Codemist Standard Lisp 3.54 for DEC Alpha: May 23 1994
Dump file created: Mon May 23 10:39:11 1994
REDUCE 3.5, 15-Oct-93 ...
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comment
Test and demonstration file for the Taylor expansion package,
by Rainer M. Schoepf. Works with version 1.8b (03-Sep-93);
showtime;
Time: 0 ms
on errcont;
% disable interruption on errors
comment Simple Taylor expansion;
xx := taylor (e**x, x, 0, 4);
1 2 1 3 1 4 5
xx := 1 + x + ---*x + ---*x + ----*x + O(x )
2 6 24
yy := taylor (e**y, y, 0, 4);
1 2 1 3 1 4 5
yy := 1 + y + ---*y + ---*y + ----*y + O(y )
2 6 24
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);
2 4 3 2 4 5
1 + 2*x + 2*x + ---*x + ---*x + O(x )
3 3
taylorcombine (ws - xx);
3 2 7 3 5 4 5
x + ---*x + ---*x + ---*x + O(x )
2 6 8
comment The result is again a Taylor kernel;
if taylorseriesp ws then write "OK";
OK
comment It is not possible to combine Taylor kernels that were
expanded with respect to different variables;
taylorcombine (xx**yy);
1 2 1 3 1 4 5
(1 + x + ---*x + ---*x + ----*x + O(x ))
2 6 24
1 2 1 3 1 4 5
**(1 + y + ---*y + ---*y + ----*y + O(y ))
2 6 24
comment But we can take the exponential or the logarithm
of a Taylor kernel;
taylorcombine (e**xx);
2 5*e 3 5*e 4 5
e + e*x + e*x + -----*x + -----*x + O(x )
6 8
taylorcombine log ws;
1 2 1 3 1 4 5
1 + x + ---*x + ---*x + ----*x + O(x )
2 6 24
comment A more complicated example;
hugo := taylor(log(1/(1-x)),x,0,5);
1 2 1 3 1 4 1 5 6
hugo := x + ---*x + ---*x + ---*x + ---*x + O(x )
2 3 4 5
taylorcombine(exp(hugo/(1+hugo)));
1 4 6
1 + x + ----*x + O(x )
12
comment We may try to expand about another point;
taylor (xx, x, 1, 2);
65 8 5 2 3
---- + ---*(x - 1) + ---*(x - 1) + O((x - 1) )
24 3 4
comment Arc tangent is one of the functions this package knows of;
xxa := taylorcombine atan ws;
xxa :=
65 1536 2933040 2 3
atan(----) + ------*(x - 1) - ----------*(x - 1) + O((x - 1) )
24 4801 23049601
comment The trigonometric functions;
taylor (tan x / x, x, 0, 2);
1 2 3
1 + ---*x + O(x )
3
taylorcombine sin ws;
cos(1) 2 3
sin(1) + --------*x + O(x )
3
taylor (cot x / x, x, 0, 4);
-2 1 1 2 2 4 5
x - --- - ----*x - -----*x + O(x )
3 45 945
comment Expansion with respect to more than one kernel is possible;
xy := taylor (e**(x+y), x, 0, 2, y, 0, 2);
1 2 3 3
xy := 1 + y + ---*y + x + y*x + (4 terms) + O(x ,y )
2
taylorcombine (ws**2);
2 3 3
1 + 2*y + 2*y + 2*x + 4*y*x + (4 terms) + O(x ,y )
comment We take the inverse and convert back to REDUCE's standard
representation;
taylorcombine (1/ws);
2 3 3
1 - 2*x + 2*x - 2*y + 4*y*x + (4 terms) + O(x ,y )
taylortostandard ws;
2 2 2 2 2 2
4*x *y - 4*x *y + 2*x - 4*x*y + 4*x*y - 2*x + 2*y - 2*y + 1
