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__all__ = ['fixed_quad','quadrature','romberg','trapz','simps','romb',
'cumtrapz','newton_cotes']
from scipy.special.orthogonal import p_roots
from scipy.special import gammaln
from numpy import sum, ones, add, diff, isinf, isscalar, \
asarray, real, trapz, arange, empty
import numpy as np
import math
import warnings
class AccuracyWarning(Warning):
pass
def fixed_quad(func,a,b,args=(),n=5):
"""
Compute a definite integral using fixed-order Gaussian quadrature.
Integrate `func` from a to b using Gaussian quadrature of order n.
Parameters
----------
func : callable
A Python function or method to integrate (must accept vector inputs).
a : float
Lower limit of integration.
b : float
Upper limit of integration.
args : tuple, optional
Extra arguments to pass to function, if any.
n : int, optional
Order of quadrature integration. Default is 5.
Returns
-------
val : float
Gaussian quadrature approximation to the integral
See Also
--------
quad : adaptive quadrature using QUADPACK
dblquad, tplquad : double and triple integrals
romberg : adaptive Romberg quadrature
quadrature : adaptive Gaussian quadrature
romb, simps, trapz : integrators for sampled data
cumtrapz : cumulative integration for sampled data
ode, odeint - ODE integrators
"""
[x,w] = p_roots(n)
x = real(x)
ainf, binf = map(isinf,(a,b))
if ainf or binf:
raise ValueError("Gaussian quadrature is only available for "
"finite limits.")
y = (b-a)*(x+1)/2.0 + a
return (b-a)/2.0*sum(w*func(y,*args),0), None
def vectorize1(func, args=(), vec_func=False):
"""Vectorize the call to a function.
This is an internal utility function used by `romberg` and
`quadrature` to create a vectorized version of a function.
If `vec_func` is True, the function `func` is assumed to take vector
arguments.
Parameters
----------
func : callable
User defined function.
args : tuple
Extra arguments for the function.
vec_func : bool
True if the function func takes vector arguments.
Returns
-------
vfunc : callable
A function that will take a vector argument and return the
result.
"""
if vec_func:
def vfunc(x):
return func(x, *args)
else:
def vfunc(x):
if isscalar(x):
return func(x, *args)
x = asarray(x)
# call with first point to get output type
y0 = func(x[0], *args)
n = len(x)
if hasattr(y0, 'dtype'):
output = empty((n,), dtype=y0.dtype)
else:
output = empty((n,), dtype=type(y0))
output[0] = y0
for i in xrange(1, n):
output[i] = func(x[i], *args)
return output
return vfunc
def quadrature(func, a, b, args=(), tol=1.49e-8, rtol=1.49e-8, maxiter=50,
vec_func=True):
"""
Compute a definite integral using fixed-tolerance Gaussian quadrature.
Integrate func from a to b using Gaussian quadrature
with absolute tolerance `tol`.
Parameters
----------
func : function
A Python function or method to integrate.
a : float
Lower limit of integration.
b : float
Upper limit of integration.
args : tuple, optional
Extra arguments to pass to function.
tol, rol : float, optional
Iteration stops when error between last two iterates is less than
`tol` OR the relative change is less than `rtol`.
maxiter : int, optional
Maximum number of iterations.
vec_func : bool, optional
True or False if func handles arrays as arguments (is
a "vector" function). Default is True.
Returns
-------
val : float
Gaussian quadrature approximation (within tolerance) to integral.
err : float
Difference between last two estimates of the integral.
See also
--------
romberg: adaptive Romberg quadrature
fixed_quad: fixed-order Gaussian quadrature
quad: adaptive quadrature using QUADPACK
dblquad: double integrals
tplquad: triple integrals
romb: integrator for sampled data
simps: integrator for sampled data
trapz: integrator for sampled data
cumtrapz: cumulative integration for sampled data
ode: ODE integrator
odeint: ODE integrator
"""
vfunc = vectorize1(func, args, vec_func=vec_func)
val = np.inf
err = np.inf
for n in xrange(1, maxiter+1):
newval = fixed_quad(vfunc, a, b, (), n)[0]
err = abs(newval-val)
val = newval
if err < tol or err < rtol*abs(val):
break
else:
warnings.warn(
"maxiter (%d) exceeded. Latest difference = %e" % (maxiter, err),
AccuracyWarning)
return val, err
def tupleset(t, i, value):
l = list(t)
l[i] = value
return tuple(l)
def cumtrapz(y, x=None, dx=1.0, axis=-1, initial=None):
"""
Cumulatively integrate y(x) using the composite trapezoidal rule.
