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integrate.c
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integrate.c
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/*
* R : A Computer Language for Statistical Data Analysis
* Copyright (C) 2001-2014 The R Core Team
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, a copy is available at
* https://www.R-project.org/Licenses/
*
* Most of this file is C translations of Fortran routines in
* QUADPACK: the latter is part of SLATEC 'and therefore in the public
* domain' (https://en.wikipedia.org/wiki/QUADPACK).
*
*
*/
#ifdef HAVE_CONFIG_H
#include <config.h>
#endif
#include <math.h>
#include <float.h>
#include <Rmath.h> /* for fmax2, fmin2, imin2 */
#include <R_ext/Applic.h> /* exporting the API , particularly */
/*--- typedef void integr_fn(double *x, int n, void *ex) ---
* vectorizing function f(x[1:n], ...) -> x[] {overwriting x[]}.
* Vectorization can be used to speed up the integrand
* instead of calling it n times.
*/
/* f2c-ed translations + modifications of QUADPACK functions */
static void rdqagie(integr_fn f, void *ex,
double *, int *, double * , double *, int *,
double *, double *, int *,
int *, double *, double *, double *, double *,
int *, int *);
static void rdqk15i(integr_fn f, void *ex,
double *, int *, double * , double *,
double *, double *, double *, double *);
static void rdqagse(integr_fn f, void *ex, double *, double *,
double *, double *, int *, double *, double *,
int *, int *, double *, double *, double *,
double *, int *, int *);
static void rdqk21(integr_fn f, void *ex,
double *, double *, double *, double *, double *, double *);
static void rdqpsrt(int *, int *, int *, double *, double *, int *, int *);
static void rdqelg(int *, double *, double *, double *, double *, int *);
/* Table of constant values */
static double c_b6 = 0.;
static double c_b7 = 1.;
void Rdqagi(integr_fn f, void *ex, double *bound, int *inf,
double *epsabs, double *epsrel,
double *result, double *abserr, int *neval, int *ier,
int *limit, int *lenw, int *last,
int *iwork, double *work)
{
int l1, l2, l3;
/*
***begin prologue dqagi
***date written 800101 (yymmdd)
***revision date 830518 (yymmdd)
***category no. h2a3a1,h2a4a1
***keywords automatic integrator, infinite intervals,
general-purpose, transformation, extrapolation,
globally adaptive
***author piessens,robert,appl. math. & progr. div. - k.u.leuven
de doncker,elise,appl. math. & progr. div. -k.u.leuven
***purpose the routine calculates an approximation result to a given
integral i = integral of f over (bound,+infinity)
or i = integral of f over (-infinity,bound)
or i = integral of f over (-infinity,+infinity)
hopefully satisfying following claim for accuracy
abs(i-result) <= max(epsabs,epsrel*abs(i)).
***description
integration over infinite intervals
standard fortran subroutine
parameters
on entry
f - double precision
function subprogram defining the integrand
function f(x). the actual name for f needs to be
declared e x t e r n a l in the driver program.
bound - double precision
finite bound of integration range
(has no meaning if interval is doubly-infinite)
inf - int
indicating the kind of integration range involved
inf = 1 corresponds to (bound,+infinity),
inf = -1 to (-infinity,bound),
inf = 2 to (-infinity,+infinity).
epsabs - double precision
absolute accuracy requested
epsrel - double precision
relative accuracy requested
if epsabs <= 0
and epsrel < max(50*rel.mach.acc.,0.5d-28),
the routine will end with ier = 6.
on return
result - double precision
approximation to the integral
abserr - double precision
estimate of the modulus of the absolute error,
which should equal or exceed abs(i-result)
neval - int
number of integrand evaluations
ier - int
ier = 0 normal and reliable termination of the
routine. it is assumed that the requested
accuracy has been achieved.