comment Some examples of Taylor kernel divsion;
xx1 := taylor (sin (x), x, 0, 4);
1 3 5
xx1 := x - ---*x + O(x )
6
taylorcombine (xx/xx1);
-1 2 1 2 3
x + 1 + ---*x + ---*x + O(x )
3 3
taylorcombine (1/xx1);
-1 1 3
x + ---*x + O(x )
6
tt1 := taylor (exp (x), x, 0, 3);
1 2 1 3 4
tt1 := 1 + x + ---*x + ---*x + O(x )
2 6
tt2 := taylor (sin (x), x, 0, 3);
1 3 4
tt2 := x - ---*x + O(x )
6
tt3 := taylor (1 + tt2, x, 0, 3);
1 3 4
tt3 := 1 + x - ---*x + O(x )
6
taylorcombine(tt1/tt2);
-1 2 2
x + 1 + ---*x + O(x )
3
taylorcombine(tt1/tt3);
1 2 1 3 4
1 + ---*x - ---*x + O(x )
2 6
taylorcombine(tt2/tt1);
2 1 3 4
x - x + ---*x + O(x )
3
taylorcombine(tt3/tt1);
1 2 1 3 4
1 - ---*x + ---*x + O(x )
2 6
comment Here's what I call homogeneous expansion;
xx := taylor (e**(x*y), {x,y}, 0, 2);
3
xx := 1 + y*x + O({x,y} )
xx1 := taylor (sin (x+y), {x,y}, 0, 2);
3
xx1 := y + x + O({x,y} )
xx2 := taylor (cos (x+y), {x,y}, 0, 2);
1 2 1 2 3
xx2 := 1 - ---*y - y*x - ---*x + O({x,y} )
2 2
temp := taylorcombine (xx/xx2);
1 2 1 2 3
temp := 1 + ---*y + 2*y*x + ---*x + O({x,y} )
2 2
taylorcombine (ws*xx2);
3
1 + y*x + O({x,y} )
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);
***** Not a unit in argument to invtaylor
comment Substitution in Taylor expressions is possible;
sub (x=z, xy);
1 2 3 3
1 + y + ---*y + z + y*z + (4 terms) + O(z ,y )
2
comment Expression dependency in substitution is detected;
sub (x=y, xy);
***** Substitution of dependent variables y y
comment It is possible to replace a Taylor variable by a constant;
sub (x=4, xy);
13 2 3
13 + 13*y + ----*y + O(y )
2
sub (x=4, xx1);
3
4 + y + O(y )
comment This package has three switches:
TAYLORKEEPORIGINAL, TAYLORAUTOEXPAND, and TAYLORAUTOCOMBINE;
on taylorkeeporiginal;
temp := taylor (e**(x+y), x, 0, 5);
y y y
y y e 2 e 3 e 4 6
temp := e + e *x + ----*x + ----*x + ----*x + (1 term) + O(x )
2 6 24
taylorcombine (log (temp));
6
y + x + O(x )
taylororiginal ws;
x + y
taylorcombine (temp * e**x);
y y y
x y y e 2 e 3 e 4 6
e *(e + e *x + ----*x + ----*x + ----*x + (1 term) + O(x ))
2 6 24
on taylorautoexpand;
taylorcombine ws;
y y
y y y 2 4*e 3 2*e 4 6
e + 2*e *x + 2*e *x + ------*x + ------*x + (1 term) + O(x )
3 3
taylororiginal ws;
2*x + y
e
taylorcombine (xx1 / x);
-1 2
y*x + 1 + O({x,y} )
on taylorautocombine;
xx / xx2;
1 2 1 2 3
1 + ---*y + 2*y*x + ---*x + O({x,y} )
2 2
ws * xx2;
3
1 + y*x + O({x,y} )
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;
taylorprintterms := all
p := taylor (x**2 + 2, x, 0, 10);
2 11
p := 2 + x + O(x )
p - x**2;
2 11 2
(2 + x + O(x )) - x
p - taylor (x**2, x, 0, 5);
6
2 + O(x )
taylor (p - x**2, x, 0, 6);
7
2 + O(x )
off taylorautocombine;
