Sat Jun 29 14:12:23 PDT 1991
REDUCE 3.4, 15-Jul-91 ...
1: 1:
2: 2:
3: 3: comment
Test and demonstration file for the Taylor expansion package,
by Rainer M. Schoepf. Works with version 1.3 (31-Jan-91);
showtime;
Time: 17 ms
on errcont;
% disable interruption on errors
comment Simple Taylor expansion;
xx := taylor (e**x, x, 0, 4);
1 2 1 3 1 4
XX := 1 + X + ---*X + ---*X + ----*X + ...
2 6 24
yy := taylor (e**y, y, 0, 4);
1 2 1 3 1 4
YY := 1 + Y + ---*Y + ---*Y + ----*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
1 + 2*X + 2*X + ---*X + ---*X + ...
3 3
taylorcombine (ws - xx);
3 2 7 3 5 4
X + ---*X + ---*X + ---*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
(1 + X + ---*X + ---*X + ----*X + ...)
2 6 24
1 2 1 3 1 4
**(1 + Y + ---*Y + ---*Y + ----*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
E + E*X + E*X + -----*X + -----*X + ...
6 8
taylorcombine log ws;
1 2 1 3 1 4
1 + X + ---*X + ---*X + ----*X + ...
2 6 24
comment We may try to expand about another point;
taylor (xx, x, 1, 2);
65 8 5 2
---- + ---*(X - 1) + ---*(X - 1) + ...
24 3 4
comment Arc tangent is one of the functions this package knows of;
xxa := taylorcombine atan ws;
65 1536 - 2933040 2
XXA := ATAN(----) + ------*(X - 1) + ------------*(X - 1) + ...
24 4801 23049601
comment Expansion with respect to more than one kernel is possible;
xy := taylor (e**(x+y), x, 0, 2, y, 0, 2);
1 2 1 2 1 2 1 2
XY := 1 + Y + ---*Y + X + Y*X + ---*Y *X + ---*X + ---*Y*X
2 2 2 2
1 2 2
+ ---*Y *X + ...
4
taylorcombine (ws**2);
2 2 2 2 2 2
1 + 2*Y + 2*Y + 2*X + 4*Y*X + 4*Y *X + 2*X + 4*Y*X + 4*Y *X + ...
comment We take the inverse and convert back to REDUCE's standard
representation;
taylorcombine (1/ws);
2 2 2 2 2 2
1 - 2*X + 2*X - 2*Y + 4*Y*X - 4*Y*X + 2*Y - 4*Y *X + 4*Y *X + ...
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 An example of Taylor kernel divsion;
xx1 := taylor (sin (x), x, 0, 4);
- 1 3
XX1 := X + ------*X + ...
6
taylorcombine (xx/xx1);
-1 2
X + 1 + ---*X + ...
3
taylorcombine (1/xx1);
-1 1 1 3
X + ---*X + ----*X + ...
6 36
comment Here's what I call homogeneous expansion;
xx := taylor (e**(x*y), {x,y}, 0, 2);
XX := 1 + Y*X + ...
xx1 := taylor (sin (x+y), {x,y}, 0, 2);
XX1 := Y + X + ...
xx2 := taylor (cos (x+y), {x,y}, 0, 2);
- 1 2 - 1 2
XX2 := 1 + ------*Y - Y*X + ------*X + ...
2 2
temp := taylorcombine (xx/xx2);
1 2 1 2
TEMP := 1 + ---*Y + 2*Y*X + ---*X + ...
2 2
taylorcombine (ws*xx2);
1 + Y*X + ...
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 1 2 1 2 1 2 1 2 2
1 + Y + ---*Y + Z + Y*Z + ---*Y *Z + ---*Z + ---*Y*Z + ---*Y *Z
2 2 2 2 4
+ ...
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
13 + 13*Y + ----*Y + ...
2
sub (x=4, xx1);
4 + 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 Y E 2 E 3 E 4 E 5
TEMP := E + E *X + ----*X + ----*X + ----*X + -----*X + ...
