - integer 00060 c ier = 0 normal and reliable termination of the 00061 c routine. it is assumed that the requested 00062 c accuracy has been achieved. 00063 c - ier.gt.0 abnormal termination of the routine. the 00064 c estimates for result and error are less 00065 c reliable. it is assumed that the requested 00066 c accuracy has not been achieved. 00067 c error messages 00068 c ier = 1 maximum number of subdivisions allowed 00069 c has been achieved. one can allow more 00070 c subdivisions by increasing the value of 00071 c limit (and taking the according dimension 00072 c adjustments into account). however, if 00073 c this yields no improvement it is advised 00074 c to analyze the integrand in order to 00075 c determine the integration difficulties. if 00076 c the position of a local difficulty can be 00077 c determined (e.g. singularity, 00078 c discontinuity within the interval) one 00079 c will probably gain from splitting up the 00080 c interval at this point and calling the 00081 c integrator on the subranges. if possible, 00082 c an appropriate special-purpose integrator 00083 c should be used, which is designed for 00084 c handling the type of difficulty involved. 00085 c = 2 the occurrence of roundoff error is 00086 c detected, which prevents the requested 00087 c tolerance from being achieved. 00088 c the error may be under-estimated. 00089 c = 3 extremely bad integrand behaviour occurs 00090 c at some points of the integration 00091 c interval. 00092 c = 4 the algorithm does not converge. 00093 c roundoff error is detected in the 00094 c extrapolation table. 00095 c it is assumed that the requested tolerance 00096 c cannot be achieved, and that the returned 00097 c result is the best which can be obtained. 00098 c = 5 the integral is probably divergent, or 00099 c slowly convergent. it must be noted that 00100 c divergence can occur with any other value 00101 c of ier. 00102 c = 6 the input is invalid, because 00103 c (epsabs.le.0 and 00104 c epsrel.lt.max(50*rel.mach.acc.,0.5d-28)) 00105 c or limit.lt.1 or leniw.lt.limit*4. 00106 c result, abserr, neval, last are set to 00107 c zero. exept when limit or leniw is 00108 c invalid, iwork(1), work(limit*2+1) and 00109 c work(limit*3+1) are set to zero, work(1) 00110 c is set to a and work(limit+1) to b. 00111 c 00112 c dimensioning parameters 00113 c limit - integer 00114 c dimensioning parameter for iwork 00115 c limit determines the maximum number of subintervals 00116 c in the partition of the given integration interval 00117 c (a,b), limit.ge.1. 00118 c if limit.lt.1, the routine will end with ier = 6. 00119 c 00120 c lenw - integer 00121 c dimensioning parameter for work 00122 c lenw must be at least limit*4. 00123 c if lenw.lt.limit*4, the routine will end 00124 c with ier = 6. 00125 c 00126 c last - integer 00127 c on return, last equals the number of subintervals 00128 c produced in the subdivision process, which 00129 c determines the number of significant elements 00130 c actually in the work arrays. 00131 c 00132 c work arrays 00133 c iwork - integer 00134 c vector of dimension at least limit, the first 00135 c k elements of which contain pointers 00136 c to the error estimates over the subintervals, 00137 c such that work(limit*3+iwork(1)),... , 00138 c work(limit*3+iwork(k)) form a decreasing 00139 c sequence, with k = last if last.le.(limit/2+2), and 00140 c k = limit+1-last otherwise 00141 c 00142 c work - real 00143 c vector of dimension at least lenw 00144 c on return 00145 c work(1), ..., work(last) contain the left 00146 c end points of the subintervals in the 00147 c partition of (a,b), 00148 c work(limit+1), ..., work(limit+last) contain 00149 c the right end points, 00150 c work(limit*2+1), ...,work(limit*2+last) contain the 00151 c integral approximations over the subintervals, 00152 c work(limit*3+1), ..., work(limit*3) 00153 c contain the error estimates. 00154 c***references (none) 00155 c***routines called qagie,xerror 00156 c***end prologue qagi 00157 c 00158 real abserr, epsabs,epsrel,result,work 00159 integer ier,iwork, lenw,limit,lvl,l1,l2,l3,neval 00160 c 00161 dimension iwork(limit),work(lenw) 00162 c 00163 external f 00164 c 00165 c check validity of limit and lenw. 00166 c 00167 c***first executable statement qagi 00168 ier = 6 00169 neval = 0 00170 last = 0 00171 result = 0.0e+00 00172 abserr = 0.0e+00 00173 if(limit.lt.1.or.lenw.lt.limit*4) go to 10 00174 c 00175 c prepare call for qagie. 00176 c 00177 l1 = limit+1 00178 l2 = limit+l1 00179 l3 = limit+l2 00180 c 00181 call qagie(f,bound,inf,epsabs,epsrel,limit,result,abserr, 00182 * neval,ier,work(1),work(l1),work(l2),work(l3),iwork,last) 00183 c 00184 c call error handler if necessary. 00185 c 00186 lvl = 0 00187 10 if(ier.eq.6) lvl = 1 00188 if(ier.ne.0) call xerror('abnormal return from qagi',26,ier,lvl) 00189 return 00190 end
1.7.1
