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;