a Ым=єzhsteps..)┌get┌precrZmag┌ldexp┌convert┌signr)rrrr r┌optionsrrr┌workprec┌origZ hextramagZsteps┌norm┌valuesr rr┌hstepss8 ■r'csfd}ztИГ}tИГЙd}Wnty.Yn0|rVЗfddДИDГЙtИИИ||ГS|аddб}ИdkrК|dkrК|аd бsКИИаИбГSИj}z╞|dkr╘tИИИИ|fi|дО\} } }|И_Иа| Иб| И}nz|dkРrBИjd 7_Иа|аddббЙЗЗЗЗЗfd dД} Иа| ddИj gб}|Иа ИбdИj }ntd|ГВW|И_n|И_0| S)a╠ Numerically computes the derivative of `f`, `f'(x)`, or generally for an integer `n \ge 0`, the `n`-th derivative `f^{(n)}(x)`. A few basic examples are:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> diff(lambda x: x**2 + x, 1.0) 3.0 >>> diff(lambda x: x**2 + x, 1.0, 2) 2.0 >>> diff(lambda x: x**2 + x, 1.0, 3) 0.0 >>> nprint([diff(exp, 3, n) for n in range(5)]) # exp'(x) = exp(x) [20.0855, 20.0855, 20.0855, 20.0855, 20.0855] Even more generally, given a tuple of arguments `(x_1, \ldots, x_k)` and order `(n_1, \ldots, n_k)`, the partial derivative `f^{(n_1,\ldots,n_k)}(x_1,\ldots,x_k)` is evaluated. For example:: >>> diff(lambda x,y: 3*x*y + 2*y - x, (0.25, 0.5), (0,1)) 2.75 >>> diff(lambda x,y: 3*x*y + 2*y - x, (0.25, 0.5), (1,1)) 3.0 **Options** The following optional keyword arguments are recognized: ``method`` Supported methods are ``'step'`` or ``'quad'``: derivatives may be computed using either a finite difference with a small step size `h` (default), or numerical quadrature. ``direction`` Direction of finite difference: can be -1 for a left difference, 0 for a central difference (default), or +1 for a right difference; more generally can be any complex number. ``addprec`` Extra precision for `h` used to account for the function's sensitivity to perturbations (default = 10). ``relative`` Choose `h` relative to the magnitude of `x`, rather than an absolute value; useful for large or tiny `x` (default = False). ``h`` As an alternative to ``addprec`` and ``relative``, manually select the step size `h`. ``singular`` If True, evaluation exactly at the point `x` is avoided; this is useful for differentiating functions with removable singularities. Default = False. ``radius`` Radius of integration contour (with ``method = 'quad'``). Default = 0.25. A larger radius typically is faster and more accurate, but it must be chosen so that `f` has no singularities within the radius from the evaluation point. A finite difference requires `n+1` function evaluations and must be performed at `(n+1)` times the target precision. Accordingly, `f` must support fast evaluation at high precision. With integration, a larger number of function evaluations is required, but not much extra precision is required. For high order derivatives, this method may thus be faster if f is very expensive to evaluate at high precision. **Further examples** The direction option is useful for computing left- or right-sided derivatives of nonsmooth functions:: >>> diff(abs, 0, direction=0) 0.0 >>> diff(abs, 0, direction=1) 1.0 >>> diff(abs, 0, direction=-1) -1.0 More generally, if the direction is nonzero, a right difference is computed where the step size is multiplied by sign(direction). For example, with direction=+j, the derivative from the positive imaginary direction will be computed:: >>> diff(abs, 0, direction=j) (0.0 - 1.0j) With integration, the result may have a small imaginary part even even if the result is purely real:: >>> diff(sqrt, 1, method='quad') # doctest:+ELLIPSIS (0.5 - 4.59...e-26j) >>> chop(_) 0.5 Adding precision to obtain an accurate value:: >>> diff(cos, 1e-30) 0.0 >>> diff(cos, 1e-30, h=0.0001) -9.99999998328279e-31 >>> diff(cos, 1e-30, addprec=100) -1.0e-30 FTcsg|]}Иа|бСqSr )r )r┌_йrr rr┤rzdiff..