- ier > 0 abnormal termination of the routine. the
estimates for result and error are less
reliable. it is assumed that the requested
accuracy has not been achieved.
error messages
ier = 1 maximum number of subdivisions allowed
has been achieved. one can allow more
subdivisions by increasing the value of
limit (and taking the according dimension
adjustments into account). however, if
this yields no improvement it is advised
to analyze the integrand in order to
determine the integration difficulties. if
the position of a local difficulty can be
determined (e.g. singularity,
discontinuity within the interval) one
will probably gain from splitting up the
interval at this point and calling the
integrator on the subranges. if possible,
an appropriate special-purpose integrator
should be used, which is designed for
handling the type of difficulty involved.
= 2 the occurrence of roundoff error is
detected, which prevents the requested
tolerance from being achieved.
the error may be under-estimated.
= 3 extremely bad integrand behaviour occurs
at some points of the integration
interval.
= 4 the algorithm does not converge.
roundoff error is detected in the
extrapolation table.
it is assumed that the requested tolerance
cannot be achieved, and that the returned
result is the best which can be obtained.
= 5 the integral is probably divergent, or
slowly convergent. it must be noted that
divergence can occur with any other value
of ier.
= 6 the input is invalid, because
(epsabs <= 0 and
epsrel < max(50*rel.mach.acc.,0.5d-28))
or limit < 1 or leniw < limit*4.
result, abserr, neval, last are set to
zero. exept when limit or leniw is
invalid, iwork(1), work(limit*2+1) and
work(limit*3+1) are set to zero, work(1)
is set to a and work(limit+1) to b.
dimensioning parameters
limit - int
dimensioning parameter for iwork
limit determines the maximum number of subintervals
in the partition of the given integration interval
(a,b), limit >= 1.
if limit < 1, the routine will end with ier = 6.
lenw - int
dimensioning parameter for work
lenw must be at least limit*4.
if lenw < limit*4, the routine will end
with ier = 6.
last - int
on return, last equals the number of subintervals
produced in the subdivision process, which
determines the number of significant elements
actually in the work arrays.
work arrays
iwork - int
vector of dimension at least limit, the first
k elements of which contain pointers
to the error estimates over the subintervals,
such that work(limit*3+iwork(1)),... ,
work(limit*3+iwork(k)) form a decreasing
sequence, with k = last if last <= (limit/2+2), and
k = limit+1-last otherwise
work - double precision
vector of dimension at least lenw
on return
work(1), ..., work(last) contain the left
end points of the subintervals in the
partition of (a,b),
work(limit+1), ..., work(limit+last) contain
the right end points,
work(limit*2+1), ...,work(limit*2+last) contain the
integral approximations over the subintervals,
work(limit*3+1), ..., work(limit*3)
contain the error estimates.
***routines called dqagie
***end prologue dqagi */
*ier = 6;
*neval = 0;
*last = 0;
*result = 0.;
*abserr = 0.;
if (*limit < 1 || *lenw < *limit << 2) return;
l1 = *limit;
l2 = *limit + l1;
l3 = *limit + l2;
rdqagie(f, ex, bound, inf, epsabs, epsrel, limit, result, abserr, neval, ier,
work, &work[l1], &work[l2], &work[l3], iwork, last);
return;
} /* Rdqagi */
static
void rdqagie(integr_fn f, void *ex, double *bound, int *inf, double *
epsabs, double *epsrel, int *limit, double *result,
double *abserr, int *neval, int *ier, double *alist,
double *blist, double *rlist, double *elist, int *
iord, int *last)
{
/* Local variables */
double area, dres;
int ksgn;
double boun;
int nres;
double area1, area2, area12;
int k;
double small = 0.0, erro12;
int ierro;
double a1, a2, b1, b2, defab1, defab2, oflow;
int ktmin, nrmax;
double uflow;
Rboolean noext;
int iroff1, iroff2, iroff3;
double res3la[3], error1, error2;
int id;
double rlist2[52];
int numrl2;
double defabs, epmach, erlarg = 0.0, abseps, correc = 0.0, errbnd, resabs;
int jupbnd;
double erlast, errmax;
int maxerr;
double reseps;
Rboolean extrap;
double ertest = 0.0, errsum;
/**begin prologue dqagie
***date written 800101 (yymmdd)
***revision date 830518 (yymmdd)
***category no. h2a3a1,h2a4a1
***keywords automatic integrator, infinite intervals,
general-purpose, transformation, extrapolation,
globally adaptive
***author piessens,robert,appl. math & progr. div - k.u.leuven
de doncker,elise,appl. math & progr. div - k.u.leuven
***purpose the routine calculates an approximation result to a given
integral i = integral of f over (bound,+infinity)
or i = integral of f over (-infinity,bound)
or i = integral of f over (-infinity,+infinity),
hopefully satisfying following claim for accuracy
abs(i-result) <= max(epsabs,epsrel*abs(i))
***description
integration over infinite intervals
standard fortran subroutine
f - double precision
function subprogram defining the integrand
function f(x). the actual name for f needs to be
declared e x t e r n a l in the driver program.
