REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...
depend y,x;
generic_function f(x,y);
df(f(),x);
df(y,x)*f (x,y) + f (x,y)
y x
df(f(x,y),x);
df(y,x)*f (x,y) + f (x,y)
y x
df(f(x,x**3),x);
3 3 2
f (x,x ) + 3*f (x,x )*x
x y
df(f(x,z**3),x);
3
f (x,z )
x
df(a*f(x,y),x);
a*(df(y,x)*f (x,y) + f (x,y))
y x
dfp(a*f(x,y),x);
f (x,y)*a
x
df(f(x,y),x,2);
2
df(y,x,2)*f (x,y) + df(y,x) *f (x,y) + df(y,x)*f (x,y) + df(y,x)*f (x,y)
y yy xy yx
+ f (x,y)
xx
df(dfp(f(x,y),x),x);
df(y,x)*f (x,y) + f (x,y)
xy xx
df(dfp(f(x,x**3),x),x);
3 3 2
f (x,x ) + 3*f (x,x )*x
xx xy
% using a generic fucntion with commutative derivatives
generic_function u(x,y);
dfp_commute u(x,y);
df(u(x,y),x,x);
2
df(y,x,2)*u (x,y) + df(y,x) *u (x,y) + 2*df(y,x)*u (x,y) + u (x,y)
y yy xy xx
% explicitly declare 1st and second derivative commutative
generic_function v(x,y);
let dfp(v(~a,~b),{y,x}) => dfp(v(a,b),{x,y});
df(v(),x,2);
2
df(y,x,2)*v (x,y) + df(y,x) *v (x,y) + 2*df(y,x)*v (x,y) + v (x,y)
y yy xy xx
% substitute expressions for the arguments
w:=df(f(),x,2);
2
w := df(y,x,2)*f (x,y) + df(y,x) *f (x,y) + df(y,x)*f (x,y) + df(y,x)*f (x,y)
y yy xy yx
+ f (x,y)
xx
sub(x=0,y=x,w);
f (0,x) + f (0,x) + f (0,x) + f (0,x)
xx xy yx yy
% composite generic functions
generic_function g(x,y);
generic_function h(y,z);
depend z,x;
w:=df(g()*h(),x);
w :=
df(y,x)*g (x,y)*h() + df(y,x)*h (y,z)*g() + df(z,x)*h (y,z)*g() + g (x,y)*h()
y y z x
sub(y=0,w);
df(z,x)*h (0,z)*g(x,0) + g (x,0)*h(0,z)
z x
% substituting g*h for f in a partial derivative of f,
% inheriting the arguments of f. Here no derivative of h
% appears because h does not depend of x.
sub(f=g*h,dfp(f(a,b),x));
g (a,b)*h(b,z)
x
% indexes.
% in the following total differential the partial
% derivatives wrt i and j do not appear because i and
% j do not depend of x.
generic_function m(i,j,x,y);
df(m(i,j,x,y),x);
df(y,x)*m (i,j,x,y) + m (i,j,x,y)
y x
% computation with a differential equation.
generic_function f(x,y);
operator y;
let df(y(~x),x) => f(x,y(x));
% some derivatives
df(y(x),x);
f(x,y(x))
df(y(x),x,2);
f (x,y(x)) + f (x,y(x))*f(x,y(x))
x y
df(y(x),x,3);
f (x,y(x)) + f (x,y(x))*f(x,y(x)) + f (x,y(x))*f (x,y(x))
xx xy x y
2 2
+ f (x,y(x))*f(x,y(x)) + f (x,y(x))*f(x,y(x)) + f (x,y(x)) *f(x,y(x))
yx yy y
sub(x=22,ws);
f (22,y(22)) + f (22,y(22))*f(22,y(22)) + f (22,y(22))*f (22,y(22))
xx xy x y
2
+ f (22,y(22))*f(22,y(22)) + f (22,y(22))*f(22,y(22))
yx yy
2
+ f (22,y(22)) *f(22,y(22))
y
% taylor expansion for y
load_package taylor;
