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r38/packages/taylor/taylor.tst
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comment Test and demonstration file for the Taylor expansion package, by Rainer M. Schoepf. Works with version 2.2a (01-Apr-2000); %%% showtime; on errcont; % disable interruption on errors comment Simple Taylor expansion; xx := taylor (e**x, x, 0, 4); yy := taylor (e**y, y, 0, 4); comment Basic operations, i.e. addition, subtraction, multiplication, and division are possible: this is not done automatically if the switch TAYLORAUTOCOMBINE is OFF. In this case it is necessary to use taylorcombine; taylorcombine (xx**2); taylorcombine (ws - xx); taylorcombine (xx**3); comment The result is again a Taylor kernel; if taylorseriesp ws then write "OK"; comment It is not possible to combine Taylor kernels that were expanded with respect to different variables; taylorcombine (xx**yy); comment But we can take the exponential or the logarithm of a Taylor kernel; taylorcombine (e**xx); taylorcombine log ws; comment A more complicated example; hugo := taylor(log(1/(1-x)),x,0,5); taylorcombine(exp(hugo/(1+hugo))); comment We may try to expand about another point; taylor (xx, x, 1, 2); comment Arc tangent is one of the functions this package knows of; xxa := taylorcombine atan ws; comment The trigonometric functions; taylor (tan x / x, x, 0, 2); taylorcombine sin ws; taylor (cot x / x, x, 0, 4); comment The poles of these functions are correctly handled; taylor(tan x,x,pi/2,0); taylor(tan x,x,pi/2,3); comment Expansion with respect to more than one kernel is possible; xy := taylor (e**(x+y), x, 0, 2, y, 0, 2); taylorcombine (ws**2); comment We take the inverse and convert back to REDUCE's standard representation; taylorcombine (1/ws); taylortostandard ws; comment Some examples of Taylor kernel divsion; xx1 := taylor (sin (x), x, 0, 4); taylorcombine (xx/xx1); taylorcombine (1/xx1); tt1 := taylor (exp (x), x, 0, 3); tt2 := taylor (sin (x), x, 0, 3); tt3 := taylor (1 + tt2, x, 0, 3); taylorcombine(tt1/tt2); taylorcombine(tt1/tt3); taylorcombine(tt2/tt1); taylorcombine(tt3/tt1); comment Here's what I call homogeneous expansion; xx := taylor (e**(x*y), {x,y}, 0, 2); xx1 := taylor (sin (x+y), {x,y}, 0, 2); xx2 := taylor (cos (x+y), {x,y}, 0, 2); temp := taylorcombine (xx/xx2); taylorcombine (ws*xx2); comment The following shows a principal difficulty: since xx1 is symmetric in x and y but has no constant term it is impossible to compute 1/xx1; taylorcombine (1/xx1); comment Substitution in Taylor expressions is possible; sub (x=z, xy); comment Expression dependency in substitution is detected; sub (x=y, xy); comment It is possible to replace a Taylor variable by a constant; sub (x=4, xy); sub (x=4, xx1); sub (y=0, ws); comment This package has three switches: TAYLORKEEPORIGINAL, TAYLORAUTOEXPAND, and TAYLORAUTOCOMBINE; on taylorkeeporiginal; temp := taylor (e**(x+y), x, 0, 5); taylorcombine (log (temp)); taylororiginal ws; taylorcombine (temp * e**x); on taylorautoexpand; taylorcombine ws; taylororiginal ws; taylorcombine (xx1 / x); on taylorautocombine; xx / xx2; ws * xx2; comment