qpint1d.pro
60.6 KB
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;+
; NAME:
; QPINT1D
;
; AUTHOR:
; Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
; craigm@lheamail.gsfc.nasa.gov
; UPDATED VERSIONs can be found on my WEB PAGE:
; http://cow.physics.wisc.edu/~craigm/idl/idl.html
;
; PURPOSE:
; One dimensional numerical integration of IDL function or expression
;
; MAJOR TOPICS:
; Numerical Analysis.
;
; CALLING SEQUENCE:
; value = QPINT1D(FUNCT, A, B, [ PRIVATE, /EXPRESSION, FUNCTARGS=,
; ERROR=error, NFEV=nfev, STATUS=status, NSUBINTERVALS=nsub,
; EPSABS=, EPSREL=, LIMIT=, BREAKPOINTS=, NPOINTS=
; /SYMMETRIC, SYM_AXIS= ] )
;
; DESCRIPTION:
;
; QPINT1D adaptively calculates an approximation result to a given
; definite integral
;
; result = Integral[ f(x) dx ] over [a,b]
;
; hopefully satisfying a constraint on the accuracy of the solution.
; QPINT1D is based on the QUADPACK fortran package originally by
; Piessens, de Doncker, Ueberhuber and Kahaner (and implements
; equivalents to the QAGSE, QAGPE, QAGIE, and DQKxx fortran routines).
;
; The returned result is intended to satisfy the following claim for
; accuracy: ABS(result-value) LE MAX([epsabs, epsrel*ABS(value)]),
; where VALUE is the true value of the integral, and EPSABS and
; EPSREL are the absolute and relative error tolerances defined
; below. An estimate of the error is returned in the ERROR keyword.
; Either A or B may be finite or infinite (i.e., an improper
; integral).
;
; QPINT1D is "adaptive" in the sense that it locates regions of the
; integration interval which contain the highest error, and focusses
; its efforts on those regions. The algorithm locates these regions
; by successively bisecting the starting interval. Each subinterval
; is assigned an error estimate, and the region with the largest
; error estimate is subdivided further, until each subinterval
; carries approximately the same amount of error. Convergence of the
; procedure may be accelerated by the Epsilon algorithm due to Wynn.
;
; The estimate of the integral and the estimate of the error in each
; subinterval are computed using Gauss Kronrod quadrature.
; Integrators based on the 15-, 21-, 31-, 41-, 51- and 61-point
; Gauss-Kronrod rule are available, and selected using the NPOINTS
; keyword. Generally, the more points the greater the precision,
; especially for rapidly varying functions. However the default
; value of 21 is often sufficient, especially because of the adaptive
; nature of QPINT1D.
;
; In the following sections the requirements for the form of the
; integrand are established. Also, a description of how QPINT1D
; handles singularities and discontinuities is presented.
;
; INTEGRAND is a FUNCTION
;
; The integrand can be specified in two forms, either as a standard
; IDL function, or as an IDL expression. If integrating a function,
; then the FUNCT should be a string naming the function. The
; function must be declared as following:
;
; FUNCTION MYFUNCT, X, P, KEYWORDS=...
; RETURN, (compute function of X and P)
; END
;
; The function must accept at least one, but optionally two,
; parameters. The first, 'X', is a vector of abcissae where the
; function is to be computed. The function must return the same
; number of function values as abcissae passed. The second
; positional parameter, 'P', is a purely optional PRIVATE parameter
; as described below. MYFUNCT may accept more positional parameters,
; but QPINT1D will not use them. The difference between X and P is
; that X is the variable of integration, while P contains any other
; information expected to remain essentially constant over the
; integration.
;
; INTEGRAND is an EXPRESSION
;
; The integrand can also be specfied as an IDL expression and setting
; the EXPRESSION keyword. Any expression that can accept a vector of
; abcissae named 'X' and produce a corresponding vector of output is
; a valid expression. Here is an example:
;
; RESULT = QPINT1D('X^2 * EXP(-X)', /EXPRESSION, 0D, 10D)
;
; It is important to note that the variable of integration must
; always be named 'X', and the expression must be vectorizable. The
; expression may also use the PRIVATE data, and as above, it would be
; referred to according to the variable 'P'. For example, if the
; exponential decay constant is parameterized by PRIVATE(0), then the
; expression would be:
;
; RESULT = QPINT1D('X^2 * EXP(-X/P(0))', /EXPRESSION, 0D, 10D, PRIVATE)
;
; The user is solely responsible for defining and using the PRIVATE
; data. QPINT1D does not access or modify PRIVATE / P; it only
; passes it on to the user routine for convenience.
;
; IMPROPER INTEGRALS and DISCONTINUITIES
;
; QPINT1D computes improper integrals, as well as integrands with
; discontinuities or singularities.
;
; Improper integrals are integrals where one or both of the limits of
; integration are "infinity." (Formally, these integrals are defined
; by taking the limit as the integration limit tends to infinity).
; QPINT1D handles a small class of such integrals, generally for
; integrands that are convergent and monotonic (i.e.,
; non-oscillatory, and falling off as 1/ABS(X)^2 or steeper). Such
; integrals are handled by a transformation of the original interval
; into the interval [0,1].
;
; Integrals from negative infinity to positive infinity are done in
; two subintervals. By default the interval is split at X EQ 0,
; however this can be controlled by using the SYM_AXIS keyword.
; Users should note that if the first subinterval fails the second is
; not attempted, and thus the return value VALUE should not be
; trusted in those cases.
;
; Infinite integration limits are specified by using the standard
; values !VALUES.F_INFINITY or !VALUES.D_INFINITY. No other special
; invocation syntax is required.
;
; The integration routine is able to handle integrands which have
; integrable singularities at the endpoints. For example, the
; integral:
;
; RESULT = QPINT1D('2*sqrt((1-x)/(1+x))/(1-x^2)', 0.0d, 1d, /expr)
;
; has a singularity at a value of X EQ 1. Still, the singularity is
; integrable, and the value returned is a correct value of 2.
;
; If known singularities are present within the interval of
; integration, then users should pass the BREAKPOINTS keyword to list
; the locations of these points. QPINT1D will then integrate each
; subinterval separately, while still maintaining an overall error
; budget.
;
; If known discontinuities exist in the integrand, then the user may
; additionally list those points using the BREAKPOINTS keyword.
;
; It should be noted that the algorithm used is different, depending
; on whether the BREAKPOINTS keyword has been specified or not (this
; is the difference between the QAGSE vs. QAGPE routines in the
; original FORTRAN). The algorithm *without* BREAKPOINTS is
; generally thought to be more precise than *with*. Thus, it may be
; worth splitting the original integration interval manually and
; invoking QPINT1D without BREAKPOINTS.
;
;
; INPUTS:
;
; FUNCT - by default, a scalar string containing the name of an IDL
; function to be integrated. See above for the formal
; definition of MYFUNCT. (No default).
;
; If the EXPRESSION keyword is set, then FUNCT is a scalar
; string containing an IDL expression to be evaluated, as
; described above.
;
; A, B - a scalar number indicating the lower and upper limits of the
; interval of integration (i.e., [A, B] is the interval of
; integration).
;
; PRIVATE - any optional variable to be passed on to the function to
; be integrated. For functions, PRIVATE is passed as the
; second positional parameter; for expressions, PRIVATE can
; be referenced by the variable 'P'. QPINT1D does not
; examine or alter PRIVATE.
;
; RETURNS:
;
; The value of the integral. If either A or B are double precision,
; then the integral is computed in double precision; otherwise the
; result is returned in single precision floating point.