┌method┌stepr┌quadrr┌radiusg╨?cs&ИИа|б}И|}И|Г|ИSйN)Zexpj)┌tZrei┌zйrrr r-rr r┌g┬szdiff..grzunknown method: %r)┌list┌ TypeError┌ _partial_diffrr rr'rZquadts┌pi┌ factorial┌ ValueError)rrrr r"┌partial┌ordersr*rr&r%r#┌vr2r r r1r┌diffCs8i r<csp|s ИГSt|ГsИ|ОSdЙtt|ГГD]Й|Иr*q.fdiff_inner..innerйr<)r?rAйrrr@r"┌order)r?r┌fdiff_inner╪sz"_partial_diff..fdiff_inner)┌sum┌range┌lenr5)rr┌xsr:r"rEr rCrr5╬sr5Nc ks^|dur|j}nt|Г}|аddбdkr^d}||dkrZ|j|||fi|дОV|d7}q,dS|аdб}|rВ|j||dddНVn||а|бГV|dkrЮdS||jkr▓d \}}nd|d}}|j} t||||| fi|дО\} }}t||ГD]H}z$||_|а| |б||} W| |_n| |_0| V||krюdSqю|t|d dГ}}t ||Г}q└dS)ad Returns a generator that yields the sequence of derivatives .. math :: f(x), f'(x), f''(x), \ldots, f^{(k)}(x), \ldots With ``method='step'``, :func:`~mpmath.diffs` uses only `O(k)` function evaluations to generate the first `k` derivatives, rather than the roughly `O(k^2)` evaluations required if one calls :func:`~mpmath.diff` `k` separate times. With `n < \infty`, the generator stops as soon as the `n`-th derivative has been generated. If the exact number of needed derivatives is known in advance, this is further slightly more efficient. Options are the same as for :func:`~mpmath.diff`. **Examples** >>> from mpmath import * >>> mp.dps = 15 >>> nprint(list(diffs(cos, 1, 5))) [0.540302, -0.841471, -0.540302, 0.841471, 0.540302, -0.841471] >>> for i, d in zip(range(6), diffs(cos, 1)): ... print("%s %s" % (i, d)) ... 0 0.54030230586814 1 -0.841470984807897 2 -0.54030230586814 3 0.841470984807897 4 0.54030230586814 5 -0.841470984807897 Nr*r+rrrT)r)rrgffffffЎ?) ┌infrrr<r rr'rr┌min)rrrr r"rr┌A┌BZcallprec┌yr%r#r r r r┌diffs▀s<& rOcstИГЙgЙЗЗfddД}|S)Ncs.ttИГ|dГD]}ИаtИГбqИ|Sr=)rrH┌append┌next)rr@й┌data┌genr rr,sziterable_to_function..f)┌iter)rTrr rRr┌iterable_to_function)srVc cs╩t|Г}|dkr&|dD] }|Vqnаt|а|d|dЕбГ}t|а||ddЕбГ}d}||Г|dГ}d}td|dГD]4} ||| d| }||||| Г|| Г7}qА|V|d7}q^dS)aV Given a list of `N` iterables or generators yielding `f_k(x), f'_k(x), f''_k(x), \ldots` for `k = 1, \ldots, N`, generate `g(x), g'(x), g''(x), \ldots` where `g(x) = f_1(x) f_2(x) \cdots f_N(x)`. At high precision and for large orders, this is typically more efficient than numerical differentiation if the derivatives of each `f_k(x)` admit direct computation. Note: This function does not increase the working precision internally, so guard digits may have to be added externally for full accuracy. **Examples** >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> f = lambda x: exp(x)*cos(x)*sin(x) >>> u = diffs(f, 1) >>> v = mp.diffs_prod([diffs(exp,1), diffs(cos,1), diffs(sin,1)]) >>> next(u); next(v) 1.23586333600241 1.23586333600241 >>> next(u); next(v) 0.104658952245596 0.104658952245596 >>> next(u); next(v) -5.96999877552086 -5.96999877552086 >>> next(u); next(v) -12.4632923122697 -12.4632923122697 rrNr)rHrV┌ diffs_prodr) rZfactors┌N┌c┌ur;r r┌arr r rrW2s$ rWc Cs6||vr||S|s ddi|d<t|dГ}tddДt|ГDГГ}i}t|ГD]D\}}|ddf|ddЕ}||vrК|||7<qN|||<qNt|ГD]И\}}t|ГsоqЬt|ГD]l\}}|r╢|d|Е|d||ddf||ddЕ} | |vРr|| ||7<q╢|||| <q╢qЬ|||<||S)zд