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de_quadrature.py
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de_quadrature.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Mon May 29 12:00:25 2017
Porting
Ooura's implementation of DE-Quadrature
http://www.kurims.kyoto-u.ac.jp/~ooura/intde.html
@author: Vladimir Kryzhniy
"""
import math
def intde(f, a, b, eps):
"""
intde I = integral of f(x) over (a,b)
f : integrand f(x)
a : lower limit of integration (double)
b : upper limit of integration (double)
eps : relative error requested (double)
i : approximation to the integral
err : estimate of the absolute error
function f(x) needs to be analytic over (a,b).
relative error
eps is relative error requested excluding
cancellation of significant digits.
i.e. eps means : (absolute error) /
(integral_a^b |f(x)| dx).
eps does not mean : (absolute error) / I.
error message
err >= 0 : normal termination.
err < 0 : abnormal termination (m >= mmax).
i.e. convergent error is detected :
1. f(x) or (d/dx)^n f(x) has
discontinuous points or sharp
peaks over (a,b).
you must divide the interval
(a,b) at this points.
2. relative error of f(x) is
greater than eps.
3. f(x) has oscillatory factor
and frequency of the oscillation
is very high.
"""
# ---- adjustable parameter ----
mmax = 256
efs = 0.1
hoff = 8.5
# int m;
# double pi2, epsln, epsh, h0, ehp, ehm, epst, ba, ir, h, iback,
# irback, t, ep, em, xw, xa, wg, fa, fb, errt, errh, errd;
pi2 = math.pi/2
epsln = 1 - math.log(efs * eps)
epsh = math.sqrt(efs * eps)
h0 = hoff / epsln
ehp = math.exp(h0)
ehm = 1 / ehp
epst = math.exp(-ehm * epsln)
ba = b - a
ir = f((a + b) * 0.5) * (ba * 0.25)
i = ir * math.pi
err = abs(i) * epst
h = 2 * h0
m = 1
while 1:
iback = i
irback = ir
t = h * 0.5
while 1:
em = math.exp(t)
ep = pi2 * em
em = pi2 / em
while 1:
xw = 1. / (1 + math.exp(ep - em))
xa = ba * xw
wg = xa * (1 - xw)
fa = f(a + xa) * wg
fb = f(b - xa) * wg
ir += fa + fb
i += (fa + fb) * (ep + em)
errt = (abs(fa) + abs(fb)) * (ep + em)
if m == 1:
err += errt * epst
ep *= ehp
em *= ehm
if not ((errt > err) | (xw > epsh)):
break
t += h
if not t < h0:
break
if m == 1:
errh = (err / epst) * epsh * h0
errd = 1 + 2 * errh
else:
errd = h * (abs(i - 2 * iback) + 4 * abs(ir - 2 * irback))
h *= 0.5
m *= 2
if not ((errd > errh) & (m < mmax)):
break
i *= h
if (errd > errh):
err = -errd * m
else:
err = errh * epsh * m / (2 * efs)
return i, err
def intdei(f, a, eps):
"""
intdei
I = integral of f(x) over (a,infinity),
f(x) has not oscillatory factor.
f : integrand f(x)
a : lower limit of integration (double)
eps : relative error requested (double)
i : approximation to the integral
err : estimate of the absolute error
function
f(x) needs to be analytic over (a,infinity).
relative error
eps is relative error requested excluding
cancellation of significant digits.
i.e. eps means : (absolute error) /
(integral_a^infinity |f(x)| dx).
eps does not mean : (absolute error) / I.
error message
err >= 0 : normal termination.
err < 0 : abnormal termination (m >= mmax).
i.e. convergent error is detected :
1. f(x) or (d/dx)^n f(x) has
discontinuous points or sharp
peaks over (a,infinity).
you must divide the interval
(a,infinity) at this points.
2. relative error of f(x) is
greater than eps.
3. f(x) has oscillatory factor
and decay of f(x) is very slow
as x -> infinity
"""
# ---- adjustable parameter ----
mmax = 256
efs = 0.1; hoff = 11.0
# int m;
# double pi4, epsln, epsh, h0, ehp, ehm, epst, ir, h, iback, irback,
# t, ep, em, xp, xm, fp, fm, errt, errh, errd;
pi4 = math.pi/4
epsln = 1 - math.log(efs * eps)
epsh = math.sqrt(efs * eps)
h0 = hoff / epsln
ehp = math.exp(h0)
ehm = 1 / ehp
epst = math.exp(-ehm * epsln)
ir = f(a + 1)
i = ir * (2 * pi4)
err = abs(i) * epst
h = 2 * h0
m = 1
while 1:
iback = i
irback = ir
t = h * 0.5
while 1:
em = math.exp(t)
ep = pi4 * em
em = pi4 / em
while 1:
xp = math.exp(ep - em)
xm = 1 / xp
fp = f(a + xp) * xp
fm = f(a + xm) * xm
ir += fp + fm
i += (fp + fm) * (ep + em)
errt = (abs(fp) + abs(fm)) * (ep + em)
if m == 1:
err += errt * epst
ep *= ehp
em *= ehm
if not ((errt > err) | (xm > epsh)):
break
t += h
if not t < h0:
break
if m == 1:
errh = (err / epst) * epsh * h0
errd = 1 + 2 * errh
else:
errd = h * (abs(i - 2 * iback) + 4 * abs(ir - 2 * irback))
h *= 0.5;
m *= 2;
if not ((errd > errh) & (m < mmax)):
break
i *= h
if errd > errh:
err = -errd * m
else:
err = errh * epsh * m / (2 * efs)
return i, err
def intdeo(f, a, omega, eps):
"""
intdeo
I = integral of f(x) over (a,infinity),
f(x) has oscillatory factor :
f(x) = g(x) * sin(omega * x + theta) as x -> infinity.