Parameters
----------
y : array_like
Values to integrate.
x : array_like, optional
The coordinate to integrate along. If None (default), use spacing `dx`
between consecutive elements in `y`.
dx : int, optional
Spacing between elements of `y`. Only used if `x` is None.
axis : int, optional
Specifies the axis to cumulate. Default is -1 (last axis).
initial : scalar, optional
If given, uses this value as the first value in the returned result.
Typically this value should be 0. Default is None, which means no
value at ``x[0]`` is returned and `res` has one element less than `y`
along the axis of integration.
Returns
-------
res : ndarray
The result of cumulative integration of `y` along `axis`.
If `initial` is None, the shape is such that the axis of integration
has one less value than `y`. If `initial` is given, the shape is equal
to that of `y`.
See Also
--------
numpy.cumsum, numpy.cumprod
quad: adaptive quadrature using QUADPACK
romberg: adaptive Romberg quadrature
quadrature: adaptive Gaussian quadrature
fixed_quad: fixed-order Gaussian quadrature
dblquad: double integrals
tplquad: triple integrals
romb: integrators for sampled data
ode: ODE integrators
odeint: ODE integrators
Examples
--------
>>> from scipy import integrate
>>> x = np.linspace(-2, 2, num=20)
>>> y = x
>>> y_int = integrate.cumtrapz(y, x, initial=0)
>>> plt.plot(x, y_int, 'ro', x, y[0] + 0.5 * x**2, 'b-')
>>> plt.show()
"""
y = asarray(y)
if x is None:
d = dx
else:
d = diff(x, axis=axis)
nd = len(y.shape)
slice1 = tupleset((slice(None),)*nd, axis, slice(1, None))
slice2 = tupleset((slice(None),)*nd, axis, slice(None, -1))
res = add.accumulate(d * (y[slice1] + y[slice2]) / 2.0, axis)
if initial is not None:
if not np.isscalar(initial):
raise ValueError("`initial` parameter should be a scalar.")
shape = list(res.shape)
shape[axis] = 1
res = np.concatenate([np.ones(shape, dtype=res.dtype) * initial, res],
axis=axis)
return res
def _basic_simps(y,start,stop,x,dx,axis):
nd = len(y.shape)
if start is None:
start = 0
step = 2
all = (slice(None),)*nd
slice0 = tupleset(all, axis, slice(start, stop, step))
slice1 = tupleset(all, axis, slice(start+1, stop+1, step))
slice2 = tupleset(all, axis, slice(start+2, stop+2, step))
if x is None: # Even spaced Simpson's rule.
result = add.reduce(dx/3.0* (y[slice0]+4*y[slice1]+y[slice2]),
axis)
else:
# Account for possibly different spacings.
# Simpson's rule changes a bit.
h = diff(x,axis=axis)
sl0 = tupleset(all, axis, slice(start, stop, step))
sl1 = tupleset(all, axis, slice(start+1, stop+1, step))
h0 = h[sl0]
h1 = h[sl1]
hsum = h0 + h1
hprod = h0 * h1
h0divh1 = h0 / h1
result = add.reduce(hsum/6.0*(y[slice0]*(2-1.0/h0divh1) + \
y[slice1]*hsum*hsum/hprod + \
y[slice2]*(2-h0divh1)),axis)
return result
def simps(y, x=None, dx=1, axis=-1, even='avg'):
"""
Integrate y(x) using samples along the given axis and the composite
Simpson's rule. If x is None, spacing of dx is assumed.
If there are an even number of samples, N, then there are an odd
number of intervals (N-1), but Simpson's rule requires an even number
of intervals. The parameter 'even' controls how this is handled.