Another example that shows truncation if Taylor kernels of different expansion order are combined; comment First we increase the number of terms to be printed; taylorprintterms := all; p := taylor (x**2 + 2, x, 0, 10); p - x**2; p - taylor (x**2, x, 0, 5); taylor (p - x**2, x, 0, 6); off taylorautocombine; taylorcombine(p-x**2); taylorcombine(p - taylor(x**2,x,0,5)); comment Switch back to finite number of terms; taylorprintterms := 6; comment Some more examples; taylor(1/(1+y^4+x^2*y^2+x^4),{x,y},0,6); taylor ((1 + x)**n, x, 0, 3); taylor (e**(-a*t) * (1 + sin(t)), t, 0, 4); operator f; taylor (1 + f(t), t, 0, 3); taylor(f(sqrt(x^2+y^2)),x,x0,4,y,y0,4); clear f; taylor (sqrt(1 + a*x + sin(x)), x, 0, 3); taylorcombine (ws**2); taylor (sqrt(1 + x), x, 0, 5); taylor ((cos(x) - sec(x))^3, x, 0, 5); taylor ((cos(x) - sec(x))^-3, x, 0, 5); taylor (sqrt(1 - k^2*sin(x)^2), x, 0, 6); taylor (sin(x + y), x, 0, 3, y, 0, 3); taylor (e^x - 1 - x,x,0,6); taylorcombine sqrt ws; taylor(sin(x)/x,x,1,2); taylor((sqrt(4+h)-2)/h,h,0,5); taylor((sqrt(x)-2)/(4-x),x,4,2); taylor((sqrt(y+4)-2)/(-y),y,0,2); taylor(x*tanh(x)/(sqrt(1-x^2)-1),x,0,3); taylor((e^(5*x)-2*x)^(1/x),x,0,2); taylor(sin x/cos x,x,pi/2,3); taylor(log x*sin(x^2)/(x*sinh x),x,0,2); taylor(1/x-1/sin x,x,0,2); taylor(tan x/log cos x,x,pi/2,2); taylor(log(x^2/(x^2-a)),x,0,3); comment Three more complicated examples contributed by Stan Kameny; zz2 := (z*(z-2*pi*i)*(z-pi*i/2)^2)/(sinh z-i); dz2 := df(zz2,z); z0 := pi*i/2; taylor(dz2,z,z0,6); zz3:=(z*(z-2*pi)*(z-pi/2)^2)/(sin z-1); dz3 := df(zz3,z); z1 := pi/2; taylor(dz3,z,z1,6); taylor((sin tan x-tan sin x)/(asin atan x-atan asin x),x,0,6); comment If the expansion point is not constant, it has to be taken care of in differentation, as the following examples show; taylor(sin(x+a),x,a,8); df(ws,a); taylor(cos(x+a),x,a,7); comment A problem are non-analytical terms: rational powers and logarithmic terms can be handled, but other types of essential singularities cannot; taylor(sqrt(x),x,0,2); taylor(asinh(1/x),x,0,5); taylor(e**(1/x),x,0,2); comment Another example for non-integer powers; sub (y = sqrt (x), yy); comment Expansion about infinity is possible in principle...; taylor (e**(1/x), x, infinity, 5); xi := taylor (sin (1/x), x, infinity, 5); y1 := taylor(x/(x-1), x, infinity, 3); z := df(y1, x); comment ...but far from being perfect; taylor (1 / sin (x), x, infinity, 5); clear z; comment You may access the expansion with the PART operator; part(yy,0); part(yy,1); part(yy,4); part(yy,6); comment The template of a Taylor kernel can be extracted; taylortemplate yy; taylortemplate xxa; taylortemplate xi; taylortemplate xy; taylortemplate xx1; comment Here is a slightly less trivial example; exp := (sin (x) * sin (y) / (x * y))**2; taylor (exp, x, 0, 1, y, 0, 1); taylor (exp, x, 0, 2, y, 0, 2); tt := taylor (exp, {x,y}, 0, 2); comment An example that uses factorization; on factor; ff := y**5 - 1; zz := sub (y = taylor(e**x, x, 0, 3), ff); on exp; zz; comment A simple example of Taylor kernel differentiation; hugo := taylor(e^x,x,0,5); df(hugo^2,x); comment The following shows the (limited) capabilities to integrate Taylor kernels. Only simple cases are supported, otherwise a warning is printed and the Taylor kernels are converted to standard representation; zz := taylor (sin x, x, 0, 5); ww := taylor (cos y, y, 0, 5); int (zz, x); int (ww, x); int (zz + ww, x); comment And here we present Taylor series reversion. We start with the example given by Knuth for the algorithm; taylor (t - t**2, t, 0, 5); taylorrevert (ws, t, x); tan!