taylor(y(x0+h),h,0,3);
f (x0,y(x0)) + f (x0,y(x0))*f(x0,y(x0))
x y 2
y(x0) + f(x0,y(x0))*h + -----------------------------------------*h + (
2
f (x0,y(x0)) + f (x0,y(x0))*f(x0,y(x0)) + f (x0,y(x0))*f (x0,y(x0))
xx xy x y
2
+ f (x0,y(x0))*f(x0,y(x0)) + f (x0,y(x0))*f(x0,y(x0))
yx yy
2 3 4
+ f (x0,y(x0)) *f(x0,y(x0)))/6*h + O(h )
y
clear w;
%------------------------ Runge Kutta -------------------------
% computing Runge Kutta formulas for ODE systems Y'=F(x,y(x));
% forms corresponding to Ralston Rabinowitz
load_package taylor;
operator alpha,beta,w,k;
% s= order of Runge Kutta formula
s:=3;
s := 3
generic_function f(x,y);
operator y;
*** y already defined as operator
% introduce ODE
let df(y(~x),x)=>f(x,y(x));
% formal series for solution
y1_form := taylor(y(x0+h),h,0,s);
f (x0,y(x0)) + f (x0,y(x0))*f(x0,y(x0))
x y 2
y1_form := y(x0) + f(x0,y(x0))*h + -----------------------------------------*h
2
+ (f (x0,y(x0)) + f (x0,y(x0))*f(x0,y(x0))
xx xy
+ f (x0,y(x0))*f (x0,y(x0)) + f (x0,y(x0))*f(x0,y(x0))
x y yx
2 2 3
+ f (x0,y(x0))*f(x0,y(x0)) + f (x0,y(x0)) *f(x0,y(x0)))/6*h
yy y
4
+ O(h )
% Runge-Kutta Ansatz:
let alpha(1)=>0;
for i:=1:s do
let k(i) => h*f(x0 + alpha(i)*h,
y(x0) + for j:=1:(i-1) sum beta(i,j)*k(j));
y1_ansatz:= y(x0) + for i:=1:s sum w(i)*k(i);
y1_ansatz := f(alpha(3)*h + x0,
beta(3,2)*f(alpha(2)*h + x0,beta(2,1)*f(x0,y(x0))*h + y(x0))*h
+ beta(3,1)*f(x0,y(x0))*h + y(x0))*w(3)*h
+ f(alpha(2)*h + x0,beta(2,1)*f(x0,y(x0))*h + y(x0))*w(2)*h
+ f(x0,y(x0))*w(1)*h + y(x0)
y1_ansatz := taylor(y1_ansatz,h,0,s);
y1_ansatz := y(x0) + f(x0,y(x0))*(w(3) + w(2) + w(1))*h + (
alpha(3)*f (x0,y(x0))*w(3) + alpha(2)*f (x0,y(x0))*w(2)
x x
+ beta(3,2)*f (x0,y(x0))*f(x0,y(x0))*w(3)
y
+ beta(3,1)*f (x0,y(x0))*f(x0,y(x0))*w(3)
y
2
+ beta(2,1)*f (x0,y(x0))*f(x0,y(x0))*w(2))*h + (
y
2
alpha(3) *f (x0,y(x0))*w(3)
xx
+ alpha(3)*beta(3,2)*f (x0,y(x0))*f(x0,y(x0))*w(3)
xy
+ alpha(3)*beta(3,2)*f (x0,y(x0))*f(x0,y(x0))*w(3)
yx
+ alpha(3)*beta(3,1)*f (x0,y(x0))*f(x0,y(x0))*w(3)
xy
+ alpha(3)*beta(3,1)*f (x0,y(x0))*f(x0,y(x0))*w(3)
yx
2
+ alpha(2) *f (x0,y(x0))*w(2)
xx
+ 2*alpha(2)*beta(3,2)*f (x0,y(x0))*f (x0,y(x0))*w(3)
x y
+ alpha(2)*beta(2,1)*f (x0,y(x0))*f(x0,y(x0))*w(2)
xy
+ alpha(2)*beta(2,1)*f (x0,y(x0))*f(x0,y(x0))*w(2)
yx
2 2
+ beta(3,2) *f (x0,y(x0))*f(x0,y(x0)) *w(3)
yy
2
+ 2*beta(3,2)*beta(3,1)*f (x0,y(x0))*f(x0,y(x0)) *w(3)
yy
2
+ 2*beta(3,2)*beta(2,1)*f (x0,y(x0)) *f(x0,y(x0))*w(3)
y
2 2
+ beta(3,1) *f (x0,y(x0))*f(x0,y(x0)) *w(3)
yy
2 2 3 4