bound - double precision
finite bound of integration range
(has no meaning if interval is doubly-infinite)
inf - double precision
indicating the kind of integration range involved
inf = 1 corresponds to (bound,+infinity),
inf = -1 to (-infinity,bound),
inf = 2 to (-infinity,+infinity).
epsabs - double precision
absolute accuracy requested
epsrel - double precision
relative accuracy requested
if epsabs <= 0
and epsrel < max(50*rel.mach.acc.,0.5d-28),
the routine will end with ier = 6.
limit - int
gives an upper bound on the number of subintervals
in the partition of (a,b), limit >= 1
on return
result - double precision
approximation to the integral
abserr - double precision
estimate of the modulus of the absolute error,
which should equal or exceed abs(i-result)
neval - int
number of integrand evaluations
ier - int
ier = 0 normal and reliable termination of the
routine. it is assumed that the requested
accuracy has been achieved.
- ier > 0 abnormal termination of the routine. the
estimates for result and error are less
reliable. it is assumed that the requested
accuracy has not been achieved.
error messages
ier = 1 maximum number of subdivisions allowed
has been achieved. one can allow more
subdivisions by increasing the value of
limit (and taking the according dimension
adjustments into account). however,if
this yields no improvement it is advised
to analyze the integrand in order to
determine the integration difficulties.
if the position of a local difficulty can
be determined (e.g. singularity,
discontinuity within the interval) one
will probably gain from splitting up the
interval at this point and calling the
integrator on the subranges. if possible,
an appropriate special-purpose integrator
should be used, which is designed for
handling the type of difficulty involved.
= 2 the occurrence of roundoff error is
detected, which prevents the requested
tolerance from being achieved.
the error may be under-estimated.
= 3 extremely bad integrand behaviour occurs
at some points of the integration
interval.
= 4 the algorithm does not converge.
roundoff error is detected in the
extrapolation table.
it is assumed that the requested tolerance
cannot be achieved, and that the returned
result is the best which can be obtained.
= 5 the integral is probably divergent, or
slowly convergent. it must be noted that
divergence can occur with any other value
of ier.
= 6 the input is invalid, because
(epsabs <= 0 and
epsrel < max(50*rel.mach.acc.,0.5d-28),
result, abserr, neval, last, rlist(1),
elist(1) and iord(1) are set to zero.
alist(1) and blist(1) are set to 0
and 1 respectively.
alist - double precision
vector of dimension at least limit, the first
last elements of which are the left
end points of the subintervals in the partition
of the transformed integration range (0,1).
blist - double precision
vector of dimension at least limit, the first
last elements of which are the right
end points of the subintervals in the partition
of the transformed integration range (0,1).
