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quad.py
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quad.py
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"""
Defining various quadrature routines.
Based on the quadrature routines found in the CompEcon toolbox by
Miranda and Fackler.
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational Economics
and Finance, MIT Press, 2002.
"""
import math
import numpy as np
import scipy.linalg as la
from numba import jit, vectorize
from .ce_util import ckron, gridmake
from .util import check_random_state
__all__ = ['qnwcheb', 'qnwequi', 'qnwlege', 'qnwnorm', 'qnwlogn',
'qnwsimp', 'qnwtrap', 'qnwunif', 'quadrect', 'qnwbeta',
'qnwgamma']
@vectorize(nopython=True)
def gammaln(x):
return math.lgamma(x)
@vectorize(nopython=True)
def fix(x):
if x < 0:
return math.ceil(x)
else:
return math.floor(x)
# ------------------ #
# Exported Functions #
# ------------------ #
def qnwcheb(n, a=1, b=1):
"""
Computes multivariate Guass-Checbychev quadrature nodes and weights.
Parameters
----------
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
a : scalar or array_like(float)
A length-d iterable of lower endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
b : scalar or array_like(float)
A length-d iterable of upper endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwcheb`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
return _make_multidim_func(_qnwcheb1, n, a, b)
def qnwequi(n, a, b, kind="N", equidist_pp=None, random_state=None):
"""
Generates equidistributed sequences with property that averages
value of integrable function evaluated over the sequence converges
to the integral as n goes to infinity.
Parameters
----------
n : int
Number of sequence points
a : scalar or array_like(float)
A length-d iterable of lower endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
b : scalar or array_like(float)
A length-d iterable of upper endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
kind : string, optional(default="N")
One of the following:
- N - Neiderreiter (default)
- W - Weyl
- H - Haber
- R - pseudo Random
equidist_pp : array_like, optional(default=None)
TODO: I don't know what this does
random_state : int or np.random.RandomState, optional
Random seed (integer) or np.random.RandomState instance to set
the initial state of the random number generator for
reproducibility. If None, a randomly initialized RandomState is
used.
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwequi`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
random_state = check_random_state(random_state)
if equidist_pp is None:
import sympy as sym
equidist_pp = np.sqrt(np.array(list(sym.primerange(0, 7920))))
n, a, b = list(map(np.atleast_1d, list(map(np.asarray, [n, a, b]))))
d = max(list(map(len, [n, a, b])))
n = np.prod(n)
if a.size == 1:
a = np.repeat(a, d)
if b.size == 1:
b = np.repeat(b, d)
i = np.arange(1, n + 1)
if kind.upper() == "N": # Neiderreiter
j = 2.0 ** (np.arange(1, d+1) / (d+1))
nodes = np.outer(i, j)
nodes = (nodes - fix(nodes)).squeeze()
elif kind.upper() == "W": # Weyl
j = equidist_pp[:d]
nodes = np.outer(i, j)
nodes = (nodes - fix(nodes)).squeeze()
elif kind.upper() == "H": # Haber
j = equidist_pp[:d]
nodes = np.outer(i * (i+1) / 2, j)
nodes = (nodes - fix(nodes)).squeeze()
elif kind.upper() == "R": # pseudo-random
nodes = random_state.rand(n, d).squeeze()
else:
raise ValueError("Unknown sequence requested")
# compute nodes and weights
r = b - a
nodes = a + nodes * r
weights = (np.prod(r) / n) * np.ones(n)
return nodes, weights
def qnwlege(n, a, b):
"""
Computes multivariate Guass-Legendre quadrature nodes and weights.
Parameters
----------
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
a : scalar or array_like(float)
A length-d iterable of lower endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
b : scalar or array_like(float)
A length-d iterable of upper endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwlege`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
return _make_multidim_func(_qnwlege1, n, a, b)
def qnwnorm(n, mu=None, sig2=None, usesqrtm=False):
"""
Computes nodes and weights for multivariate normal distribution
Parameters
----------
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
mu : scalar or array_like(float), optional(default=zeros(d))
The means of each dimension of the random variable. If a scalar
is given, that constant is repeated d times, where d is the
number of dimensions
sig2 : array_like(float), optional(default=eye(d))
A d x d array representing the variance-covariance matrix of the
multivariate normal distribution.