Parameters
----------
y : array_like
Array to be integrated.
x : array_like, optional
If given, the points at which `y` is sampled.
dx : int, optional
Spacing of integration points along axis of `y`. Only used when
`x` is None. Default is 1.
axis : int, optional
Axis along which to integrate. Default is the last axis.
even : {'avg', 'first', 'str'}, optional
'avg' : Average two results:1) use the first N-2 intervals with
a trapezoidal rule on the last interval and 2) use the last
N-2 intervals with a trapezoidal rule on the first interval.
'first' : Use Simpson's rule for the first N-2 intervals with
a trapezoidal rule on the last interval.
'last' : Use Simpson's rule for the last N-2 intervals with a
trapezoidal rule on the first interval.
See Also
--------
quad: adaptive quadrature using QUADPACK
romberg: adaptive Romberg quadrature
quadrature: adaptive Gaussian quadrature
fixed_quad: fixed-order Gaussian quadrature
dblquad: double integrals
tplquad: triple integrals
romb: integrators for sampled data
trapz: integrators for sampled data
cumtrapz: cumulative integration for sampled data
ode: ODE integrators
odeint: ODE integrators
Notes
-----
For an odd number of samples that are equally spaced the result is
exact if the function is a polynomial of order 3 or less. If
the samples are not equally spaced, then the result is exact only
if the function is a polynomial of order 2 or less.
"""
y = asarray(y)
nd = len(y.shape)
N = y.shape[axis]
last_dx = dx
first_dx = dx
returnshape = 0
if not x is None:
x = asarray(x)
if len(x.shape) == 1:
shapex = ones(nd)
shapex[axis] = x.shape[0]
saveshape = x.shape
returnshape = 1
x=x.reshape(tuple(shapex))
elif len(x.shape) != len(y.shape):
raise ValueError("If given, shape of x must be 1-d or the "
"same as y.")
if x.shape[axis] != N:
raise ValueError("If given, length of x along axis must be the "
"same as y.")
if N % 2 == 0:
val = 0.0
result = 0.0
slice1 = (slice(None),)*nd
slice2 = (slice(None),)*nd
if not even in ['avg', 'last', 'first']:
raise ValueError("Parameter 'even' must be 'avg', 'last', or 'first'.")
# Compute using Simpson's rule on first intervals
if even in ['avg', 'first']:
slice1 = tupleset(slice1, axis, -1)
slice2 = tupleset(slice2, axis, -2)
if not x is None:
last_dx = x[slice1] - x[slice2]
val += 0.5*last_dx*(y[slice1]+y[slice2])
result = _basic_simps(y,0,N-3,x,dx,axis)
# Compute using Simpson's rule on last set of intervals
if even in ['avg', 'last']:
slice1 = tupleset(slice1, axis, 0)
slice2 = tupleset(slice2, axis, 1)
if not x is None:
first_dx = x[tuple(slice2)] - x[tuple(slice1)]
val += 0.5*first_dx*(y[slice2]+y[slice1])
result += _basic_simps(y,1,N-2,x,dx,axis)
if even == 'avg':
val /= 2.0
result /= 2.0
result = result + val
else:
result = _basic_simps(y,0,N-2,x,dx,axis)
if returnshape:
x = x.reshape(saveshape)
return result
def romb(y, dx=1.0, axis=-1, show=False):
"""
Romberg integration using samples of a function.
Parameters
-----------
y : array_like
A vector of ``2**k + 1`` equally-spaced samples of a function.
dx : array_like, optional
The sample spacing. Default is 1.
axis : array_like?, optional
The axis along which to integrate. Default is -1 (last axis).
show : bool, optional
When y is a single 1-D array, then if this argument is True
print the table showing Richardson extrapolation from the
samples. Default is False.
Returns
-------
ret : array_like?
The integrated result for each axis.
See also
--------
quad - adaptive quadrature using QUADPACK
romberg - adaptive Romberg quadrature
quadrature - adaptive Gaussian quadrature
fixed_quad - fixed-order Gaussian quadrature
dblquad, tplquad - double and triple integrals
simps, trapz - integrators for sampled data
cumtrapz - cumulative integration for sampled data
ode, odeint - ODE integrators
"""
y = asarray(y)
nd = len(y.shape)
Nsamps = y.shape[axis]
Ninterv = Nsamps-1
n = 1
k = 0
while n < Ninterv:
n <<= 1
k += 1
if n != Ninterv:
raise ValueError("Number of samples must be one plus a "
"non-negative power of 2.")