;
; KEYWORD PARAMETERS:
;
; BREAKPOINTS - an array of numbers specifying points within the
; integration interval where the integrand is
; discontinuous or singular. Out of bounds points are
; ignored.
; Default: undefined, i.e., no such points
;
; EPSABS - a scalar number, the absolute error tolerance requested
; in the integral computation. If less than or equal to
; zero, then the value is ignored.
; Default: 0
;
; EPSREL - a scalar number, the relative (i.e., fractional) error
; tolerance in the integral computation. If less than or
; equal to zero, then the value is ignored.
; Default: 1e-4 for float, 1d-6 for double
;
; EXPRESSION - if set, then FUNCT is an IDL expression. Otherwise,
; FUNCT is an IDL function.
;
; ERROR - upon return, this keyword contains an estimate of the
; error in the computation.
;
; FUNCTARGS - A structure which contains the parameters to be passed
; to the user-supplied function specified by FUNCT via
; the _EXTRA mechanism. This is the way you can pass
; additional data to your user-supplied function without
; using common blocks. By default, no extra parameters
; are passed to the user-supplied function.
;
; LIMIT - a scalar, the maximum number of subintervals to create
; before terminating execution. Upon return, a STATUS value
; of 1 indicates such an overflow occurred.
; Default: 100
;
; NFEV - upon return, this keyword contains the number of function
; calls executed (i.e., the number of abcissae).
;
; NPOINTS - a scalar, the number of Gauss Kronrod points to use in
; computing the integral over a subinterval. A larger
; number of points can in principle increase the precision
; of the integral, but also makes the computation take
; longer. Possible values are 15, 21, 31, 41, 51, and 61.
; NPOINTS is rounded up to the next nearest available set,
; with a maximum of 61.
; Default: 21
;
; NSUBINTERVALS - upon return, this keyword contains the number of
; subintervals the integration interval was divided
; into.
;
; STATUS - upon return, the status of the integration operation is
; returned in this keyword as an integer value. A value of
; zero indicates success; otherwise an abnormal condition
; has occurred and the returned value should be considered
; erroneous or less reliable according to STATUS:
;
; any negative number - outright failure (reserved for
; future use).
;
; -1 - the input parameters are invalid, because
; epsabs LE 0 and epsrel LT max([50*EPS,0.5d-28]),
; where EPS is the machine precision, or if LIMIT
; is smaller than the number of BREAKPOINTS.
;
; 0 - success.
;
; 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 (i.e. singularity,
; discontinuity within the interval), it should
; be supplied to the routine as an element of the
; vector BREAKPOINTS.
;
; 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 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 only
; slowly convergent. It must be noted that
; divergence can occur with any other value of
; ier GT 0.
;
; SYM_AXIS - a scalar number, the bisection point of the real line
; for improper integrals from negative infinity to
; positive infinity. Otherwise ignored.
; Default: 0.
;
;
; EXAMPLES:
;
; Shows how function and expression can be used for exponential
; integrand:
;
; IDL> print, qpint1d('EXP(X)', 0D, 10D, /expr)
; 22025.466
; IDL> print, qpint1d('EXP', 0D, 10D)
; 22025.466
;
; Normal definite integral, and then parameterized using a PRIVATE
; value of 2.
; IDL> print, qpint1d('X^2*EXP(-X)', 0D, 10D, /expr)
; 1.9944612
; IDL> print, qpint1d('X^2*EXP(-X/P(0))', 0D, 10D, 2D, /expr)
; 14.005568
;
; Improper integrals of the gaussian function
; IDL> inf = !values.d_infinity
; IDL> print, qpint1d('EXP(-X^2)', 0D, +inf, 2D, /expr)
; 0.88622693
; IDL> print, qpint1d('EXP(-X^2)', -inf, +inf, 2D, /expr), sqrt(!dpi)
; 1.7724539 1.7724539
; The second integral shows the comparison to the analytic value of
; SQRT(!DPI).
;
; COMMON BLOCKS:
;
; COMMON QPINT1D_MACHAR
; COMMON QPINT1D_PROFILE_COMMON
; COMMON QPINT1D_QKEVAL_COMMON
;
; These common blocks are used internally only and should not be
; accessed or modified.
;
; REFERENCES:
;
; R. Piessens, E. deDoncker-Kapenga, C. Uberhuber, D. Kahaner
; Quadpack: a Subroutine Package for Automatic Integration
; Springer Verlag, 1983. Series in Computational Mathematics v.1
; 515.43/Q1S 100394Z
;
; Netlib repository: http://www.netlib.org/quadpack/
;
; SLATEC Common Mathematical Library, Version 4.1, July 1993
; a comprehensive software library containing over
; 1400 general purpose mathematical and statistical routines
; written in Fortran 77. (http://www.netlib.org/slatec/)
;
; MODIFICATION HISTORY:
; Written, Feb-Jun, 2001, CM
; Documented, 04 Jun, 2001, CM
; Add usage message, error checking, 15 Mar 2002, CM
; Correct usage message, 28 Apr 2002, CM
; More error checking when user EXPRession fails, 10 Jun 2009, CM
;
; $Id: qpint1d.pro,v 1.14 2009/06/10 22:03:34 craigm Exp $
;-
; Copyright (C) 2001, 2002, 2009, Craig Markwardt
; This software is provided as is without any warranty whatsoever.
; Permission to use, copy, modify, and distribute modified or
; unmodified copies is granted, provided this copyright and disclaimer
; are included unchanged.
;-
FORWARD_FUNCTION qpint1d_eval, qpint1d
; --------------------------------------------------------------
; Some functions used in testing the algorithm
;
; Test functions
; .COMP
; FUNCTION BESTEST, X, N
; RETURN, BESELJ(ABS(X-10)+0.001, N)
; END
; print, qpint1d('BESTEST', .1d, 35d, 1L, nev=nev, nsub=nsub, status=ier)
; .comp
; function invsq, x
; return, 1/(abs(x-10)+0.0001)^2
; end
; print, qpint1d('INVSQ', 0d, 50d, nev=nev, nsub=nsub, status=ier, error=err)
; .comp
; function broken, x
; f = x*0
; wh = where(x LT 0, ct)
; if ct GT 0 then f(wh) = exp(x(wh))
; wh = where(x GE 0 AND x LT 10, ct)
; if ct GT 0 then f(wh) = x(wh)^2
; wh = where(x GE 10, ct)
; if ct GT 0 then f(wh) = exp(-x(wh))
; return, f
; end
; print, qpint1d('BROKEN', -10d, 50d, nev=nev, nsub=nsub, status=ier, error=err) & print, err, ier, nsub
;; Function used in testing speed
pro qpint1d_profile, clear=clear
common qpint1d_profile_common, profvals
; if n_elements(profvals) EQ 0 OR keyword_set(clear) then $
; profvals = {functime: 0D, tottime: 0D, srttime: 0D, acctime: 0D, $
; qktime: 0D}
end
;; Following are machine constants that can be loaded once. I have
;; found that bizarre underflow messages can be produced in each call
;; to MACHAR(), so this structure minimizes the number of calls to
;; one.
pro qpint1d_setmachar, double=isdouble
common qpint1d_machar, qpint1d_machar_vals
;; In earlier versions of IDL, MACHAR itself could produce a load of
;; error messages. We try to mask some of that out here.
if (!version.release) LT 5 then dummy = check_math(1, 1)
mch = 0.