2 6 24 120
taylorcombine (log (temp));
Y + X + ...
taylororiginal ws;
X + Y
taylorcombine (temp * e**x);
Y Y Y Y
X Y Y E 2 E 3 E 4 E 5
E *(E + E *X + ----*X + ----*X + ----*X + -----*X + ...)
2 6 24 120
on taylorautoexpand;
taylorcombine ws;
Y Y Y
Y Y Y 2 4*E 3 2*E 4 4*E 5
E + (2*E )*X + (2*E )*X + ------*X + ------*X + ------*X + ...
3 3 15
taylororiginal ws;
2*X + Y
E
taylorcombine (xx1 / x);
-1 -1
X + Y*X + 1 + ...
on taylorautocombine;
xx / xx2;
1 2 1 2
1 + ---*Y + 2*Y*X + ---*X + ...
2 2
ws * xx2;
1 + Y*X + ...
comment Another example that shows truncation if Taylor kernels
of different expansion order are combined;
p := taylor (x**2 + 2, x, 0, 10);
2
P := 2 + X + ...
p - x**2;
2 2
(2 + X + ...) - X
p - taylor (x**2, x, 0, 5);
2 + ...
taylor (p - x**2, x, 0, 6);
2 + ...
off taylorautocombine;
taylorcombine(p-x**2);
2 + ...
taylorcombine(p - taylor(x**2,x,0,5));
2 + ...
comment A problem are non-analytic terms: there are no precautions
taken to detect or handle them;
taylor (sqrt (x), x, 0, 2);
***** Zero divisor
***** Error during expansion (possible singularity!)
taylor (e**(1/x), x, 0, 2);
***** Zero divisor
***** Error during expansion (possible singularity!)
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
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 + --- + ---*---- + ---*---- + ----*---- + -----*---- + ...
X 2 2 6 3 24 4 120 5
X X X X
xi := taylor (sin (1/x), x, infinity, 5);
1 - 1 1 1 1
XI := --- + ------*---- + -----*---- + ...
X 6 3 120 5
X X
y1 := taylor(x/(x-1), x, infinity, 3);
1 1 1
Y1 := 1 + --- + ---- + ---- + ...
X 2 3
X X
z := df(y1, x);
1 1 1
Z := - ---- - 2*---- - 3*---- + ...
2 3 4
X X X
comment ...but far from being perfect;
taylor (1 / sin (x), x, infinity, 5);
***** Zero divisor
***** Error during expansion (possible singularity!)
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);
1 + ...
taylor (exp, x, 0, 2, y, 0, 2);
- 1 2 - 1 2 1 2 2
1 + ------*Y + ------*X + ---*Y *X + ...
3 3 9
tt := taylor (exp, {x,y}, 0, 2);
- 1 2 - 1 2
TT := 1 + ------*Y + ------*X + ...
3 3
comment An application is the problem posed by Prof. Stanley:
we prove that the finite difference expression below
corresponds to the given derivative expression;
comment We use gg to avoid conflicts with the predefined g operator;
define g=gg;
operator diff,a,f,g;
for all f,arg let diff(f,arg) = df(f,arg);
derivative!_expression :=
diff(a(x,y)*diff(g(x,y),x)*diff(g(x,y),y)*diff(f(x,y),y),x) +
diff(a(x,y)*diff(g(x,y),x)*diff(g(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)*g(x+dx,y+dy)^2/(32*dx^2*dy^2)
+a(x+dx,y)*f(x+dx,y+dy)*g(x+dx,y+dy)^2/(32*dx^2*dy^2)
+a(x,y+dy)*f(x+dx,y+dy)*g(x+dx,y+dy)^2/(32*dx^2*dy^2)
