# Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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### Modified Euler Method: region of absolute stability

I am having trouble finding the region of absolute stability for modified Euler method: \begin{align} w^*_{i+1}&=w_i+hf(t_i,w_i) \\ w_{i+1}&=w_i+\frac{h}2\left[f(t_i,w_i)+f(t_{i+1},w^*_{i+1})\...
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### Reference for Shooting Method

Consider the following setup. We have a second order boundary value problem: $$\dfrac{d^2y}{dx^2}=F(x,y,dy/dx);\qquad y(x_0)=y_0,\quad y(x_f)=y_f.$$ A numerical approach is to almost first write as ...
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### Derive Lanczos algorithm by imitating the derivation of Arnoldi iteration algorithm.

If A is Hermitian, then everything above simplifies (e.g., Hessenberg matrices turn into tridiagonal), and we get what is know in the literature separately as Lanczos iteration. My attempt:- ...
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### Recommended iterative numeric solver for generalized eigenvalue problem?

I have a generalized eigenvalue problem of the form $Av=\lambda Bv$, with the following conditions: $A$ is symmetric, but not sure if it is positive-definite. $B$ is diagonal with positive entries $A$...
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### Two SDEs that share a Brownian motion

I have the system \begin{align} dX_t & = \beta(\alpha - X_t)dt + Y_t dt + dB^1_t + dB^2_t \newline dY_t & = \beta(\alpha - Y_t)dt + X_t dt + dB^2_t + dB^3_t \end{align} Now, I am creating a ...
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### Fit of a (discrete) Pareto distribution for prices of goods

I have goods in a price range from 200 up to a few 10 Mio. I have price bins $$P_1 = [200,500[, \quad P_2 = [500, 3000[, \quad P_3 = [3000, \infty[$$ I would restrict the latter interval to ...
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### Can changing Gradient Descent step size/learing rate from constant 1/L to Armijo or exact line search change the convergence rate?

If instead of the classical $1/L$ constant step size we have adaptive step sizes chosen with exact line search or Armijo (let's say) can this alter the Big-O complexity of the convergence rate? Here: ...
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### For the given function, find each fixed point and decide whether fixed point iteration is locally convergent to it

A theorem in my book states: Let $g$ be a function and $r$ a number fixed by the function (i.e. $g(r) = r$). Assume $g$ is continuously differentiable, $g(r) = r$ and $|g'(r)| < 1$, then the ...
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### Computing Hessian in Python using finite differences

I am computing the Hessian of a scalar field, and tried using numdifftools. This seems to work, but was quite slow so I wrote my own approach using finite differences. Here is my code for the Hessian:...
1 vote
All I want to do is numerically map the upper-half plane $\mathbb H:=\{z\in\mathbb C:\Im(z)>0\}$ to the unit square $[0,1)^2$. How this can be done is described in the Wikipedia article of the ...