taylorcombine(p-x**2);
11
2 + O(x )
taylorcombine(p - taylor(x**2,x,0,5));
6
2 + O(x )
comment Switch back to finite number of terms;
taylorprintterms := 6;
taylorprintterms := 6
comment Some more examples;
taylor ((1 + x)**n, x, 0, 3);
2
n*(n - 1) 2 n*(n - 3*n + 2) 3 4
1 + n*x + -----------*x + ------------------*x + O(x )
2 6
taylor (e**(-a*t) * (1 + sin(t)), t, 0, 4);
3 2
a*(a - 2) 2 - a + 3*a - 1 3
1 + ( - a + 1)*t + -----------*t + ------------------*t
2 6
3 2
a*(a - 4*a + 4) 4 5
+ -------------------*t + O(t )
24
operator f;
taylor (1 + f(t), t, 0, 3);
sub(t=0,df(f(t),t,2)) 2
(f(0) + 1) + sub(t=0,df(f(t),t))*t + -----------------------*t
2
sub(t=0,df(f(t),t,3)) 3 4
+ -----------------------*t + O(t )
6
clear f;
taylor (sqrt(1 + a*x + sin(x)), x, 0, 3);
2 3 2
a + 1 - a - 2*a - 1 2 3*a + 9*a + 9*a - 1 3
1 + -------*x + -----------------*x + -----------------------*x
2 8 48
4
+ O(x )
taylorcombine (ws**2);
1 3 4
1 + (a + 1)*x - ---*x + O(x )
6
taylor (sqrt(1 + x), x, 0, 5);
1 1 2 1 3 5 4 7 5 6
1 + ---*x - ---*x + ----*x - -----*x + -----*x + O(x )
2 8 16 128 256
taylor ((cos(x) - sec(x))^3, x, 0, 5);
6
0 + O(x )
taylor ((cos(x) - sec(x))^-3, x, 0, 5);
-6 1 -4 11 -2 347 6767 2 15377 4
- x + ---*x + -----*x - ------- - --------*x - ---------*x
2 120 15120 604800 7983360
6
+ O(x )
taylor (sqrt(1 - k^2*sin(x)^2), x, 0, 6);
2 2 2 2 4 2
k 2 k *( - 3*k + 4) 4 k *( - 45*k + 60*k - 16) 6
1 - ----*x + ------------------*x + ----------------------------*x
2 24 720
7
+ O(x )
taylor (sin(x + y), x, 0, 3, y, 0, 3);
1 3 1 2 1 2 1 2 3
x - ---*x + y - ---*y*x - ---*y *x + ----*y *x + (2 terms)
6 2 2 12
4 4
+ O(x ,y )
taylor (e^x - 1 - x,x,0,6);
1 2 1 3 1 4 1 5 1 6 7
---*x + ---*x + ----*x + -----*x + -----*x + O(x )
2 6 24 120 720
taylorcombine sqrt ws;
1 1 2 1 3 1 4
---------*x + -----------*x + ------------*x + -------------*x
sqrt(2) 6*sqrt(2) 36*sqrt(2) 270*sqrt(2)
1 5 6
+ --------------*x + O(x )
2592*sqrt(2)
comment A more complicated example contributed by Stan Kameny;
zz2 := (z*(z-2*pi*i)*(z-pi*i/2)^2)/(sinh z-i);
3 2 2 3
z*(2*i*pi - 12*i*pi*z - 9*pi *z + 4*z )
zz2 := -------------------------------------------
4*(sinh(z) - i)
dz2 := df(zz2,z);
3 3 2 2
dz2 := ( - 2*cosh(z)*i*pi *z + 12*cosh(z)*i*pi*z + 9*cosh(z)*pi *z
4 3 2
- 4*cosh(z)*z + 2*sinh(z)*i*pi - 36*sinh(z)*i*pi*z
2 3 2 3
- 18*sinh(z)*pi *z + 16*sinh(z)*z + 18*i*pi *z - 16*i*z
3 2 2
+ 2*pi - 36*pi*z )/(4*(sinh(z) - 2*sinh(z)*i - 1))
z0 := pi*i/2;
i*pi
z0 := ------
2
taylor(dz2,z,z0,6);
2
- pi + 16 i*pi pi i*pi 2
- 2*pi + -------------*(z - ------) + ----*(z - ------)
4*i 2 2 2
2
3*pi - 80 i*pi 3 pi i*pi 4
+ ------------*(z - ------) - ----*(z - ------)
120*i 2 24 2
2