+a(x,y)*f(x+dx,y+dy)*g(x+dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x+dx,y+dy)*g(x+dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x+dx,y)*g(x+dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x,y+dy)*g(x+dx,y+dy)^2/(32*dx^2*dy^2)
-a(x,y)*f(x,y)*g(x+dx,y+dy)^2/(32*dx^2*dy^2)
-g(x,y)*a(x+dx,y+dy)*f(x+dx,y+dy)*g(x+dx,y+dy)/(16*dx^2*dy^2)
-g(x,y)*a(x+dx,y)*f(x+dx,y+dy)*g(x+dx,y+dy)/(16*dx^2*dy^2)
-g(x,y)*a(x,y+dy)*f(x+dx,y+dy)*g(x+dx,y+dy)/(16*dx^2*dy^2)
-a(x,y)*g(x,y)*f(x+dx,y+dy)*g(x+dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x+dx,y+dy)*g(x+dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x+dx,y)*g(x+dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x,y+dy)*g(x+dx,y+dy)/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*g(x,y)*g(x+dx,y+dy)/(16*dx^2*dy^2)
-g(x+dx,y)^2*a(x+dx,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2)
+g(x,y+dy)*g(x+dx,y)*a(x+dx,y+dy)*f(x+dx,y+dy)/(16*dx^2*dy^2)
-g(x,y+dy)^2*a(x+dx,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x+dx,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x+dx,y)*g(x+dx,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x,y+dy)*g(x+dx,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x,y)*g(x+dx,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
+g(x,y+dy)*a(x+dx,y)*g(x+dx,y)*f(x+dx,y+dy)/(16*dx^2*dy^2)
+a(x,y+dy)*g(x,y+dy)*g(x+dx,y)*f(x+dx,y+dy)/(16*dx^2*dy^2)
+a(x,y)*g(x,y+dy)*g(x+dx,y)*f(x+dx,y+dy)/(16*dx^2*dy^2)
-g(x,y+dy)^2*a(x+dx,y)*f(x+dx,y+dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x+dx,y)*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x,y+dy)*g(x,y+dy)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x,y)*g(x,y+dy)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2)
+a(x,y)*g(x,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
+f(x,y)*g(x+dx,y)^2*a(x+dx,y+dy)/(32*dx^2*dy^2)
-f(x,y)*g(x,y+dy)*g(x+dx,y)*a(x+dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y+dy)^2*a(x+dx,y+dy)/(32*dx^2*dy^2)
-f(x,y)*g(x,y)^2*a(x+dx,y+dy)/(32*dx^2*dy^2)
+a(x+dx,y-dy)*f(x+dx,y-dy)*g(x+dx,y-dy)^2/(32*dx^2*dy^2)
+a(x+dx,y)*f(x+dx,y-dy)*g(x+dx,y-dy)^2/(32*dx^2*dy^2)
+a(x,y-dy)*f(x+dx,y-dy)*g(x+dx,y-dy)^2/(32*dx^2*dy^2)
+a(x,y)*f(x+dx,y-dy)*g(x+dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x+dx,y-dy)*g(x+dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x+dx,y)*g(x+dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x,y-dy)*g(x+dx,y-dy)^2/(32*dx^2*dy^2)
-a(x,y)*f(x,y)*g(x+dx,y-dy)^2/(32*dx^2*dy^2)
-g(x,y)*a(x+dx,y-dy)*f(x+dx,y-dy)*g(x+dx,y-dy)/(16*dx^2*dy^2)
-g(x,y)*a(x+dx,y)*f(x+dx,y-dy)*g(x+dx,y-dy)/(16*dx^2*dy^2)
-g(x,y)*a(x,y-dy)*f(x+dx,y-dy)*g(x+dx,y-dy)/(16*dx^2*dy^2)