mch = machar(double=keyword_set(isdouble))
dmachep = mch.eps
dmaxnum = mch.xmax
dminnum = mch.xmin
dmaxlog = alog(mch.xmax)
dminlog = alog(mch.xmin)
if keyword_set(isdouble) then $
dmaxgam = 171.624376956302725D $
else $
dmaxgam = 171.624376956302725
drdwarf = sqrt(dminnum*1.5) * 10
drgiant = sqrt(dmaxnum) * 0.1
qpint1d_machar_vals = {machep: dmachep, maxnum: dmaxnum, minnum: dminnum, $
maxlog: dmaxlog, minlog: dminlog, maxgam: dmaxgam, $
rdwarf: drdwarf, rgiant: drgiant}
if (!version.release) LT 5 then dummy = check_math(0, 0)
return
end
;; --------------------------------------------------------
;; Main workhorse routine
pro qpint1d_qagse, f, a0, b0, result, abserr, private, functargs=fa, $
epsabs=epsabs, epsrel=epsrel, npoints=npts0, $
status=ier, limit=limit, neval=neval, nsubintervals=last, $
breakpoints=bpoints0, isdouble=isdouble, $
alist=alist, blist=blist, rlist=rlist, $
elist=elist, iord=iord
;; Derived from QUADPACK QAGSE
;;***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).LE.MAX(EPSABS,EPSREL*ABS(I)).
;;***LIBRARY SLATEC (QUADPACK)
common qpint1d_machar, machvals
r1mach1 = machvals.minnum
r1mach2 = machvals.maxnum
r1mach4 = machvals.machep
;; Establish the precision we will be computing in, base on the
;; integration limits.
a = a0
b = b0
zero = a*0.
if n_elements(isdouble) EQ 0 then begin
sz = size(zero)
if sz(sz(0)+1) EQ 5 then isdouble = 1 else isdouble = 0
endif
isdouble = isdouble(0)
;; Establish default values
if n_elements(neval) EQ 0 then neval = 0L
if n_elements(limit) EQ 0 then limit = 100L
if n_elements(epsabs) EQ 0 then epsabs = zero
if n_elements(epsrel) EQ 0 then epsrel = zero + 1e-6
if n_elements(npts0) EQ 0 then npoints = 21L $
else npoints = floor(npts0(0))
rlist2 = make_array(52, value=zero)
res3la = rlist2(0:2)
;; These are the work arrays which manage the subintervals
alist = make_array(limit, value=zero)
blist = alist
rlist = alist
elist = alist
iord = lonarr(limit)
level = iord
epmach = r1mach4
; TEST ON VALIDITY OF PARAMETERS
ier = 0L
neval = 0
last = 0
result = zero
abserr = zero
;; Extract the valid breakpoints
nbp0 = n_elements(bpoints0)
nbp = nbp0
if nbp GT 0 then begin
abmin = min([a,b], max=abmax)
wh = where(bpoints0 GT abmin(0) AND bpoints0 LT abmax(0), nbp)
if nbp GT 0 then begin
;; Valid breakpoints were found. Sort them into place.
bpoints = bpoints0(wh)
if a LE b then bpoints = [a, bpoints(sort(bpoints)), b] $
else bpoints = [a, bpoints(sort(-bpoints)), b]
nbp = nbp + 2
if nbp GE limit then goto, INPUT_ERROR
endif
endif
if nbp EQ 0 then begin
;; By default, we have two breakpoints, namely the start and
;; stop of the interval.
bpoints = [a, b]
nbp = 2L
endif
;; Search for infinite bounds
dirsign = 1
if finite(b) EQ 0 then begin
bound = a
dirsign = 2*(b GT 0) - 1
a = zero
b = zero + 1
;; Transform the breakpoints into the [0,1] interval
bpoints = rotate(1/((bpoints-bound(0))*dirsign + 1),2)
endif
alist(0) = a
blist(0) = b
rlist(0) = zero
elist(0) = zero
xnum = (a(0)*b(0)*0.)
sz = size(xnum)
if sz(sz(0)+1) EQ 5 then xlim = 0.5d-28 else xlim = 0.5d-14
if (epsabs LE 0 AND epsrel LT max([0.5e2*epmach,xlim])) then begin
INPUT_ERROR:
ier = 6L
goto, LAB999
endif
nint = nbp-1
npts = nbp-2
npts2 = nbp
; FIRST APPROXIMATION TO THE INTEGRAL
uflow = r1mach1
oflow = r1mach2
resabs = zero
a1 = bpoints(0)
ndin = iord * 0
for i = 0L,nint-1 do begin
b1 = bpoints(i+1)
qpint1d_qkeval, f,a1,b1,area1,error1,defabs,resa, private, $
neval=neval, functargs=fa, inflow=bound, dirsign=dirsign, $
isdouble=isdouble, npoints=npoints
abserr = abserr+error1
result = result+area1
ndin(i) = (error1 EQ resa AND error1 NE zero)
resabs = resabs+defabs
level(i) = 0
elist(i) = error1
alist(i) = a1
blist(i) = b1
rlist(i) = area1
iord(i) = i+1
a1 = b1
endfor
if nbp0 GT 0 then begin
wh = where(ndin EQ 1, ct)
if ct GT 0 then elist(wh) = abserr
endif
errsum = total(elist(0:nint-1))
; TEST ON ACCURACY.
last = nint
dres = abs(result)
errbnd = max([epsabs,epsrel*dres])
if (abserr LE 1.0e+02*epmach*resabs) AND (abserr GT errbnd) then $
ier = 2L
;; Sort in descending order
iord(0:nint-1) = sort(-elist(0:nint-1)) + 1
if (limit LT npts2) then ier = 1L
if (ier NE 0 OR abserr LE errbnd) then goto, LAB999
; INITIALIZATION
rlist2(1-1) = result
maxerr = iord(0)
errmax = elist(maxerr-1)
area = result
nrmax = 1L
nres = 0L
if nbp0 EQ 0 then begin
numrl2 = 2L
errsum = abserr
endif else begin
numrl2 = 1L
erlarg = errsum
endelse
ktmin = 0
extrap = 0
noext = 0
ertest = errbnd
levmax = 1L
iroff1 = 0
iroff2 = 0
iroff3 = 0
ierro = 0
abserr = oflow
ksgn = -1
if (dres GE (0.1E+01-0.5E+02*epmach)*resabs) then ksgn = 1
; MAIN DO-LOOP
for last = npts2, limit do begin
; BISECT THE SUBINTERVAL WITH THE NRMAX-TH LARGEST
; ERROR ESTIMATE.
levcur = level(maxerr)+1
a1 = alist(maxerr-1)
b1 = 0.5e+00*(alist(maxerr-1)+blist(maxerr-1))
a2 = b1
b2 = blist(maxerr-1)
erlast = errmax
qpint1d_qkeval, f,a1,b1,area1,error1,resa,defab1, private, $
neval=neval, functargs=fa, inflow=bound, dirsign=dirsign, $
isdouble=isdouble, npoints=npoints
qpint1d_qkeval, f,a2,b2,area2,error2,resa,defab2, private, $
neval=neval, functargs=fa, inflow=bound, dirsign=dirsign, $
isdouble=isdouble, npoints=npoints
; 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-1))
if NOT (defab1 EQ error1 OR defab2 EQ error2) then begin
if NOT (abs(rlist(maxerr-1)-area12) GT 0.1e-04*abs(area12) $
OR erro12 LT 0.99e+00*errmax) then begin
if (extrap) then iroff2 = iroff2+1
if (NOT extrap) then iroff1 = iroff1+1
endif
if(last GT 10 AND erro12 GT errmax) then iroff3 = iroff3+1
endif
level(maxerr-1) = levcur
level(last-1) = levcur
rlist(maxerr-1) = area1
rlist(last-1) = area2
errbnd = max([epsabs,epsrel*abs(area)])
; TEST FOR ROUNDOFF ERROR AND EVENTUALLY
; SET ERROR FLAG.
