Source code for mpmath.calculus.quadrature
import math
from ..libmp.backend import xrange
[docs]class QuadratureRule(object):
"""
Quadrature rules are implemented using this class, in order to
simplify the code and provide a common infrastructure
for tasks such as error estimation and node caching.
You can implement a custom quadrature rule by subclassing
:class:`QuadratureRule` and implementing the appropriate
methods. The subclass can then be used by :func:`~mpmath.quad` by
passing it as the *method* argument.
:class:`QuadratureRule` instances are supposed to be singletons.
:class:`QuadratureRule` therefore implements instance caching
in :func:`~mpmath.__new__`.
"""
def __init__(self, ctx):
self.ctx = ctx
self.standard_cache = {}
self.transformed_cache = {}
self.interval_count = {}
[docs] def clear(self):
"""
Delete cached node data.
"""
self.standard_cache = {}
self.transformed_cache = {}
self.interval_count = {}
[docs] def calc_nodes(self, degree, prec, verbose=False):
r"""
Compute nodes for the standard interval `[-1, 1]`. Subclasses
should probably implement only this method, and use
:func:`~mpmath.get_nodes` method to retrieve the nodes.
"""
raise NotImplementedError
[docs] def get_nodes(self, a, b, degree, prec, verbose=False):
"""
Return nodes for given interval, degree and precision. The
nodes are retrieved from a cache if already computed;
otherwise they are computed by calling :func:`~mpmath.calc_nodes`
and are then cached.
Subclasses should probably not implement this method,
but just implement :func:`~mpmath.calc_nodes` for the actual
node computation.
"""
key = (a, b, degree, prec)
if key in self.transformed_cache:
return self.transformed_cache[key]
orig = self.ctx.prec
try:
self.ctx.prec = prec+20
# Get nodes on standard interval
if (degree, prec) in self.standard_cache:
nodes = self.standard_cache[degree, prec]
else:
nodes = self.calc_nodes(degree, prec, verbose)
self.standard_cache[degree, prec] = nodes
# Transform to general interval
nodes = self.transform_nodes(nodes, a, b, verbose)
if key in self.interval_count:
self.transformed_cache[key] = nodes
else:
self.interval_count[key] = True
finally:
self.ctx.prec = orig
return nodes
[docs] def transform_nodes(self, nodes, a, b, verbose=False):
r"""
Rescale standardized nodes (for `[-1, 1]`) to a general
interval `[a, b]`. For a finite interval, a simple linear
change of variables is used. Otherwise, the following
transformations are used:
.. math ::
[a, \infty] : t = \frac{1}{x} + (a-1)
[-\infty, b] : t = (b+1) - \frac{1}{x}
[-\infty, \infty] : t = \frac{x}{\sqrt{1-x^2}}
"""
ctx = self.ctx
a = ctx.convert(a)
b = ctx.convert(b)
one = ctx.one
if (a, b) == (-one, one):
return nodes
half = ctx.mpf(0.5)
new_nodes = []
if ctx.isinf(a) or ctx.isinf(b):
if (a, b) == (ctx.ninf, ctx.inf):
p05 = -half
for x, w in nodes:
x2 = x*x
px1 = one-x2
spx1 = px1**p05
x = x*spx1
w *= spx1/px1
new_nodes.append((x, w))
elif a == ctx.ninf:
b1 = b+1
for x, w in nodes:
u = 2/(x+one)
x = b1-u
w *= half*u**2
new_nodes.append((x, w))
elif b == ctx.inf:
a1 = a-1
for x, w in nodes:
u = 2/(x+one)
x = a1+u
w *= half*u**2
new_nodes.append((x, w))
elif a == ctx.inf or b == ctx.ninf:
return [(x,-w) for (x,w) in self.transform_nodes(nodes, b, a, verbose)]
else:
raise NotImplementedError
else:
# Simple linear change of variables
C = (b-a)/2
D = (b+a)/2
for x, w in nodes:
new_nodes.append((D+C*x, C*w))
return new_nodes
[docs] def guess_degree(self, prec):
"""
Given a desired precision `p` in bits, estimate the degree `m`
of the quadrature required to accomplish full accuracy for
typical integrals. By default, :func:`~mpmath.quad` will perform up
to `m` iterations. The value of `m` should be a slight
overestimate, so that "slightly bad" integrals can be dealt
with automatically using a few extra iterations. On the
other hand, it should not be too big, so :func:`~mpmath.quad` can
quit within a reasonable amount of time when it is given
an "unsolvable" integral.