-a(x,y)*g(x,y)*f(x+dx,y-dy)*g(x+dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x+dx,y-dy)*g(x+dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x+dx,y)*g(x+dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x,y-dy)*g(x+dx,y-dy)/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*g(x,y)*g(x+dx,y-dy)/(16*dx^2*dy^2)
-g(x+dx,y)^2*a(x+dx,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2)
+g(x,y-dy)*g(x+dx,y)*a(x+dx,y-dy)*f(x+dx,y-dy)/(16*dx^2*dy^2)
-g(x,y-dy)^2*a(x+dx,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x+dx,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x+dx,y)*g(x+dx,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x,y-dy)*g(x+dx,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x,y)*g(x+dx,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
+g(x,y-dy)*a(x+dx,y)*g(x+dx,y)*f(x+dx,y-dy)/(16*dx^2*dy^2)
+a(x,y-dy)*g(x,y-dy)*g(x+dx,y)*f(x+dx,y-dy)/(16*dx^2*dy^2)
+a(x,y)*g(x,y-dy)*g(x+dx,y)*f(x+dx,y-dy)/(16*dx^2*dy^2)
-g(x,y-dy)^2*a(x+dx,y)*f(x+dx,y-dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x+dx,y)*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x,y-dy)*g(x,y-dy)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x,y)*g(x,y-dy)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2)
+a(x,y)*g(x,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
+f(x,y)*g(x+dx,y)^2*a(x+dx,y-dy)/(32*dx^2*dy^2)
-f(x,y)*g(x,y-dy)*g(x+dx,y)*a(x+dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y-dy)^2*a(x+dx,y-dy)/(32*dx^2*dy^2)
-f(x,y)*g(x,y)^2*a(x+dx,y-dy)/(32*dx^2*dy^2)
+f(x,y)*a(x+dx,y)*g(x+dx,y)^2/(16*dx^2*dy^2)
+f(x,y)*a(x,y+dy)*g(x+dx,y)^2/(32*dx^2*dy^2)
+f(x,y)*a(x,y-dy)*g(x+dx,y)^2/(32*dx^2*dy^2)
+a(x,y)*f(x,y)*g(x+dx,y)^2/(16*dx^2*dy^2)
-f(x,y)*g(x,y+dy)*a(x+dx,y)*g(x+dx,y)/(16*dx^2*dy^2)
-f(x,y)*g(x,y-dy)*a(x+dx,y)*g(x+dx,y)/(16*dx^2*dy^2)
-f(x,y)*a(x,y+dy)*g(x,y+dy)*g(x+dx,y)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*g(x,y+dy)*g(x+dx,y)/(16*dx^2*dy^2)
-f(x,y)*a(x,y-dy)*g(x,y-dy)*g(x+dx,y)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*g(x,y-dy)*g(x+dx,y)/(16*dx^2*dy^2)
+f(x,y)*g(x,y+dy)^2*a(x+dx,y)/(32*dx^2*dy^2)
+f(x,y)*g(x,y-dy)^2*a(x+dx,y)/(32*dx^2*dy^2)
-f(x,y)*g(x,y)^2*a(x+dx,y)/(16*dx^2*dy^2)
+a(x-dx,y+dy)*f(x-dx,y+dy)*g(x-dx,y+dy)^2/(32*dx^2*dy^2)
+a(x-dx,y)*f(x-dx,y+dy)*g(x-dx,y+dy)^2/(32*dx^2*dy^2)
+a(x,y+dy)*f(x-dx,y+dy)*g(x-dx,y+dy)^2/(32*dx^2*dy^2)
+a(x,y)*f(x-dx,y+dy)*g(x-dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x-dx,y+dy)*g(x-dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x-dx,y)*g(x-dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x,y+dy)*g(x-dx,y+dy)^2/(32*dx^2*dy^2)
-a(x,y)*f(x,y)*g(x-dx,y+dy)^2/(32*dx^2*dy^2)
-g(x,y)*a(x-dx,y+dy)*f(x-dx,y+dy)*g(x-dx,y+dy)/(16*dx^2*dy^2)
-g(x,y)*a(x-dx,y)*f(x-dx,y+dy)*g(x-dx,y+dy)/(16*dx^2*dy^2)
-g(x,y)*a(x,y+dy)*f(x-dx,y+dy)*g(x-dx,y+dy)/(16*dx^2*dy^2)