if (iroff1+iroff2 GE 10 OR iroff3 GE 20) then ier = 2L
if (iroff2 GE 5) then ierro = 3
; SET ERROR FLAG IN THE CASE THAT THE NUMBER OF
; SUBINTERVALS EQUALS LIMIT.
if last EQ limit then ier = 1L
; SET ERROR FLAG IN THE CASE OF BAD INTEGRAND BEHAVIOUR
; AT A POINT OF THE INTEGRATION RANGE.
if (max([abs(a1),abs(b2)]) LE (0.1e+01+0.1e+03*epmach)* $
(abs(a2)+0.1e+04*uflow)) then ier = 4L
; APPEND THE NEWLY-CREATED INTERVALS TO THE LIST.
if (error2 LE error1) then begin ;; IDL OK
alist(last-1) = a2
blist(maxerr-1) = b1
blist(last-1) = b2
elist(maxerr-1) = error1
elist(last-1) = error2
endif else begin
alist(maxerr-1) = a2
alist(last-1) = a1
blist(last-1) = b1
rlist(maxerr-1) = area2
rlist(last-1) = area1
elist(maxerr-1) = error2
elist(last-1) = error1
endelse
; CALL SUBROUTINE QPSRT 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).
qpint1d_qpsrt, limit,last,maxerr,errmax,elist,iord,nrmax
; ***JUMP OUT OF DO-LOOP
if (errsum LE errbnd) then goto, LAB115
; ***JUMP OUT OF DO-LOOP
if (ier NE 0) then goto, LAB100
if (nbp0 EQ 0 AND last EQ 2) then goto, LAB80
if (noext) then goto, LAB90
erlarg = erlarg-erlast
if (nbp0 EQ 0) then begin
if (abs(b1-a1) GT small) then erlarg = erlarg+erro12
endif else begin
if (levcur+1 LE levmax) then erlarg = erlarg+erro12
endelse
if NOT (extrap) then begin
bool = 0
if nbp0 EQ 0 then begin
if (abs(blist(maxerr-1)-alist(maxerr-1)) GT small) then $
goto, LAB90
endif else begin
if (level(maxerr-1)+1 LE levmax) then goto, LAB90
endelse
extrap = 1
nrmax = 2L
endif
if NOT (ierro EQ 3 OR erlarg LE ertest) then begin
;
; 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 GT (2+limit/2)) then jupbnd = limit+3-last
for k = id, jupbnd do begin
maxerr = iord(nrmax-1)
errmax = elist(maxerr-1)
; ***JUMP OUT OF DO-LOOP
bool = 0
if nbp0 EQ 0 then begin
if (abs(blist(maxerr-1)-alist(maxerr-1)) GT small) then $
goto, LAB90
endif else begin
if (level(maxerr-1)+1 LE levmax) then goto, LAB90
endelse
nrmax = nrmax+1
endfor
endif
; PERFORM EXTRAPOLATION.
numrl2 = numrl2+1
rlist2(numrl2-1) = area
if (nbp0 GT 0 AND numrl2 LE 2) then goto, LAB72
qpint1d_qelg, numrl2,rlist2,reseps,abseps,res3la,nres
ktmin = ktmin+1
if (ktmin GT 5 AND abserr LT 0.1e-02*errsum) then ier = 5L
if NOT (abseps GE abserr) then begin
ktmin = 0
abserr = abseps
result = reseps
correc = erlarg
ertest = max([epsabs,epsrel*abs(reseps)])
; ***JUMP OUT OF DO-LOOP
if (abserr LE ertest) then goto, LAB100
endif
; PREPARE BISECTION OF THE SMALLEST INTERVAL.
if (numrl2 EQ 1) then noext = 1
if (ier EQ 5) then goto, LAB100
LAB72:
maxerr = iord(0)
errmax = elist(maxerr-1)
nrmax = 1L
extrap = 0
if nbp0 EQ 0 then small = small*0.5e+00
levmax = levmax + 1
erlarg = errsum
goto, LAB90
LAB80:
small = abs(b-a)*0.375e+00
erlarg = errsum
ertest = errbnd
rlist2(2-1) = area
LAB90:
endfor
LAB100:
; SET FINAL RESULT AND ERROR ESTIMATE.
if abserr EQ oflow then goto, LAB115
if ier+ierro EQ 0 then goto, LAB110
if ierro EQ 3 then abserr = abserr+correc
if ier EQ 0 then ier = 3L
if NOT (result NE zero AND area NE zero) then begin
if abserr GT errsum then goto, LAB115
if area EQ 0 then goto, LAB130
goto, LAB110
endif
if(abserr/abs(result) GT errsum/abs(area)) then goto, LAB115
; TEST ON DIVERGENCE.
LAB110:
if (ksgn EQ (-1) AND max([abs(result),abs(area)]) LE $
resabs*0.1e-01) then goto, LAB130
if (0.1e-01 GT (result/area) OR (result/area) GT 0.1e+03 $
OR errsum GT abs(area)) then ier = 6L
goto, LAB130
LAB115:
result = total(rlist(0:last-1))
abserr = errsum
LAB130:
if (ier GT 2) then ier = ier-1
LAB999:
result = result * dirsign
return
end
;; Originally DQK21.F, this routine performs Gauss-Kronrod quadrature
;; using 15, 21, 31, 41, 51 or 61 points.
pro qpint1d_qkeval, f, a, b, result, abserr, resabs, resasc, priv, $
functargs=fa, neval=neval, reset=reset, isdouble=isdouble, $
inflow=bound, dirsign=dirsign, npoints=npts0
;; Derived from QUADPACK QK21-QK61, and QK15I
;;***PURPOSE To compute I = Integral of F over (A,B), with error
;; estimate
;; J = Integral of ABS(F) over (A,B)
;;***LIBRARY SLATEC (QUADPACK)
common qpint1d_qkeval_common, wg, wgk, xgk, ig, nptsreq, nptsact, prec
if n_elements(npts0) EQ 0 then npts0 = 21L
if n_elements(isdouble) EQ 0 then isdouble = 1
;; Determine the number of points "requested", versus the number
;; "actually" granted.
if n_elements(nptsreq) EQ 0 OR keyword_set(reset) then begin
nptsreq = 0L
prec = -1L
endif
if npts0(0) NE nptsreq OR isdouble(0) NE prec then begin
;; If this is the first time around, or if we need to use a
;; different set of GK points, then request the new set of
;; points from GKWEIGHTS.
nptsreq = npts0(0)
prec = isdouble(0) EQ 1
qpint1d_gkweights, wg, wgk, xgk, ig, nptsreq, nptsact, prec
if keyword_set(reset) then return
endif
common qpint1d_machar, machvals
r1mach1 = machvals.minnum
r1mach2 = machvals.maxnum
r1mach4 = machvals.machep
common qpint1d_profile_common, profvals
; prof_start1 = systime(1)
if n_elements(neval) EQ 0 then neval = 0L
epmach = r1mach4
uflow = r1mach1
zero = a*b*0.
