# Author: Travis Oliphant 2001
__all__ = ['quad', 'dblquad', 'tplquad', 'quad_explain', 'Inf','inf']
import _quadpack
import sys
import numpy
from numpy import inf, Inf
error = _quadpack.error
def quad_explain(output=sys.stdout):
"""
Print extra information about integrate.quad() parameters and returns.
Parameters
----------
output : instance with "write" method
Information about `quad` is passed to ``output.write()``.
Default is ``sys.stdout``.
Returns
-------
None
"""
output.write("""
Extra information for quad() inputs and outputs:
If full_output is non-zero, then the third output argument (infodict)
is a dictionary with entries as tabulated below. For infinite limits, the
range is transformed to (0,1) and the optional outputs are given with
respect to this transformed range. Let M be the input argument limit and
let K be infodict['last']. The entries are:
'neval' : The number of function evaluations.
'last' : The number, K, of subintervals produced in the subdivision process.
'alist' : A rank-1 array of length M, the first K elements of which are the
left end points of the subintervals in the partition of the
integration range.
'blist' : A rank-1 array of length M, the first K elements of which are the
right end points of the subintervals.
'rlist' : A rank-1 array of length M, the first K elements of which are the
integral approximations on the subintervals.
'elist' : A rank-1 array of length M, the first K elements of which are the
moduli of the absolute error estimates on the subintervals.
'iord' : A rank-1 integer array of length M, the first L elements of
which are pointers to the error estimates over the subintervals
with L=K if K<=M/2+2 or L=M+1-K otherwise. Let I be the sequence
infodict['iord'] and let E be the sequence infodict['elist'].
Then E[I[1]], ..., E[I[L]] forms a decreasing sequence.
If the input argument points is provided (i.e. it is not None), the
following additional outputs are placed in the output dictionary. Assume the
points sequence is of length P.
'pts' : A rank-1 array of length P+2 containing the integration limits
and the break points of the intervals in ascending order.
This is an array giving the subintervals over which integration
will occur.
'level' : A rank-1 integer array of length M (=limit), containing the
subdivision levels of the subintervals, i.e., if (aa,bb) is a
subinterval of (pts[1], pts[2]) where pts[0] and pts[2] are
adjacent elements of infodict['pts'], then (aa,bb) has level l if
|bb-aa|=|pts[2]-pts[1]| * 2**(-l).
'ndin' : A rank-1 integer array of length P+2. After the first integration
over the intervals (pts[1], pts[2]), the error estimates over some
of the intervals may have been increased artificially in order to
put their subdivision forward. This array has ones in slots
corresponding to the subintervals for which this happens.
Weighting the integrand:
The input variables, weight and wvar, are used to weight the integrand by
a select list of functions. Different integration methods are used
to compute the integral with these weighting functions. The possible values
of weight and the corresponding weighting functions are.
'cos' : cos(w*x) : wvar = w
'sin' : sin(w*x) : wvar = w
'alg' : g(x) = ((x-a)**alpha)*((b-x)**beta) : wvar = (alpha, beta)
'alg-loga': g(x)*log(x-a) : wvar = (alpha, beta)
'alg-logb': g(x)*log(b-x) : wvar = (alpha, beta)
'alg-log' : g(x)*log(x-a)*log(b-x) : wvar = (alpha, beta)
'cauchy' : 1/(x-c) : wvar = c
wvar holds the parameter w, (alpha, beta), or c depending on the weight
selected. In these expressions, a and b are the integration limits.
For the 'cos' and 'sin' weighting, additional inputs and outputs are
available.
For finite integration limits, the integration is performed using a
Clenshaw-Curtis method which uses Chebyshev moments. For repeated
calculations, these moments are saved in the output dictionary:
'momcom' : The maximum level of Chebyshev moments that have been computed,
i.e., if M_c is infodict['momcom'] then the moments have been
computed for intervals of length |b-a|* 2**(-l), l=0,1,...,M_c.
'nnlog' : A rank-1 integer array of length M(=limit), containing the
subdivision levels of the subintervals, i.e., an element of this
array is equal to l if the corresponding subinterval is
|b-a|* 2**(-l).
'chebmo' : A rank-2 array of shape (25, maxp1) containing the computed
Chebyshev moments. These can be passed on to an integration
over the same interval by passing this array as the second
element of the sequence wopts and passing infodict['momcom'] as
the first element.
If one of the integration limits is infinite, then a Fourier integral is
computed (assuming w neq 0). If full_output is 1 and a numerical error
is encountered, besides the error message attached to the output tuple,
a dictionary is also appended to the output tuple which translates the
error codes in the array info['ierlst'] to English messages. The output
information dictionary contains the following entries instead of 'last',
'alist', 'blist', 'rlist', and 'elist':
'lst' : The number of subintervals needed for the integration (call it K_f).