-a(x,y)*g(x,y)*f(x-dx,y+dy)*g(x-dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x-dx,y+dy)*g(x-dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x-dx,y)*g(x-dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x,y+dy)*g(x-dx,y+dy)/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*g(x,y)*g(x-dx,y+dy)/(16*dx^2*dy^2)
-g(x-dx,y)^2*a(x-dx,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2)
+g(x,y+dy)*g(x-dx,y)*a(x-dx,y+dy)*f(x-dx,y+dy)/(16*dx^2*dy^2)
-g(x,y+dy)^2*a(x-dx,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x-dx,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x-dx,y)*g(x-dx,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x,y+dy)*g(x-dx,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x,y)*g(x-dx,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
+g(x,y+dy)*a(x-dx,y)*g(x-dx,y)*f(x-dx,y+dy)/(16*dx^2*dy^2)
+a(x,y+dy)*g(x,y+dy)*g(x-dx,y)*f(x-dx,y+dy)/(16*dx^2*dy^2)
+a(x,y)*g(x,y+dy)*g(x-dx,y)*f(x-dx,y+dy)/(16*dx^2*dy^2)
-g(x,y+dy)^2*a(x-dx,y)*f(x-dx,y+dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x-dx,y)*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x,y+dy)*g(x,y+dy)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x,y)*g(x,y+dy)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2)
+a(x,y)*g(x,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
+f(x,y)*g(x-dx,y)^2*a(x-dx,y+dy)/(32*dx^2*dy^2)
-f(x,y)*g(x,y+dy)*g(x-dx,y)*a(x-dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y+dy)^2*a(x-dx,y+dy)/(32*dx^2*dy^2)
-f(x,y)*g(x,y)^2*a(x-dx,y+dy)/(32*dx^2*dy^2)
+a(x-dx,y-dy)*f(x-dx,y-dy)*g(x-dx,y-dy)^2/(32*dx^2*dy^2)
+a(x-dx,y)*f(x-dx,y-dy)*g(x-dx,y-dy)^2/(32*dx^2*dy^2)
+a(x,y-dy)*f(x-dx,y-dy)*g(x-dx,y-dy)^2/(32*dx^2*dy^2)
+a(x,y)*f(x-dx,y-dy)*g(x-dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x-dx,y-dy)*g(x-dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x-dx,y)*g(x-dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x,y-dy)*g(x-dx,y-dy)^2/(32*dx^2*dy^2)
-a(x,y)*f(x,y)*g(x-dx,y-dy)^2/(32*dx^2*dy^2)
-g(x,y)*a(x-dx,y-dy)*f(x-dx,y-dy)*g(x-dx,y-dy)/(16*dx^2*dy^2)
-g(x,y)*a(x-dx,y)*f(x-dx,y-dy)*g(x-dx,y-dy)/(16*dx^2*dy^2)
-g(x,y)*a(x,y-dy)*f(x-dx,y-dy)*g(x-dx,y-dy)/(16*dx^2*dy^2)
-a(x,y)*g(x,y)*f(x-dx,y-dy)*g(x-dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x-dx,y-dy)*g(x-dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x-dx,y)*g(x-dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y)*a(x,y-dy)*g(x-dx,y-dy)/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*g(x,y)*g(x-dx,y-dy)/(16*dx^2*dy^2)
-g(x-dx,y)^2*a(x-dx,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2)
+g(x,y-dy)*g(x-dx,y)*a(x-dx,y-dy)*f(x-dx,y-dy)/(16*dx^2*dy^2)
