o Eb(@sddlmZddlmZmZmZmZmZmZddl Z ddl Z ddl Z ddl Z ddlZddl mZddlmZddlmZgdZdd ZeeZejrSejd d e_dddZGdddeZerqddlmZGdddeZneZdddZeddZee_ dddZ!dd!d"Z" $ &dd'd(Z#d)d*Z$dd+d,Z%dd-d.Z&d/d0Z'dd2d3Z(dd4d5Z)dd6d7Z*d8d9Z+d:d;Z,d>> from scipy import integrate >>> f = lambda x: x**8 >>> integrate.fixed_quad(f, 0.0, 1.0, n=4) (0.1110884353741496, None) >>> integrate.fixed_quad(f, 0.0, 1.0, n=5) (0.11111111111111102, None) >>> print(1/9.0) # analytical result 0.1111111111111111 >>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=4) (0.9999999771971152, None) >>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=5) (1.000000000039565, None) >>> np.sin(np.pi/2)-np.sin(0) # analytical result 1.0 z8Gaussian quadrature is only available for finite limits.@r(r+N)r7nprealisinf ValueErrorsum)r3abargsr8r)wr,r$r$r%r Ss > .r Fcs(|r fdd}|Sfdd}|S)aoVectorize the call to a function. This is an internal utility function used by `romberg` and `quadrature` to create a vectorized version of a function. If `vec_func` is True, the function `func` is assumed to take vector arguments. Parameters ---------- func : callable User defined function. args : tuple, optional Extra arguments for the function. vec_func : bool, optional True if the function func takes vector arguments. Returns ------- vfunc : callable A function that will take a vector argument and return the result. cs|gRSr5r$)r)rDr3r$r%vfuncszvectorize1..vfunccst|r |gRSt|}|dgR}t|}t|dt|}tj|f|d}||d<td|D]}||gR||<q9|S)NrdtyperHr:)r=isscalarasarraylengetattrtypeemptyrange)r)Zy0r8rHoutputirFr$r%rGs  r$)r3rDvec_funcrGr$rFr% vectorize1s  rT"\O>2Tr:c Cst|ts|f}t|||d} tj} tj} t|d|}t||dD]%} t| ||d| d} t| | } | } | |ksC| |t| krH| | fSq#t d|| ft | | fS)a 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 simpson: integrator for sampled data cumulative_trapezoid: cumulative integration for sampled data ode: ODE integrator odeint: ODE integrator Examples -------- >>> from scipy import integrate >>> f = lambda x: x**8 >>> integrate.quadrature(f, 0.0, 1.0) (0.11111111111111106, 4.163336342344337e-17) >>> print(1/9.0) # analytical result 0.1111111111111111 >>> integrate.quadrature(np.cos, 0.0, np.pi/2) (0.9999999999999536, 3.9611425250996035e-11) >>> np.sin(np.pi/2)-np.sin(0) # analytical result 1.0 rSr:r$rz-maxiter (%d) exceeded. Latest difference = %e) isinstancetuplerTr=infmaxrPr abswarningswarnr)r3rBrCrDtolrtolmaxiterrSZminiterrGvalerrr8Znewvalr$r$r%r s& @  r cCst|}|||<t|Sr5)listrY)trRvaluelr$r$r%tuplesetsrhcCt|||||dS)z`An alias of `cumulative_trapezoid`. `cumtrapz` is kept for backwards compatibility. For new code, prefer `cumulative_trapezoid` instead. )r)r*r+initial)r)r,r)r*r+rjr$r$r%r#rc CsTt|}|dur |}nDt|}|jdkr+t|}dg|j}d||<||}nt|jt|jkr9tdtj||d}|j||j|dkrPtdt|j}tt df||t dd}tt df||t dd} tj ||||| d|d} |durt |stdt | j}d||<tj tj||| jd | g|d} | S) a Cumulatively integrate y(x) using the composite trapezoidal rule. Parameters ---------- y : array_like Values to integrate. x : array_like, optional The coordinate to integrate along. If None (default), use spacing `dx` between consecutive elements in `y`. dx : float, optional Spacing between elements of `y`. Only used if `x` is None. axis : int, optional Specifies the axis to cumulate. Default is -1 (last axis). initial : scalar, optional If given, insert this value at the beginning of the returned result. Typically this value should be 0. Default is None, which means no value at ``x[0]`` is returned and `res` has one element less than `y` along the axis of