The default formula used by :func:`~mpmath.guess_degree` is tuned
for both :class:`TanhSinh` and :class:`GaussLegendre`.
The output is roughly as follows:
+---------+---------+
| `p` | `m` |
+=========+=========+
| 50 | 6 |
+---------+---------+
| 100 | 7 |
+---------+---------+
| 500 | 10 |
+---------+---------+
| 3000 | 12 |
+---------+---------+
This formula is based purely on a limited amount of
experimentation and will sometimes be wrong.
"""
# Expected degree
# XXX: use mag
g = int(4 + max(0, self.ctx.log(prec/30.0, 2)))
# Reasonable "worst case"
g += 2
return g
[docs] def estimate_error(self, results, prec, epsilon):
r"""
Given results from integrations `[I_1, I_2, \ldots, I_k]` done
with a quadrature of rule of degree `1, 2, \ldots, k`, estimate
the error of `I_k`.
For `k = 2`, we estimate `|I_{\infty}-I_2|` as `|I_2-I_1|`.
For `k > 2`, we extrapolate `|I_{\infty}-I_k| \approx |I_{k+1}-I_k|`
from `|I_k-I_{k-1}|` and `|I_k-I_{k-2}|` under the assumption
that each degree increment roughly doubles the accuracy of
the quadrature rule (this is true for both :class:`TanhSinh`
and :class:`GaussLegendre`). The extrapolation formula is given
by Borwein, Bailey & Girgensohn. Although not very conservative,
this method seems to be very robust in practice.
"""
if len(results) == 2:
return abs(results[0]-results[1])
try:
if results[-1] == results[-2] == results[-3]:
return self.ctx.zero
D1 = self.ctx.log(abs(results[-1]-results[-2]), 10)
D2 = self.ctx.log(abs(results[-1]-results[-3]), 10)
except ValueError:
return epsilon
D3 = -prec
D4 = min(0, max(D1**2/D2, 2*D1, D3))
return self.ctx.mpf(10) ** int(D4)
[docs] def summation(self, f, points, prec, epsilon, max_degree, verbose=False):
"""
Main integration function. Computes the 1D integral over
the interval specified by *points*. For each subinterval,
performs quadrature of degree from 1 up to *max_degree*
until :func:`~mpmath.estimate_error` signals convergence.
:func:`~mpmath.summation` transforms each subintegration to
the standard interval and then calls :func:`~mpmath.sum_next`.
"""
ctx = self.ctx
I = err = ctx.zero
for i in xrange(len(points)-1):
a, b = points[i], points[i+1]
if a == b:
continue
# XXX: we could use a single variable transformation,
# but this is not good in practice. We get better accuracy
# by having 0 as an endpoint.
if (a, b) == (ctx.ninf, ctx.inf):
_f = f
f = lambda x: _f(-x) + _f(x)
a, b = (ctx.zero, ctx.inf)
results = []
for degree in xrange(1, max_degree+1):
nodes = self.get_nodes(a, b, degree, prec, verbose)
if verbose:
print("Integrating from %s to %s (degree %s of %s)" % \
(ctx.nstr(a), ctx.nstr(b), degree, max_degree))
results.append(self.sum_next(f, nodes, degree, prec, results, verbose))
if degree > 1:
err = self.estimate_error(results, prec, epsilon)
if err <= epsilon:
break
if verbose:
print("Estimated error:", ctx.nstr(err))
I += results[-1]
if err > epsilon:
if verbose:
print("Failed to reach full accuracy. Estimated error:", ctx.nstr(err))
return I, err
[docs] def sum_next(self, f, nodes, degree, prec, previous, verbose=False):
r"""
Evaluates the step sum `\sum w_k f(x_k)` where the *nodes* list
contains the `(w_k, x_k)` pairs.
:func:`~mpmath.summation` will supply the list *results* of
values computed by :func:`~mpmath.sum_next` at previous degrees, in
case the quadrature rule is able to reuse them.
"""
return self.ctx.fdot((w, f(x)) for (x,w) in nodes)
[docs]class TanhSinh(QuadratureRule):
r"""
This class implements "tanh-sinh" or "doubly exponential"
quadrature. This quadrature rule is based on the Euler-Maclaurin
integral formula. By performing a change of variables involving
nested exponentials / hyperbolic functions (hence the name), the
derivatives at the endpoints vanish rapidly. Since the error term
in the Euler-Maclaurin formula depends on the derivatives at the
endpoints, a simple step sum becomes extremely accurate. In
practice, this means that doubling the number of evaluation
points roughly doubles the number of accurate digits.