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwnorm`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
n = np.atleast_1d(n)
d = n.size
if mu is None:
mu = np.zeros(d)
else:
mu = np.atleast_1d(mu)
if sig2 is None:
sig2 = np.eye(d)
else:
sig2 = np.atleast_1d(sig2).reshape(d, d)
if all([x.size == 1 for x in [n, mu, sig2]]):
nodes, weights = _qnwnorm1(n[0])
else:
nodes = []
weights = []
for i in range(d):
_1d = _qnwnorm1(n[i])
nodes.append(_1d[0])
weights.append(_1d[1])
nodes = gridmake(*nodes)
weights = ckron(*weights[::-1])
if usesqrtm:
new_sig2 = la.sqrtm(sig2)
else: # cholesky
new_sig2 = la.cholesky(sig2)
if d > 1:
nodes = nodes.dot(new_sig2) + mu # Broadcast ok
else: # nodes.dot(sig) will not be aligned in scalar case.
nodes = nodes * new_sig2 + mu
return nodes.squeeze(), weights
def qnwlogn(n, mu=None, sig2=None):
"""
Computes nodes and weights for multivariate lognormal distribution
Parameters
----------
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
mu : scalar or array_like(float), optional(default=zeros(d))
The means of each dimension of the random variable. If a scalar
is given, that constant is repeated d times, where d is the
number of dimensions
sig2 : array_like(float), optional(default=eye(d))
A d x d array representing the variance-covariance matrix of the
multivariate normal distribution.
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwlogn`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
nodes, weights = qnwnorm(n, mu, sig2)
return np.exp(nodes), weights
def qnwsimp(n, a, b):
"""
Computes multivariate Simpson quadrature nodes and weights.
Parameters
----------
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
a : scalar or array_like(float)
A length-d iterable of lower endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
b : scalar or array_like(float)
A length-d iterable of upper endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwsimp`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
return _make_multidim_func(_qnwsimp1, n, a, b)
def qnwtrap(n, a, b):
"""
Computes multivariate trapezoid rule quadrature nodes and weights.
Parameters
----------
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
a : scalar or array_like(float)
A length-d iterable of lower endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
b : scalar or array_like(float)
A length-d iterable of upper endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwtrap`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
return _make_multidim_func(_qnwtrap1, n, a, b)
def qnwunif(n, a, b):
"""
Computes quadrature nodes and weights for multivariate uniform
distribution
Parameters
----------
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
a : scalar or array_like(float)
A length-d iterable of lower endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
b : scalar or array_like(float)
A length-d iterable of upper endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwunif`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
n, a, b = list(map(np.asarray, [n, a, b]))
nodes, weights = qnwlege(n, a, b)
weights = weights / np.prod(b - a)
return nodes, weights
def quadrect(f, n, a, b, kind='lege', *args, **kwargs):
"""
Integrate the d-dimensional function f on a rectangle with lower and
upper bound for dimension i defined by a[i] and b[i], respectively;
using n[i] points.
Parameters
----------
f : function
The function to integrate over. This should be a function
that accepts as its first argument a matrix representing points
along each dimension (each dimension is a column). Other
arguments that need to be passed to the function are caught by
`*args` and `**kwargs`
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
a : scalar or array_like(float)
A length-d iterable of lower endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
b : scalar or array_like(float)
A length-d iterable of upper endpoints. If a scalar is given,
that constant is repeated d times, where d is the number of
dimensions
kind : string, optional(default='lege')
Specifies which type of integration to perform. Valid
values are:
lege - Gauss-Legendre
cheb - Gauss-Chebyshev
trap - trapezoid rule
simp - Simpson rule
N - Neiderreiter equidistributed sequence
W - Weyl equidistributed sequence
H - Haber equidistributed sequence
R - Monte Carlo
*args, **kwargs :
Other arguments passed to the function f
Returns
-------
out : scalar (float)
The value of the integral on the region [a, b]
Notes
-----
Based of original function ``quadrect`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
if kind.lower() == "lege":
nodes, weights = qnwlege(n, a, b)
elif kind.lower() == "cheb":
nodes, weights = qnwcheb(n, a, b)
elif kind.lower() == "trap":
nodes, weights = qnwtrap(n, a, b)
elif kind.lower() == "simp":
nodes, weights = qnwsimp(n, a, b)
else:
nodes, weights = qnwequi(n, a, b, kind)
out = weights.dot(f(nodes, *args, **kwargs))
return out
def qnwbeta(n, a=1.0, b=1.0):
"""
Computes nodes and weights for beta distribution
Parameters
----------
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
a : scalar or array_like(float), optional(default=1.0)
A length-d
b : array_like(float), optional(default=1.0)
A d x d array representing the variance-covariance matrix of the
multivariate normal distribution.