integration. Returns ------- res : ndarray The result of cumulative integration of `y` along `axis`. If `initial` is None, the shape is such that the axis of integration has one less value than `y`. If `initial` is given, the shape is equal to that of `y`. See Also -------- numpy.cumsum, numpy.cumprod quad: adaptive quadrature using QUADPACK romberg: adaptive Romberg quadrature quadrature: adaptive Gaussian quadrature fixed_quad: fixed-order Gaussian quadrature dblquad: double integrals tplquad: triple integrals romb: integrators for sampled data ode: ODE integrators odeint: ODE integrators Examples -------- >>> from scipy import integrate >>> import matplotlib.pyplot as plt >>> x = np.linspace(-2, 2, num=20) >>> y = x >>> y_int = integrate.cumulative_trapezoid(y, x, initial=0) >>> plt.plot(x, y_int, 'ro', x, y[0] + 0.5 * x**2, 'b-') >>> plt.show() Nr:r(2If given, shape of x must be 1-D or the same as y.r<7If given, length of x along axis must be the same as y.r;z'`initial` parameter should be a scalar.rI)r=rKndimdiffreshaperLshaper@rhsliceZcumsumrJrdZ concatenateZfullrH) r,r)r*r+rjdrqndslice1slice2resr$r$r%r,s6 6      "  rcCsjt|j}|dur d}d}tdf|}t||t|||} t||t|d|d|} t||t|d|d|} |durYtj|| d|| || |d} | |d9} | Stj||d} t||t|||}t||t|d|d|}| |}| |}||}||}||}|d|| dd||| ||||| d|}tj||d} | S) Nrr:r<@g@r')rLrqrrrhr=rAro)r,startstopr)r*r+rtstep slice_allslice0rurvresulthZsl0Zsl1Zh0Zh1ZhsumZhprodZh0divh1tmpr$r$r%_basic_simpsons6 & ravgcCri)zz`An alias of `simpson`. `simps` is kept for backwards compatibility. For new code, prefer `simpson` instead. )r)r*r+even)r)r,r)r*r+rr$r$r%rrkrcCst|}t|j}|j|}|}|}d} |durWt|}t|jdkr>dg|} |jd| |<|j} d} |t| }nt|jt|jkrLtd|j||krWtd|ddkrd} d} tdf|}tdf|}|dvrwtd |d vrt||d }t||d }|dur||||}| d |||||7} t |d|d|||} |dvrt||d}t||d}|dur|t||t|}| d |||||7} | t |d|d|||7} |dkr| d} | d} | | } n t |d|d|||} | r || }| S)a 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 : float, optional Spacing of integration points along axis of `x`. 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 cumulative_trapezoid: 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. Examples -------- >>> from scipy import integrate >>> x = np.arange(0, 10) >>> y = np.arange(0, 10) >>> integrate.simpson(y, x) 40.5 >>> y = np.power(x, 3) >>> integrate.simpson(y, x) 1642.5 >>> integrate.quad(lambda x: x**3, 0, 9)[0] 1640.25 >>> integrate.simpson(y, x, even='first') 1644.5 rNr:rlrmrxg)rlastfirstz3Parameter 'even' must be 'avg', 'last', or 'first'.)rrr(?)rrrr;) r=rKrLrqrprYr@rrrhr)r,r)r*r+rrtNZlast_dxZfirst_dxZ returnshapeZshapexZ saveshaperbrrurvr$r$r%rs^ F           rc Cst|}t|j}|j|}|d}d}d}||kr'|dK}|d7}||ks||kr/tdi} tdf|} t| |d} t| |d} |tj|td} || || d| | d<| }|}}}td|dD]\}|dL}t||t|||}|dL}d | |ddf| ||j |d | |df<td|dD]$}| ||df}||| |d|dfdd |>d| ||f<q| d} qj|r>> from scipy import integrate >>> x = np.arange(10, 14.25, 0.25) >>> y = np.arange(3, 12) >>> integrate.romb(y) 56.0 >>> y = np.sin(np.power(x, 2.5)) >>> integrate.romb(y) -0.742561336672229 >>> integrate.romb(y, show=True) Richardson Extrapolation Table for Romberg Integration ==================================================================== -0.81576 4.63862 6.45674 -1.10581 -3.02062 -3.65245 -2.57379 -3.06311 -3.06595 -3.05664 -1.34093 -0.92997 -0.78776 -0.75160 -0.74256 ==================================================================== -0.742561336672229 r:rz=Number of samples must be one plus a non-negative power of 2.Nr(rIr;)rrrr<rxzE*** Printing table only supported for integrals of a single data set.r9z%%%d.