[hq"`SSKJr SSKJr SrS Sjr\S Sj5r\S5r\S5rg ) )xrange)defuncURS5nURS5n[SUS-5HMnURX- U- 5n Xa"XU- -U-X2-U--5URXHU- -U- 5-- nMO SU-U- $)N?rr)mpfrangecospi) ctxfabjNshkts U/var/www/auris/envauris/lib/python3.13/site-packages/mpmath/calculus/approximation.py chebcoeffrs  A  A 1ac] IIqsAg  QqA#wqyAC7" #ciiQ3 &: :: Q3q5Lc## S/nUv X!/nUv S/UVs/sH nSU-U-PM sn-n[U5HupxXg==SU-U-- ss'M [U5HupxXg==U-ss'M Xdp4Mks snf7f)Nrrr) enumerate) r rrTbTarTmpics rchebTr s B H B cB'BqQqSUB''bMDA36QqSU?6MbMDA36Q;6MB (sA>A9AA>c tURU5upVURnU[US-5-S-Ul[U5Vs/sHn[ XXVX5PM n nUR /U-n U S*S- U S'UR S5n [XR S5Xe- - UR S5Xe--Xe- - 5n [[U5U 5H3up[[U 55HnX==XX-- ss'M M5 U SSS2n UR n[U5HbnURURU-U- 5Xe- -U -Xe-U --n[U[U"U5URU U5- 55nMd XplU(aX74$U $s snf!Xplf=f)a8 Computes a polynomial of degree `N-1` that approximates the given function `f` on the interval `[a, b]`. With ``error=True``, :func:`~mpmath.chebyfit` also returns an accurate estimate of the maximum absolute error; that is, the maximum value of `|f(x) - P(x)|` for `x \in [a, b]`. :func:`~mpmath.chebyfit` uses the Chebyshev approximation formula, which gives a nearly optimal solution: that is, the maximum error of the approximating polynomial is very close to the smallest possible for any polynomial of the same degree. Chebyshev approximation is very useful if one needs repeated evaluation of an expensive function, such as function defined implicitly by an integral or a differential equation. (For example, it could be used to turn a slow mpmath function into a fast machine-precision version of the same.) **Examples** Here we use :func:`~mpmath.chebyfit` to generate a low-degree approximation of `f(x) = \cos(x)`, valid on the interval `[1, 2]`:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> poly, err = chebyfit(cos, [1, 2], 5, error=True) >>> nprint(poly) [0.00291682, 0.146166, -0.732491, 0.174141, 0.949553] >>> nprint(err, 12) 1.61351758081e-5 The polynomial can be evaluated using ``polyval``:: >>> nprint(polyval(poly, 1.6), 12) -0.0291858904138 >>> nprint(cos(1.6), 12) -0.0291995223013 Sampling the true error at 1000 points shows that the error estimate generated by ``chebyfit`` is remarkably good:: >>> error = lambda x: abs(cos(x) - polyval(poly, x)) >>> nprint(max([error(1+n/1000.) for n in range(1000)]), 12) 1.61349954245e-5 **Choice of degree** The degree `N` can be set arbitrarily high, to obtain an arbitrarily good approximation. As a rule of thumb, an `N`-term Chebyshev approximation is good to `N/(b-a)` decimal places on a unit interval (although this depends on how well-behaved `f` is). The cost grows accordingly: ``chebyfit`` evaluates the function `(N^2)/2` times to compute the coefficients and an additional `N` times to estimate the error. **Possible issues** One should be careful to use a sufficiently high working precision both when calling ``chebyfit`` and when evaluating the resulting polynomial, as the polynomial is sometimes ill-conditioned. It is for example difficult to reach 15-digit accuracy when evaluating the polynomial using machine precision floats, no matter the theoretical accuracy of the polynomial. (The option to return the coefficients in Chebyshev form should be made available in the future.) It is important to note the Chebyshev approximation works poorly if `f` is not smooth. A function containing singularities, rapid oscillation, etc can be approximated more effectively by multiplying it by a weight function