centr = 0.5e+00*(a+b)
hlgth = 0.5e+00*(b-a)
dhlgth = abs(hlgth)
x = centr+hlgth*xgk
if n_elements(bound) GT 0 then begin
;; Transformation for infinite integrals
u = temporary(x)
x = bound(0) + dirsign*(1/u - 1)
endif
; prof_start2 = systime(1)
;; Call with or without PRIVATE and _EXTRA keywords
if n_elements(priv) GT 0 then begin
if n_elements(fa) GT 0 then fv = call_function(f, x, priv, _EXTRA=fa) $
else fv = call_function(f, x, priv)
endif else begin
if n_elements(fa) GT 0 then fv = call_function(f, x, _EXTRA=fa) $
else fv = call_function(f, x)
endelse
; profvals.functime = profvals.functime + (systime(1) - prof_start2)
neval = neval + n_elements(x)
if n_elements(fv) NE n_elements(x) then $
message, 'ERROR: Integrand function '+strupcase(f)+$
' must return a vector of values'
if n_elements(bound) GT 0 then begin
;; Complete the transformation for infinite integrals
fv = fv / u^2
endif
resk = total(wgk*fv)
resabs = total(wgk*abs(fv))
resg = total(wg*fv(ig))
resasc = total(wgk*abs(fv-resk*0.5e+00))
result = resk*hlgth
resabs = resabs*dhlgth
resasc = resasc*dhlgth
abserr = abs((resk-resg)*hlgth)
if (resasc NE 0.0e+00 AND abserr NE 0.0e+00) then $
abserr = resasc*min([0.1e+01,(0.2e+03*abserr/resasc)^1.5])
if (resabs GT uflow/(0.5e+02*epmach)) then $
abserr = max([(epmach*0.5e+02)*resabs,abserr])
; profvals.qktime = profvals.qktime + (systime(1) - prof_start1)
return
end
pro qpint1d_qelg, n, epstab, result, abserr, res3la, nres
;; Derived from QUADPACK QELG
;;***PURPOSE The routine determines the limit of a given sequence of
;; approximations, by means of the Epsilon algorithm of
;; P. Wynn. An estimate of the absolute error is also given.
;; The condensed Epsilon table is computed. Only those
;; elements needed for the computation of the next diagonal
;; are preserved.
;;***LIBRARY SLATEC
common qpint1d_machar, machvals
r1mach1 = machvals.minnum
r1mach2 = machvals.maxnum
r1mach4 = machvals.machep
common qpint1d_profile_common, profvals
; prof_start = systime(1)
epmach = r1mach4
oflow = r1mach2
nres = nres+1
abserr = oflow
result = epstab(n-1)
if (n LT 3) then goto, LAB100
limexp = 50
epstab(n+2-1) = epstab(n-1)
newelm = (n-1)/2
epstab(n-1) = oflow
num = n
k1 = n-1 ;; OK IDL
for i = 1, newelm do begin
res = epstab(k1+2) ;; OK IDL
e0 = epstab(k1-2) ;; OK IDL
e1 = epstab(k1-1) ;; OK IDL
e2 = res
e1abs = abs(e1)
delta2 = e2-e1
err2 = abs(delta2)
tol2 = max([abs(e2),e1abs])*epmach
delta3 = e1-e0
err3 = abs(delta3)
tol3 = max([e1abs,abs(e0)])*epmach
if NOT (err2 GT tol2 OR err3 GT tol3) then begin
; IF E0, E1 AND E2 ARE EQUAL TO WITHIN MACHINE
; ACCURACY, CONVERGENCE IS ASSUMED.
result = res
abserr = err2+err3
; ***JUMP OUT OF DO-LOOP
goto, LAB100
endif
e3 = epstab(k1)
epstab(k1) = e1
delta1 = e1-e3
err1 = abs(delta1)
tol1 = max([e1abs,abs(e3)])*epmach
; IF TWO ELEMENTS ARE VERY CLOSE TO EACH OTHER, OMIT
; A PART OF THE TABLE BY ADJUSTING THE VALUE OF N
if NOT (err1 LE tol1 OR err2 LE tol2 OR err3 LE tol3) then begin
;; Avoid underflow errors
if abs(delta1) GE 0.5*abs(oflow) then odelta1 = delta1*0 $
else odelta1 = 1/delta1
ss = odelta1+0.1e+01/delta2-0.1e+01/delta3
epsinf = abs(ss*e1)
; TEST TO DETECT IRREGULAR BEHAVIOUR IN THE TABLE, AND
; EVENTUALLY OMIT A PART OF THE TABLE ADJUSTING THE VALUE
; OF N.
if (epsinf GT 0.1e-03) then goto, LAB30
endif
n = i+i-1
; ***JUMP OUT OF DO-LOOP
goto, LAB50
; COMPUTE A NEW ELEMENT AND EVENTUALLY ADJUST
; THE VALUE OF RESULT.
LAB30:
res = e1+0.1e+01/ss
epstab(k1) = res
k1 = k1-2
error = err2+abs(res-e2)+err3
if NOT (error GT abserr) then begin
abserr = error
result = res
endif
endfor
; SHIFT THE TABLE.
LAB50:
if (n EQ limexp) then n = 2*(limexp/2)-1
ib = 1
if ((num/2)*2 EQ num) then ib = 2
ie = newelm+1
ii = ib-1+lindgen(ie)*2
epstab(ii) = epstab(ii+2) ;; OK IDL
if NOT (num EQ n) then begin
indx = num-n+1
ii = lindgen(n)
epstab(ii) = epstab(indx-1+ii) ;; OK IDL
endif
if NOT (nres GE 4) then begin
res3la(nres-1) = result
abserr = oflow
endif else begin
; COMPUTE ERROR ESTIMATE - OK IDL
abserr = (abs(result-res3la(2))+abs(result-res3la(1)) $
+abs(result-res3la(0)))
res3la(0) = res3la(1)
res3la(1) = res3la(2)
res3la(2) = result
endelse
LAB100:
abserr = max([abserr,0.5e+01*epmach*abs(result)])
; profvals.acctime = profvals.acctime + (systime(1) - prof_start)
return
end
pro qpint1d_qpsrt, limit, last, maxerr, ermax, elist, iord, nrmax
;; OOOOOO OOOOO OOOO OOOOO
;; Derived from QUADPACK QPSRT
;;***PURPOSE Subsidiary to QAGE, QAGIE, QAGPE, QAGSE, QAWCE, QAWOE and
;; QAWSE
;;***LIBRARY SLATEC
;; ELIST - list of unsorted errors
;; IORD - (O) list of indices which sort ELIST in descending order
;; LAST - index of last element (new elt) in ELIST
;; LIMIT - maximum size of ELIST
;; NRMAX - (O) position in IORD of the maximum error
;; MAXERR - (O) position in ELIST of the maximum error
;; ERMAX - (O) amount of maximum error
common qpint1d_profile_common, profvals
; prof_start = systime(1)
if (last LE 2) then begin
iord(1-1) = 1
iord(2-1) = 2
goto, LAB90
endif
; THIS PART OF THE ROUTINE IS ONLY EXECUTED
; IF, DUE TO A DIFFICULT INTEGRAND, SUBDIVISION
; INCREASED THE ERROR ESTIMATE. IN THE NORMAL CASE
; THE INSERT PROCEDURE SHOULD START AFTER THE
; NRMAX-TH LARGEST ERROR ESTIMATE.
errmax = elist(maxerr-1)
if (nrmax NE 1) then begin
ido = nrmax-1
for i = 1, ido do begin
isucc = iord(nrmax-1-1)
; ***JUMP OUT OF DO-LOOP
if (errmax LE elist(isucc-1)) then goto, LAB30
iord(nrmax-1) = isucc
nrmax = nrmax-1
endfor
endif
LAB30:
jupbn = last
if (last GT (limit/2+2)) then jupbn = limit+3-last
errmin = elist(last-1)
; INSERT ERRMAX BY TRAVERSING THE LIST TOP-DOWN,
; STARTING COMPARISON FROM THE ELEMENT ELIST(IORD(NRMAX+1)).
jbnd = jupbn-1
ibeg = nrmax+1
if (ibeg LE jbnd) then begin
for i = ibeg, jbnd do begin
isucc = iord(i-1)
; ***JUMP OUT OF DO-LOOP
if (errmax GE elist(isucc-1)) then goto, LAB60
iord(i-1-1) = isucc
endfor
endif
iord(jbnd-1) = maxerr
iord(jupbn-1) = last
goto, LAB90
; INSERT ERRMIN BY TRAVERSING THE LIST BOTTOM-UP.