R = {}
all = (slice(None),) * nd
slice0 = tupleset(all, axis, 0)
slicem1 = tupleset(all, axis, -1)
h = Ninterv*asarray(dx)*1.0
R[(1,1)] = (y[slice0] + y[slicem1])/2.0*h
slice_R = all
start = stop = step = Ninterv
for i in range(2,k+1):
start >>= 1
slice_R = tupleset(slice_R, axis, slice(start,stop,step))
step >>= 1
R[(i,1)] = 0.5*(R[(i-1,1)] + h*add.reduce(y[slice_R],axis))
for j in range(2,i+1):
R[(i,j)] = R[(i,j-1)] + \
(R[(i,j-1)]-R[(i-1,j-1)]) / ((1 << (2*(j-1)))-1)
h = h / 2.0
if show:
if not isscalar(R[(1,1)]):
print "*** Printing table only supported for integrals" + \
" of a single data set."
else:
try:
precis = show[0]
except (TypeError, IndexError):
precis = 5
try:
width = show[1]
except (TypeError, IndexError):
width = 8
formstr = "%" + str(width) + '.' + str(precis)+'f'
print "\n Richardson Extrapolation Table for Romberg Integration "
print "===================================================================="
for i in range(1,k+1):
for j in range(1,i+1):
print formstr % R[(i,j)],
print
print "====================================================================\n"
return R[(k,k)]
# Romberg quadratures for numeric integration.
#
# Written by Scott M. Ransom <ransom@cfa.harvard.edu>
# last revision: 14 Nov 98
#
# Cosmetic changes by Konrad Hinsen <hinsen@cnrs-orleans.fr>
# last revision: 1999-7-21
#
# Adapted to scipy by Travis Oliphant <oliphant.travis@ieee.org>
# last revision: Dec 2001
def _difftrap(function, interval, numtraps):
"""
Perform part of the trapezoidal rule to integrate a function.
Assume that we had called difftrap with all lower powers-of-2
starting with 1. Calling difftrap only returns the summation
of the new ordinates. It does _not_ multiply by the width
of the trapezoids. This must be performed by the caller.
'function' is the function to evaluate (must accept vector arguments).
'interval' is a sequence with lower and upper limits
of integration.
'numtraps' is the number of trapezoids to use (must be a
power-of-2).
"""
if numtraps <= 0:
raise ValueError("numtraps must be > 0 in difftrap().")
elif numtraps == 1:
return 0.5*(function(interval[0])+function(interval[1]))
else:
numtosum = numtraps/2
h = float(interval[1]-interval[0])/numtosum
lox = interval[0] + 0.5 * h;
points = lox + h * arange(0, numtosum)
s = sum(function(points),0)
return s
def _romberg_diff(b, c, k):
"""
Compute the differences for the Romberg quadrature corrections.
See Forman Acton's "Real Computing Made Real," p 143.
"""
tmp = 4.0**k
return (tmp * c - b)/(tmp - 1.0)
def _printresmat(function, interval, resmat):
# Print the Romberg result matrix.
i = j = 0
print 'Romberg integration of', `function`,
print 'from', interval
print ''
print '%6s %9s %9s' % ('Steps', 'StepSize', 'Results')
for i in range(len(resmat)):
print '%6d %9f' % (2**i, (interval[1]-interval[0])/(2.**i)),
for j in range(i+1):
print '%9f' % (resmat[i][j]),
print ''
print ''
print 'The final result is', resmat[i][j],
print 'after', 2**(len(resmat)-1)+1, 'function evaluations.'
def romberg(function, a, b, args=(), tol=1.48e-8, rtol=1.48e-8, show=False,
divmax=10, vec_func=False):
"""
Romberg integration of a callable function or method.
Returns the integral of `function` (a function of one variable)
over the interval (`a`, `b`).
If `show` is 1, the triangular array of the intermediate results
will be printed. If `vec_func` is True (default is False), then `function` is
assumed to support vector arguments.
Parameters
----------
function : callable
Function to be integrated.
a : float
Lower limit of integration.
b : float
Upper limit of integration.
Returns
--------
results : float
Result of the integration.