+ beta(2,1) *f (x0,y(x0))*f(x0,y(x0)) *w(2))/2*h + O(h )
yy
% compute y1_form - y1_ans and collect coeffients of powers of h
y1_diff := num(taylortostandard(y1_ansatz)-taylortostandard(y1_form))$
cl := coeff(y1_diff,h);
cl := {0,
6*f(x0,y(x0))*(w(3) + w(2) + w(1) - 1),
3*(2*alpha(3)*f (x0,y(x0))*w(3) + 2*alpha(2)*f (x0,y(x0))*w(2)
x x
+ 2*beta(3,2)*f (x0,y(x0))*f(x0,y(x0))*w(3)
y
+ 2*beta(3,1)*f (x0,y(x0))*f(x0,y(x0))*w(3)
y
+ 2*beta(2,1)*f (x0,y(x0))*f(x0,y(x0))*w(2) - f (x0,y(x0))
y x
- f (x0,y(x0))*f(x0,y(x0))),
y
2
3*alpha(3) *f (x0,y(x0))*w(3)
xx
+ 3*alpha(3)*beta(3,2)*f (x0,y(x0))*f(x0,y(x0))*w(3)
xy
+ 3*alpha(3)*beta(3,2)*f (x0,y(x0))*f(x0,y(x0))*w(3)
yx
+ 3*alpha(3)*beta(3,1)*f (x0,y(x0))*f(x0,y(x0))*w(3)
xy
+ 3*alpha(3)*beta(3,1)*f (x0,y(x0))*f(x0,y(x0))*w(3)
yx
2
+ 3*alpha(2) *f (x0,y(x0))*w(2)
xx
+ 6*alpha(2)*beta(3,2)*f (x0,y(x0))*f (x0,y(x0))*w(3)
x y
+ 3*alpha(2)*beta(2,1)*f (x0,y(x0))*f(x0,y(x0))*w(2)
xy
+ 3*alpha(2)*beta(2,1)*f (x0,y(x0))*f(x0,y(x0))*w(2)
yx
2 2
+ 3*beta(3,2) *f (x0,y(x0))*f(x0,y(x0)) *w(3)
yy
2
+ 6*beta(3,2)*beta(3,1)*f (x0,y(x0))*f(x0,y(x0)) *w(3)
yy
2
+ 6*beta(3,2)*beta(2,1)*f (x0,y(x0)) *f(x0,y(x0))*w(3)
y
2 2
+ 3*beta(3,1) *f (x0,y(x0))*f(x0,y(x0)) *w(3)
yy
2 2
+ 3*beta(2,1) *f (x0,y(x0))*f(x0,y(x0)) *w(2) - f (x0,y(x0))
yy xx
- f (x0,y(x0))*f(x0,y(x0)) - f (x0,y(x0))*f (x0,y(x0))
xy x y
2
- f (x0,y(x0))*f(x0,y(x0)) - f (x0,y(x0))*f(x0,y(x0))
yx yy
2
- f (x0,y(x0)) *f(x0,y(x0))}
y
% f_forms: forms of f and its derivatives which occur in cl
f_forms :=q := {f(x0,y(x0))}$
for i:=1:(s-1) do
<<q:= for each r in q join {dfp(r,x),dfp(r,y)};
f_forms := append(f_forms,q);
>>;
f_forms;
{f(x0,y(x0)),
f (x0,y(x0)),
x
f (x0,y(x0)),
y
f (x0,y(x0)),
xx
f (x0,y(x0)),
xy
f (x0,y(x0)),
yx
f (x0,y(x0))}
yy
% extract coefficients of the f_forms in cl
sys := cl$
for each fr in f_forms do
sys:=for each c in sys join coeff(c,fr);
% and eliminate zeros
sys := for each c in sys join if c neq 0 then {c} else {};
sys := {6*(w(3) + w(2) + w(1) - 1),
3*(2*alpha(3)*w(3) + 2*alpha(2)*w(2) - 1),
3*(2*beta(3,2)*w(3) + 2*beta(3,1)*w(3) + 2*beta(2,1)*w(2) - 1),
2 2
3*alpha(3) *w(3) + 3*alpha(2) *w(2) - 1,
6*alpha(2)*beta(3,2)*w(3) - 1,
3*alpha(3)*beta(3,2)*w(3) + 3*alpha(3)*beta(3,1)*w(3)
+ 3*alpha(2)*beta(2,1)*w(2) - 1,
3*alpha(3)*beta(3,2)*w(3) + 3*alpha(3)*beta(3,1)*w(3)
+ 3*alpha(2)*beta(2,1)*w(2) - 1,
6*beta(3,2)*beta(2,1)*w(3) - 1,
2 2
3*beta(3,2) *w(3) + 6*beta(3,2)*beta(3,1)*w(3) + 3*beta(3,1) *w(3)
2
+ 3*beta(2,1) *w(2) - 1}
end;
(TIME: dfpart 2390 2580)