that cancels out the nonsmooth features, or by dividing the interval into several segments. rrrN) _as_pointsprecintr rzeror r ziplencospimaxabspolyval)r r intervalrerrorrrorigrrdrTTkrerrxs rchebyfitr7&sX >>( #DA 88D#af+%*/4Qx 8x!YsQ %x 8 XXJN!uQw! GGCL #wwqz13'ac):AC)@ A58Q'GQ3r7^RU "$( ddGhhqAq #qs+A-a7Ac3qtckk!Q&7789C $w' 9s&F/F*D9F/*F//F7c^^^ TRU5nUSnUSnXT- n/n/nTRS-n [US-5Hn SU -TR-U- m STR UUU 4SjU5-U- n STR UUU 4SjU5-U- n U S:XaU S-n [ U 5U :a TR n [ U 5U :a TR n URU 5 URU 5 M Xx4$)ac Computes the Fourier series of degree `N` of the given function on the interval `[a, b]`. More precisely, :func:`~mpmath.fourier` returns two lists `(c, s)` of coefficients (the cosine series and sine series, respectively), such that .. math :: f(x) \sim \sum_{k=0}^N c_k \cos(k m x) + s_k \sin(k m x) where `m = 2 \pi / (b-a)`. Note that many texts define the first coefficient as `2 c_0` instead of `c_0`. The easiest way to evaluate the computed series correctly is to pass it to :func:`~mpmath.fourierval`. **Examples** The function `f(x) = x` has a simple Fourier series on the standard interval `[-\pi, \pi]`. The cosine coefficients are all zero (because the function has odd symmetry), and the sine coefficients are rational numbers:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> c, s = fourier(lambda x: x, [-pi, pi], 5) >>> nprint(c) [0.0, 0.0, 0.0, 0.0, 0.0, 0.0] >>> nprint(s) [0.0, 2.0, -1.0, 0.666667, -0.5, 0.4] This computes a Fourier series of a nonsymmetric function on a nonstandard interval:: >>> I = [-1, 1.5] >>> f = lambda x: x**2 - 4*x + 1 >>> cs = fourier(f, I, 4) >>> nprint(cs[0]) [0.583333, 1.12479, -1.27552, 0.904708, -0.441296] >>> nprint(cs[1]) [0.0, -2.6255, 0.580905, 0.219974, -0.540057] It is instructive to plot a function along with its truncated Fourier series:: >>> plot([f, lambda x: fourierval(cs, I, x)], I) #doctest: +SKIP Fourier series generally converge slowly (and may not converge pointwise). For example, if `f(x) = \cosh(x)`, a 10-term Fourier series gives an `L^2` error corresponding to 2-digit accuracy:: >>> I = [-1, 1] >>> cs = fourier(cosh, I, 9) >>> g = lambda x: (cosh(x) - fourierval(cs, I, x))**2 >>> nprint(sqrt(quad(g, I))) 0.00467963 :func:`~mpmath.fourier` uses numerical quadrature. For nonsmooth functions, the accuracy (and speed) can be improved by including all singular points in the interval specification:: >>> nprint(fourier(abs, [-1, 1], 0), 10) ([0.5000441648], [0.0]) >>> nprint(fourier(abs, [-1, 0, 1], 0), 10) ([0.5], [0.0]) rr# rrc>>T"U5TRTU-5-$Nr*rr r ms rfourier..AaD1$5rc>>T"U5TRTU-5-$r;sinr=s rr?r@rAr)r$epsrr+quadglr-r'append)r r r/rrrL cos_series sin_seriescutoffnanbnr>s`` @rfourierrOsL~~h'H A A AJJ WWRZF AaC[ aCJqL szz5x@ @ B szz5x@ @ B 6 !GB r7V #((R r7V #((R""  !!rc j^^^^ ^ Uumm TRU5nUSnUSnSTR-USUS- - m TRnUTRUUU U4Sj[ [ T5555- nUTRUU U U4Sj[ [ T 5555- nU$)a  Evaluates a Fourier series (in the format computed by by :func:`~mpmath.fourier` for the given interval) at the point `x`. The series should be a pair `(c, s)` where `c` is the cosine series and `s` is the sine series. The two lists need not have the same length. rr#rc3v># UH.nTU(dMTUTRTU-T-5-v M0 g7fr;r<).0rLcsr r>r6s r fourierval..3I12a5&"Q%!A&9$9c3v># UH.nTU(dMTUTRTU-T-5-v M0 g7fr;rC)rRrLr r>ssr6s rrTrUrVrW)r$r+r'fsumrr)) r seriesr/r6abrrrrSr>rYs ` ` @@@r fouriervalr]sFB  !B A A #&&"R&A,A AIs2wI IIAIs2wI IIA HrN)rr)F) libmp.backendrcalculusrrr r7rOr]rrrasW" bbHV"V"p  r