rlist - double precision
vector of dimension at least limit, the first
last elements of which are the integral
approximations on the subintervals
elist - double precision
vector of dimension at least limit, the first
last elements of which are the moduli of the
absolute error estimates on the subintervals
iord - int
vector of dimension limit, the first k
elements of which are pointers to the
error estimates over the subintervals,
such that elist(iord(1)), ..., elist(iord(k))
form a decreasing sequence, with k = last
if last <= (limit/2+2), and k = limit+1-last
otherwise
last - int
number of subintervals actually produced
in the subdivision process
***routines called dqelg,dqk15i,dqpsrt
***end prologue dqagie
the dimension of rlist2 is determined by the value of
limexp in subroutine dqelg.
list of major variables
-----------------------
alist - list of left end points of all subintervals
considered up to now
blist - list of right end points of all subintervals
considered up to now
rlist(i) - approximation to the integral over
(alist(i),blist(i))
rlist2 - array of dimension at least (limexp+2),
containing the part of the epsilon table
wich is still needed for further computations
elist(i) - error estimate applying to rlist(i)
maxerr - pointer to the interval with largest error
estimate
errmax - elist(maxerr)
erlast - error on the interval currently subdivided
(before that subdivision has taken place)
area - sum of the integrals over the subintervals
errsum - sum of the errors over the subintervals
errbnd - requested accuracy max(epsabs,epsrel*
abs(result))
*****1 - variable for the left subinterval
*****2 - variable for the right subinterval
last - index for subdivision
nres - number of calls to the extrapolation routine
numrl2 - number of elements currently in rlist2. if an
appropriate approximation to the compounded
integral has been obtained, it is put in
rlist2(numrl2) after numrl2 has been increased
by one.
small - length of the smallest interval considered up
to now, multiplied by 1.5
erlarg - sum of the errors over the intervals larger
than the smallest interval considered up to now
extrap - logical variable denoting that the routine
is attempting to perform extrapolation. i.e.
before subdividing the smallest interval we
try to decrease the value of erlarg.
noext - logical variable denoting that extrapolation
is no longer allowed (true-value)
machine dependent constants
---------------------------
epmach is the largest relative spacing.
uflow is the smallest positive magnitude.
oflow is the largest positive magnitude. */
/* ***first executable statement dqagie */
/* Parameter adjustments */
--iord;
--elist;
--rlist;
--blist;
--alist;
/* Function Body */
epmach = DBL_EPSILON;
/* test on validity of parameters */
/* ----------------------------- */
*ier = 0;
*neval = 0;
*last = 0;
*result = 0.;
*abserr = 0.;
alist[1] = 0.;
blist[1] = 1.;
rlist[1] = 0.;
elist[1] = 0.;
iord[1] = 0;
if (*epsabs <= 0. && (*epsrel < fmax2(epmach * 50., 5e-29))) *ier = 6;
if (*ier == 6) return;
/* first approximation to the integral */
/* ----------------------------------- */
/* determine the interval to be mapped onto (0,1).
if inf = 2 the integral is computed as i = i1+i2, where
i1 = integral of f over (-infinity,0),
i2 = integral of f over (0,+infinity). */
boun = *bound;