Comparison to Gauss-Legendre:
* Initial computation of nodes is usually faster
* Handles endpoint singularities better
* Handles infinite integration intervals better
* Is slower for smooth integrands once nodes have been computed
The implementation of the tanh-sinh algorithm is based on the
description given in Borwein, Bailey & Girgensohn, "Experimentation
in Mathematics - Computational Paths to Discovery", A K Peters,
2003, pages 312-313. In the present implementation, a few
improvements have been made:
* A more efficient scheme is used to compute nodes (exploiting
recurrence for the exponential function)
* The nodes are computed successively instead of all at once
Various documents describing the algorithm are available online, e.g.:
* http://crd.lbl.gov/~dhbailey/dhbpapers/dhb-tanh-sinh.pdf
* http://users.cs.dal.ca/~jborwein/tanh-sinh.pdf
"""
[docs] def sum_next(self, f, nodes, degree, prec, previous, verbose=False):
"""
Step sum for tanh-sinh quadrature of degree `m`. We exploit the
fact that half of the abscissas at degree `m` are precisely the
abscissas from degree `m-1`. Thus reusing the result from
the previous level allows a 2x speedup.
"""
h = self.ctx.mpf(2)**(-degree)
# Abscissas overlap, so reusing saves half of the time
if previous:
S = previous[-1]/(h*2)
else:
S = self.ctx.zero
S += self.ctx.fdot((w,f(x)) for (x,w) in nodes)
return h*S
[docs] def calc_nodes(self, degree, prec, verbose=False):
r"""
The abscissas and weights for tanh-sinh quadrature of degree
`m` are given by
.. math::
x_k = \tanh(\pi/2 \sinh(t_k))
w_k = \pi/2 \cosh(t_k) / \cosh(\pi/2 \sinh(t_k))^2
where `t_k = t_0 + hk` for a step length `h \sim 2^{-m}`. The
list of nodes is actually infinite, but the weights die off so
rapidly that only a few are needed.
"""
ctx = self.ctx
nodes = []
extra = 20
ctx.prec += extra
tol = ctx.ldexp(1, -prec-10)
pi4 = ctx.pi/4
# For simplicity, we work in steps h = 1/2^n, with the first point
# offset so that we can reuse the sum from the previous degree
# We define degree 1 to include the "degree 0" steps, including
# the point x = 0. (It doesn't work well otherwise; not sure why.)
t0 = ctx.ldexp(1, -degree)
if degree == 1:
#nodes.append((mpf(0), pi4))
#nodes.append((-mpf(0), pi4))
nodes.append((ctx.zero, ctx.pi/2))
h = t0
else:
h = t0*2
# Since h is fixed, we can compute the next exponential
# by simply multiplying by exp(h)
expt0 = ctx.exp(t0)
a = pi4 * expt0
b = pi4 / expt0
udelta = ctx.exp(h)
urdelta = 1/udelta
for k in xrange(0, 20*2**degree+1):
# Reference implementation:
# t = t0 + k*h
# x = tanh(pi/2 * sinh(t))
# w = pi/2 * cosh(t) / cosh(pi/2 * sinh(t))**2
# Fast implementation. Note that c = exp(pi/2 * sinh(t))
c = ctx.exp(a-b)
d = 1/c
co = (c+d)/2
si = (c-d)/2
x = si / co
w = (a+b) / co**2
diff = abs(x-1)
if diff <= tol:
break
nodes.append((x, w))
nodes.append((-x, w))
a *= udelta
b *= urdelta
if verbose and k % 300 == 150:
# Note: the number displayed is rather arbitrary. Should
# figure out how to print something that looks more like a
# percentage
print("Calculating nodes:", ctx.nstr(-ctx.log(diff, 10) / prec))
ctx.prec -= extra
return nodes
[docs]class GaussLegendre(QuadratureRule):
"""
This class implements Gauss-Legendre quadrature, which is
exceptionally efficient for polynomials and polynomial-like (i.e.
very smooth) integrands.
The abscissas and weights are given by roots and values of
Legendre polynomials, which are the orthogonal polynomials
on `[-1, 1]` with respect to the unit weight
(see :func:`~mpmath.legendre`).