- 5*pi + 168 i*pi 5 i*pi 7
+ ----------------*(z - ------) + (1 term) + O((z - ------) )
3360*i 2 2
comment A problem are non-analytic terms: there are no precautions
taken to detect or handle them;
taylor (sqrt (x), x, 0, 2);
taylor(sqrt(x),x,0,2)
taylor (e**(1/x), x, 0, 2);
1/x
taylor(e ,x,0,2)
comment Even worse: you can substitute a non analytical kernel;
sub (y = sqrt (x), yy);
1 2 1 3 1 4
1 + sqrt(x) + ---*sqrt(x) + ---*sqrt(x) + ----*sqrt(x)
2 6 24
5
+ O(sqrt(x) )
comment Expansion about infinity is possible in principle...;
taylor (e**(1/x), x, infinity, 5);
1 1 1 1 1 1 1 1 1 1
1 + --- + ---*---- + ---*---- + ----*---- + -----*---- + O(----)
x 2 2 6 3 24 4 120 5 6
x x x x x
xi := taylor (sin (1/x), x, infinity, 5);
1 1 1 1 1 1
xi := --- - ---*---- + -----*---- + O(----)
x 6 3 120 5 6
x x x
y1 := taylor(x/(x-1), x, infinity, 3);
1 1 1 1
y1 := 1 + --- + ---- + ---- + O(----)
x 2 3 4
x x x
z := df(y1, x);
1 1 1 1
z := - ---- - 2*---- - 3*---- + O(----)
2 3 4 5
x x x x
comment ...but far from being perfect;
taylor (1 / sin (x), x, infinity, 5);
1
taylor(--------,x,infinity,5)
sin(x)
clear z;
comment The template of a Taylor kernel can be extracted;
taylortemplate yy;
{{y,0,4}}
taylortemplate xxa;
{{x,1,2}}
taylortemplate xi;
{{x,infinity,5}}
taylortemplate xy;
{{x,0,2},{y,0,2}}
taylortemplate xx1;
{{{x,y},0,2}}
comment Here is a slightly less trivial example;
exp := (sin (x) * sin (y) / (x * y))**2;
2 2
sin(x) *sin(y)
exp := -----------------
2 2
x *y
taylor (exp, x, 0, 1, y, 0, 1);
2 2
1 + O(x ,y )
taylor (exp, x, 0, 2, y, 0, 2);
1 2 1 2 1 2 2 3 3
1 - ---*x - ---*y + ---*y *x + O(x ,y )
3 3 9
tt := taylor (exp, {x,y}, 0, 2);
1 2 1 2 3
tt := 1 - ---*y - ---*x + O({x,y} )
3 3
comment An example that uses factorization;
on factor;
ff := y**5 - 1;
4 3 2
ff := (y + y + y + y + 1)*(y - 1)
zz := sub (y = taylor(e**x, x, 0, 3), ff);
1 2 1 3 4 4
zz := ((1 + x + ---*x + ---*x + O(x ))
2 6
1 2 1 3 4 3
+ (1 + x + ---*x + ---*x + O(x ))
2 6
1 2 1 3 4 2
+ (1 + x + ---*x + ---*x + O(x ))
2 6
1 2 1 3 4
+ (1 + x + ---*x + ---*x + O(x )) + 1)
2 6
1 2 1 3 4
*((1 + x + ---*x + ---*x + O(x )) - 1)
2 6
on exp;
zz;
1 2 1 3 4 5
(1 + x + ---*x + ---*x + O(x )) - 1
2 6
comment A simple example of Taylor kernel differentiation;
hugo := taylor(e^x,x,0,5);
1 2 1 3 1 4 1 5 6
hugo := 1 + x + ---*x + ---*x + ----*x + -----*x + O(x )
2 6 24 120
df(hugo^2,x);
2 8 3 4 4 5
2 + 4*x + 4*x + ---*x + ---*x + O(x )
3 3
comment The following shows the (limited) capabilities to integrate
Taylor kernels. Only a toplevel Taylor kernel is supported,
in all other cases a warning is printed and the Taylor kernels
are converted to standard representation;