-g(x,y-dy)^2*a(x-dx,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x-dx,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x-dx,y)*g(x-dx,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x,y-dy)*g(x-dx,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x,y)*g(x-dx,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
+g(x,y-dy)*a(x-dx,y)*g(x-dx,y)*f(x-dx,y-dy)/(16*dx^2*dy^2)
+a(x,y-dy)*g(x,y-dy)*g(x-dx,y)*f(x-dx,y-dy)/(16*dx^2*dy^2)
+a(x,y)*g(x,y-dy)*g(x-dx,y)*f(x-dx,y-dy)/(16*dx^2*dy^2)
-g(x,y-dy)^2*a(x-dx,y)*f(x-dx,y-dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x-dx,y)*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x,y-dy)*g(x,y-dy)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x,y)*g(x,y-dy)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
+g(x,y)^2*a(x,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2)
+a(x,y)*g(x,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
+f(x,y)*g(x-dx,y)^2*a(x-dx,y-dy)/(32*dx^2*dy^2)
-f(x,y)*g(x,y-dy)*g(x-dx,y)*a(x-dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*g(x,y-dy)^2*a(x-dx,y-dy)/(32*dx^2*dy^2)
-f(x,y)*g(x,y)^2*a(x-dx,y-dy)/(32*dx^2*dy^2)
+f(x,y)*a(x-dx,y)*g(x-dx,y)^2/(16*dx^2*dy^2)
+f(x,y)*a(x,y+dy)*g(x-dx,y)^2/(32*dx^2*dy^2)
+f(x,y)*a(x,y-dy)*g(x-dx,y)^2/(32*dx^2*dy^2)
+a(x,y)*f(x,y)*g(x-dx,y)^2/(16*dx^2*dy^2)
-f(x,y)*g(x,y+dy)*a(x-dx,y)*g(x-dx,y)/(16*dx^2*dy^2)
-f(x,y)*g(x,y-dy)*a(x-dx,y)*g(x-dx,y)/(16*dx^2*dy^2)
-f(x,y)*a(x,y+dy)*g(x,y+dy)*g(x-dx,y)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*g(x,y+dy)*g(x-dx,y)/(16*dx^2*dy^2)
-f(x,y)*a(x,y-dy)*g(x,y-dy)*g(x-dx,y)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*g(x,y-dy)*g(x-dx,y)/(16*dx^2*dy^2)
+f(x,y)*g(x,y+dy)^2*a(x-dx,y)/(32*dx^2*dy^2)
+f(x,y)*g(x,y-dy)^2*a(x-dx,y)/(32*dx^2*dy^2)
-f(x,y)*g(x,y)^2*a(x-dx,y)/(16*dx^2*dy^2)
+f(x,y)*a(x,y+dy)*g(x,y+dy)^2/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*g(x,y+dy)^2/(16*dx^2*dy^2)
-f(x,y)*g(x,y)^2*a(x,y+dy)/(16*dx^2*dy^2)
+f(x,y)*a(x,y-dy)*g(x,y-dy)^2/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*g(x,y-dy)^2/(16*dx^2*dy^2)
-f(x,y)*g(x,y)^2*a(x,y-dy)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*g(x,y)^2/(8*dx^2*dy^2)$
comment We define abbreviations for the partial derivatives;
operator ax,ay,fx,fy,gx,gy;
for all x,y let df(a(x,y),x) = ax(x,y);
for all x,y let df(a(x,y),y) = ay(x,y);
for all x,y let df(f(x,y),x) = fx(x,y);
for all x,y let df(f(x,y),y) = fy(x,y);
for all x,y let df(g(x,y),x) = gx(x,y);
for all x,y let df(g(x,y),y) = gy(x,y);
operator axx,axy,ayy,fxx,fxy,fyy,gxx,gxy,gyy;
for all x,y let df(ax(x,y),x) = axx(x,y);
for all x,y let df(ax(x,y),y) = axy(x,y);
for all x,y let df(ay(x,y),x) = axy(x,y);
for all x,y let df(ay(x,y),y) = ayy(x,y);
for all x,y let df(fx(x,y),x) = fxx(x,y);
for all x,y let df(fx(x,y),y) = fxy(x,y);