In this implementation, we take the "degree" `m` of the quadrature
to denote a Gauss-Legendre rule of degree `3 \cdot 2^m` (following
Borwein, Bailey & Girgensohn). This way we get quadratic, rather
than linear, convergence as the degree is incremented.
Comparison to tanh-sinh quadrature:
* Is faster for smooth integrands once nodes have been computed
* Initial computation of nodes is usually slower
* Handles endpoint singularities worse
* Handles infinite integration intervals worse
"""
[docs] def calc_nodes(self, degree, prec, verbose=False):
"""
Calculates the abscissas and weights for Gauss-Legendre
quadrature of degree of given degree (actually `3 \cdot 2^m`).
"""
ctx = self.ctx
# It is important that the epsilon is set lower than the
# "real" epsilon
epsilon = ctx.ldexp(1, -prec-8)
# Fairly high precision might be required for accurate
# evaluation of the roots
orig = ctx.prec
ctx.prec = int(prec*1.5)
if degree == 1:
x = ctx.mpf(3)/5
w = ctx.mpf(5)/9
nodes = [(-x,w),(ctx.zero,ctx.mpf(8)/9),(x,w)]
ctx.prec = orig
return nodes
nodes = []
n = 3*2**(degree-1)
upto = n//2 + 1
for j in xrange(1, upto):
# Asymptotic formula for the roots
r = ctx.mpf(math.cos(math.pi*(j-0.25)/(n+0.5)))
# Newton iteration
while 1:
t1, t2 = 1, 0
# Evaluates the Legendre polynomial using its defining
# recurrence relation
for j1 in xrange(1,n+1):
t3, t2, t1 = t2, t1, ((2*j1-1)*r*t1 - (j1-1)*t2)/j1
t4 = n*(r*t1- t2)/(r**2-1)
t5 = r
a = t1/t4
r = r - a
if abs(a) < epsilon:
break
x = r
w = 2/((1-r**2)*t4**2)
if verbose and j % 30 == 15:
print("Computing nodes (%i of %i)" % (j, upto))
nodes.append((x, w))
nodes.append((-x, w))
ctx.prec = orig
return nodes
class QuadratureMethods(object):
def __init__(ctx, *args, **kwargs):
ctx._gauss_legendre = GaussLegendre(ctx)
ctx._tanh_sinh = TanhSinh(ctx)
def quad(ctx, f, *points, **kwargs):
r"""
Computes a single, double or triple integral over a given
1D interval, 2D rectangle, or 3D cuboid. A basic example::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> quad(sin, [0, pi])
2.0
A basic 2D integral::
>>> f = lambda x, y: cos(x+y/2)
>>> quad(f, [-pi/2, pi/2], [0, pi])
4.0
**Interval format**
The integration range for each dimension may be specified
using a list or tuple. Arguments are interpreted as follows:
``quad(f, [x1, x2])`` -- calculates
`\int_{x_1}^{x_2} f(x) \, dx`
``quad(f, [x1, x2], [y1, y2])`` -- calculates
`\int_{x_1}^{x_2} \int_{y_1}^{y_2} f(x,y) \, dy \, dx`
``quad(f, [x1, x2], [y1, y2], [z1, z2])`` -- calculates
`\int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} f(x,y,z)
\, dz \, dy \, dx`
Endpoints may be finite or infinite. An interval descriptor
may also contain more than two points. In this
case, the integration is split into subintervals, between
each pair of consecutive points. This is useful for
dealing with mid-interval discontinuities, or integrating
over large intervals where the function is irregular or
oscillates.
**Options**
:func:`~mpmath.quad` recognizes the following keyword arguments:
*method*
Chooses integration algorithm (described below).
*error*
If set to true, :func:`~mpmath.quad` returns `(v, e)` where `v` is the
integral and `e` is the estimated error.
*maxdegree*
Maximum degree of the quadrature rule to try before
quitting.
*verbose*
Print details about progress.
**Algorithms**
Mpmath presently implements two integration algorithms: tanh-sinh
quadrature and Gauss-Legendre quadrature. These can be selected
using *method='tanh-sinh'* or *method='gauss-legendre'* or by
passing the classes *method=TanhSinh*, *method=GaussLegendre*.
The functions :func:`~mpmath.quadts` and :func:`~mpmath.quadgl` are also available
as shortcuts.
