0
votes
1answer
47 views

Show that the iteration $x_{n+1} = x_n - 2\frac{f(x_n)}{f'(x_n)}$ converges quadratically to $x_*$ provided $x_0$ is sufficiently close to $x_*$

We have the following conditions for the above slightly-modified Newton's method iteration: $f$ is a real function of one real variable $f''$ is Lipschitz continuous $f(x_*) = f'(x_*) = 0$ I also ...
3
votes
1answer
112 views

How to solve this system of 3 equations with 3 variables?

I stumbled upon this system with constants $a_{i,j}>0$ that I want to solve for $x,y,z \in\mathbb{R}$: \begin{align} a_{2,1}y+a_{3,1}z=& x(y+z) \\ a_{1,2}x+a_{3,2}z=& y(x+z) \\ ...
1
vote
2answers
127 views

Showing that $A=B+\alpha \cdot I$ is an invertible matrix

Let $B$ be a non-zero random $n\times n$ matrix generated using the matlab command $B=rand(n,n)$. I need to show that $A=B+\alpha \cdot I$ is an invertible matrix, where $\alpha=\|B\|_{\infty}$. I ...
2
votes
1answer
77 views

SOR and Gauss-Seidel Method - Confusion

Can anyone explain to me the SOR Method for finding the root(s) of a function? Its supposedly very similar to the Gauss-Seidel method. The Gauss-Seidel method, from my understanding, is similar to ...
2
votes
1answer
59 views

Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$.

I want to Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$. Where $A$ is a matrix. I have a proof that involves Jordan Blocks. The proof is long and involved but it not ...
2
votes
1answer
238 views

Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
0
votes
0answers
73 views

relative error relation

Let $x$ be a non-null quantity. Let $\hat{x}$ be its approximation. I am trying to find the relation between: $\frac{\left | x-\hat{x} \right |}{\left | x \right |}$ and $ \frac{\left | x-\hat{x} ...
2
votes
1answer
95 views

Matrix inversion of an analytical function

Following problem: I have a function $f(x_1,x_2)$ and Im looking for the inverse $finv(x_1,x_2)$ of the function which is defined through: $\int f(x_1,y)\cdot finv(y,x_2) d y =\delta(x_1,x_2) $ ...
0
votes
1answer
482 views

Could explain me how eigenvector helps with compute gradient and how make differentiate operation on decrete space like digital image?

Could you explain me how eigenvector helps with compute gradient and how make differentiate operation on descrete space like digital image? I know that this question is so connected with computer ...