for all x,y let df(fy(x,y),x) = fxy(x,y);
for all x,y let df(fy(x,y),y) = fyy(x,y);
for all x,y let df(gx(x,y),x) = gxx(x,y);
for all x,y let df(gx(x,y),y) = gxy(x,y);
for all x,y let df(gy(x,y),x) = gxy(x,y);
for all x,y let df(gy(x,y),y) = gyy(x,y);
operator axxx,axxy,axyy,ayyy,fxxx,fxxy,fxyy,fyyy,gxxx,gxxy,gxyy,gyyy;
for all x,y let df(axx(x,y),x) = axxx(x,y);
for all x,y let df(axy(x,y),x) = axxy(x,y);
for all x,y let df(ayy(x,y),x) = axyy(x,y);
for all x,y let df(ayy(x,y),y) = ayyy(x,y);
for all x,y let df(fxx(x,y),x) = fxxx(x,y);
for all x,y let df(fxy(x,y),x) = fxxy(x,y);
for all x,y let df(fxy(x,y),y) = fxyy(x,y);
for all x,y let df(fyy(x,y),x) = fxyy(x,y);
for all x,y let df(fyy(x,y),y) = fyyy(x,y);
for all x,y let df(gxx(x,y),x) = gxxx(x,y);
for all x,y let df(gxx(x,y),y) = gxxy(x,y);
for all x,y let df(gxy(x,y),x) = gxxy(x,y);
for all x,y let df(gxy(x,y),y) = gxyy(x,y);
for all x,y let df(gyy(x,y),x) = gxyy(x,y);
for all x,y let df(gyy(x,y),y) = gyyy(x,y);
operator axxxy,axxyy,axyyy,fxxxy,fxxyy,fxyyy,
gxxxx,gxxxy,gxxyy,gxyyy,gyyyy;
for all x,y let df(axyy(x,y),x) = axxyy(x,y);
for all x,y let df(axxy(x,y),x) = axxxy(x,y);
for all x,y let df(ayyy(x,y),x) = axyyy(x,y);
for all x,y let df(fxxy(x,y),x) = fxxxy(x,y);
for all x,y let df(fxyy(x,y),x) = fxxyy(x,y);
for all x,y let df(fyyy(x,y),x) = fxyyy(x,y);
for all x,y let df(gxxx(x,y),x) = gxxxx(x,y);
for all x,y let df(gxxy(x,y),x) = gxxxy(x,y);
for all x,y let df(gxyy(x,y),x) = gxxyy(x,y);
for all x,y let df(gyyy(x,y),x) = gxyyy(x,y);
for all x,y let df(gyyy(x,y),y) = gyyyy(x,y);
operator axxxyy,axxyyy,fxxyyy,fxxxyy,gxxxxy,gxxxyy,gxxyyy,gxyyyy;
for all x,y let df(axxyy(x,y),x) = axxxyy(x,y);
for all x,y let df(axyyy(x,y),x) = axxyyy(x,y);
for all x,y let df(fxxyy(x,y),x) = fxxxyy(x,y);
for all x,y let df(fxyyy(x,y),x) = fxxyyy(x,y);
for all x,y let df(gxxxy(x,y),x) = gxxxxy(x,y);
for all x,y let df(gxxyy(x,y),x) = gxxxyy(x,y);
for all x,y let df(gxyyy(x,y),x) = gxxyyy(x,y);
for all x,y let df(gyyyy(x,y),x) = gxyyyy(x,y);
operator gxxxxyy,gxxxyyy,gxxyyyy;
for all x,y let df(gxxxyy(x,y),x) = gxxxxyy(x,y);
for all x,y let df(gxxyyy(x,y),x) = gxxxyyy(x,y);
for all x,y let df(gxyyyy(x,y),x) = gxxyyyy(x,y);
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)*GY(X,Y)*GXY(X,Y)
+ A(X,Y)*FY(X,Y)*GX(X,Y)*GXY(X,Y)
+ A(X,Y)*FY(X,Y)*GY(X,Y)*GXX(X,Y)
+ 2*A(X,Y)*GX(X,Y)*GY(X,Y)*FXY(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) + ...
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)*GY(X,Y)*GXY(X,Y)
+ A(X,Y)*FY(X,Y)*GX(X,Y)*GXY(X,Y)
+ A(X,Y)*FY(X,Y)*GY(X,Y)*GXX(X,Y)
+ 2*A(X,Y)*GX(X,Y)*GY(X,Y)*FXY(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
comment That's all, folks;
showtime;
Time: 19924 ms
end;
4: 4:
Quitting
Sat Jun 29 14:12:56 PDT 1991