Both algorithms have the property that doubling the number of
evaluation points roughly doubles the accuracy, so both are ideal
for high precision quadrature (hundreds or thousands of digits).
At high precision, computing the nodes and weights for the
integration can be expensive (more expensive than computing the
function values). To make repeated integrations fast, nodes
are automatically cached.
The advantages of the tanh-sinh algorithm are that it tends to
handle endpoint singularities well, and that the nodes are cheap
to compute on the first run. For these reasons, it is used by
:func:`~mpmath.quad` as the default algorithm.
Gauss-Legendre quadrature often requires fewer function
evaluations, and is therefore often faster for repeated use, but
the algorithm does not handle endpoint singularities as well and
the nodes are more expensive to compute. Gauss-Legendre quadrature
can be a better choice if the integrand is smooth and repeated
integrations are required (e.g. for multiple integrals).
See the documentation for :class:`TanhSinh` and
:class:`GaussLegendre` for additional details.
**Examples of 1D integrals**
Intervals may be infinite or half-infinite. The following two
examples evaluate the limits of the inverse tangent function
(`\int 1/(1+x^2) = \tan^{-1} x`), and the Gaussian integral
`\int_{\infty}^{\infty} \exp(-x^2)\,dx = \sqrt{\pi}`::
>>> mp.dps = 15
>>> quad(lambda x: 2/(x**2+1), [0, inf])
3.14159265358979
>>> quad(lambda x: exp(-x**2), [-inf, inf])**2
3.14159265358979
Integrals can typically be resolved to high precision.
The following computes 50 digits of `\pi` by integrating the
area of the half-circle defined by `x^2 + y^2 \le 1`,
`-1 \le x \le 1`, `y \ge 0`::
>>> mp.dps = 50
>>> 2*quad(lambda x: sqrt(1-x**2), [-1, 1])
3.1415926535897932384626433832795028841971693993751
One can just as well compute 1000 digits (output truncated)::
>>> mp.dps = 1000
>>> 2*quad(lambda x: sqrt(1-x**2), [-1, 1]) #doctest:+ELLIPSIS
3.141592653589793238462643383279502884...216420198
Complex integrals are supported. The following computes
a residue at `z = 0` by integrating counterclockwise along the
diamond-shaped path from `1` to `+i` to `-1` to `-i` to `1`::
>>> mp.dps = 15
>>> chop(quad(lambda z: 1/z, [1,j,-1,-j,1]))
(0.0 + 6.28318530717959j)
**Examples of 2D and 3D integrals**
Here are several nice examples of analytically solvable
2D integrals (taken from MathWorld [1]) that can be evaluated
to high precision fairly rapidly by :func:`~mpmath.quad`::
>>> mp.dps = 30
>>> f = lambda x, y: (x-1)/((1-x*y)*log(x*y))
>>> quad(f, [0, 1], [0, 1])
0.577215664901532860606512090082
>>> +euler
0.577215664901532860606512090082
>>> f = lambda x, y: 1/sqrt(1+x**2+y**2)
>>> quad(f, [-1, 1], [-1, 1])
3.17343648530607134219175646705
>>> 4*log(2+sqrt(3))-2*pi/3
3.17343648530607134219175646705
>>> f = lambda x, y: 1/(1-x**2 * y**2)
>>> quad(f, [0, 1], [0, 1])
1.23370055013616982735431137498
>>> pi**2 / 8
1.23370055013616982735431137498
>>> quad(lambda x, y: 1/(1-x*y), [0, 1], [0, 1])
1.64493406684822643647241516665
>>> pi**2 / 6
1.64493406684822643647241516665
Multiple integrals may be done over infinite ranges::
>>> mp.dps = 15
>>> print(quad(lambda x,y: exp(-x-y), [0, inf], [1, inf]))
0.367879441171442
>>> print(1/e)
0.367879441171442
For nonrectangular areas, one can call :func:`~mpmath.quad` recursively.
For example, we can replicate the earlier example of calculating
`\pi` by integrating over the unit-circle, and actually use double
quadrature to actually measure the area circle::
>>> f = lambda x: quad(lambda y: 1, [-sqrt(1-x**2), sqrt(1-x**2)])
>>> quad(f, [-1, 1])
3.14159265358979
Here is a simple triple integral::
>>> mp.dps = 15
>>> f = lambda x,y,z: x*y/(1+z)
>>> quad(f, [0,1], [0,1], [1,2], method='gauss-legendre')
0.101366277027041
>>> (log(3)-log(2))/4
0.101366277027041
**Singularities**
Both tanh-sinh and Gauss-Legendre quadrature are designed to
integrate smooth (infinitely differentiable) functions. Neither
algorithm copes well with mid-interval singularities (such as
mid-interval discontinuities in `f(x)` or `f'(x)`).
