Inversion in Fourier coefficients estimation problems

Question

Let us suppose we have a coefficient vetor:

coeff = {a0, a1, b1, a2, b2, ..., an, bn} 

with a total of (2*n + 1) elements. Those coefficients are applied in a Fourier series, such as:

I need to calculate both Jacobian (J) and Hessian (H) matrices from the coefficients, in the form:

Transpose[J] = 

and

H = 

both evaluated in a range of points x = {1, ... M}. Could anyone help me build the matrices J and H?

Answer 1

This is pretty easy. Note, I am using indexed variables instead of subscripts because these give less problems than subscripted variables:

For an example I use n=2, m=3. We first create the matrices without the special first row:

n = 2; m = 3;
rows0 = Table[0.5, m];
rows1 = Table[Cos [i x[j]], {i, n}, {j, m}];
rows2 = Table[Sin[i x[j]], {i, n}, {j, m}];
jac = Join[rows1, rows2];

hess = Table[ jac[[;; m, i]] . jac[[;; m, i]] + 
    jac[[m + 1 ;;, i]] . jac[[m + 1 ;;, i]], {i, 1, m}];
hess = DiagonalMatrix[hess] ;

Now we add the special rows and columns:

PrependTo[hess, Table[0, m]];
hess = Transpose[Prepend[Transpose@hess, Table[0, m + 1]]];
hess[[1, 1]] = rows0 . rows0;
hess // MatrixForm

PrependTo[jac, rows0] // MatrixForm