if (*inf == 2) {
boun = 0.;
}
rdqk15i(f, ex, &boun, inf, &c_b6, &c_b7, result, abserr, &defabs, &resabs);
/* test on accuracy */
*last = 1;
rlist[1] = *result;
elist[1] = *abserr;
iord[1] = 1;
dres = fabs(*result);
errbnd = fmax2(*epsabs, *epsrel * dres);
if (*abserr <= epmach * 100. * defabs && *abserr > errbnd) *ier = 2;
if (*limit == 1) *ier = 1;
if (*ier != 0 || (*abserr <= errbnd && *abserr != resabs)
|| *abserr == 0.) goto L130;
/* initialization */
/* -------------- */
uflow = DBL_MIN;
oflow = DBL_MAX;
rlist2[0] = *result;
errmax = *abserr;
maxerr = 1;
area = *result;
errsum = *abserr;
*abserr = oflow;
nrmax = 1;
nres = 0;
ktmin = 0;
numrl2 = 2;
extrap = FALSE;
noext = FALSE;
ierro = 0;
iroff1 = 0;
iroff2 = 0;
iroff3 = 0;
ksgn = -1;
if (dres >= (1. - epmach * 50.) * defabs) {
ksgn = 1;
}
/* main do-loop */
/* ------------ */
for (*last = 2; *last <= *limit; ++(*last)) {
/* bisect the subinterval with nrmax-th largest error estimate. */
a1 = alist[maxerr];
b1 = (alist[maxerr] + blist[maxerr]) * .5;
a2 = b1;
b2 = blist[maxerr];
erlast = errmax;
rdqk15i(f, ex, &boun, inf, &a1, &b1, &area1, &error1, &resabs, &defab1);
rdqk15i(f, ex, &boun, inf, &a2, &b2, &area2, &error2, &resabs, &defab2);
/* improve previous approximations to integral
and error and test for accuracy. */
area12 = area1 + area2;
erro12 = error1 + error2;
errsum = errsum + erro12 - errmax;
area = area + area12 - rlist[maxerr];
if (!(defab1 == error1 || defab2 == error2)) {
if (fabs(rlist[maxerr] - area12) <= fabs(area12) * 1e-5 &&
erro12 >= errmax * .99) {
if (extrap)
++iroff2;
else /* if (! extrap) */
++iroff1;
}
if (*last > 10 && erro12 > errmax)
++iroff3;
}
rlist[maxerr] = area1;
rlist[*last] = area2;
errbnd = fmax2(*epsabs, *epsrel * fabs(area));
/* test for roundoff error and eventually set error flag. */
if (iroff1 + iroff2 >= 10 || iroff3 >= 20)
*ier = 2;
if (iroff2 >= 5)
ierro = 3;
/* set error flag in the case that the number of
subintervals equals limit. */
if (*last == *limit)
*ier = 1;
/* set error flag in the case of bad integrand behaviour
at some points of the integration range. */
if (fmax2(fabs(a1), fabs(b2)) <=
(epmach * 100. + 1.) * (fabs(a2) + uflow * 1e3))
{
*ier = 4;
}
/* append the newly-created intervals to the list. */
if (error2 <= error1) {
alist[*last] = a2;
blist[maxerr] = b1;
blist[*last] = b2;
elist[maxerr] = error1;
elist[*last] = error2;
}
else {
alist[maxerr] = a2;
alist[*last] = a1;
blist[*last] = b1;
rlist[maxerr] = area2;
rlist[*last] = area1;
elist[maxerr] = error2;
elist[*last] = error1;
}
/* call subroutine dqpsrt to maintain the descending ordering
in the list of error estimates and select the subinterval
with nrmax-th largest error estimate (to be bisected next). */
rdqpsrt(limit, last, &maxerr, &errmax, &elist[1], &iord[1], &nrmax);
if (errsum <= errbnd) {
goto L115;
}
if (*ier != 0) break;
if (*last == 2) { /* L80: */
small = .375;
erlarg = errsum;
ertest = errbnd;
rlist2[1] = area; continue;
}
if (noext) continue;
erlarg -= erlast;
if (fabs(b1 - a1) > small) {
erlarg += erro12;
}
if (!extrap) {
/* test whether the interval to be bisected next is the
smallest interval. */
if (fabs(blist[maxerr] - alist[maxerr]) > small) {
continue;
}
extrap = TRUE;
nrmax = 2;
}
if (ierro != 3 && erlarg > ertest) {
/* the smallest interval has the largest error.