nth differentiation polynomial for exp (Faa di Bruno's formula). TODO: most exponents are zero, so maybe a sparse representation would be better. йrrrcss|]\}}|d|fVqdS)r\Nr )rrYr;r r r┌ trzdpoly..Nr)┌dpoly┌dict┌ iteritemsrF┌ enumerate) r ┌_cache┌RZRa┌powers┌countZpowers1r┌pZpowers2r r rr^hs. 4 r^c#s|t|ГЙ|аИdГб}|Vd}|аdб}tt|ГГD],\}}|||аЗfddДt|ГDГб7}q6||V|d7}q dS)a╨ Given an iterable or generator yielding `f(x), f'(x), f''(x), \ldots` generate `g(x), g'(x), g''(x), \ldots` where `g(x) = \exp(f(x))`. At high precision and for large orders, this is typically more efficient than numerical differentiation if the derivatives of `f(x)` admit direct computation. Note: This function does not increase the working precision internally, so guard digits may have to be added externally for full accuracy. **Examples** The derivatives of the gamma function can be computed using logarithmic differentiation:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> >>> def diffs_loggamma(x): ... yield loggamma(x) ... i = 0 ... while 1: ... yield psi(i,x) ... i += 1 ... >>> u = diffs_exp(diffs_loggamma(3)) >>> v = diffs(gamma, 3) >>> next(u); next(v) 2.0 2.0 >>> next(u); next(v) 1.84556867019693 1.84556867019693 >>> next(u); next(v) 2.49292999190269 2.49292999190269 >>> next(u); next(v) 3.44996501352367 3.44996501352367 rrc3s&|]\}}|rИ|dГ|VqdS)rNr )rrrfй┌fnr rr]╝rzdiffs_exp..N)rV┌expZmpfr`r^Zfprodra)rZfdiffsZf0r@rrdrYr rgr┌ diffs_expЙs, & rjrcsXttИаИа|ббГddГ}||dЙЗЗЗЗfddД}Иа|||бИа||бS)aЕ Calculates the Riemann-Liouville differintegral, or fractional derivative, defined by .. math :: \,_{x_0}{\mathbb{D}}^n_xf(x) = \frac{1}{\Gamma(m-n)} \frac{d^m}{dx^m} \int_{x_0}^{x}(x-t)^{m-n-1}f(t)dt where `f` is a given (presumably well-behaved) function, `x` is the evaluation point, `n` is the order, and `x_0` is the reference point of integration (`m` is an arbitrary parameter selected automatically). With `n = 1`, this is just the standard derivative `f'(x)`; with `n = 2`, the second derivative `f''(x)`, etc. With `n = -1`, it gives `\int_{x_0}^x f(t) dt`, with `n = -2` it gives `\int_{x_0}^x \left( \int_{x_0}^t f(u) du \right) dt`, etc. As `n` is permitted to be any number, this operator generalizes iterated differentiation and iterated integration to a single operator with a continuous order parameter. **Examples** There is an exact formula for the fractional derivative of a monomial `x^p`, which may be used as a reference. For example, the following gives a half-derivative (order 0.5):: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> x = mpf(3); p = 2; n = 0.5 >>> differint(lambda t: t**p, x, n) 7.81764019044672 >>> gamma(p+1)/gamma(p-n+1) * x**(p-n) 7.81764019044672 Another useful test function is the exponential function, whose integration / differentiation formula easy generalizes to arbitrary order. Here we first compute a third derivative, and then a triply nested integral. (The reference point `x_0` is set to `-\infty` to avoid nonzero endpoint terms.):: >>> differint(lambda x: exp(pi*x), -1.5, 3) 0.278538406900792 >>> exp(pi*-1.5) * pi**3 0.278538406900792 >>> differint(lambda x: exp(pi*x), 3.5, -3, -inf) 1922.50563031149 >>> exp(pi*3.5) / pi**3 1922.50563031149 