-series := taylor (tan x, x, 0, 5); taylorrevert (tan!-series, x, y); atan!-series:=taylor (atan y, y, 0, 5); tmp := taylor (e**x, x, 0, 5); taylorrevert (tmp, x, y); taylor (log y, y, 1, 5); comment The following example calculates the perturbation expansion of the root x = 20 of the following polynomial in terms of EPS, in ROUNDED mode; poly := for r := 1 : 20 product (x - r); on rounded; tpoly := taylor (poly, x, 20, 4); taylorrevert (tpoly, x, eps); comment Some more examples using rounded mode; taylor(sin x/x,x,0,4); taylor(sin x,x,pi/2,4); taylor(tan x,x,pi/2,4); off rounded; comment An example that involves computing limits of type 0/0 if expansion is done via differentiation; taylor(sqrt((e^x - 1)/x),x,0,15); comment An example that involves intermediate non-analytical terms which cancel entirely; taylor(x^(5/2)/(log(x+1)*tan(x^(3/2))),x,0,5); comment Other examples involving non-analytical terms; taylor(log(e^x-1),x,0,5); taylor(e^(1/x)*(e^x-1),x,0,5); taylor(log(x)*x^10,x,0,5); taylor(log(x)*x^10,x,0,11); taylor(log(x-a)/((a-b)*(a-c)) + log(2(x-b))/((b-c)*(b-a)) + log(x-c)/((c-a)*(c-b)),x,infinity,2); ss := (sqrt(x^(2/5) +1) - x^(1/3)-1)/x^(1/3); taylor(exp ss,x,0,2); taylor(exp sub(x=x^15,ss),x,0,2); taylor(dilog(x),x,0,4); taylor(ei(x),x,0,4); comment In the following we demonstrate the possibiblity to compute the expansion of a function which is given by a simple first order differential equation: the function myexp(x) is exp(-x^2); operator myexp,myerf; let {df(myexp(~x),~x) => -2*x*myexp(x), myexp(0) => 1, df(myerf(~x),~x) => 2/sqrt(pi)*myexp(x), myerf(0) => 0}; taylor(myexp(x),x,0,5); taylor(myerf(x),x,0,5); clear {df(myexp(~x),~x) => -2*x*myexp(x), myexp(0) => 1, df(myerf(~x),~x) => 2/sqrt(pi)*myexp(x), myerf(0) => 0}; clear myexp,erf; %%% showtime; comment There are two special operators, implicit_taylor and inverse_taylor, to compute the Taylor expansion of implicit or inverse functions; implicit_taylor(x^2 + y^2 - 1,x,y,0,1,5); implicit_taylor(x^2 + y^2 - 1,x,y,0,1,20); implicit_taylor(x+y^3-y,x,y,0,0,8); implicit_taylor(x+y^3-y,x,y,0,1,5); implicit_taylor(x+y^3-y,x,y,0,-1,5); implicit_taylor(y*e^y-x,x,y,0,0,5); comment This is the function exp(-1/x^2), which has an essential singularity at the point 0; implicit_taylor(x^2*log y+1,x,y,0,0,3); inverse_taylor(exp(x)-1,x,y,0,8); inverse_taylor(exp(x),x,y,0,5); inverse_taylor(sqrt(x),x,y,0,5); inverse_taylor(log(1+x),x,y,0,5); inverse_taylor((e^x-e^(-x))/2,x,y,0,5); comment In the next two cases the inverse functions have a branch point, therefore the computation fails; inverse_taylor((e^x+e^(-x))/2,x,y,0,5); inverse_taylor(exp(x^2-1),x,y,0,5); inverse_taylor(exp(sqrt(x))-1,x,y,0,5); inverse_taylor(x*exp(x),x,y,0,5); %%% showtime; comment