LAB60:
iord(i-1-1) = maxerr
k = jbnd
for j=i, jbnd do begin
isucc = iord(k-1)
; ***JUMP OUT OF DO-LOOP
if (errmin LT elist(isucc-1)) then goto, LAB80
iord(k+1-1) = isucc
k = k-1
endfor
iord(i-1) = last
goto, LAB90
LAB80:
iord(k+1-1) = last
; SET MAXERR AND ERMAX.
LAB90:
maxerr = iord(nrmax-1)
ermax = elist(maxerr-1)
; profvals.srttime = profvals.srttime + (systime(1) - prof_start)
return
end
pro qpint1d_gkweights, wg, wgk, xgk, ig, nptsreq, nptsact, prec
;; The user has requested NPTSREQ points, however that number may
;; not be available. Filter the number and record the *actual*
;; number of points used, NPTSACT
; THE ABSCISSAE AND WEIGHTS ARE GIVEN FOR THE INTERVAL (-1,1).
; BECAUSE OF SYMMETRY ONLY THE POSITIVE ABSCISSAE AND THEIR
; CORRESPONDING WEIGHTS ARE GIVEN.
;
; XGK - ABSCISSAE OF THE 21-POINT KRONROD RULE
; XGK(2), XGK(4), ... ABSCISSAE OF THE 10-POINT
; GAUSS RULE
; XGK(1), XGK(3), ... ABSCISSAE WHICH ARE OPTIMALLY
; ADDED TO THE 10-POINT GAUSS RULE
;
; WGK - WEIGHTS OF THE 21-POINT KRONROD RULE
;
; WG - WEIGHTS OF THE 10-POINT GAUSS RULE
;; gauss quadrature weights and kronron quadrature abscissae and
;; weights as evaluated with 80 decimal digit arithmetic by
;; l. w. fullerton, bell labs, nov. 1981.
if nptsreq LE 15 then begin
nptsact = 15
wg = [ $
0.129484966168869693270611432679082d0, $
0.279705391489276667901467771423780d0, $
0.381830050505118944950369775488975d0, $
0.417959183673469387755102040816327d0 ]
xgk = [ $
0.991455371120812639206854697526329d0, $
0.949107912342758524526189684047851d0, $
0.864864423359769072789712788640926d0, $
0.741531185599394439863864773280788d0, $
0.586087235467691130294144838258730d0, $
0.405845151377397166906606412076961d0, $
0.207784955007898467600689403773245d0, $
0.000000000000000000000000000000000d0 ]
wgk = [ $
0.022935322010529224963732008058970d0, $
0.063092092629978553290700663189204d0, $
0.104790010322250183839876322541518d0, $
0.140653259715525918745189590510238d0, $
0.169004726639267902826583426598550d0, $
0.190350578064785409913256402421014d0, $
0.204432940075298892414161999234649d0, $
0.209482141084727828012999174891714d0 ]
endif else if nptsreq LE 21 then begin
nptsact = 21
wg = [ 0.066671344308688137593568809893332d0, $
0.149451349150580593145776339657697d0, $
0.219086362515982043995534934228163d0, $
0.269266719309996355091226921569469d0, $
0.295524224714752870173892994651338d0 ]
xgk = [ 0.995657163025808080735527280689003d0, $
0.973906528517171720077964012084452d0, $
0.930157491355708226001207180059508d0, $
0.865063366688984510732096688423493d0, $
0.780817726586416897063717578345042d0, $
0.679409568299024406234327365114874d0, $
0.562757134668604683339000099272694d0, $
0.433395394129247190799265943165784d0, $
0.294392862701460198131126603103866d0, $
0.148874338981631210884826001129720d0, $
0.000000000000000000000000000000000d0 ]
wgk = [ 0.011694638867371874278064396062192d0, $
0.032558162307964727478818972459390d0, $
0.054755896574351996031381300244580d0, $
0.075039674810919952767043140916190d0, $
0.093125454583697605535065465083366d0, $
0.109387158802297641899210590325805d0, $
0.123491976262065851077958109831074d0, $
0.134709217311473325928054001771707d0, $
0.142775938577060080797094273138717d0, $
0.147739104901338491374841515972068d0, $
0.149445554002916905664936468389821d0 ]
endif else if nptsreq LE 31 then begin
nptsact = 31
wg = [ $
0.03075324199611726834628393577204d0, $
0.07036604748810812479267416450667d0, $
0.10715922046717193501869546685869d0, $
0.13957067792615431447804794511028d0, $
0.16626920581699393353200860481209d0, $
0.18616100001556221106800561866423d0, $
0.19843148532711157646118326443839d0, $
0.20257824192556127280620199967519d0 ]
xgk = [ $
0.99800229869339706025172840152271d0, $
0.98799251802048542849565718586613d0, $
0.96773907567913913427347978784337d0, $
0.93727339240070590437758947710209d0, $
0.89726453234408190082509656454496d0, $
0.84820658341042721620648320774217d0, $
0.79041850144246593297649294817947d0, $
0.72441773136017004746186054613938d0, $
0.65099674129741697053735895313275d0, $
0.57097217260853884757226737253911d0, $
0.48508186364023968063655740232351d0, $
0.39415134707756336987207370981045d0, $
0.29918000715316881216780024266389d0, $
0.20119409399743452230628303394596d0, $
0.10114206691871749907074231447392d0, $
0.00000000000000000000000000000000d0 ]
wgk = [ $
0.00537747987292334897792051430128d0, $
0.01500794732931612258374763075807d0, $
0.02546084732671532016874001019653d0, $
0.03534636079137584622037948478360d0, $
0.04458975132476487668227299373280d0, $
0.05348152469092808725343147239430d0, $
0.06200956780067064025139230960803d0, $
0.06985412131872825879520077099147d0, $
0.07684968075772037884432777482659d0, $
0.08308050282313302108289247286104d0, $
0.08856444305621177067275443693774d0, $
0.09312659817082532125486872747346d0, $
0.09664272698362367855179907627589d0, $
0.09917359872179195932393173484603d0, $
0.10076984552387559504946662617570d0, $
0.10133000701479154907374792767493d0 ]
endif else if nptsreq LE 41 then begin
nptsact = 41
wg = [ $
0.017614007139152118311861962351853d0, $
0.040601429800386941331039952274932d0, $
0.062672048334109063569506535187042d0, $
0.083276741576704748724758143222046d0, $
0.101930119817240435036750135480350d0, $
0.118194531961518417312377377711382d0, $
0.131688638449176626898494499748163d0, $
0.142096109318382051329298325067165d0, $
0.149172986472603746787828737001969d0, $
0.152753387130725850698084331955098d0 ]
xgk = [ $
0.998859031588277663838315576545863d0, $
0.993128599185094924786122388471320d0, $
0.981507877450250259193342994720217d0, $
0.963971927277913791267666131197277d0, $
0.940822633831754753519982722212443d0, $
0.912234428251325905867752441203298d0, $
0.878276811252281976077442995113078d0, $
0.839116971822218823394529061701521d0, $
0.795041428837551198350638833272788d0, $
0.746331906460150792614305070355642d0, $
0.693237656334751384805490711845932d0, $
0.636053680726515025452836696226286d0, $
0.575140446819710315342946036586425d0, $
0.510867001950827098004364050955251d0, $
0.443593175238725103199992213492640d0, $
0.373706088715419560672548177024927d0, $
0.301627868114913004320555356858592d0, $
0.227785851141645078080496195368575d0, $
0.152605465240922675505220241022678d0, $
0.076526521133497333754640409398838d0, $
0.000000000000000000000000000000000d0 ]
wgk = [ $
0.003073583718520531501218293246031d0, $
0.008600269855642942198661787950102d0, $
0.014626169256971252983787960308868d0, $
0.020388373461266523598010231432755d0, $
0.025882133604951158834505067096153d0, $
0.031287306777032798958543119323801d0, $
0.036600169758200798030557240707211d0, $
0.041668873327973686263788305936895d0, $
0.046434821867497674720231880926108d0, $
0.050944573923728691932707670050345d0, $
0.055195105348285994744832372419777d0, $
0.059111400880639572374967220648594d0, $
0.062653237554781168025870122174255d0, $
0.065834597133618422111563556969398d0, $
0.068648672928521619345623411885368d0, $
