vv

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9 years, 146 days

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These are answers submitted by vv

You have a sum of a series, not an equation.

The sum cannot be computed exactly, but you can approximate it with any precision: just use

evalf[20](sum((factorial(n)/factorial(2*n))^n, n = 1 .. infinity));

to get 20 decimals.

Note that your series converges very fast: the first 4 terms are enough for 20 correct digits!

 

I'd say that the solution is straightforward.

a*(x-m)+b*(y-n)=0 is a general equation of a line thru [m,n]; it can be normalized by a^2+b^2=1
and one may suppose f(0,0)>=0 (because f=0 and -f=0 produce the same line).

The distance from (0,0) to the line is exactly |f(0,0)|/sqrt(a^2+b^2) = f(0,0). So, the tangency condition is just f(0,0)=r. That's all.

Happy New Year!

 

Why not the exact solution?

sol1:=dsolve({ode,ics}):
plot(rhs(sol1),x=0..1);

a:=5: b:=3:
F := proc(u)
   plots[display](
   plot([sqrt(a^2*cos(u)^2+b^2*sin(u)^2)+ a*cos(t)*cos(u)-b*sin(t)*sin(u),
             sqrt(a^2*sin(u)^2+b^2*cos(u)^2)+ a*cos(t)*sin(u)+b*sin(t)*cos(u), t=0..2*Pi]),
   plot([sqrt(a^2*cos(v)^2+b^2*sin(v)^2),sqrt(a^2*sin(v)^2+b^2*cos(v)^2),v=0..u],color=blue));
end proc:
plots[animate](F, [u], u = 0 .. Pi,  scaling = constrained );

Strangely, if you simply remove the assumptions in your sheet, it works.

To compute the (linear!) fit, you don't need the presence in memory of the vectors X,Y.
It is enugh to have X,Y in a text file (each line containing x_i, y_i)  and read it line by line, e.g.

f:="c:/temp/xy.txt";
A:=Vector(5);B:=Vector(3);
line:=readline(f):
while line<>0 do
x,y:=op(sscanf(line,"%f%f"));
A := A + Vector([1,x,x^2,x^3,x^4]);
B := B + Vector([y,x*y,x^2*y]);
line:=readline(f);
od:
p:= <A[1],A[2],A[3];A[2],A[3],A[4];A[3],A[4],A[5]>^(-1).B:
p[1]+p[2]*t+p[3]*t^2;  # the linear fit (parabola)


You have in two places subs({r=x, s=t} instead of subs({x=r, t=s}.

Note that it is not a good idea to use exclusively ":" instead of ";". You will have no chance to debug the code.

Compute the antiderivative first.

J:=Int((4/3)*t^(3/2)/sqrt(x-t), t = 0 .. x);

int((4/3)*t^(3/2)/sqrt(x-t), t):
J = limit(%,t=x,left) - limit(%,t=0,right) assuming x>0;



Edit. Actually,

value(J) assuming x>0;

also works.

 

 

It is obvious that b should not be allowed to be too large, otherwise the graph may become almost "dense" and will fit any data.

For example, DirectSearch ==>

DataFit(a*sin(b*x+c)+d, [b<=2,b>=0], Vector(X), Vector(Y), x, fitmethod = lms, strategy = globalsearch, method = quadratic);

gave

which seems resonable.

The arctan function seems to be not very "smart".

For example,

arctan( cos(t/2),sin(t/2)) assuming t>0,t<Pi/2;

simplifies to Pi/2 - t/2

but equivalent expressions

arctan( 1+cos(t),sin(t)) assuming t>0,t<Pi/2;

simplify(  arctan( 1+cos(2*t),sin(2*t))  ) assuming t>0,t<Pi/4;

do not.

In such circumstances I think that it would be better to use an inert variant; at least Maple will not try to simplify.

roundcoeffs:=proc(p,x)  local t;
 add(round~([coeffs(p,x,t)]) *~ [t])
end:

p:=3.5*x^2+3.2*x-6.5+88.3*x*y-y^3+a*y:
roundcoeffs(p,[x,y]);

Use

f := unapply(evalf(x^2/(sin(x)+x+1)), x);

 

restart;
f:=(x,y)->evalf(2*x*Int(sqrt(1+y^2*(t*x-1)^2/(1-(t*x-1)^2)), t = 0 .. 1)):
m:=19;n:=9;
G:=Matrix(m*n,3,datatype=float);
k:=0;
for i to m  do   for j to n do
k:=k+1;
G[k,1]:=2*i/(m+1); G[k,2]:=j/(n+1); G[k,3]:=f(2*i/(m+1),j/(n+1))
od od:


f_fit:=a1*x*sqrt(1+y^2*((a2*x-a3)^2/(1-(a4*x-1)^2)));
Statistics[NonlinearFit](f_fit, G, [x, y], initialvalues = [a1 = 1, a2 = 0, a3 = 0, a4=1/2],
output = [leastsquaresfunction, residuals]);
f1:=unapply(%[1],x,y);

printf("x    y    f(x,y)    f_fit(x,y)  \n");    
for x_ from 0.1 to 1.9 by 0.1 do
for y_ from 0.1 to 0.9 by 0.1 do
printf("%2.1f  %2.1f  %6f  %6f\n",x_,y_,f(x_,y_), f1(x_,y_)      );
od; od;

You may replace arcsin by wrongarcsin where:

wrongarcsin := x -> piecewise(x<0, arcsin(x)+2*Pi, arcsin(x));

You may simply "cut" the polynomial, or use:

hor:= (f,x,i) -> numapprox[hornerform](select(u -> degree(u,x)<=i, f) ,x);

f:=add( (10*i+1)*x^i, i=0..6);
hor(f,x,3);

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