I use usually Mathematica in my scientific work. It possesses two interesting functions Series[f,{x,x0,n}] and InverseSeries[s]. The first one generates a power series expansion for a given function f(x) about the point x=x0 to order (x-x0)^n, and the second takes the series s, and gives a series for the inverse of the function represented by s.
Of course we can implement appropriate algorithm in any mathematical software but I am looking for such programs which have a built in one as standard library (package).
I tried to find this feature in Matlab but I did not find any information on this subject in manuals?
As far as I know the best option is Mathematica and has the InverseSeries[] command as you said, however some other options are: "flint" with command: fmpz_poly_revert_series; "Maple" with command: reversion; "SAGE" with command: reversion; or tical see "http://www.ticalc.org/archives/files/fileinfo/424/42499.html"
maple will help you , coefftayl
I write for you example I expand the function sin(x+y) in taylor about (0,0) and I find the coefficient of x, y ,x^2y, xy^2,
restart:
f:=(x,y)-> sin(x+y);
e:=mtaylor(f(x,y), [x, y], 8);
coeftayl(e, [x, y] = [0, 0], [1, 0]);
coeftayl(e, [x, y] = [0, 0], [0, 1]);
coeftayl(e, [x, y] = [0, 0], [1, 1]);
coeftayl(e, [x, y] = [0, 0], [2, 1]);
coeftayl(e, [x, y] = [0, 0], [1, 2]);
coeftayl(e, [x, y] = [0, 0], [2, 2]);
Dear Fredy,
Thank you for your interesting tips. I have to admit that I do not know the library FLINTS for C language and it seems very efficient for some numerical problems. Do you have any materials which can explain the implemented algorithms on inverse series in this library?
Dear Ayad,
Thank you for your example. What about inverse series? Could you also give a simple example?
I write simple example to solve inverse series , in this example H(x) is the function and h(x) its inverse this example not in general case you should enter the number of terms N, if this suitable for you I will try to write a general case
restart:
N:=2;
h:=x-> sum(a[n]*x^n,n=0..N);
H:=x-> sum(b[k]*x^k,k=0..N);
e:=expand(h(H(x))-x);
eq[0]:=eval(e,x=0);
for i from 1 to N do
eq[i]:=coeff(e, x^i);
od;
y:=seq(a[i], i = 0 .. N):
E:=seq(eq[i], i = 0 .. N):
sol:=solve({E},{y}):
for j from 1 to N+1 do;
a[j-1]:=rhs(sol[j]);
od;
# the inverse of H(x) is h(x) which given by;
Dear Mounesha,
could you send me a simple example with solution of inversion series in Matlab?
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