New answers tagged matlab
5
Q1: No. Here's a counter-example:
>> A = eye(4)*1e-300
A =
1.0e-300 *
1.0000 0 0 0
0 1.0000 0 0
0 0 1.0000 0
0 0 0 1.0000
>> rank(A)
ans =
4
>>
>> rank([A, ones(4, 1)])
ans =
1
>>
so, if the added column ...
1
Adding another answer to the second part of your question:
"Besides a constant step size, is there an easy variable-step size I could implement and play with?"
An easy way to implement some variable step size would be the following algorithm:
Consider your cost function $\Phi(x)$ you would like to minimize.
Choose an initial step size $\alpha$
...
3
Suppose you want to minimize
$$\Phi(x)=\frac{1}{2}||Ax-b||^2$$
The gradient is
$$\frac{\partial \Phi}{\partial x} = A^T(Ax-b)$$
The step size to guarantee convergence is
$$\alpha=||A^TA||^{-1}$$
Why? The direct solution to the problem is:
$$x_{opt}=(A^TA)^{-1}A^Tb$$
This can be achieved iteratively if we look at the update on the estimate $x_k$. Suppose we ...
3
You lose convergence order, or in the worst case convergence altogether. You can try this out: Take
$$
f(x) = \begin{cases} 0 & \text{if $x<0$} \\ x^2 & \text{if $x\ge 0$}.\end{cases}
$$
The function is differentiable, but not twice differentiable at $x=0$. It's exact derivative is $f'(0)=0$. Now compute the (second-order) symmetric finite ...
2
Newton's method can refer either to a method for solving $f(x)=0$ where $f: R^{n} \rightarrow R^{n}$, or to a method for minimizing/maximizing a function $g: R^{n} \rightarrow R$ by solving the system of equations $\nabla g(x)=0$.
Your function $h$ maps $R^{2}$ to $R$ and you want to find a zero of the function. This is typically done by minimizing
$\min h(...
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