0.071054423553444068305790361723210d0, $
0.073030690332786667495189417658913d0, $
0.074582875400499188986581418362488d0, $
0.075704497684556674659542775376617d0, $
0.076377867672080736705502835038061d0, $
0.076600711917999656445049901530102d0 ]
endif else if nptsreq LE 51 then begin
nptsact = 51
wg = [ $
0.011393798501026287947902964113235d0, $
0.026354986615032137261901815295299d0, $
0.040939156701306312655623487711646d0, $
0.054904695975835191925936891540473d0, $
0.068038333812356917207187185656708d0, $
0.080140700335001018013234959669111d0, $
0.091028261982963649811497220702892d0, $
0.100535949067050644202206890392686d0, $
0.108519624474263653116093957050117d0, $
0.114858259145711648339325545869556d0, $
0.119455763535784772228178126512901d0, $
0.122242442990310041688959518945852d0, $
0.123176053726715451203902873079050d0 ]
xgk = [ $
0.999262104992609834193457486540341d0, $
0.995556969790498097908784946893902d0, $
0.988035794534077247637331014577406d0, $
0.976663921459517511498315386479594d0, $
0.961614986425842512418130033660167d0, $
0.942974571228974339414011169658471d0, $
0.920747115281701561746346084546331d0, $
0.894991997878275368851042006782805d0, $
0.865847065293275595448996969588340d0, $
0.833442628760834001421021108693570d0, $
0.797873797998500059410410904994307d0, $
0.759259263037357630577282865204361d0, $
0.717766406813084388186654079773298d0, $
0.673566368473468364485120633247622d0, $
0.626810099010317412788122681624518d0, $
0.577662930241222967723689841612654d0, $
0.526325284334719182599623778158010d0, $
0.473002731445714960522182115009192d0, $
0.417885382193037748851814394594572d0, $
0.361172305809387837735821730127641d0, $
0.303089538931107830167478909980339d0, $
0.243866883720988432045190362797452d0, $
0.183718939421048892015969888759528d0, $
0.122864692610710396387359818808037d0, $
0.061544483005685078886546392366797d0, $
0.000000000000000000000000000000000d0 ]
wgk = [ $
0.001987383892330315926507851882843d0, $
0.005561932135356713758040236901066d0, $
0.009473973386174151607207710523655d0, $
0.013236229195571674813656405846976d0, $
0.016847817709128298231516667536336d0, $
0.020435371145882835456568292235939d0, $
0.024009945606953216220092489164881d0, $
0.027475317587851737802948455517811d0, $
0.030792300167387488891109020215229d0, $
0.034002130274329337836748795229551d0, $
0.037116271483415543560330625367620d0, $
0.040083825504032382074839284467076d0, $
0.042872845020170049476895792439495d0, $
0.045502913049921788909870584752660d0, $
0.047982537138836713906392255756915d0, $
0.050277679080715671963325259433440d0, $
0.052362885806407475864366712137873d0, $
0.054251129888545490144543370459876d0, $
0.055950811220412317308240686382747d0, $
0.057437116361567832853582693939506d0, $
0.058689680022394207961974175856788d0, $
0.059720340324174059979099291932562d0, $
0.060539455376045862945360267517565d0, $
0.061128509717053048305859030416293d0, $
0.061471189871425316661544131965264d0, $
0.061580818067832935078759824240066d0 ]
; note: wgk (26) was calculated from the values of wgk(1..25)
endif else begin
nptsact = 61
wg = [ $
0.007968192496166605615465883474674d0, $
0.018466468311090959142302131912047d0, $
0.028784707883323369349719179611292d0, $
0.038799192569627049596801936446348d0, $
0.048402672830594052902938140422808d0, $
0.057493156217619066481721689402056d0, $
0.065974229882180495128128515115962d0, $
0.073755974737705206268243850022191d0, $
0.080755895229420215354694938460530d0, $
0.086899787201082979802387530715126d0, $
0.092122522237786128717632707087619d0, $
0.096368737174644259639468626351810d0, $
0.099593420586795267062780282103569d0, $
0.101762389748405504596428952168554d0, $
0.102852652893558840341285636705415d0 ]
xgk = [ $
0.999484410050490637571325895705811d0, $
0.996893484074649540271630050918695d0, $
0.991630996870404594858628366109486d0, $
0.983668123279747209970032581605663d0, $
0.973116322501126268374693868423707d0, $
0.960021864968307512216871025581798d0, $
0.944374444748559979415831324037439d0, $
0.926200047429274325879324277080474d0, $
0.905573307699907798546522558925958d0, $
0.882560535792052681543116462530226d0, $
0.857205233546061098958658510658944d0, $
0.829565762382768397442898119732502d0, $
0.799727835821839083013668942322683d0, $
0.767777432104826194917977340974503d0, $
0.733790062453226804726171131369528d0, $
0.697850494793315796932292388026640d0, $
0.660061064126626961370053668149271d0, $
0.620526182989242861140477556431189d0, $
0.579345235826361691756024932172540d0, $
0.536624148142019899264169793311073d0, $
0.492480467861778574993693061207709d0, $
0.447033769538089176780609900322854d0, $
0.400401254830394392535476211542661d0, $
0.352704725530878113471037207089374d0, $
0.304073202273625077372677107199257d0, $
0.254636926167889846439805129817805d0, $
0.204525116682309891438957671002025d0, $
0.153869913608583546963794672743256d0, $
0.102806937966737030147096751318001d0, $
0.051471842555317695833025213166723d0, $
0.000000000000000000000000000000000d0 ]
wgk = [ $
0.001389013698677007624551591226760d0, $
0.003890461127099884051267201844516d0, $
0.006630703915931292173319826369750d0, $
0.009273279659517763428441146892024d0, $
0.011823015253496341742232898853251d0, $
0.014369729507045804812451432443580d0, $
0.016920889189053272627572289420322d0, $
0.019414141193942381173408951050128d0, $
0.021828035821609192297167485738339d0, $
0.024191162078080601365686370725232d0, $
0.026509954882333101610601709335075d0, $
0.028754048765041292843978785354334d0, $
0.030907257562387762472884252943092d0, $
0.032981447057483726031814191016854d0, $
0.034979338028060024137499670731468d0, $
0.036882364651821229223911065617136d0, $
0.038678945624727592950348651532281d0, $
0.040374538951535959111995279752468d0, $
0.041969810215164246147147541285970d0, $
0.043452539701356069316831728117073d0, $
0.044814800133162663192355551616723d0, $
0.046059238271006988116271735559374d0, $
0.047185546569299153945261478181099d0, $
0.048185861757087129140779492298305d0, $
0.049055434555029778887528165367238d0, $
0.049795683427074206357811569379942d0, $
0.050405921402782346840893085653585d0, $
0.050881795898749606492297473049805d0, $
0.051221547849258772170656282604944d0, $
0.051426128537459025933862879215781d0, $
0.051494729429451567558340433647099d0 ]
endelse
nhalf = (nptsact-1)/2
nquart = (nptsact-1)/4
wgk = [ wgk(0:nhalf-1), rotate(wgk,2) ]
xgk = [ -xgk(0:nhalf-1), rotate(xgk,2) ]
wg = [wg(0:nquart-1), rotate(wg,2)]
ig = lindgen(nhalf)*2 + 1
;; Convert to float from double if requested
if prec EQ 0 then begin
wgk = float(wgk)
xgk = float(xgk)
wg = float(wg)
endif
end
;; Evaluate a user-supplied expression
function qpint1d_eval, x, p, expression=expr
y = 0
cmd = 'Y = '+expr
dummy = execute(cmd)
if dummy NE 1 then begin
err_string = ''+!error_state.msg
message, 'ERROR: failed to execute expression "'+expr+'" ('+err_string+')'
endif
return, y
end
;; The outer routine which does most of the preparation and special
;; cases.