An application is the problem posed by Prof. Stanley: we prove that the finite difference expression below corresponds to the given derivative expression; operator diff,a,f,gg; % We use gg to avoid conflict with high energy % physics operator. let diff(~f,~arg) => df(f,arg); derivative_expression := diff(a(x,y)*diff(gg(x,y),x)*diff(gg(x,y),y)*diff(f(x,y),y),x) + diff(a(x,y)*diff(gg(x,y),x)*diff(gg(x,y),y)*diff(f(x,y),x),y) ; finite_difference_expression := +a(x+dx,y+dy)*f(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2) +a(x+dx,y)*f(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2) +a(x,y+dy)*f(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2) +a(x,y)*f(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x+dx,y)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2) -a(x,y)*f(x,y)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2) -gg(x,y)*a(x+dx,y+dy)*f(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2) -gg(x,y)*a(x+dx,y)*f(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2) -gg(x,y)*a(x,y+dy)*f(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2) -a(x,y)*gg(x,y)*f(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x+dx,y)*gg(x+dx,y+dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2) +a(x,y)*f(x,y)*gg(x,y)*gg(x+dx,y+dy)/(16*dx^2*dy^2) -gg(x+dx,y)^2*a(x+dx,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2) +gg(x,y+dy)*gg(x+dx,y)*a(x+dx,y+dy)*f(x+dx,y+dy)/(16*dx^2*dy^2) -gg(x,y+dy)^2*a(x+dx,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x+dx,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2) -a(x+dx,y)*gg(x+dx,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2) -a(x,y+dy)*gg(x+dx,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2) -a(x,y)*gg(x+dx,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2) +gg(x,y+dy)*a(x+dx,y)*gg(x+dx,y)*f(x+dx,y+dy)/(16*dx^2*dy^2) +a(x,y+dy)*gg(x,y+dy)*gg(x+dx,y)*f(x+dx,y+dy)/(16*dx^2*dy^2) +a(x,y)*gg(x,y+dy)*gg(x+dx,y)*f(x+dx,y+dy)/(16*dx^2*dy^2) -gg(x,y+dy)^2*a(x+dx,y)*f(x+dx,y+dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x+dx,y)*f(x+dx,y+dy)/(32*dx^2*dy^2) -a(x,y+dy)*gg(x,y+dy)^2*f(x+dx,y+dy)/(32*dx^2*dy^2) -a(x,y)*gg(x,y+dy)^2*f(x+dx,y+dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2) +a(x,y)*gg(x,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2) +f(x,y)*gg(x+dx,y)^2*a(x+dx,y+dy)/(32*dx^2*dy^2) -f(x,y)*gg(x,y+dy)*gg(x+dx,y)*a(x+dx,y+dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y+dy)^2*a(x+dx,y+dy)/(32*dx^2*dy^2) -f(x,y)*gg(x,y)^2*a(x+dx,y+dy)/(32*dx^2*dy^2) +a(x+dx,y-dy)*f(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2) +a(x+dx,y)*f(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2) +a(x,y-dy)*f(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2) +a(x,y)*f(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x+dx,y)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2) -a(x,y)*f(x,y)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2) -gg(x,y)*a(x+dx,y-dy)*f(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2) -gg(x,y)*a(x+dx,y)*f(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2) -gg(x,y)*a(x,y-dy)*f(