2
votes
1answer
40 views

Existence of solution for matrix equation $ (I - \alpha A) \bar{x}=\bar{b}$

This is my first question in here and I would be really thankful if someone could help me with understanding the matter. I am solving a matrix equation $(I-\alpha A) \bar{x} = \bar{b}$ for a positive ...
1
vote
0answers
43 views

Which (approximative) methods are there to compute the inverse of a complicated function?

I have a complicated function $f(x)$ for which I want to compute the inverse $f^{-1}$ over a certain range $R(f): a \leq f(x) \leq b$. The only way to find the inverse I can think of is power series ...
1
vote
1answer
64 views

Power series and their inverses (radius of convergence of each)

Suppose I have a power series approximation $y$ to an invertible function $f(x)$, and I know that $y$ convergences around $x$ on an interval $(-R,R)$, $R$ being the radius of convergence. How are the ...
3
votes
2answers
88 views

Does $\exp(\ln(I+A))=I+A$ when $\|A\|<1$?

For matrices, I know certain equalities like $e^{A+B}=e^Ae^B$ aren't always true. I'm curious, do $\exp$ and $\ln$ serve as inverses? I saw earlier that if $\|A\|<1$, then $\ln(I+A)$ converges. My ...
0
votes
1answer
157 views

What's the limit of coefficient ratio for a reciprocating power series?

I have a question about the coefficient in the inverse of the power series. Assume $$ f=1-\sum_{i=1}^{\infty}(ck_i)x^i, $$ where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...
0
votes
2answers
154 views

Domain of convergence of $f^{-1}: \mathbb R ^N \mapsto \mathbb R^N$ taylor series

In another question, I ask about the topology of the singular manifold of the Jacobian. What i want to ask in here is about the radius of convergence of a Taylor series expansion of the inverse ...