'rslst' : A rank-1 array of length M_f=limlst, whose first K_f elements
contain the integral contribution over the interval (a+(k-1)c,
a+kc) where c = (2*floor(|w|) + 1) * pi / |w| and k=1,2,...,K_f.
'erlst' : A rank-1 array of length M_f containing the error estimate
corresponding to the interval in the same position in
infodict['rslist'].
'ierlst' : A rank-1 integer array of length M_f containing an error flag
corresponding to the interval in the same position in
infodict['rslist']. See the explanation dictionary (last entry
in the output tuple) for the meaning of the codes.
""")
return
def quad(func, a, b, args=(), full_output=0, epsabs=1.49e-8, epsrel=1.49e-8,
limit=50, points=None, weight=None, wvar=None, wopts=None, maxp1=50,
limlst=50):
"""
Compute a definite integral.
Integrate func from a to b (possibly infinite interval) using a technique
from the Fortran library QUADPACK.
If func takes many arguments, it is integrated along the axis corresponding
to the first argument. Use the keyword argument `args` to pass the other
arguments.
Run scipy.integrate.quad_explain() for more information on the
more esoteric inputs and outputs.
Parameters
----------
func : function
A Python function or method to integrate.
a : float
Lower limit of integration (use -scipy.integrate.Inf for -infinity).
b : float
Upper limit of integration (use scipy.integrate.Inf for +infinity).
args : tuple, optional
extra arguments to pass to func
full_output : int
Non-zero to return a dictionary of integration information.
If non-zero, warning messages are also suppressed and the
message is appended to the output tuple.
Returns
-------
y : float
The integral of func from a to b.
abserr : float
an estimate of the absolute error in the result.
infodict : dict
a dictionary containing additional information.
Run scipy.integrate.quad_explain() for more information.
message :
a convergence message.
explain :
appended only with 'cos' or 'sin' weighting and infinite
integration limits, it contains an explanation of the codes in
infodict['ierlst']
Other Parameters
----------------
epsabs :
absolute error tolerance.
epsrel :
relative error tolerance.
limit :
an upper bound on the number of subintervals used in the adaptive
algorithm.
points :
a sequence of break points in the bounded integration interval
where local difficulties of the integrand may occur (e.g.,
singularities, discontinuities). The sequence does not have
to be sorted.
weight :
string indicating weighting function.
wvar :
variables for use with weighting functions.
limlst :
Upper bound on the number of cylces (>=3) for use with a sinusoidal
weighting and an infinite end-point.
wopts :
Optional input for reusing Chebyshev moments.
maxp1 :
An upper bound on the number of Chebyshev moments.
See Also
--------
dblquad, tplquad : double and triple integrals
fixed_quad : fixed-order Gaussian quadrature
quadrature : adaptive Gaussian quadrature
odeint, ode : ODE integrators
simps, trapz, romb : integrators for sampled data
scipy.special : for coefficients and roots of orthogonal polynomials
Examples
--------
Calculate :math:`\\int^4_0 x^2 dx` and compare with an analytic result
>>> from scipy import integrate
>>> x2 = lambda x: x**2
>>> integrate.quad(x,0.,4.)
(21.333333333333332, 2.3684757858670003e-13)
>> print 4.**3/3
21.3333333333
Calculate :math:`\\int^\\infty_0 e^{-x} dx`
>>> invexp = lambda x: exp(-x)
>>> integrate.quad(invexp,0,inf)
(0.99999999999999989, 5.8426061711142159e-11)
>>> f = lambda x,a : a*x
>>> y, err = integrate.quad(f, 0, 1, args=(1,))
>>> y
0.5
>>> y, err = integrate.quad(f, 0, 1, args=(3,))
>>> y
1.5
"""
if type(args) != type(()): args = (args,)
if (weight is None):
retval = _quad(func,a,b,args,full_output,epsabs,epsrel,limit,points)
else:
retval = _quad_weight(func,a,b,args,full_output,epsabs,epsrel,limlst,limit,maxp1,weight,wvar,wopts)
ier = retval[-1]
if ier == 0:
return retval[:-1]
msgs = {80: "A Python error occurred possibly while calling the function.",
1: "The maximum number of subdivisions (%d) has been achieved.\n If increasing the limit yields no improvement it is advised to analyze \n the integrand in order to determine the difficulties. If the position of a \n local difficulty can be determined (singularity, discontinuity) one will \n probably gain from splitting up the interval and calling the integrator \n on the subranges. Perhaps a special-purpose integrator should be used." % limit,
2: "The ocurrence of roundoff error is detected, which prevents \n the requested tolerance from being achieved. The error may be \n underestimated.",
3: "Extremely bad integrand behavior occurs at some points of the\n integration interval.",
4: "The algorithm does not converge. Roundoff error is detected\n in the extrapolation table. It is assumed that the requested tolerance\n cannot be achieved, and that the returned result (if full_output = 1) is \n the best which can be obtained.",
5: "The integral is probably divergent, or slowly convergent.",
6: "The input is invalid.",
7: "Abnormal termination of the routine. The estimates for result\n and error are less reliable. It is assumed that the requested accuracy\n has not been achieved.",
'unknown': "Unknown error."}
if weight in ['cos','sin'] and (b == Inf or a == -Inf):
msgs[1] = "The maximum number of cycles allowed has been achieved., e.e.\n of subintervals (a+(k-1)c, a+kc) where c = (2*int(abs(omega)+1))\n *pi/abs(omega), for k = 1, 2, ..., lst. One can allow more cycles by increasing the value of limlst. Look at info['ierlst'] with full_output=1."