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2) -a(x,y)*gg(x,y)*f(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x+dx,y)*gg(x+dx,y-dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2) +a(x,y)*f(x,y)*gg(x,y)*gg(x+dx,y-dy)/(16*dx^2*dy^2) -gg(x+dx,y)^2*a(x+dx,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2) +gg(x,y-dy)*gg(x+dx,y)*a(x+dx,y-dy)*f(x+dx,y-dy)/(16*dx^2*dy^2) -gg(x,y-dy)^2*a(x+dx,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x+dx,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2) -a(x+dx,y)*gg(x+dx,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2) -a(x,y-dy)*gg(x+dx,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2) -a(x,y)*gg(x+dx,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2) +gg(x,y-dy)*a(x+dx,y)*gg(x+dx,y)*f(x+dx,y-dy)/(16*dx^2*dy^2) +a(x,y-dy)*gg(x,y-dy)*gg(x+dx,y)*f(x+dx,y-dy)/(16*dx^2*dy^2) +a(x,y)*gg(x,y-dy)*gg(x+dx,y)*f(x+dx,y-dy)/(16*dx^2*dy^2) -gg(x,y-dy)^2*a(x+dx,y)*f(x+dx,y-dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x+dx,y)*f(x+dx,y-dy)/(32*dx^2*dy^2) -a(x,y-dy)*gg(x,y-dy)^2*f(x+dx,y-dy)/(32*dx^2*dy^2) -a(x,y)*gg(x,y-dy)^2*f(x+dx,y-dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2) +a(x,y)*gg(x,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2) +f(x,y)*gg(x+dx,y)^2*a(x+dx,y-dy)/(32*dx^2*dy^2) -f(x,y)*gg(x,y-dy)*gg(x+dx,y)*a(x+dx,y-dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y-dy)^2*a(x+dx,y-dy)/(32*dx^2*dy^2) -f(x,y)*gg(x,y)^2*a(x+dx,y-dy)/(32*dx^2*dy^2) +f(x,y)*a(x+dx,y)*gg(x+dx,y)^2/(16*dx^2*dy^2) +f(x,y)*a(x,y+dy)*gg(x+dx,y)^2/(32*dx^2*dy^2) +f(x,y)*a(x,y-dy)*gg(x+dx,y)^2/(32*dx^2*dy^2) +a(x,y)*f(x,y)*gg(x+dx,y)^2/(16*dx^2*dy^2) -f(x,y)*gg(x,y+dy)*a(x+dx,y)*gg(x+dx,y)/(16*dx^2*dy^2) -f(x,y)*gg(x,y-dy)*a(x+dx,y)*gg(x+dx,y)/(16*dx^2*dy^2) -f(x,y)*a(x,y+dy)*gg(x,y+dy)*gg(x+dx,y)/(16*dx^2*dy^2) -a(x,y)*f(x,y)*gg(x,y+dy)*gg(x+dx,y)/(16*dx^2*dy^2) -f(x,y)*a(x,y-dy)*gg(x,y-dy)*gg(x+dx,y)/(16*dx^2*dy^2) -a(x,y)*f(x,y)*gg(x,y-dy)*gg(x+dx,y)/(16*dx^2*dy^2) +f(x,y)*gg(x,y+dy)^2*a(x+dx,y)/(32*dx^2*dy^2) +f(x,y)*gg(x,y-dy)^2*a(x+dx,y)/(32*dx^2*dy^2) -f(x,y)*gg(x,y)^2*a(x+dx,y)/(16*dx^2*dy^2) +a(x-dx,y+dy)*f(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2) +a(x-dx,y)*f(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2) +a(x,y+dy)*f(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2) +a(x,y)*f(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x-dx,y)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2) -a(x,y)*f(x,y)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2) -gg(x,y)*a(x-dx,y+dy)*f(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2) -gg(x,y)*a(x-dx,y)*f(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2) -gg(x,y)*a(x,y+dy)*f(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2) -a(x,y)*gg(x,y)*f(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x-dx,y)*gg(x-dx,y+dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2) +a(x,y)*f(x,y)*gg(x,y)*gg(x-dx,y+dy)/(16*dx^2*dy^2) -gg(x-dx,y)^2*a(x-dx,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2) +gg(x,y+dy)*gg(x-dx,y)*a(x-dx,y+dy)*f(x-dx,y+dy)/(16*dx^2*dy^2) -gg(x,y+dy)^2*a(x-dx,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