The best solution is to split the integral into parts::
>>> mp.dps = 15
>>> quad(lambda x: abs(sin(x)), [0, 2*pi]) # Bad
3.99900894176779
>>> quad(lambda x: abs(sin(x)), [0, pi, 2*pi]) # Good
4.0
The tanh-sinh rule often works well for integrands having a
singularity at one or both endpoints::
>>> mp.dps = 15
>>> quad(log, [0, 1], method='tanh-sinh') # Good
-1.0
>>> quad(log, [0, 1], method='gauss-legendre') # Bad
-0.999932197413801
However, the result may still be inaccurate for some functions::
>>> quad(lambda x: 1/sqrt(x), [0, 1], method='tanh-sinh')
1.99999999946942
This problem is not due to the quadrature rule per se, but to
numerical amplification of errors in the nodes. The problem can be
circumvented by temporarily increasing the precision::
>>> mp.dps = 30
>>> a = quad(lambda x: 1/sqrt(x), [0, 1], method='tanh-sinh')
>>> mp.dps = 15
>>> +a
2.0
**Highly variable functions**
For functions that are smooth (in the sense of being infinitely
differentiable) but contain sharp mid-interval peaks or many
"bumps", :func:`~mpmath.quad` may fail to provide full accuracy. For
example, with default settings, :func:`~mpmath.quad` is able to integrate
`\sin(x)` accurately over an interval of length 100 but not over
length 1000::
>>> quad(sin, [0, 100]); 1-cos(100) # Good
0.137681127712316
0.137681127712316
>>> quad(sin, [0, 1000]); 1-cos(1000) # Bad
-37.8587612408485
0.437620923709297
One solution is to break the integration into 10 intervals of
length 100::
>>> quad(sin, linspace(0, 1000, 10)) # Good
0.437620923709297
Another is to increase the degree of the quadrature::
>>> quad(sin, [0, 1000], maxdegree=10) # Also good
0.437620923709297
Whether splitting the interval or increasing the degree is
more efficient differs from case to case. Another example is the
function `1/(1+x^2)`, which has a sharp peak centered around
`x = 0`::
>>> f = lambda x: 1/(1+x**2)
>>> quad(f, [-100, 100]) # Bad
3.64804647105268
>>> quad(f, [-100, 100], maxdegree=10) # Good
3.12159332021646
>>> quad(f, [-100, 0, 100]) # Also good
3.12159332021646
**References**
1. http://mathworld.wolfram.com/DoubleIntegral.html
"""
rule = kwargs.get('method', 'tanh-sinh')
if type(rule) is str:
if rule == 'tanh-sinh':
rule = ctx._tanh_sinh
elif rule == 'gauss-legendre':
rule = ctx._gauss_legendre
else:
raise ValueError("unknown quadrature rule: %s" % rule)
else:
rule = rule(ctx)
verbose = kwargs.get('verbose')
dim = len(points)
orig = prec = ctx.prec
epsilon = ctx.eps/8
m = kwargs.get('maxdegree') or rule.guess_degree(prec)
points = [ctx._as_points(p) for p in points]
try:
ctx.prec += 20
if dim == 1:
v, err = rule.summation(f, points[0], prec, epsilon, m, verbose)
elif dim == 2:
v, err = rule.summation(lambda x: \
rule.summation(lambda y: f(x,y), \
points[1], prec, epsilon, m)[0],
points[0], prec, epsilon, m, verbose)
elif dim == 3:
v, err = rule.summation(lambda x: \
rule.summation(lambda y: \
rule.summation(lambda z: f(x,y,z), \
points[2], prec, epsilon, m)[0],
points[1], prec, epsilon, m)[0],
points[0], prec, epsilon, m, verbose)
else:
raise NotImplementedError("quadrature must have dim 1, 2 or 3")
finally:
ctx.prec = orig
if kwargs.get("error"):
return +v, err
return +v
def quadts(ctx, *args, **kwargs):
"""
Performs tanh-sinh quadrature. The call
quadts(func, *points, ...)
is simply a shortcut for:
quad(func, *points, ..., method=TanhSinh)
For example, a single integral and a double integral:
quadts(lambda x: exp(cos(x)), [0, 1])
quadts(lambda x, y: exp(cos(x+y)), [0, 1], [0, 1])
See the documentation for quad for information about how points
arguments and keyword arguments are parsed.