msgs[4] = "The extrapolation table constructed for convergence acceleration\n of the series formed by the integral contributions over the cycles, \n does not converge to within the requested accuracy. Look at \n info['ierlst'] with full_output=1."
msgs[7] = "Bad integrand behavior occurs within one or more of the cycles.\n Location and type of the difficulty involved can be determined from \n the vector info['ierlist'] obtained with full_output=1."
explain = {1: "The maximum number of subdivisions (= limit) has been \n achieved on this cycle.",
2: "The occurrence of roundoff error is detected and prevents\n the tolerance imposed on this cycle from being achieved.",
3: "Extremely bad integrand behavior occurs at some points of\n this cycle.",
4: "The integral over this cycle does not converge (to within the required accuracy) due ot roundoff in the extrapolation procedure invoked on this cycle. It is assumed that the result on this interval is the best which can be obtained.",
5: "The integral over this cycle is probably divergent or slowly convergent."}
try:
msg = msgs[ier]
except KeyError:
msg = msgs['unknown']
if ier in [1,2,3,4,5,7]:
if full_output:
if weight in ['cos','sin'] and (b == Inf or a == Inf):
return retval[:-1] + (msg, explain)
else:
return retval[:-1] + (msg,)
else:
print "Warning: " + msg
return retval[:-1]
else:
raise ValueError(msg)
def _quad(func,a,b,args,full_output,epsabs,epsrel,limit,points):
infbounds = 0
if (b != Inf and a != -Inf):
pass # standard integration
elif (b == Inf and a != -Inf):
infbounds = 1
bound = a
elif (b == Inf and a == -Inf):
infbounds = 2
bound = 0 # ignored
elif (b != Inf and a == -Inf):
infbounds = -1
bound = b
else:
raise RuntimeError("Infinity comparisons don't work for you.")
if points is None:
if infbounds == 0:
return _quadpack._qagse(func,a,b,args,full_output,epsabs,epsrel,limit)
else:
return _quadpack._qagie(func,bound,infbounds,args,full_output,epsabs,epsrel,limit)
else:
if infbounds !=0:
raise ValueError("Infinity inputs cannot be used with break points.")
else:
nl = len(points)
the_points = numpy.zeros((nl+2,), float)
the_points[:nl] = points
return _quadpack._qagpe(func,a,b,the_points,args,full_output,epsabs,epsrel,limit)
def _quad_weight(func,a,b,args,full_output,epsabs,epsrel,limlst,limit,maxp1,weight,wvar,wopts):
if weight not in ['cos','sin','alg','alg-loga','alg-logb','alg-log','cauchy']:
raise ValueError("%s not a recognized weighting function." % weight)
strdict = {'cos':1,'sin':2,'alg':1,'alg-loga':2,'alg-logb':3,'alg-log':4}
if weight in ['cos','sin']:
integr = strdict[weight]
if (b != Inf and a != -Inf): # finite limits
if wopts is None: # no precomputed chebyshev moments
return _quadpack._qawoe(func,a,b,wvar,integr,args,full_output,epsabs,epsrel,limit,maxp1,1)
else: # precomputed chebyshev moments
momcom = wopts[0]
chebcom = wopts[1]
return _quadpack._qawoe(func,a,b,wvar,integr,args,full_output,epsabs,epsrel,limit,maxp1,2,momcom,chebcom)
elif (b == Inf and a != -Inf):
return _quadpack._qawfe(func,a,wvar,integr,args,full_output,epsabs,limlst,limit,maxp1)
elif (b != Inf and a == -Inf): # remap function and interval
if weight == 'cos':
def thefunc(x,*myargs):
y = -x
func = myargs[0]
myargs = (y,) + myargs[1:]
return apply(func,myargs)
else:
def thefunc(x,*myargs):
y = -x
func = myargs[0]
myargs = (y,) + myargs[1:]
return -apply(func,myargs)
args = (func,) + args
return _quadpack._qawfe(thefunc,-b,wvar,integr,args,full_output,epsabs,limlst,limit,maxp1)
else:
raise ValueError("Cannot integrate with this weight from -Inf to +Inf.")