x-dx,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2) -a(x-dx,y)*gg(x-dx,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2) -a(x,y+dy)*gg(x-dx,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2) -a(x,y)*gg(x-dx,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2) +gg(x,y+dy)*a(x-dx,y)*gg(x-dx,y)*f(x-dx,y+dy)/(16*dx^2*dy^2) +a(x,y+dy)*gg(x,y+dy)*gg(x-dx,y)*f(x-dx,y+dy)/(16*dx^2*dy^2) +a(x,y)*gg(x,y+dy)*gg(x-dx,y)*f(x-dx,y+dy)/(16*dx^2*dy^2) -gg(x,y+dy)^2*a(x-dx,y)*f(x-dx,y+dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x-dx,y)*f(x-dx,y+dy)/(32*dx^2*dy^2) -a(x,y+dy)*gg(x,y+dy)^2*f(x-dx,y+dy)/(32*dx^2*dy^2) -a(x,y)*gg(x,y+dy)^2*f(x-dx,y+dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2) +a(x,y)*gg(x,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2) +f(x,y)*gg(x-dx,y)^2*a(x-dx,y+dy)/(32*dx^2*dy^2) -f(x,y)*gg(x,y+dy)*gg(x-dx,y)*a(x-dx,y+dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y+dy)^2*a(x-dx,y+dy)/(32*dx^2*dy^2) -f(x,y)*gg(x,y)^2*a(x-dx,y+dy)/(32*dx^2*dy^2) +a(x-dx,y-dy)*f(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2) +a(x-dx,y)*f(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2) +a(x,y-dy)*f(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2) +a(x,y)*f(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x-dx,y)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2) -f(x,y)*a(x,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2) -a(x,y)*f(x,y)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2) -gg(x,y)*a(x-dx,y-dy)*f(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2) -gg(x,y)*a(x-dx,y)*f(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2) -gg(x,y)*a(x,y-dy)*f(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2) -a(x,y)*gg(x,y)*f(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x-dx,y)*gg(x-dx,y-dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y)*a(x,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2) +a(x,y)*f(x,y)*gg(x,y)*gg(x-dx,y-dy)/(16*dx^2*dy^2) -gg(x-dx,y)^2*a(x-dx,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2) +gg(x,y-dy)*gg(x-dx,y)*a(x-dx,y-dy)*f(x-dx,y-dy)/(16*dx^2*dy^2) -gg(x,y-dy)^2*a(x-dx,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x-dx,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2) -a(x-dx,y)*gg(x-dx,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2) -a(x,y-dy)*gg(x-dx,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2) -a(x,y)*gg(x-dx,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2) +gg(x,y-dy)*a(x-dx,y)*gg(x-dx,y)*f(x-dx,y-dy)/(16*dx^2*dy^2) +a(x,y-dy)*gg(x,y-dy)*gg(x-dx,y)*f(x-dx,y-dy)/(16*dx^2*dy^2) +a(x,y)*gg(x,y-dy)*gg(x-dx,y)*f(x-dx,y-dy)/(16*dx^2*dy^2) -gg(x,y-dy)^2*a(x-dx,y)*f(x-dx,y-dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x-dx,y)*f(x-dx,y-dy)/(32*dx^2*dy^2) -a(x,y-dy)*gg(x,y-dy)^2*f(x-dx,y-dy)/(32*dx^2*dy^2) -a(x,y)*gg(x,y-dy)^2*f(x-dx,y-dy)/(32*dx^2*dy^2) +gg(x,y)^2*a(x,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2) +a(x,y)*gg(x,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2) +f(x,y)*gg(x-dx,y)^2*a(x-dx,y-dy)/(32*dx^2*dy^2) -f(x,y)*gg(x,y-dy)*gg(x-dx,y)*a(x-dx,y-dy)/(16*dx^2*dy^2) +f(x,y)*gg