Other Parameters
----------------
args : tuple, optional
Extra arguments to pass to function. Each element of `args` will
be passed as a single argument to `func`. Default is to pass no
extra arguments.
tol, rtol : float, optional
The desired absolute and relative tolerances. Defaults are 1.48e-8.
show : bool, optional
Whether to print the results. Default is False.
divmax : int, optional
Maximum order of extrapolation. Default is 10.
vec_func : bool, optional
Whether `func` handles arrays as arguments (i.e whether it is a
"vector" function). Default is False.
See Also
--------
fixed_quad : Fixed-order Gaussian quadrature.
quad : Adaptive quadrature using QUADPACK.
dblquad, tplquad : Double and triple integrals.
romb, simps, trapz : Integrators for sampled data.
cumtrapz : Cumulative integration for sampled data.
ode, odeint : ODE integrators.
References
----------
.. [1] 'Romberg's method' http://en.wikipedia.org/wiki/Romberg%27s_method
Examples
--------
Integrate a gaussian from 0 to 1 and compare to the error function.
>>> from scipy.special import erf
>>> gaussian = lambda x: 1/np.sqrt(np.pi) * np.exp(-x**2)
>>> result = romberg(gaussian, 0, 1, show=True)
Romberg integration of <function vfunc at 0x101eceaa0> from [0, 1]
::
Steps StepSize Results
1 1.000000 0.385872
2 0.500000 0.412631 0.421551
4 0.250000 0.419184 0.421368 0.421356
8 0.125000 0.420810 0.421352 0.421350 0.421350
16 0.062500 0.421215 0.421350 0.421350 0.421350 0.421350
32 0.031250 0.421317 0.421350 0.421350 0.421350 0.421350 0.421350
The final result is 0.421350396475 after 33 function evaluations.
>>> print 2*result,erf(1)
0.84270079295 0.84270079295
"""
if isinf(a) or isinf(b):
raise ValueError("Romberg integration only available for finite limits.")
vfunc = vectorize1(function, args, vec_func=vec_func)
n = 1
interval = [a,b]
intrange = b-a
ordsum = _difftrap(vfunc, interval, n)
result = intrange * ordsum
resmat = [[result]]
err = np.inf
for i in xrange(1, divmax+1):
n = n * 2
ordsum = ordsum + _difftrap(vfunc, interval, n)
resmat.append([])
resmat[i].append(intrange * ordsum / n)
for k in range(i):
resmat[i].append(_romberg_diff(resmat[i-1][k], resmat[i][k], k+1))
result = resmat[i][i]
lastresult = resmat[i-1][i-1]
err = abs(result - lastresult)
if err < tol or err < rtol*abs(result):
break
else:
warnings.warn(
"divmax (%d) exceeded. Latest difference = %e" % (divmax, err),
AccuracyWarning)
if show:
_printresmat(vfunc, interval, resmat)