before bisecting decrease the sum of the errors over the
larger intervals (erlarg) and perform extrapolation. */
id = nrmax;
jupbnd = *last;
if (*last > *limit / 2 + 2) {
jupbnd = *limit + 3 - *last;
}
for (k = id; k <= jupbnd; ++k) {
maxerr = iord[nrmax];
errmax = elist[maxerr];
if (fabs(blist[maxerr] - alist[maxerr]) > small) {
goto L90;
}
++nrmax;
/* L50: */
}
}
/* perform extrapolation. L60: */
++numrl2;
rlist2[numrl2 - 1] = area;
rdqelg(&numrl2, rlist2, &reseps, &abseps, res3la, &nres);
++ktmin;
if (ktmin > 5 && *abserr < errsum * .001) {
*ier = 5;
}
if (abseps >= *abserr) {
goto L70;
}
ktmin = 0;
*abserr = abseps;
*result = reseps;
correc = erlarg;
ertest = fmax2(*epsabs, *epsrel * fabs(reseps));
if (*abserr <= ertest) {
break;
}
/* prepare bisection of the smallest interval. */
L70:
if (numrl2 == 1) {
noext = TRUE;
}
if (*ier == 5) {
break;
}
maxerr = iord[1];
errmax = elist[maxerr];
nrmax = 1;
extrap = FALSE;
small *= .5;
erlarg = errsum;
L90:
;
}
/* L100: set final result and error estimate. */
/* ------------------------------------ */
if (*abserr == oflow) {
goto L115;
}
if (*ier + ierro == 0) {
goto L110;
}
if (ierro == 3) {
*abserr += correc;
}
if (*ier == 0) {
*ier = 3;
}
if (*result == 0. || area == 0.) {
if (*abserr > errsum)
goto L115;
if (area == 0.)
goto L130;
}
else { /* L105: */
if (*abserr / fabs(*result) > errsum / fabs(area)) {
goto L115;
}
}
/* test on divergence */
L110:
if (ksgn == -1 && fmax2(fabs(*result), fabs(area)) <= defabs * .01) {
goto L130;
}
if (.01 > *result / area || *result / area > 100. || errsum > fabs(area)) {
*ier = 6;
}
goto L130;
/* compute global integral sum. */
L115:
*result = 0.;
for (k = 1; k <= *last; ++k)
*result += rlist[k];
*abserr = errsum;
L130:
*neval = *last * 30 - 15;
if (*inf == 2) {
*neval <<= 1;
}
if (*ier > 2) {
--(*ier);
}
return;
} /* rdqagie_ */
void Rdqags(integr_fn f, void *ex, double *a, double *b,
double *epsabs, double *epsrel,
double *result, double *abserr, int *neval, int *ier,
int *limit, int *lenw, int *last, int *iwork, double *work)
{
int l1, l2, l3;
/*
***begin prologue dqags
***date written 800101 (yymmdd)
***revision date 830518 (yymmdd)
***category no. h2a1a1
***keywords automatic integrator, general-purpose,
(end-point) singularities, extrapolation,
globally adaptive
***author piessens,robert,appl. math. & progr. div. - k.u.leuven
de doncker,elise,appl. math. & prog. div. - k.u.leuven
***purpose the routine calculates an approximation result to a given
definite integral i = integral of f over (a,b),
hopefully satisfying following claim for accuracy
abs(i-result) <= max(epsabs,epsrel*abs(i)).
***description
computation of a definite integral
standard fortran subroutine
double precision version
parameters
on entry
f - double precision
function subprogram defining the integrand
function f(x). the actual name for f needs to be
declared e x t e r n a l in the driver program.
a - double precision
lower limit of integration
b - double precision
upper limit of integration
epsabs - double precision
absolute accuracy requested
epsrel - double precision
relative accuracy requested
if epsabs <= 0
and epsrel < max(50*rel.mach.acc.,0.5d-28),
the routine will end with ier = 6.
on return
result - double precision
approximation to the integral
abserr - double precision
estimate of the modulus of the absolute error,
which should equal or exceed abs(i-result)
neval - int
number of integrand evaluations
ier - int
ier = 0 normal and reliable termination of the
routine. it is assumed that the requested
accuracy has been achieved.
ier > 0 abnormal termination of the routine
the estimates for integral and error are
less reliable. it is assumed that the
requested accuracy has not been achieved.
error messages
ier = 1 maximum number of subdivisions allowed
has been achieved. one can allow more sub-
divisions by increasing the value of limit
(and taking the according dimension
adjustments into account. however, if
this yields no improvement it is advised
to analyze the integrand in order to
determine the integration difficulties. if
the position of a local difficulty can be
determined (e.g. singularity,
discontinuity within the interval) one
will probably gain from splitting up the
interval at this point and calling the
integrator on the subranges. if possible,
an appropriate special-purpose integrator
should be used, which is designed for
handling the type of difficulty involved.