(x,y-dy)^2*a(x-dx,y-dy)/(32*dx^2*dy^2) -f(x,y)*gg(x,y)^2*a(x-dx,y-dy)/(32*dx^2*dy^2) +f(x,y)*a(x-dx,y)*gg(x-dx,y)^2/(16*dx^2*dy^2) +f(x,y)*a(x,y+dy)*gg(x-dx,y)^2/(32*dx^2*dy^2) +f(x,y)*a(x,y-dy)*gg(x-dx,y)^2/(32*dx^2*dy^2) +a(x,y)*f(x,y)*gg(x-dx,y)^2/(16*dx^2*dy^2) -f(x,y)*gg(x,y+dy)*a(x-dx,y)*gg(x-dx,y)/(16*dx^2*dy^2) -f(x,y)*gg(x,y-dy)*a(x-dx,y)*gg(x-dx,y)/(16*dx^2*dy^2) -f(x,y)*a(x,y+dy)*gg(x,y+dy)*gg(x-dx,y)/(16*dx^2*dy^2) -a(x,y)*f(x,y)*gg(x,y+dy)*gg(x-dx,y)/(16*dx^2*dy^2) -f(x,y)*a(x,y-dy)*gg(x,y-dy)*gg(x-dx,y)/(16*dx^2*dy^2) -a(x,y)*f(x,y)*gg(x,y-dy)*gg(x-dx,y)/(16*dx^2*dy^2) +f(x,y)*gg(x,y+dy)^2*a(x-dx,y)/(32*dx^2*dy^2) +f(x,y)*gg(x,y-dy)^2*a(x-dx,y)/(32*dx^2*dy^2) -f(x,y)*gg(x,y)^2*a(x-dx,y)/(16*dx^2*dy^2) +f(x,y)*a(x,y+dy)*gg(x,y+dy)^2/(16*dx^2*dy^2) +a(x,y)*f(x,y)*gg(x,y+dy)^2/(16*dx^2*dy^2) -f(x,y)*gg(x,y)^2*a(x,y+dy)/(16*dx^2*dy^2) +f(x,y)*a(x,y-dy)*gg(x,y-dy)^2/(16*dx^2*dy^2) +a(x,y)*f(x,y)*gg(x,y-dy)^2/(16*dx^2*dy^2) -f(x,y)*gg(x,y)^2*a(x,y-dy)/(16*dx^2*dy^2) -a(x,y)*f(x,y)*gg(x,y)^2/(8*dx^2*dy^2)$ comment We define abbreviations for the partial derivatives; operator ax,ay,fx,fy,gx,gy; operator axx,axy,ayy,fxx,fxy,fyy,gxx,gxy,gyy; operator axxx,axxy,axyy,ayyy,fxxx,fxxy,fxyy,fyyy,gxxx,gxxy,gxyy,gyyy; operator axxxy,axxyy,axyyy,fxxxy,fxxyy,fxyyy, gxxxx,gxxxy,gxxyy,gxyyy,gyyyy; operator axxxyy,axxyyy,fxxyyy,fxxxyy,gxxxxy,gxxxyy,gxxyyy,gxyyyy; operator gxxxxyy,gxxxyyy,gxxyyyy; operator_diff_rules := { df(a(~x,~y),~x) => ax(x,y), df(a(~x,~y),~y) => ay(x,y), df(f(~x,~y),~x) => fx(x,y), df(f(~x,~y),~y) => fy(x,y), df(gg(~x,~y),~x) => gx(x,y), df(gg(~x,~y),~y) => gy(x,y), df(ax(~x,~y),~x) => axx(x,y), df(ax(~x,~y),~y) => axy(x,y), df(ay(~x,~y),~x) => axy(x,y), df(ay(~x,~y),~y) => ayy(x,y), df(fx(~x,~y),~x) => fxx(x,y), df(fx(~x,~y),~y) => fxy(x,y), df(fy(~x,~y),~x) => fxy(x,y), df(fy(~x,~y),~y) => fyy(x,y), df(gx(~x,~y),~x) => gxx(x,y), df(gx(~x,~y),~y) => gxy(x,y), df(gy(~x,~y),~x) => gxy(x,y), df(gy(~x,~y),~y) => gyy(x,y), df(axx(~x,~y),~x) => axxx(x,y), df(axy(~x,~y),~x) => axxy(x,y), df(ayy(~x,~y),~x) => axyy(x,y), df(ayy(~x,~y),~y) => ayyy(x,y), df(fxx(~x,~y),~x) => fxxx(x,y), df(fxy(~x,~y),~x) => fxxy(x,y), df(fxy(~x,~y),~y) => fxyy(x,y), df(fyy(~x,~y),~x) => fxyy(x,y), df(fyy(~x,~y),~y) => fyyy(x,y), df(gxx(~x,~y),~x) => gxxx(x,y), df(gxx(~x,~y),~y) => gxxy(x,y), df(gxy(~x,~y),~x) => gxxy(x,y), df(gxy(~x,~y),~y) => gxyy(x,y), df(gyy(~x,~y),~x) => gxyy(x,y), df(gyy(~x,~y),~y) => gyyy(x,y), df(axyy(~x,~y),~x) => axxyy(x,y), df(axxy(~x,~y),~x) => axxxy(x,y), df(ayyy(~x,~y),~x) => axyyy(x,y), df(fxxy(~x,~y),~x) => fxxxy(x,y), df(fxyy(~x,~y),~x) => fxxyy(x,y), df(fyyy(~x,~y),~x) => fxyyy(x,y), df(gxxx(~x,~y),~x) => gxxxx(x,y), df(gxxy(~x,~y),~x) => gxxxy(x,y), df(gxyy(~x,~y),~x) => gxxyy(x,y), df(gyyy(~x,~y),~x) => gxyyy(x,y), df(gyyy(~x,~y),~y) => gyyyy(x,y), df(axxyy(~x,~y),~x) => axxxyy(x,y), df(axyyy(~x,~y),~x) => axxyyy(x,y), df(fxxyy(~x,~y),~x) => fxxxyy(x,y), df(fxyyy(~x,~y),~x) => fxxyyy(x,y), df(gxxxy(~x,~y),~x) => gxxxxy(x,y), df(gxxyy(~x,~y),~x) => gxxxyy(x,y), df(gxyyy(~x,~y),~x) => gxxyyy(x,y), df(gyyyy(~x,~y),~x) => gxyyyy(x,y), df(gxxxyy(~x,~y),~x) => gxxxxyy(x,y), df(gxxyyy(~x,~y),~x) => gxxxyyy(x,y), df(gxyyyy(~x,~y),~x) => gxxyyyy(x,y) }; let