else:
if a in [-Inf,Inf] or b in [-Inf,Inf]:
raise ValueError("Cannot integrate with this weight over an infinite interval.")
if weight[:3] == 'alg':
integr = strdict[weight]
return _quadpack._qawse(func,a,b,wvar,integr,args,full_output,epsabs,epsrel,limit)
else: # weight == 'cauchy'
return _quadpack._qawce(func,a,b,wvar,args,full_output,epsabs,epsrel,limit)
def _infunc(x,func,gfun,hfun,more_args):
a = gfun(x)
b = hfun(x)
myargs = (x,) + more_args
return quad(func,a,b,args=myargs)[0]
def dblquad(func, a, b, gfun, hfun, args=(), epsabs=1.49e-8, epsrel=1.49e-8):
"""
Compute a double integral.
Return the double (definite) integral of func(y,x) from x=a..b and
y=gfun(x)..hfun(x).
Parameters
-----------
func : callable
A Python function or method of at least two variables: y must be the
first argument and x the second argument.
(a,b) : tuple
The limits of integration in x: a < b
gfun : callable
The lower boundary curve in y which is a function taking a single
floating point argument (x) and returning a floating point result: a
lambda function can be useful here.
hfun : callable
The upper boundary curve in y (same requirements as `gfun`).
args : sequence, optional
Extra arguments to pass to `func2d`.
epsabs : float, optional
Absolute tolerance passed directly to the inner 1-D quadrature
integration. Default is 1.49e-8.
epsrel : float
Relative tolerance of the inner 1-D integrals. Default is 1.49e-8.
Returns
-------
y : float
The resultant integral.
abserr : float
An estimate of the error.
See also
--------
quad : single integral
tplquad : triple integral
fixed_quad : fixed-order Gaussian quadrature
quadrature : adaptive Gaussian quadrature
odeint, ode : ODE integrators
simps, trapz, romb : integrators for sampled data
scipy.special : for coefficients and roots of orthogonal polynomials
"""
return quad(_infunc,a,b,(func,gfun,hfun,args),epsabs=epsabs,epsrel=epsrel)
def _infunc2(y,x,func,qfun,rfun,more_args):
a2 = qfun(x,y)
b2 = rfun(x,y)
myargs = (y,x) + more_args
return quad(func,a2,b2,args=myargs)[0]
def tplquad(func, a, b, gfun, hfun, qfun, rfun, args=(), epsabs=1.49e-8,
epsrel=1.49e-8):
"""
Compute a triple (definite) integral.
Return the triple integral of func(z, y, x) from
x=a..b, y=gfun(x)..hfun(x), and z=qfun(x,y)..rfun(x,y)
Parameters
----------
func : function
A Python function or method of at least three variables in the
order (z, y, x).
(a,b) : tuple
The limits of integration in x: a < b
gfun : function
The lower boundary curve in y which is a function taking a single
floating point argument (x) and returning a floating point result:
a lambda function can be useful here.
hfun : function
The upper boundary curve in y (same requirements as gfun).
qfun : function
The lower boundary surface in z. It must be a function that takes
two floats in the order (x, y) and returns a float.
rfun : function
The upper boundary surface in z. (Same requirements as qfun.)
args : Arguments
Extra arguments to pass to func3d.
epsabs : float, optional
Absolute tolerance passed directly to the innermost 1-D quadrature
integration. Default is 1.49e-8.
epsrel : float, optional
Relative tolerance of the innermost 1-D integrals. Default is 1.49e-8.
Returns
-------
y : float
The resultant integral.
abserr : float
An estimate of the error.
See Also
--------
quad: Adaptive quadrature using QUADPACK
quadrature: Adaptive Gaussian quadrature
fixed_quad: Fixed-order Gaussian quadrature
dblquad: Double integrals
romb: Integrators for sampled data
trapz: Integrators for sampled data
simps: Integrators for sampled data
ode: ODE integrators
odeint: ODE integrators
scipy.special: For coefficients and roots of orthogonal polynomials
"""
return dblquad(_infunc2,a,b,gfun,hfun,(func,qfun,rfun,args),epsabs=epsabs,epsrel=epsrel)