However, for noninteger `n`, the differentiation formula for the exponential function must be modified to give the same result as the Riemann-Liouville differintegral:: >>> x = mpf(3.5) >>> c = pi >>> n = 1+2*j >>> differint(lambda x: exp(c*x), x, n) (-123295.005390743 + 140955.117867654j) >>> x**(-n) * exp(c)**x * (x*c)**n * gammainc(-n, 0, x*c) / gamma(-n) (-123295.005390743 + 140955.117867654j) rcsИаЗЗЗfddДИИgбS)NcsИ|ИИ|ГSr.r r>)r┌rrr r┌rz-differint....)r,йrйrrrk┌x0rmrrlrzdifferint..)┌maxr┌ceil┌rer<┌gamma)rrrr ro┌mr2r rnr┌ differint└sDrucs"ИdkrИSЗЗЗЗfddД}|S)a3 Given a function `f`, returns a function `g(x)` that evaluates the nth derivative `f^{(n)}(x)`:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> cos2 = diffun(sin) >>> sin2 = diffun(sin, 4) >>> cos(1.3), cos2(1.3) (0.267498828624587, 0.267498828624587) >>> sin(1.3), sin2(1.3) (0.963558185417193, 0.963558185417193) The function `f` must support arbitrary precision evaluation. See :func:`~mpmath.diff` for additional details and supported keyword options. rcsИjИ|ИfiИдОSr.rBrmйrrr r"r rr2szdiffun..gr )rrr r"r2r rvr┌diffun srwcsNtИj|||fi|дОГ}|аddбr8ЗfddД|DГSЗfddД|DГSdS)aз Produces a degree-`n` Taylor polynomial around the point `x` of the given function `f`. The coefficients are returned as a list. >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> nprint(chop(taylor(sin, 0, 5))) [0.0, 1.0, 0.0, -0.166667, 0.0, 0.00833333] The coefficients are computed using high-order numerical differentiation. The function must be possible to evaluate to arbitrary precision. See :func:`~mpmath.diff` for additional details and supported keyword options. Note that to evaluate the Taylor polynomial as an approximation of `f`, e.g. with :func:`~mpmath.polyval`, the coefficients must be reversed, and the point of the Taylor expansion must be subtracted from the argument: >>> p = taylor(exp, 2.0, 10) >>> polyval(p[::-1], 2.5 - 2.0) 12.1824939606092 >>> exp(2.5) 12.1824939607035 ┌chopTcs$g|]\}}Иа|бИа|бСqSr )rxr7йrr@r r)r rr@rztaylor..csg|]\}}|Иа|бСqSr )r7ryr)r rrBrN)rarOr)rrrr r"rTr r)r┌taylor"srzcCsHt|Г||dkrtdГВ|dkrT|dkr<|jg|jgfS|d|dЕ|jgfS|а|б}t|ГD]8}tt|||dГГD]}|||||||f<qАqf|а||d||dЕб}|а||б}|jgt|Г} dg|d} t|dГD]H}||}tdt||ГdГD]}|| ||||7}Рq|| |<qЎ| | fS)aф Computes a Pade approximation of degree `(L, M)` to a function. Given at least `L+M+1` Taylor coefficients `a` approximating a function `A(x)`, :func:`~mpmath.pade` returns coefficients of polynomials `P, Q` satisfying .. math :: P = \sum_{k=0}^L p_k x^k Q = \sum_{k=0}^M q_k x^k Q_0 = 1 A(x) Q(x) = P(x) + O(x^{L+M+1}) `P(x)/Q(x)` can provide a good approximation to an analytic function beyond the radius of convergence of its Taylor series (example from G.A. Baker 'Essentials of Pade Approximants' Academic Press, Ch.1A):: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> one = mpf(1) >>> def f(x): ... return sqrt((one + 2*x)/(one + x)) ... >>> a = taylor(f, 0, 6) >>> p, q = pade(a, 3, 3) >>> x = 10 >>> polyval(p[::-1], x)/polyval(q[::-1], x) 1.38169105566806 >>> f(x) 1.38169855941551 rz%L+M+1 Coefficients should be providedrN)rHr8ZoneZmatrixrGrKZlu_solver3)rr[┌L┌MrL┌jr@r;r┌qrfrr r r┌padeDs(( r)r)N)rr)r)Z libmp.backendrZcalculusrr_r`┌AttributeError┌itemsrr'r<r5rOrVrWr^rjrurwrzrr r r r┌s8 $I 5! 6H !