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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