function qpint1d, f0, a0, b0, private, npoints=npoints, expression=expr, $
epsabs=epsabs, epsrel=epsrel, error=abserr, nfev=neval, $
status=ier, functargs=fa, limit=limit, nsubintervals=nsub, $
sym_axis=symaxis, symmetric=symmetric, breakpoints=bpoints0,$
alist=alist, blist=blist, rlist=rlist, $
elist=elist, iord=iord, nocatch=nocatch
;; Derived from QUADPACK QAGS
;;***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).LE.MAX(EPSABS,EPSREL*ABS(I)).
;;***LIBRARY SLATEC (QUADPACK)
if keyword_set(query) then return, 1
common qpint1d_profile_common, profvals
qpint1d_profile
if n_params() EQ 0 then begin
USAGE:
message, 'USAGE:', /info
message, ' G = QPINT1D(FUNCNAME, A, B, $', /info
message, ' [EPSABS=, EPSREL=, ERROR=, STATUS=])', /info
message, ' (or)', /info
message, ' G = QPINT1D(EXPR, A, B, /EXPRESSION, $', /info
message, ' [EPSABS=, EPSREL=, ERROR=, STATUS=])', /info
return, !values.d_nan
endif
if n_elements(f0) EQ 0 OR n_elements(a0) EQ 0 OR n_elements(b0) EQ 0 then $
goto, USAGE
sz = size(f0)
if sz(sz(0)+1) NE 7 OR n_elements(f0) NE 1 then begin
message, 'ERROR: FUNCT must be a scalar string', /info
return, !values.d_nan
endif
;; Handle error conditions gracefully
if NOT keyword_set(nocatch) then begin
catch, catcherror
if catcherror NE 0 then begin
catch, /cancel
message, 'Error detected while integrating "'+f0(0)+'"', /info
message, !err_string, /info
errmsg = 0
if NOT keyword_set(expr) then begin
f1 = byte(strupcase(strtrim(f0(0),2)))
ca = (byte('A'))(0)
cz = (byte('Z'))(0)
c0 = (byte('0'))(0)
c9 = (byte('9'))(0)
c_ = (byte('_'))(0)
wh = where((f1 GE ca AND f1 LE cz) EQ 0 AND f1 NE c_ $
AND (f1 GE c0 AND f1 LE c9) EQ 0, ct)
if ct GT 0 OR (f1(0) GE c0 AND f1(0) LE c9) then begin
message, ('FUNCT appears to be an expression. Did you '+$
'intend to pass the /EXPRESSION keyword?'), /info
errmsg = 1
endif
endif
if errmsg EQ 0 then $
message, ('Please verify that function works and conforms to '+$
'the documentation'), /info
ier = -1L
return, !values.d_nan
endif
endif
; prof_start = systime(1)
neval = 0L
sza = size(a0)
szb = size(b0)
;; Determine the data precision, i.e. single or double precision
if sza(sza(0)+1) EQ 5 OR szb(szb(0)+1) EQ 5 then begin
isdouble = 1
zero = 0D
reldef = 1e-6
a = double(a0(0))
b = double(b0(0))
endif else begin
isdouble = 0
zero = 0.
reldef = 1e-4
a = float(a0(0))
b = float(b0(0))
endelse
qpint1d_setmachar, double=isdouble
;; Default values
if n_elements(limit) EQ 0 then limit = 100L
if n_elements(epsrel) EQ 0 then epsrel = zero + reldef
if n_elements(epsabs) EQ 0 then epsabs = zero
;; Establish the initial return values return
abserr = zero
result = zero
if a EQ b then begin
ier = 0L
return, result
endif
;; Now prepare for potentially one or two integrals, depending on
;; whether the integral is fully infinite, or only partially
infa = finite(a) EQ 0
infb = finite(b) EQ 0
if infa AND infb then begin
;; Fully infinite: break it into two partial integrals
if n_elements(symaxis0) EQ 0 then symaxis = zero $
else symaxis = zero + symaxis0(0)
a1 = symaxis & b1 = b
a2 = symaxis & b2 = a
ef = 2. ;; Each integral contributes half of error budget
endif else if infa then begin
;; Partially infinite, [a,inf]: do only second partial integral
a1 = zero & b1 = zero
a2 = b & b2 = a
ef = 1.
endif else begin
;; Partially infinite, [inf,b]: do only first partial integral
a1 = a & b1 = b
a2 = zero & b2 = zero
ef = 1.
endelse
;; Prepare for EXPRESSION if requested
if keyword_set(expr) then begin
f = 'QPINT1D_EVAL'
fa = {expression: strtrim(f0(0),2)}
endif else begin
f = strtrim(f0(0),2)
endelse
;; Call first partial integral if requested
ier = 0L
neval = 0L
nsub = 0L
if a1 NE b1 then begin
qpint1d_qagse, f, a1, b1, result, abserr, private, functargs=fa, $
epsabs=epsabs/ef, epsrel=epsrel/ef, breakpoints=bpoints0, $
status=ier, limit=limit, neval=neval, nsubintervals=nsub, $
alist=alist, blist=blist, rlist=rlist, elist=elist, iord=iord, $
npoints=npoints
endif
;; Return if an error condition was detected
if ier EQ 6 then ier = -1L
if ier EQ -1 OR ier EQ 3 then return, result
;; Now call the second partial integral if requested
if a2 NE b2 then begin
qpint1d_qagse, f, a2, b2, result2, abserr2, private, functargs=fa, $
epsabs=epsabs/ef, epsrel=epsrel/ef, breakpoints=bpoints0, $
status=ier, limit=limit, neval=neval2, nsubintervals=nsub2, $
alist=alist, blist=blist, rlist=rlist, elist=elist, iord=iord, $
npoints=npoints
;; Merge the two results together
result = result - result2
abserr = abserr + abserr2
neval = neval + neval2
nsub = nsub + nsub2
endif
; profvals.tottime = profvals.tottime + (systime(1) - prof_start)
;; Convert from QUADPACK to QPINT1D errors
if ier EQ 6 then ier = -1L
return, result
end