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwbeta`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
return _make_multidim_func(_qnwbeta1, n, a, b)
def qnwgamma(n, a=1.0, b=1.0, tol=3e-14):
"""
Computes nodes and weights for gamma distribution
Parameters
----------
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
a : scalar or array_like(float) : optional(default=ones(d))
Shape parameter of the gamma distribution parameter. Must be positive
b : scalar or array_like(float) : optional(default=ones(d))
Scale parameter of the gamma distribution parameter. Must be positive
tol : scalar or array_like(float) : optional(default=ones(d) * 3e-14)
Tolerance parameter for newton iterations for each node
Returns
-------
nodes : np.ndarray(dtype=float)
Quadrature nodes
weights : np.ndarray(dtype=float)
Weights for quadrature nodes
Notes
-----
Based of original function ``qnwgamma`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
return _make_multidim_func(_qnwgamma1, n, a, b, tol)
# ------------------ #
# Internal Functions #
# ------------------ #
def _make_multidim_func(one_d_func, n, *args):
"""
A helper function to cut down on code repetition. Almost all of the
code in qnwcheb, qnwlege, qnwsimp, qnwtrap is just dealing
various forms of input arguments and then shelling out to the
corresponding 1d version of the function.
This routine does all the argument checking and passes things
through the appropriate 1d function before using a tensor product
to combine weights and nodes.
Parameters
----------
one_d_func : function
The 1d function to be called along each dimension
n : int or array_like(float)
A length-d iterable of the number of nodes in each dimension
args :
These are the arguments to various qnw____ functions. For the
majority of the functions this is just a and b, but some differ.
Returns
-------
func : function
The multi-dimensional version of the parameter ``one_d_func``
"""
_args = list(args)
n = np.atleast_1d(n)
args = list(map(np.atleast_1d, _args))
if all([x.size == 1 for x in [n] + args]):
return one_d_func(n[0], *_args)
d = n.size
for i in range(len(args)):
if args[i].size == 1:
args[i] = np.repeat(args[i], d)
nodes = []
weights = []
for i in range(d):
ai = [x[i] for x in args]
_1d = one_d_func(n[i], *ai)
nodes.append(_1d[0])
weights.append(_1d[1])
weights = ckron(*weights[::-1]) # reverse ordered tensor product
nodes = gridmake(*nodes)
return nodes, weights
@jit(nopython=True)
def _qnwcheb1(n, a, b):
"""
Compute univariate Guass-Checbychev quadrature nodes and weights
Parameters
----------
n : int
The number of nodes
a : int
The lower endpoint
b : int
The upper endpoint
Returns
-------
nodes : np.ndarray(dtype=float)
An n element array of nodes
nodes : np.ndarray(dtype=float)
An n element array of weights
Notes
-----
Based of original function ``qnwcheb1`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
nodes = (b+a)/2 - (b-a)/2 * np.cos(np.pi/n * np.linspace(0.5, n-0.5, n))
# Create temporary arrays to be used in computing weights
t1 = np.arange(1, n+1) - 0.5
t2 = np.arange(0.0, n, 2)
t3 = np.concatenate((np.array([1.0]),
-2.0/(np.arange(1.0, n-1, 2)*np.arange(3.0, n+1, 2))))
# compute weights and return
weights = ((b-a)/n)*np.cos(np.pi/n*np.outer(t1, t2)) @ t3
return nodes, weights
@jit(nopython=True)
def _qnwlege1(n, a, b):
"""
Compute univariate Guass-Legendre quadrature nodes and weights
Parameters
----------
n : int
The number of nodes
a : int
The lower endpoint
b : int
The upper endpoint
Returns
-------
nodes : np.ndarray(dtype=float)
An n element array of nodes
nodes : np.ndarray(dtype=float)
An n element array of weights
Notes
-----
Based of original function ``qnwlege1`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
# import ipdb; ipdb.set_trace()
maxit = 100
m = int(fix((n + 1) / 2.0))
xm = 0.5 * (b + a)
xl = 0.5 * (b - a)
nodes = np.zeros(n)
weights = nodes.copy()
i = np.arange(m)
