g&ddlmZddlmZGddeZdZd d Zee_ed krddlZej dSdS) )bisect)xrangeceZdZdS) ODEMethodsN)__name__ __module__ __qualname__P/var/www/html/ai-engine/env/lib/python3.11/site-packages/mpmath/calculus/odes.pyrrsDr rc L|d| x}t|}|g}|g} |} ||j} | d|zz|_t|D]g} || fdt tD| z } || | hdt|D} t|dzD]}dg|z}d|dzz}d}t|dzD]G} t|D]!}||xx|| | |zz cc<"|||z dzz| z}|dz }H| z||z }t|D]1}|||z||<| |||2 | |_n #| |_wxYw|j}| D]D}|dr:t|| |t|dz |}E|dz}| ||zfS)Nc8g|]}||zzSr r ).0ifxyhys r zode_taylor..s)77711aAh777r cg|]}gSr r )rds r rzode_taylor..s&&&ar&&&r rr) ldexplenprecrangerappendfaconeminnthrootabs)ctxderivsx0y0tol_precntoldimxsysxorigrserjsbkrscaleradiustsrrrs @@@r ode_taylorr8siiH9%%%A b''C B B A A 8D1:q  A&A,,C777777s1vv777A FA IIaLLL IIaLLLL&&5::&&&qs $ $ACAQAA1Q3ZZ  s))AaDDDA1aL(DDDD!A#a%[qb)QGcggajj(E3ZZ $ $te|!A ad#### $ $4WFBB b6 BSRV_a!@!@AAF aKF 6 >s E9G G NtaylorFc  |r(t|d dzn jdzpdtdjzdz zjdz t |dn#t $r fd|g}dYnwxYwt |\}} | g|| fgfd   f d  fd } | S) a Returns a function `y(x) = [y_0(x), y_1(x), \ldots, y_n(x)]` that is a numerical solution of the `n+1`-dimensional first-order ordinary differential equation (ODE) system .. math :: y_0'(x) = F_0(x, [y_0(x), y_1(x), \ldots, y_n(x)]) y_1'(x) = F_1(x, [y_0(x), y_1(x), \ldots, y_n(x)]) \vdots y_n'(x) = F_n(x, [y_0(x), y_1(x), \ldots, y_n(x)]) The derivatives are specified by the vector-valued function *F* that evaluates `[y_0', \ldots, y_n'] = F(x, [y_0, \ldots, y_n])`. The initial point `x_0` is specified by the scalar argument *x0*, and the initial value `y(x_0) = [y_0(x_0), \ldots, y_n(x_0)]` is specified by the vector argument *y0*. For convenience, if the system is one-dimensional, you may optionally provide just a scalar value for *y0*. In this case, *F* should accept a scalar *y* argument and return a scalar. The solution function *y* will return scalar values instead of length-1 vectors. Evaluation of the solution function `y(x)` is permitted for any `x \ge x_0`. A high-order ODE can be solved by transforming it into first-order vector form. This transformation is described in standard texts on ODEs. Examples will also be given below. **Options, speed and accuracy** By default, :func:`~mpmath.odefun` uses a high-order Taylor series method. For reasonably well-behaved problems, the solution will be fully accurate to within the working precision. Note that *F* must be possible to evaluate to very high precision for the generation of Taylor series to work. To get a faster but less accurate solution, you can set a large value for *tol* (which defaults roughly to *eps*). If you just want to plot the solution or perform a basic simulation, *tol = 0.01* is likely sufficient. The *degree* argument controls the degree of the solver (with *method='taylor'*, this is the degree of the Taylor series expansion). A higher degree means that a longer step can be taken before a new local solution must be generated from *F*, meaning that fewer steps are required to get from `x_0` to a given `x_1`. On the other hand, a higher degree also means that each local solution becomes more expensive (i.e., more evaluations of *F* are required per step, and at higher precision). The optimal setting therefore involves a tradeoff. Generally, decreasing the *degree* for Taylor series is likely to give faster solution at low precision, while increasing is likely to be better at higher precision. The function object returned by :func:`~mpmath.odefun` caches the solutions at all step points and uses polynomial interpolation between step points. Therefore, once `y(x_1)` has been evaluated for some `x_1`, `y(x)` can be evaluated very quickly for any `x_0 \le x \le x_1`. and continuing the evaluation up to `x_2 > x_1` is also fast. **Examples of first-order ODEs** We will solve the standard test problem `y'(x) = y(x), y(0) = 1` which has explicit solution `y(x) = \exp(x)`:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> f = odefun(lambda x, y: y, 0, 1) >>> for x in [0, 1, 2.5]: ... print((f(x), exp(x))) ... (1.0, 1.0) (2.71828182845905, 2.71828182845905) (12.1824939607035, 12.1824939607035) The solution with high precision:: >>> mp.dps = 50 >>> f = odefun(lambda x, y: y, 0, 1) >>> f(1) 2.7182818284590452353602874713526624977572470937 >>> exp(1) 2.7182818284590452353602874713526624977572470937 Using the more general vectorized form, the test problem can be input as (note that *f* returns a 1-element vector):: >>> mp.dps = 15 >>> f = odefun(lambda x, y: [y[0]], 0, [1]) >>> f(1) [2.71828182845905] :func:`~mpmath.odefun` can solve nonlinear ODEs, which are generally impossible (and at best difficult) to solve analytically. As an example of a nonlinear ODE, we will solve `y'(x) = x \sin(y(x))` for `y(0) = \pi/2`. An exact solution happens to be known for this problem, and is given by `y(x) = 2 \tan^{-1}\left(\exp\left(x^2/2\right)\right)`:: >>> f = odefun(lambda x, y: x*sin(y), 0, pi/2) >>> for x in [2, 5, 10]: ... print((f(x), 2*atan(exp(mpf(x)**2/2)))) ... (2.87255666284091, 2.87255666284091) (3.14158520028345, 3.14158520028345) (3.14159265358979, 3.14159265358979) If `F` is independent of `y`, an ODE can be solved using direct integration. We can therefore obtain a reference solution with :func:`~mpmath.quad`:: >>> f = lambda x: (1+x**2)/(1+x**3) >>> g = odefun(lambda x, y: f(x), pi, 0) >>> g(2*pi) 0.72128263801696 >>> quad(f, [pi, 2*pi]) 0.72128263801696 **Examples of second-order ODEs** We will solve the harmonic oscillator equation `y''(x) + y(x) = 0`. To do this, we introduce the helper functions `y_0 = y, y_1 = y_0'` whereby the original equation can be written as `y_1' + y_0' = 0`. Put together, we get the first-order, two-dimensional vector ODE .. math :: \begin{cases} y_0' = y_1 \\ y_1' = -y_0 \end{cases} To get a well-defined IVP, we need two initial values. With `y(0) = y_0(0) = 1` and `-y'(0) = y_1(0) = 0`, the problem will of course be solved by `y(x) = y_0(x) = \cos(x)` and `-y'(x) = y_1(x) = \sin(x)`. We check this:: >>> f = odefun(lambda x, y: [-y[1], y[0]], 0, [1, 0]) >>> for x in [0, 1, 2.5, 10]: ... nprint(f(x), 15) ... nprint([cos(x), sin(x)], 15) ... print("---") ... [1.0, 0.0] [1.0, 0.0] --- [0.54030230586814, 0.841470984807897] [0.54030230586814, 0.841470984807897] --- [-0.801143615546934, 0.598472144103957] [-0.801143615546934, 0.598472144103957] --- [-0.839071529076452, -0.54402111088937] [-0.839071529076452, -0.54402111088937] --- Note that we get both the sine and the cosine solutions simultaneously. **TODO** * Better automatic choice of degree and step size * Make determination of Taylor series convergence radius more robust * Allow solution for `x < x_0` * Allow solution for complex `x` * Test for difficult (ill-conditioned) problems * Implement Runge-Kutta and other algorithms r g@(Tc*||dgS)Nrr )r.rF_s r zodefun..s""Q!++r Fc$fd|DS)NcNg|]!}|ddd"S)Nr)polyval)rr2ar$s r rz,odefun..mpolyval..s1555A AdddGQ''555r r )r0rDr$s `r mpolyvalzodefun..mpolyvals!555555555r c |krtt |}|t kr |dz S d\}}} rtd||fz |||z }|}t || \}} | |||f||kr dS)Nrrz$Computing Taylor series for [%f, %f]) ValueErrorrrprintr8r)r.r)r0xaxbrFr$degreerEseries_boundaries series_datar(verboser&s r get_serieszodefun..get_seriess r66  $a ( ( s$%% % %qs# # '%b/KCR I<BxGHHHbe$$AB aQ&AAGC  $ $R ( ( (   R} - - -Bww"2& 'r c|}j} _|\}}}|||z }|_n #|_wxYw r d|DS|d S)Ncg|]}| Sr r )ryks r rz/odefun..interpolant..s$$$BRC$$$r r)convertr) r.r/r0rIrJrr$rPrE return_vectorworkprecs r interpolantzodefun..interpolant s KKNNx CH$*Q--KCRad##ACHHtCHOOOO  $$!$$$ $aD5Ls %A A)intlogrdpsr TypeErrorr8)r$rKr&r'r*rLmethodrOr0rJrWr?rPrErUrMrNr(rVs``` ` ` @@@@@@@@r odefunr]3sf Q'((+8B;  .C#' " ---Fx"}H B   & & & &T  aR6::GCRR=/K66666'''''''''''''$          s,A>>BB__main__)NNr9F) r libmp.backendrobjectrr8r]rdoctesttestmodr r r rcs""""""        ***XggggR  zNNNGOr