f : integrand f(x)
a : lower limit of integration (double)
omega : frequency of oscillation (double)
eps : relative error requested (double)
i : approximation to the integral
err : estimate of the absolute error
function
f(x) needs to be analytic over (a,infinity).
relative error
eps is relative error requested excluding
cancellation of significant digits.
i.e. eps means : (absolute error) /
(integral_a^R |f(x)| dx).
eps does not mean : (absolute error) / I.
error message
err >= 0 : normal termination.
err < 0 : abnormal termination (m >= mmax).
i.e. convergent error is detected :
1. f(x) or (d/dx)^n f(x) has
discontinuous points or sharp
peaks over (a,infinity).
you must divide the interval
(a,infinity) at this points.
2. relative error of f(x) is
greater than eps.
"""
# ---- adjustable parameter ----
mmax = 256; lmax = 5
efs = 0.1; enoff = 0.40; pqoff = 2.9; ppoff = -0.72
# /* ------------------------------ */
# int n, m, l, k;
# double pi4, epsln, epsh, frq4, per2, pp, pq, ehp, ehm, ir, h, iback,
# irback, t, ep, em, tk, xw, wg, xa, fp, fm, errh, tn, errd;
pi4 = math.pi/4
epsln = 1 - math.log(efs * eps)
epsh = math.sqrt(efs * eps)
n = int(enoff * epsln)
frq4 = abs(omega) / (2 * pi4)
per2 = math.pi / abs(omega)
pq = pqoff / epsln
pp = ppoff - math.log(pq * pq * frq4)
ehp = math.exp(2 * pq)
ehm = 1 / ehp
xw = math.exp(pp - 2 * pi4)
i = f(a + math.sqrt(xw * (per2 * 0.5)))
ir = i * xw
i *= per2 * 0.5
err = abs(i)
h = 2
m = 1
while 1:
iback = i
irback = ir
t = h * 0.5
while 1:
em = math.exp(2 * pq * t)
ep = pi4 * em
em = pi4 / em
tk = t
while 1:
xw = math.exp(pp - ep - em)
wg = math.sqrt(frq4 * xw + tk * tk)
xa = xw / (tk + wg)
wg = (pq * xw * (ep - em) + xa) / wg
fm = f(a + xa)
fp = f(a + xa + per2 * tk)
ir += (fp + fm) * xw
fm *= wg
fp *= per2 - wg
i += fp + fm
if m == 1:
err += abs(fp) + abs(fm)
ep *= ehp
em *= ehm
tk += 1
if (ep >= epsln):
break
if (m == 1):
errh = err * epsh
err *= eps
tn = tk
while abs(fm) > err:
xw = math.exp(pp - ep - em)
xa = xw / tk * 0.5
wg = xa * (1 / tk + 2 * pq * (ep - em))
fm = f(a + xa)
ir += fm * xw
fm *= wg
i += fm
ep *= ehp
em *= ehm
tk += 1
fm = f(a + per2 * tn)
em = per2 * fm
i += em
if (abs(fp) > err) | (abs(em) > err):
l = 0
while 1:
l+=1
tn += n
em = fm
fm = f(a + per2 * tn)
xa = fm
ep = fm
em += fm
xw = 1
wg = 1
for k in range(1,n):
xw = xw * (n + 1 - k) / k
wg += xw
fp = f(a + per2 * (tn - k))
xa += fp
ep += fp * wg
em += fp * xw
wg = per2 * n / (wg * n + xw)
em = wg * abs(em)
if (em <= err) | (l >= lmax):
break
i += per2 * xa
i += wg * ep
if (em > err):
err = em
t += h
if t >= 1:
break
if m == 1:
errd = 1 + 2 * errh
else:
errd = h * (abs(i - 2 * iback) + pq * abs(ir - 2 * irback))
h *= 0.5
m *= 2
if not ((errd > errh) & (m < mmax)):
break
i *= h
if errd > errh:
err = -errd
else:
err *= m * 0.5
return i, err
if __name__ == "__main__":
# intde test 1 nfunc = 0
def intde_test1(x):
return 1/math.sqrt(x)
i,err = intde(intde_test1,0.,1.,1e-15)
print("result1 = ",i,"error1 = ",err)
#intde test 2
def intde_test2(x):
return math.sqrt(4 - x * x)
i,err = intde(intde_test2,0.,2.,1e-15)
print("result2 = ",i,"error2 = ",err)
#intdei test 1
def intdei_test1(x):
return 1. / (1 + x * x)
i,err = intdei(intdei_test1,0.,1e-15)
print("result3 = ",i,"error3 = ",err)
#intdei test 2
def intdei_test2(x):
return math.exp(-x) / math.sqrt(x)
i,err = intdei(intdei_test2,0.,1e-15)
print("result4 = ",i,"error4 = ",err)
#intdeo test 1
def intdeo_test1(x):
return math.sin(x)/x
i,err = intdeo(intdeo_test1,0.,1.,1e-15)
print("result5 = ",i,"error5 = ",err)
#intdeo test2
def intdeo_test2(x):
return math.cos(x) / math.sqrt(x)
i,err = intdeo(intdeo_test2,0.,1.,1e-15)
print("result6 = ",i,"error6= ",err)