z = np.cos(np.pi * ((i + 1.0) - 0.25) / (n + 0.5))
for its in range(maxit):
p1 = np.ones_like(z)
p2 = np.zeros_like(z)
for j in range(1, n+1):
p3 = p2
p2 = p1
p1 = ((2 * j - 1) * z * p2 - (j - 1) * p3) / j
pp = n * (z * p1 - p2)/(z * z - 1.0)
z1 = z.copy()
z = z1 - p1/pp
if np.all(np.abs(z - z1) < 1e-14):
break
if its == maxit - 1:
raise ValueError("Maximum iterations in _qnwlege1")
nodes[i] = xm - xl * z
nodes[- i - 1] = xm + xl * z
weights[i] = 2 * xl / ((1 - z * z) * pp * pp)
weights[- i - 1] = weights[i]
return nodes, weights
@jit(nopython=True)
def _qnwnorm1(n):
"""
Compute nodes and weights for quadrature of univariate standard
normal distribution
Parameters
----------
n : int
The number of nodes
Returns
-------
nodes : np.ndarray(dtype=float)
An n element array of nodes
nodes : np.ndarray(dtype=float)
An n element array of weights
Notes
-----
Based of original function ``qnwnorm1`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
maxit = 100
pim4 = 1 / np.pi**(0.25)
m = int(fix((n + 1) / 2))
nodes = np.zeros(n)
weights = np.zeros(n)
for i in range(m):
if i == 0:
z = np.sqrt(2*n+1) - 1.85575 * ((2 * n + 1)**(-1 / 6.1))
elif i == 1:
z = z - 1.14 * (n ** 0.426) / z
elif i == 2:
z = 1.86 * z + 0.86 * nodes[0]
elif i == 3:
z = 1.91 * z + 0.91 * nodes[1]
else:
z = 2 * z + nodes[i-2]
its = 0
while its < maxit:
its += 1
p1 = pim4
p2 = 0
for j in range(1, n+1):
p3 = p2
p2 = p1
p1 = z * math.sqrt(2.0/j) * p2 - math.sqrt((j - 1.0) / j) * p3
pp = math.sqrt(2 * n) * p2
z1 = z
z = z1 - p1/pp
if abs(z - z1) < 1e-14:
break
if its == maxit:
raise ValueError("Failed to converge in _qnwnorm1")
nodes[n - 1 - i] = z
nodes[i] = -z
weights[i] = 2 / (pp*pp)
weights[n - 1 - i] = weights[i]
weights /= math.sqrt(math.pi)
nodes = nodes * math.sqrt(2.0)
return nodes, weights
@jit(nopython=True)
def _qnwsimp1(n, a, b):
"""
Compute univariate Simpson quadrature nodes and weights
Parameters
----------
n : int
The number of nodes
a : int
The lower endpoint
b : int
The upper endpoint
Returns
-------
nodes : np.ndarray(dtype=float)
An n element array of nodes
nodes : np.ndarray(dtype=float)
An n element array of weights
Notes
-----
Based of original function ``qnwsimp1`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
if n % 2 == 0:
print("WARNING qnwsimp: n must be an odd integer. Increasing by 1")
n += 1
nodes = np.linspace(a, b, n)
dx = nodes[1] - nodes[0]
weights = np.kron(np.ones((n+1) // 2), np.array([2.0, 4.0]))
weights = weights[:n]
weights[0] = weights[-1] = 1
weights = (dx / 3.0) * weights
return nodes, weights
@jit(nopython=True)
def _qnwtrap1(n, a, b):
"""
Compute univariate trapezoid rule quadrature nodes and weights
Parameters
----------
n : int
The number of nodes
a : int
The lower endpoint
b : int
The upper endpoint
Returns
-------
nodes : np.ndarray(dtype=float)
An n element array of nodes
nodes : np.ndarray(dtype=float)
An n element array of weights
Notes
-----
Based of original function ``qnwtrap1`` in CompEcon toolbox by
Miranda and Fackler
References
----------
Miranda, Mario J, and Paul L Fackler. Applied Computational
Economics and Finance, MIT Press, 2002.
"""
if n < 1:
raise ValueError("n must be at least one")
nodes = np.linspace(a, b, n)
dx = nodes[1] - nodes[0]
weights = dx * np.ones(n)
weights[0] *= 0.5
weights[-1] *= 0.5
return nodes, weights
@jit(nopython=True)
def _qnwbeta1(n, a=1.0, b=1.0):
"""
Computes nodes and weights for quadrature on the beta distribution.
Default is a=b=1 which is just a uniform distribution
NOTE: For now I am just following compecon; would be much better to
find a different way since I don't know what they are doing.
Parameters
----------
n : scalar : int
The number of quadrature points
a : scalar : float, optional(default=1)
First Beta distribution parameter
b : scalar : float, optional(default=1)
Second Beta distribution parameter
Returns
-------
nodes : np.ndarray(dtype=float, ndim=1)
The quadrature points