zz := taylor (sin x, x, 0, 5);
1 3 1 5 6
zz := x - ---*x + -----*x + O(x )
6 120
ww := taylor (cos y, y, 0, 5);
1 2 1 4 6
ww := 1 - ---*y + ----*y + O(y )
2 24
int (zz, x);
1 2 1 4 1 6 7
---*x - ----*x + -----*x + O(x )
2 24 720
int (ww, x);
x 2 x 4 6
x - ---*y + ----*y + O(y )
2 24
int (zz + ww, x);
*** Converting Taylor kernels to standard representation
5 3 4 2
x*(x - 30*x + 360*x + 30*y - 360*y + 720)
-----------------------------------------------
720
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);
2 6
t - t + O(t )
taylorrevert (ws, t, x);
2 3 4 5 6
x + x + 2*x + 5*x + 14*x + O(x )
tan!-series := taylor (tan x, x, 0, 5);
1 3 2 5 6
tan-series := x + ---*x + ----*x + O(x )
3 15
taylorrevert (tan!-series, x, y);
1 3 1 5 6
y - ---*y + ---*y + O(y )
3 5
atan!-series:=taylor (atan y, y, 0, 5);
1 3 1 5 6
atan-series := y - ---*y + ---*y + O(y )
3 5
tmp := taylor (e**x, x, 0, 5);
1 2 1 3 1 4 1 5 6
tmp := 1 + x + ---*x + ---*x + ----*x + -----*x + O(x )
2 6 24 120
taylorrevert (tmp, x, y);
1 2 1 3 1 4 1 5
y - 1 - ---*(y - 1) + ---*(y - 1) - ---*(y - 1) + ---*(y - 1)
2 3 4 5
6
+ O((y - 1) )
taylor (log y, y, 1, 5);
1 2 1 3 1 4 1 5
y - 1 - ---*(y - 1) + ---*(y - 1) - ---*(y - 1) + ---*(y - 1)
2 3 4 5
6
+ O((y - 1) )
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);
20 19 18 17 16
poly := x - 210*x + 20615*x - 1256850*x + 53327946*x
15 14 13
- 1672280820*x + 40171771630*x - 756111184500*x
12 11
+ 11310276995381*x - 135585182899530*x
10 9
+ 1307535010540395*x - 10142299865511450*x
8 7
+ 63030812099294896*x - 311333643161390640*x
6 5
+ 1206647803780373360*x - 3599979517947607200*x
4 3
+ 8037811822645051776*x - 12870931245150988800*x
2
+ 13803759753640704000*x - 8752948036761600000*x
+ 2432902008176640000
on rounded;
tpoly := taylor (poly, x, 20, 4);
2
tpoly := 1.21649393692e+17*(x - 20) + 4.31564847287e+17*(x - 20)
3 4
+ 6.68609351672e+17*(x - 20) + 6.10115975015e+17*(x - 20)
5
+ O((x - 20) )
taylorrevert (tpoly, x, eps);
2
20 + 8.22034512177e-18*eps - 2.39726594661e-34*eps
3 4 5
+ 1.09290580231e-50*eps - 5.9711415946e-67*eps + O(eps )
comment Some more examples using rounded mode;
taylor(sin x/x,x,0,4);
2 4 5
1 - 0.166666666667*x + 0.00833333333333*x + O(x )
taylor(sin x,x,pi/2,4);
2
1 + 6.12323399574e-17*(x - 1.57079632679) - 0.5*(x - 1.57079632679)
3
- 1.02053899929e-17*(x - 1.57079632679)
4 5
+ 0.0416666666667*(x - 1.57079632679) + O((x - 1.57079632679) )
taylor(tan x,x,pi/2,4);
-1
- (x - 1.57079632679) + 0.333333333333*(x - 1.57079632679)
3 5
+ 0.0222222222222*(x - 1.57079632679) + O((x - 1.57079632679) )
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);
1 5 2 1 3 79 4 3 5
1 + ---*x + ----*x + -----*x + -------*x + -------*x + (10 terms)
