Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation...

[tommy1729]

serial summation
0.709801988103 towards 2'nd fixpoint: \( \sum_{h=0}^{\infty} (-1)^h * f^{\circ h}(1.0) \)
0.419756033790 towards 1'st fixpoint: \( \sum_{h=0}^{\infty} (-1)^h * f^{\circ -h}(1.0) \)

Matrix-method:
0.580243966210 towards 2'nd fixpoint // incorrect, doesn't match serial summation
0.419756033790 towards 1'st fixpoint // matches serial summation

a reason might be this : the vandermonde matrix must have a determinant <> 1 for almost all functions.

hence the determinant of f^h(x) and f^-h(x) cannot both satisfy to be in the radius ( determinant < 1 = within radius 1 ) for (1 - A)^-1.

basicly just taylor series radius argument for matrices.

have you considered this ?

if i am correct about that , the question becomes : what if the determinant of f(x) is 1 ? will the matrix method agree on both fixpoints ?