%dfz6Richardson Extrapolation Table for Romberg IntegrationDzD==================================================================== )sepend r)r=rKrLrqr@rrrhfloatrPrArJprint TypeError IndexErrorcenter)r,r*r+showrtZNsampsZNintervr8kRr~rZslicem1rZslice_Rr{r|r}rRjprevZpreciswidthZformstrtitler$r$r%r/sh ;     06       rcCs|dkrtd|dkrd||d||dS|d}t|d|d|}|dd|}||t|}tj||dd}|S)aU 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). rz#numtraps must be > 0 in difftrap().r:rrxr<)r@rr=arangerA)functionintervalZnumtrapsZnumtosumrZloxZpointssr$r$r% _difftraps rcCsd|}||||dS)z Compute the differences for the Romberg quadrature corrections. See Forman Acton's "Real Computing Made Real," p 143. g@r'r$)rCcrrr$r$r% _romberg_diffsrcCsd}}tdt|ddtd|tdtddtt|D]1}td d ||d |dd |fddt|d D]}td |||ddq@tdq"tdtd|||ddtdd t|d d ddS)NrzRomberg integration ofrrfromrz %6s %9s %9s)ZStepsZStepSizeZResultsz%6d %9frxr:r;z%9fzThe final result isZafterzfunction evaluations.)rreprrPrL)rrresmatrRrr$r$r% _printresmats  ,  r`sbO> c  CsDt|s t|rtdt|||d} d} ||g} ||} t| | | } | | }|gg}tj}|d}td|dD]R}| d9} | t| | | 7} | | | g}t|D]}|t|||||dqT||}||d}|rw||t ||}||ks||t |krn |}q;t d||ft |rt | | ||S)a 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. simpson : Integrators for sampled data. cumulative_trapezoid : Cumulative integration for sampled data. ode : ODE integrator. odeint : ODE integrator. References ---------- .. [1] 'Romberg's method' https://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 z5Romberg integration only available for finite limits.rWr:rrxz,divmax (%d) exceeded. Latest difference = %e)r=r?r@rTrrZrPappendrr\r]r^rr)rrBrCrDr_r`rZdivmaxrSrGr8rZintrangeZordsumrrrcZlast_rowrRrowrZ lastresultr$r$r%rs@T        rrx r)r:ryr:Zr)r:rrr:P-) rrriii )KrVrVrrii@/))irrriixriC) + rrrri iryi_7) `)iDrrrrii?# i^) ) }=8Krrrrriiip) ><sB(:ihrrrrriii0 i0) I"!jmirrrrrrl& l7iR0P) @7@!!Nd7ipRrrrrrri$MY(rrrrrrrl`:l@ Al@d@*)i`p`*oFg!f\a LRl@`rrrrrrrlx=l7-)r:rxrryr9rrrrrrrcCszt|d}|rt|d}n tt|dkrd}Wnty2|}t|d}d}Ynw|rU|tvrUt|\}}}}}|tj|td|}|t||fS|ddksa|d|kret d|t|} d| d} t|d} | | ddtj f} tj | } t dD]}d| | | | } qd| dddd}| dddddf||d}|ddkr|r||d }|d}n ||d}|d}|t| ||}|d}|t|t|}t|}|||fS) a 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 average 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. Examples -------- Compute the integral of sin(x) in [0, :math:`\pi`]: >>> from scipy.integrate import newton_cotes >>> def f(x): ... return np.sin(x) >>> a = 0 >>> b = np.pi >>> exact = 2 >>> for N in [2, 4, 6, 8, 10]: ... x = np.linspace(a, b, N + 1) ... an, B = newton_cotes(N, 1) ... dx = (b - a) / N ... quad = dx * np.sum(an * f(x)) ... error = abs(quad - exact) ... print('{:2d} {:10.9f} {:.5e}'.format(N, quad, error)) ... 2 2.094395102 9.43951e-02 4 1.998570732 1.42927e-03 6 2.000017814 1.78136e-05 8 1.999999835 1.64725e-07 10 2.000000001 1.14677e-09 Notes ----- Normally, the Newton-Cotes rules are used on smaller integration regions and a composite rule is used to return the total integral. r:rIrr(z1The sample positions must start at 0 and end at NrxNr;rz)rLr=rallro Exception_builtincoeffsZarrayrr@ZnewaxisZlinalginvrPdotmathlogr Zexp)ZrnZequalrZnadavinbZdbZanyiZtiZnvecCZCinvrRZvecZaiZBNZpowerZp1Zfacr$r$r%rsJ@       $     r)Nr'r()r3rr4r0)r$r9)r$F)r$rUrUrVTr:)Nr'r(N)Nr'r(r)r'r(F)r$rrFrF)r)1Z __future__rtypingrrrrrrrZnumpyr=rrr]r rZ scipy.specialr r __all__r&__doc__replaceWarningrZtyping_extensionsr/r0r6r7dictr1r rTr rhrrrrrrrrrrrrr$r$r$r%s            G- T  \ ! }         #