4 96 128 92160 40960
16
+ O(x )
comment Examples involving non-analytical terms;
taylor(log(e^x-1),x,0,5);
1 1 2 1 4 5
log(x) + (---*x + ----*x - ------*x + O(x ))
2 24 2880
taylor(e^(1/x)*(e^x-1),x,0,5);
1/x 1 2 1 3 1 4 1 5 6
e *(x + ---*x + ---*x + ----*x + -----*x + O(x ))
2 6 24 120
taylor(log(x)*x^10,x,0,5);
10 15
log(x)*(x + O(x ))
taylor(log(x)*x^10,x,0,11);
10 21
log(x)*(x + O(x ))
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);
log(2) 1 1 1
- ---------------------- - ---*---- + O(----)
2 2 2 3
a*b - a*c - b + b*c x x
ss := (sqrt(x^(2/5) +1) - x^(1/3)-1)/x^(1/3);
2/5 1/3
sqrt(x + 1) - x - 1
ss := ---------------------------
1/3
x
taylor(exp ss,x,0,2);
2/5 3
sqrt(x + (1 + O(x )))
exp(--------------------------)
1/3
x
---------------------------------
3
1 + O(x ) 3
exp(-----------)*(e + O(x ))
1/3
x
taylor(exp sub(x=x^15,ss),x,0,2);
1 1 1 2 3
--- + -----*x + -----*x + O(x )
e 2*e 8*e
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);
2 1 4 6
1 - x + ---*x + O(x )
2
taylor(myerf(x),x,0,5);
2*sqrt(pi) 2*sqrt(pi) 3 sqrt(pi) 5 6
------------*x - ------------*x + ----------*x + O(x )
pi 3*pi 5*pi
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;
Time: 13350 ms plus GC time: 534 ms
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) ;
derivative_expression :=
2*a(x,y)*df(f(x,y),x,y)*df(gg(x,y),x)*df(gg(x,y),y)
+ a(x,y)*df(f(x,y),x)*df(gg(x,y),x,y)*df(gg(x,y),y)
+ a(x,y)*df(f(x,y),x)*df(gg(x,y),x)*df(gg(x,y),y,2)
+ a(x,y)*df(f(x,y),y)*df(gg(x,y),x,y)*df(gg(x,y),x)
+ a(x,y)*df(f(x,y),y)*df(gg(x,y),x,2)*df(gg(x,y),y)
+ df(a(x,y),x)*df(f(x,y),y)*df(gg(x,y),x)*df(gg(x,y),y)
+ df(a(x,y),y)*df(f(x,y),x)*df(gg(x,y),x)*df(gg(x,y),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)
};
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);
texp := a(x,y)*fx(x,y)*gx(x,y)*gyy(x,y)
+ a(x,y)*fx(x,y)*gxy(x,y)*gy(x,y)
+ 2*a(x,y)*fxy(x,y)*gx(x,y)*gy(x,y)
+ a(x,y)*fy(x,y)*gx(x,y)*gxy(x,y)
+ a(x,y)*fy(x,y)*gxx(x,y)*gy(x,y)
+ ax(x,y)*fy(x,y)*gx(x,y)*gy(x,y)
2 2
+ ay(x,y)*fx(x,y)*gx(x,y)*gy(x,y) + O(dx ,dy )
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;
result := a(x,y)*fx(x,y)*gx(x,y)*gyy(x,y)
+ a(x,y)*fx(x,y)*gxy(x,y)*gy(x,y)
+ 2*a(x,y)*fxy(x,y)*gx(x,y)*gy(x,y)
+ a(x,y)*fy(x,y)*gx(x,y)*gxy(x,y)
+ a(x,y)*fy(x,y)*gxx(x,y)*gy(x,y)
+ ax(x,y)*fy(x,y)*gx(x,y)*gy(x,y)
+ ay(x,y)*fx(x,y)*gx(x,y)*gy(x,y)
derivative_expression - result;
0
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;
taylorprintterms := 5
off taylorautoexpand,taylorkeeporiginal;
comment That's all, folks;
showtime;
Time: 11283 ms plus GC time: 300 ms
end;
(TIME: taylor 24633 25467)
End of Lisp run after 24.64+1.58 seconds