operator_diff_rules; texp := taylor (finite_difference_expression, dx, 0, 1, dy, 0, 1); comment You may also try to expand further but this needs a lot of CPU time. Therefore the following line is commented out; %texp := taylor (finite_difference_expression, dx, 0, 2, dy, 0, 2); factor dx,dy; result := taylortostandard texp; derivative_expression - result; clear diff(~f,~arg); clearrules operator_diff_rules; clear diff,a,f,gg; clear ax,ay,fx,fy,gx,gy; clear axx,axy,ayy,fxx,fxy,fyy,gxx,gxy,gyy; clear axxx,axxy,axyy,ayyy,fxxx,fxxy,fxyy,fyyy,gxxx,gxxy,gxyy,gyyy; clear axxxy,axxyy,axyyy,fxxxy,fxxyy,fxyyy,gxxxx,gxxxy,gxxyy,gxyyy,gyyyy; clear axxxyy,axxyyy,fxxyyy,fxxxyy,gxxxxy,gxxxyy,gxxyyy,gxyyyy; clear gxxxxyy,gxxxyyy,gxxyyyy; taylorprintterms := 5; off taylorautoexpand,taylorkeeporiginal; %%% showtime; comment An example provided by Alan Barnes: elliptic functions; % Jacobi's elliptic functions % sn(x,k), cn(x,k), dn(x,k). % The modulus and complementary modulus are denoted by K and K!' % usually written mathematically as k and k' respectively % % epsilon(x,k) is the incomplete elliptic integral of the second kind % usually written mathematically as E(x,k) % % KK(k) is the complete elliptic integral of the first kind % usually written mathematically as K(k) % K(k) = arcsn(1,k) % KK!'(k) is the complementary complete integral % usually written mathematically as K'(k) % NB. K'(k) = K(k') % EE(k) is the complete elliptic integral of the second kind % usually written mathematically as E(k) % EE!'(k) is the complementary complete integral % usually written mathematically as E'(k) % NB. E'(k) = E(k') operator sn, cn, dn, epsilon; elliptic_rules := { % Differentiation rules for basic functions df(sn(~x,~k),~x) => cn(x,k)*dn(x,k), df(cn(~x,~k),~x) => -sn(x,k)*dn(x,k), df(dn(~x,~k),~x) => -k^2*sn(x,k)*cn(x,k), df(epsilon(~x,~k),~x)=> dn(x,k)^2, % k-derivatives % DF Lawden Elliptic Functions & Applications Springer (1989) df(sn(~x,~k),~k) => (k*sn(x,k)*cn(x,k)^2 -epsilon(x,k)*cn(x,k)*dn(x,k)/k)/(1-k^2) + x*cn(x,k)*dn(x,k)/k, df(cn(~x,~k),~k) => (-k*sn(x,k)^2*cn(x,k) +epsilon(x,k)*sn(x,k)*dn(x,k)/k)/(1-k^2) - x*sn(x,k)*dn(x,k)/k, df(dn(~x,~k),~k) => k*(-sn(x,k)^2*dn(x,k) +epsilon(x,k)*sn(x,k)*cn(x,k))/(1-k^2) - k*x*sn(x,k)*cn(x,k), df(epsilon(~x,~k),~k) => k*(sn(x,k)*cn(x,k)*dn(x,k) -epsilon(x,k)*cn(x,k)^2)/(1-k^2) -k*x*sn(x,k)^2, % parity properties sn(-~x,~k) => -sn(x,k), cn(-~x,~k) => cn(x,k), dn(-~x,~k) => dn(x,k), epsilon(-~x,~k) => -epsilon(x,k), sn(~x,-~k) => sn(x,k), cn(~x,-~k) => cn(x,k), dn(~x,-~k) => dn(x,k), epsilon(~x,-~k) => epsilon(x,k), % behaviour at zero sn(0,~k) => 0, cn(0,~k) => 1, dn(0,~k) => 1, epsilon(0,~k) => 0, % degenerate cases of modulus sn(~x,0) => sin(x), cn(~x,0) => cos(x), dn(~x,0) => 1, epsilon(~x,0) => x, sn(~x,1) => tanh(x), cn(~x,1) => 1/cosh(x), dn(~x,1) => 1/cosh(x), epsilon(~x,1) => tanh(x) }; let elliptic_rules; hugo := taylor(sn(x,k),k,0,6); otto := taylor(cn(x,k),k,0,6); taylorcombine(hugo^2 + otto^2); clearrules elliptic_rules; clear sn, cn, dn, epsilon; %%% showtime; comment That's all, folks; end;