# Tagged Questions

Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

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### Finding $1/x^2 + 1/x^3 + 1/x^5 + \dots$

The following function came up in my work: $$f(x)=\sum_{p\text{ prime}}\frac{1}{x^p}=\frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+\cdots.$$ Naturally, this converges for ...
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### Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
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### Why does the pressure term complicate numerical methods for Navier-Stokes Equations?

I'm looking to code a solver for the Navier-Stokes equations. I will be using finite differences with the method of lines. Two questions: What is the significance of the pressure term in the full 3-...
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### What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
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### Squarefree products of a class of primes

Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4. Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up ...
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### Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
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### Mean Absolute Deviation for a Stable Distribution as a Function of the Tail Exponent

Consider the standard Lévy-Stable (or Alpha Stable) distribution $S(\alpha,\beta, \mu, \sigma)$ where $\alpha$ is the tail exponent, $1 \leq \alpha \leq 2$. Picking the symmetric case with $0$ mean (...
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### How to compute Fourier coefficients using a cubic spline-corrected FFT?

I'm not particularly experienced in numerical analysis, and so I recently had quite a massive shock when I discovered that sampling a smooth function and computing the FFT of the result does not ...
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### A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} \psi(\alpha)-\...
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### Numerical integration of highly peaked 2D functions

I'm working on a problem that requires me to numerically integrate irregular and highly peaked functions $f:\mathbb{R}^2 \to \mathbb{R}, \, (x,y) \mapsto f(x,y)$ without a closed form. The method ...
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I am using a box-scheme for solving partial differential equation. The function is approximated with: $$f_p=\psi\cdot\left(\theta\cdot f_{j}^{k+1}+(1-\theta)\cdot f_{j}^{k} \right) + (1-\psi)\cdot \... 0answers 303 views ### Test for equivalence of algebraic expressions We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ... 0answers 186 views ### Inexact Newton method. Let's a nonlinear function  f:[-1,+1]^N\subset\mathbb{R}^N\to\mathbb{R}^N,\; N\in\mathbb{N},  such that the the sequence generated by the method of Newton-Raphson$$ x_{n+1}=x_n-[Df(x_n)]^{-1}\cdot ...
Suppose we have $N$ vectors $\vec{x}_1, \vec{x}_2,\dots,\vec{x}_N$. $\vec{x}_i$ is a $M$-dimensional vector: $\vec{x}_i = \left[ x_{i1}\;\; x_{i2}\;\; \dots \;\;x_{iM}\right]^T$ with all $x_{ij}>0$....
For our homework we should write a program, that creates Lagrange base polynomials $L_k(x)$ based on a few sampling points $x_i$. Now i am eager to develop a formula to be able to compute the $\ell$-...