D01APF is an adaptive integrator which calculates an approximation to the integral of a function
gxwx over a finite interval
a,b:
where the weight function
w has end point singularities of algebraico-logarithmic type.
| SUBROUTINE D01APF ( | G, A, B, ALFA, BETA, KEY, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) |
| INTEGER | KEY, LW, IW(LIW), LIW, IFAIL |
| double precision | G, A, B, ALFA, BETA, EPSABS, EPSREL, RESULT, ABSERR, W(LW) |
| EXTERNAL | G |
D01APF is based on the QUADPACK routine QAWSE (see
Piessens et al. (1983)) and integrates a function of the form
gxwx, where the weight function
wx may have algebraico-logarithmic singularities at the end points
a and/or
b. The strategy is a modification of that in
D01AKF. We start by bisecting the original interval and applying modified Clenshaw–Curtis integration of orders
12 and
24 to both halves. Clenshaw–Curtis integration is then used on all sub-intervals which have
a or
b as one of their end points (see
Piessens et al. (1974)). On the other sub-intervals Gauss–Kronrod (
7–
15 point) integration is carried out.
A ‘global’ acceptance criterion (as defined by
Malcolm and Simpson (1976)) is used. The local error estimation control is described in
Piessens et al. (1983).
Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature
ACM Trans. Math. Software 1 129–146
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983)
QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag
Piessens R, Mertens I and Branders M (1974) Integration of functions having end-point singularities
Angew. Inf. 16 65–68
- 1: G – double precision FUNCTION, supplied by the user.External Procedure
G must return the value of the function
g at a given point
X.
The specification of
G is:
| double precision FUNCTION G ( | X) |
| double precision | X |
- 1: X – double precisionInput
On entry: the point at which the function g must be evaluated.
G must be declared as EXTERNAL in the (sub)program from which D01APF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 2: A – double precisionInput
On entry: a, the lower limit of integration.
- 3: B – double precisionInput
On entry: b, the upper limit of integration.
Constraint:
B>A.
- 4: ALFA – double precisionInput
On entry: the parameter α in the weight function.
Constraint:
ALFA>-1.
- 5: BETA – double precisionInput
On entry: the parameter β in the weight function.
Constraint:
BETA>-1.
- 6: KEY – INTEGERInput
On entry: indicates which weight function is to be used.
- KEY=1
- wx=x-aαb-x β.
- KEY=2
- wx= x-a α b-x βlnx-a.
- KEY=3
- wx= x-a α b-x βlnb-x.
- KEY=4
- wx= x-a α b-x βlnx-alnb-x.
Constraint:
KEY=1, 2, 3 or 4.
- 7: EPSABS – double precisionInput
On entry: the absolute accuracy required. If
EPSABS is negative, the absolute value is used. See
Section 7.
- 8: EPSREL – double precisionInput
On entry: the relative accuracy required. If
EPSREL is negative, the absolute value is used. See
Section 7.
- 9: RESULT – double precisionOutput
On exit: the approximation to the integral I.
- 10: ABSERR – double precisionOutput
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for I-RESULT.
- 11: W(LW) – double precision arrayOutput
On exit: details of the computation, as described in
Section 8.
- 12: LW – INTEGERInput
On entry: the dimension of the array
W as declared in the (sub)program from which D01APF is called. The value of
LW (together with that of
LIW) imposes a bound on the number of sub-intervals into which the interval of integration may be divided by the routine. The number of sub-intervals cannot exceed
LW/4. The more difficult the integrand, the larger
LW should be.
Suggested value:
LW=800 to 2000 is adequate for most problems.
Constraint:
LW≥8.
- 13: IW(LIW) – INTEGER arrayOutput
On exit: IW1 contains the actual number of sub-intervals used. The rest of the array is used as workspace.
- 14: LIW – INTEGERInput
On entry: the dimension of the array
IW as declared in the (sub)program from which D01APF is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed
LIW.
Suggested value:
LIW=LW/4.
Constraint:
LIW≥2.
- 15: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
IFAIL≠0 on exit, the recommended value is
-1.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
D01APF cannot guarantee, but in practice usually achieves, the following accuracy:
where
and
EPSABS and
EPSREL are user-specified absolute and relative error tolerances. Moreover, it returns the quantity
ABSERR which, in normal circumstances, satisfies
The time taken by D01APF depends on the integrand and the accuracy required.
If
IFAIL≠0 on exit, then you may wish to examine the contents of the array
W, which contains the end points of the sub-intervals used by D01APF along with the integral contributions and error estimates over these sub-intervals.
Specifically, for
i=1,2,…,n, let
ri denote the approximation to the value of the integral over the sub-interval
ai,bi
in the partition of
a,b
and
ei
be the corresponding absolute error estimate. Then,
∫
ai
bi
fx
wx
dx
≃
ri
and
RESULT
=
∑
i=1
n
ri
. The value of
n is returned in
IW1,
and the values
ai,
bi,
ei and
ri are stored consecutively in the
array
W,
that is:
- ai=Wi,
- bi=Wn+i,
- ei=W2n+i and
- ri=W3n+i.
None.