See documentation for TanhSinh for algorithmic information about
tanh-sinh quadrature.
"""
kwargs['method'] = 'tanh-sinh'
return ctx.quad(*args, **kwargs)
def quadgl(ctx, *args, **kwargs):
"""
Performs Gauss-Legendre quadrature. The call
quadgl(func, *points, ...)
is simply a shortcut for:
quad(func, *points, ..., method=GaussLegendre)
For example, a single integral and a double integral:
quadgl(lambda x: exp(cos(x)), [0, 1])
quadgl(lambda x, y: exp(cos(x+y)), [0, 1], [0, 1])
See the documentation for quad for information about how points
arguments and keyword arguments are parsed.
See documentation for TanhSinh for algorithmic information about
tanh-sinh quadrature.
"""
kwargs['method'] = 'gauss-legendre'
return ctx.quad(*args, **kwargs)
def quadosc(ctx, f, interval, omega=None, period=None, zeros=None):
r"""
Calculates
.. math ::
I = \int_a^b f(x) dx
where at least one of `a` and `b` is infinite and where
`f(x) = g(x) \cos(\omega x + \phi)` for some slowly
decreasing function `g(x)`. With proper input, :func:`~mpmath.quadosc`
can also handle oscillatory integrals where the oscillation
rate is different from a pure sine or cosine wave.
In the standard case when `|a| < \infty, b = \infty`,
:func:`~mpmath.quadosc` works by evaluating the infinite series
.. math ::
I = \int_a^{x_1} f(x) dx +
\sum_{k=1}^{\infty} \int_{x_k}^{x_{k+1}} f(x) dx
where `x_k` are consecutive zeros (alternatively
some other periodic reference point) of `f(x)`.
Accordingly, :func:`~mpmath.quadosc` requires information about the
zeros of `f(x)`. For a periodic function, you can specify
the zeros by either providing the angular frequency `\omega`
(*omega*) or the *period* `2 \pi/\omega`. In general, you can
specify the `n`-th zero by providing the *zeros* arguments.
Below is an example of each::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> f = lambda x: sin(3*x)/(x**2+1)
>>> quadosc(f, [0,inf], omega=3)
0.37833007080198
>>> quadosc(f, [0,inf], period=2*pi/3)
0.37833007080198
>>> quadosc(f, [0,inf], zeros=lambda n: pi*n/3)
0.37833007080198
>>> (ei(3)*exp(-3)-exp(3)*ei(-3))/2 # Computed by Mathematica
0.37833007080198
Note that *zeros* was specified to multiply `n` by the
*half-period*, not the full period. In theory, it does not matter
whether each partial integral is done over a half period or a full
period. However, if done over half-periods, the infinite series
passed to :func:`~mpmath.nsum` becomes an *alternating series* and this
typically makes the extrapolation much more efficient.
Here is an example of an integration over the entire real line,
and a half-infinite integration starting at `-\infty`::
>>> quadosc(lambda x: cos(x)/(1+x**2), [-inf, inf], omega=1)
1.15572734979092
>>> pi/e
1.15572734979092
>>> quadosc(lambda x: cos(x)/x**2, [-inf, -1], period=2*pi)
-0.0844109505595739
>>> cos(1)+si(1)-pi/2
-0.0844109505595738
Of course, the integrand may contain a complex exponential just as
well as a real sine or cosine::
>>> quadosc(lambda x: exp(3*j*x)/(1+x**2), [-inf,inf], omega=3)
(0.156410688228254 + 0.0j)
>>> pi/e**3
0.156410688228254
>>> quadosc(lambda x: exp(3*j*x)/(2+x+x**2), [-inf,inf], omega=3)
(0.00317486988463794 - 0.0447701735209082j)
>>> 2*pi/sqrt(7)/exp(3*(j+sqrt(7))/2)
(0.00317486988463794 - 0.0447701735209082j)
**Non-periodic functions**
If `f(x) = g(x) h(x)` for some function `h(x)` that is not
strictly periodic, *omega* or *period* might not work, and it might
be necessary to use *zeros*.