return result
# Coefficients for Netwon-Cotes quadrature
#
# These are the points being used
# to construct the local interpolating polynomial
# a are the weights for Newton-Cotes integration
# B is the error coefficient.
# error in these coefficients grows as N gets larger.
# or as samples are closer and closer together
# You can use maxima to find these rational coefficients
# for equally spaced data using the commands
# a(i,N) := integrate(product(r-j,j,0,i-1) * product(r-j,j,i+1,N),r,0,N) / ((N-i)! * i!) * (-1)^(N-i);
# Be(N) := N^(N+2)/(N+2)! * (N/(N+3) - sum((i/N)^(N+2)*a(i,N),i,0,N));
# Bo(N) := N^(N+1)/(N+1)! * (N/(N+2) - sum((i/N)^(N+1)*a(i,N),i,0,N));
# B(N) := (if (mod(N,2)=0) then Be(N) else Bo(N));
#
# pre-computed for equally-spaced weights
#
# num_a, den_a, int_a, num_B, den_B = _builtincoeffs[N]
#
# a = num_a*array(int_a)/den_a
# B = num_B*1.0 / den_B
#
# integrate(f(x),x,x_0,x_N) = dx*sum(a*f(x_i)) + B*(dx)^(2k+3) f^(2k+2)(x*)
# where k = N // 2
#
_builtincoeffs = {
1:(1,2,[1,1],-1,12),
2:(1,3,[1,4,1],-1,90),
3:(3,8,[1,3,3,1],-3,80),
4:(2,45,[7,32,12,32,7],-8,945),
5:(5,288,[19,75,50,50,75,19],-275,12096),
6:(1,140,[41,216,27,272,27,216,41],-9,1400),
7:(7,17280,[751,3577,1323,2989,2989,1323,3577,751],-8183,518400),
8:(4,14175,[989,5888,-928,10496,-4540,10496,-928,5888,989],
-2368,467775),
9:(9,89600,[2857,15741,1080,19344,5778,5778,19344,1080,
15741,2857], -4671, 394240),
10:(5,299376,[16067,106300,-48525,272400,-260550,427368,
-260550,272400,-48525,106300,16067],
-673175, 163459296),
11:(11,87091200,[2171465,13486539,-3237113, 25226685,-9595542,
15493566,15493566,-9595542,25226685,-3237113,
13486539,2171465], -2224234463, 237758976000),
12:(1, 5255250, [1364651,9903168,-7587864,35725120,-51491295,
87516288,-87797136,87516288,-51491295,35725120,
-7587864,9903168,1364651], -3012, 875875),
13:(13, 402361344000,[8181904909, 56280729661, -31268252574,
156074417954,-151659573325,206683437987,
-43111992612,-43111992612,206683437987,
-151659573325,156074417954,-31268252574,
56280729661,8181904909], -2639651053,
344881152000),
14:(7, 2501928000, [90241897,710986864,-770720657,3501442784,
-6625093363,12630121616,-16802270373,19534438464,
-16802270373,12630121616,-6625093363,3501442784,
-770720657,710986864,90241897], -3740727473,
1275983280000)
}
def newton_cotes(rn, equal=0):
"""
Return weights and error coefficient for Newton-Cotes integration.
Suppose we have (N+1) samples of f at the positions
x_0, x_1, ..., x_N. Then an N-point Newton-Cotes formula for the
integral between x_0 and x_N is:
:math:`\\int_{x_0}^{x_N} f(x)dx = \\Delta x \\sum_{i=0}^{N} a_i f(x_i)
+ B_N (\\Delta x)^{N+2} f^{N+1} (\\xi)`
where :math:`\\xi \\in [x_0,x_N]` and :math:`\\Delta x = \\frac{x_N-x_0}{N}`
is the averages samples spacing.
If the samples are equally-spaced and N is even, then the error
term is :math:`B_N (\\Delta x)^{N+3} f^{N+2}(\\xi)`.
Parameters
----------
rn : int
The integer order for equally-spaced data or the relative positions of
the samples with the first sample at 0 and the last at N, where N+1 is
the length of `rn`. N is the order of the Newton-Cotes integration.
equal: int, optional
Set to 1 to enforce equally spaced data.
Returns
-------
an : ndarray
1-D array of weights to apply to the function at the provided sample
positions.
B : float
Error coefficient.
Notes
-----
Normally, the Newton-Cotes rules are used on smaller integration
regions and a composite rule is used to return the total integral.
"""
try:
N = len(rn)-1
if equal:
rn = np.arange(N+1)
elif np.all(np.diff(rn)==1):
equal = 1
except:
N = rn
rn = np.arange(N+1)
equal = 1
if equal and N in _builtincoeffs:
na, da, vi, nb, db = _builtincoeffs[N]
return na*np.array(vi,float)/da, float(nb)/db
if (rn[0] != 0) or (rn[-1] != N):
raise ValueError("The sample positions must start at 0"
" and end at N")
yi = rn / float(N)
ti = 2.0*yi - 1
nvec = np.arange(0,N+1)
C = np.mat(ti**nvec[:,np.newaxis])
Cinv = C.I
# improve precision of result
Cinv = 2*Cinv - Cinv*C*Cinv
Cinv = 2*Cinv - Cinv*C*Cinv
Cinv = Cinv.A
vec = 2.0/ (nvec[::2]+1)
ai = np.dot(Cinv[:,::2],vec) * N/2
if (N%2 == 0) and equal:
BN = N/(N+3.)
power = N+2
else:
BN = N/(N+2.)
power = N+1
BN = BN - np.dot(yi**power, ai)
p1 = power+1
fac = power*math.log(N) - gammaln(p1)
fac = math.exp(fac)
return ai, BN*fac
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