˽Xc@`s5ddlmZmZmZddlZddlZddlZddlmZddl m Z ddl m Z ddl m Z ddd d d d d dgZdefdYZdZee_dddZdedZddddeddZdZdddddZdZdddddZdded Zd!Zd"Z d#Z!dd$d$ed%ed&Z"idd'ddgdd(fd6dd)dd*dgdd+fd'6d)d,dd)d)dgd-d.fd)6d'd/d0d1d(d1d0gd2d3fd*6dd4d5d6ddd6d5gd7d8fd6dd9d:d;d<d=d<d;d:gd>d?fd@6d0dAdBdCdDdEdEdDdCdBgdFdGfd06d*dHdIdJdKdLdMdLdKdJdIg dNdOfd,6dPdQdRdSdTdUdVdVdUdTdSdRg dWdXfdP6ddYdZd[d\d]d^d_d^d]d\d[dZg d`dafd%6dbdcdddedfdgdhdididhdgdfdeddg djdkfdb6ddldmdndodpdqdrdsdrdqdpdodndmg dtdufd(6dvdwdxdydzd{d|d}d~d~d}d|d{dzdydxgddfdv6d0ddddddddddddddddgddfd6Z#ddZ$dS(i(tdivisiontprint_functiontabsolute_importN(ttrapz(troots_legendre(tgammaln(txranget fixed_quadt quadraturetrombergRtsimpstrombtcumtrapzt newton_cotestAccuracyWarningcB`seZRS((t__name__t __module__(((s9/tmp/pip-build-7oUkmx/scipy/scipy/integrate/quadrature.pyRscC`s8|tjkrtj|St|tj|i2ic C`st|ts|f}nt||d|} tj} tj} t|d|}xt||dD]^} t| ||d| d} t| | } | } | |ks| |t| kriPqiqiWt j d|| ft | | fS(s 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, rtol : 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 order of Gaussian quadrature. vec_func : bool, optional True or False if func handles arrays as arguments (is a "vector" function). Default is True. miniter : int, optional Minimum order of Gaussian quadrature. 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 cumtrapz: cumulative integration for sampled data ode: ODE integrator odeint: ODE integrator R-iis-maxiter (%d) exceeded. Latest difference = %e(( t isinstancettupleR.RtinftmaxRRtabstwarningstwarnR(RRRRttoltrtoltmaxiterR-tminiterR"tvalterrRtnewval((s9/tmp/pip-build-7oUkmx/scipy/scipy/integrate/quadrature.pyRs 2   " cC`s t|}|||>> from scipy import integrate >>> import matplotlib.pyplot as plt >>> 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() iis2If given, shape of x must be 1-d or the same as y.Rs7If given, length of x along axis must be the same as y.g@s'`initial` parameter should be a scalar.R#N(RR%RtndimtdifftreshapeR&tshapeRRAtslicetcumsumR$R=t concatenatetonesR#( R!RtdxRtinitialtdREtndtslice1tslice2tres((s9/tmp/pip-build-7oUkmx/scipy/scipy/integrate/quadrature.pyR s46   (()  ( cC`st|j}|dkr$d}nd}tdf|}t||t|||} t||t|d|d|} t||t|d|d|} |dkrtj|d|| d|| || d|} ntj|d|} t||t|||}t||t|d|d|}| |}| |}||}||}||}|d|| dd||| ||||| d|}tj|d|} | S( Niiig@iRg@g?(R&RERRFRARRRC(R!tstarttstopRRJRRMtstept slice_alltslice0RNROtresultthtsl0tsl1th0th1thsumthprodth0divh1ttmp((s9/tmp/pip-build-7oUkmx/scipy/scipy/integrate/quadrature.pyt _basic_simps-s0  && , &     'tavgcC`stj|}t|j}|j|}|}|}d} |dk rtj|}t|jdkrdg|} |jd| |<|j} d} |jt| }n-t|jt|jkrtdn|j||krtdqn|ddkrd} d} tdf|}tdf|}|dkratd n|dkrt ||d }t ||d }|dk r||||}n| d |||||7} t |d|d|||} n|dkrt ||d}t ||d}|dk rO|t||t|}n| d |||||7} | t |d|d|||7} n|dkr| d} | d} n| | } nt |d|d|||} | r|j| }n| S(s 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 : str {'avg', 'first', 'last'}, 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 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. iis2If given, shape of x must be 1-d or the same as y.s7If given, length of x along axis must be the same as y.igRatlasttfirsts3Parameter 'even' must be 'avg', 'last', or 'first'.iig?ig@N(savgslastsfirst(savgsfirst(savgslast( RR%R&RERRDR0RRFRAR`(R!RRJRtevenRMtNtlast_dxtfirst_dxt returnshapetshapext saveshapeR:RVRNRO((s9/tmp/pip-build-7oUkmx/scipy/scipy/integrate/quadrature.pyR Ls^4       "  !&    cC`sQtj|}t|j}|j|}|d}d}d}x$||krg|dK}|d7}qDW||krtdni} tdf|} t| |d} t| |d} |tj|dt} || || d| | d<| }|}}}xt d|dD]}|dL}t||t|||}|dL}d| |ddf| ||j d|| |dfd| ||f   )"  cC`s|dkrtdn|dkrGd||d||dS|d}t|d|d|}|dd|}||tj|}tj||dd}|SdS(sX 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). is#numtraps must be > 0 in difftrap().ig?iRN(RRpRtarangeR(tfunctiontintervaltnumtrapstnumtosumRWtloxtpointsts((s9/tmp/pip-build-7oUkmx/scipy/scipy/integrate/quadrature.pyt _difftrap s    cC`sd|}||||dS(s Compute the differences for the Romberg quadrature corrections. See Forman Acton's "Real Computing Made Real," p 143. g@g?((RtcRxR_((s9/tmp/pip-build-7oUkmx/scipy/scipy/integrate/quadrature.pyt _romberg_diff:s cC`s#d}}tdt|ddtd|tdtddxtt|D]y}td d ||d |dd|fddx4t|d D]"}td|||ddqWtdq[Wtdtd|||ddtdd t|d d ddS(NisRomberg integration ofRnRotfromRks %6s %9s %9stStepstStepSizetResultss%6d %9fiig@s%9fsThe final result istaftersfunction evaluations.(RRR(RqtreprRR&(RRtresmatR,R|((s9/tmp/pip-build-7oUkmx/scipy/scipy/integrate/quadrature.pyt _printresmatCs   2  g`sbO>i c  C`stj|stj|r-tdnt||d|} d} ||g} ||} t| | | } | | }|gg}tj}|d}x td|dD]}| d9} | t| | | 7} | | | g}x9t|D]+}|jt|||||dqW||}||d}|rN|j|nt ||}||ks||t |krPn|}qWt j d||ft |rt | | |n|S(so 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 : Double integrals. tplquad : Triple integrals. romb : Integrators for sampled data. simps : Integrators for sampled data. cumtrapz : Cumulative integration for sampled data. ode : ODE integrator. odeint : ODE integrator. 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 import integrate >>> from scipy.special import erf >>> gaussian = lambda x: 1/np.sqrt(np.pi) * np.exp(-x**2) >>> result = integrate.romberg(gaussian, 0, 1, show=True) Romberg integration of 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("%g %g" % (2*result, erf(1))) 0.842701 0.842701 s5Romberg integration only available for finite limits.R-iiis,divmax (%d) exceeded. 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