A notable exception can be made for Bessel functions which, though not
periodic, are "asymptotically periodic" in a sufficiently strong sense
that the sum extrapolation will work out::
>>> quadosc(j0, [0, inf], period=2*pi)
1.0
>>> quadosc(j1, [0, inf], period=2*pi)
1.0
More properly, one should provide the exact Bessel function zeros::
>>> j0zero = lambda n: findroot(j0, pi*(n-0.25))
>>> quadosc(j0, [0, inf], zeros=j0zero)
1.0
For an example where *zeros* becomes necessary, consider the
complete Fresnel integrals
.. math ::
\int_0^{\infty} \cos x^2\,dx = \int_0^{\infty} \sin x^2\,dx
= \sqrt{\frac{\pi}{8}}.
Although the integrands do not decrease in magnitude as
`x \to \infty`, the integrals are convergent since the oscillation
rate increases (causing consecutive periods to asymptotically
cancel out). These integrals are virtually impossible to calculate
to any kind of accuracy using standard quadrature rules. However,
if one provides the correct asymptotic distribution of zeros
(`x_n \sim \sqrt{n}`), :func:`~mpmath.quadosc` works::
>>> mp.dps = 30
>>> f = lambda x: cos(x**2)
>>> quadosc(f, [0,inf], zeros=lambda n:sqrt(pi*n))
0.626657068657750125603941321203
>>> f = lambda x: sin(x**2)
>>> quadosc(f, [0,inf], zeros=lambda n:sqrt(pi*n))
0.626657068657750125603941321203
>>> sqrt(pi/8)
0.626657068657750125603941321203
(Interestingly, these integrals can still be evaluated if one
places some other constant than `\pi` in the square root sign.)
In general, if `f(x) \sim g(x) \cos(h(x))`, the zeros follow
the inverse-function distribution `h^{-1}(x)`::
>>> mp.dps = 15
>>> f = lambda x: sin(exp(x))
>>> quadosc(f, [1,inf], zeros=lambda n: log(n))
-0.25024394235267
>>> pi/2-si(e)
-0.250243942352671
**Non-alternating functions**
If the integrand oscillates around a positive value, without
alternating signs, the extrapolation might fail. A simple trick
that sometimes works is to multiply or divide the frequency by 2::
>>> f = lambda x: 1/x**2+sin(x)/x**4
>>> quadosc(f, [1,inf], omega=1) # Bad
1.28642190869861
>>> quadosc(f, [1,inf], omega=0.5) # Perfect
1.28652953559617
>>> 1+(cos(1)+ci(1)+sin(1))/6
1.28652953559617
**Fast decay**
:func:`~mpmath.quadosc` is primarily useful for slowly decaying
integrands. If the integrand decreases exponentially or faster,
:func:`~mpmath.quad` will likely handle it without trouble (and generally be
much faster than :func:`~mpmath.quadosc`)::
>>> quadosc(lambda x: cos(x)/exp(x), [0, inf], omega=1)
0.5
>>> quad(lambda x: cos(x)/exp(x), [0, inf])
0.5
"""
a, b = ctx._as_points(interval)
a = ctx.convert(a)
b = ctx.convert(b)
if [omega, period, zeros].count(None) != 2:
raise ValueError( \
"must specify exactly one of omega, period, zeros")
if a == ctx.ninf and b == ctx.inf:
s1 = ctx.quadosc(f, [a, 0], omega=omega, zeros=zeros, period=period)
s2 = ctx.quadosc(f, [0, b], omega=omega, zeros=zeros, period=period)
return s1 + s2
if a == ctx.ninf:
if zeros:
return ctx.quadosc(lambda x:f(-x), [-b,-a], lambda n: zeros(-n))
else:
return ctx.quadosc(lambda x:f(-x), [-b,-a], omega=omega, period=period)
if b != ctx.inf:
raise ValueError("quadosc requires an infinite integration interval")
if not zeros:
if omega:
period = 2*ctx.pi/omega
zeros = lambda n: n*period/2
#for n in range(1,10):
# p = zeros(n)
# if p > a:
# break
#if n >= 9:
# raise ValueError("zeros do not appear to be correctly indexed")
n = 1
s = ctx.quadgl(f, [a, zeros(n)])
def term(k):
return ctx.quadgl(f, [zeros(k), zeros(k+1)])
s += ctx.nsum(term, [n, ctx.inf])
return s
if __name__ == '__main__':
import doctest
doctest.testmod()