= 2 the occurrence of roundoff error is detec-
ted, which prevents the requested
tolerance from being achieved.
the error may be under-estimated.
= 3 extremely bad integrand behaviour
occurs at some points of the integration
interval.
= 4 the algorithm does not converge.
roundoff error is detected in the
extrapolation table. it is presumed that
the requested tolerance cannot be
achieved, and that the returned result is
the best which can be obtained.
= 5 the integral is probably divergent, or
slowly convergent. it must be noted that
divergence can occur with any other value
of ier.
= 6 the input is invalid, because
(epsabs <= 0 and
epsrel < max(50*rel.mach.acc.,0.5d-28)
or limit < 1 or lenw < limit*4.
result, abserr, neval, last are set to
zero.except when limit or lenw is invalid,
iwork(1), work(limit*2+1) and
work(limit*3+1) are set to zero, work(1)
is set to a and work(limit+1) to b.
dimensioning parameters
limit - int
dimensioning parameter for iwork
limit determines the maximum number of subintervals
in the partition of the given integration interval
(a,b), limit >= 1.
if limit < 1, the routine will end with ier = 6.
lenw - int
dimensioning parameter for work
lenw must be at least limit*4.
if lenw < limit*4, the routine will end
with ier = 6.
last - int
on return, last equals the number of subintervals
produced in the subdivision process, detemines the
number of significant elements actually in the work
arrays.
work arrays
iwork - int
vector of dimension at least limit, the first k
elements of which contain pointers
to the error estimates over the subintervals
such that work(limit*3+iwork(1)),... ,
work(limit*3+iwork(k)) form a decreasing
sequence, with k = last if last <= (limit/2+2),
and k = limit+1-last otherwise
work - double precision
vector of dimension at least lenw
on return
work(1), ..., work(last) contain the left
end-points of the subintervals in the
partition of (a,b),
work(limit+1), ..., work(limit+last) contain
the right end-points,
work(limit*2+1), ..., work(limit*2+last) contain
the integral approximations over the subintervals,
work(limit*3+1), ..., work(limit*3+last)
contain the error estimates.
***routines called dqagse
***end prologue dqags */
/* check validity of limit and lenw. */
*ier = 6;
*neval = 0;
*last = 0;
*result = 0.;
*abserr = 0.;
if (*limit < 1 || *lenw < *limit *4) return;
/* prepare call for dqagse. */
l1 = *limit;
l2 = *limit + l1;
l3 = *limit + l2;
rdqagse(f, ex, a, b, epsabs, epsrel, limit, result, abserr, neval, ier,
work, &work[l1], &work[l2], &work[l3], iwork, last);
return;
} /* rdqags_ */
static
void rdqagse(integr_fn f, void *ex, double *a, double *b, double *
epsabs, double *epsrel, int *limit, double *result,
double *abserr, int *neval, int *ier, double *alist,
double *blist, double *rlist, double *elist, int *
iord, int *last)
{
/* Local variables */
Rboolean noext, extrap;
int k,ksgn, nres;
int ierro;
int ktmin, nrmax;
int iroff1, iroff2, iroff3;
int id;
int numrl2;
int jupbnd;
int maxerr;
double res3la[3];
double rlist2[52];
double abseps, area, area1, area2, area12, dres, epmach;
double a1, a2, b1, b2, defabs, defab1, defab2, oflow, uflow, resabs, reseps;
double error1, error2, erro12, errbnd, erlast, errmax, errsum;
double correc = 0.0, erlarg = 0.0, ertest = 0.0, small = 0.0;
/*
***begin prologue dqagse
***date written 800101 (yymmdd)
***revision date 830518 (yymmdd)