# Tagged Questions

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

22 views

### Could anybody please clarify the relationship between numerical stability and accuracy?

I was reading a paper and came up with this statement. Stability merely avoids uncontrolled error growth but cannot guarantee actual numerical accuracy. From what I understood from the concept of ...
20 views

### Newton's method for nth roots of complex numbers

Is it possible to use Newton's method to compute roots of complex numbers, say $\sqrt[n]{a+ib}$ to any desired accuracy? If yes,for what initial values will converge?
12 views

19 views

### Principal Component Analysis to find correlations in data

first of all, I'm really not a fan of stochastic methods and so I didn't spent much time to get used to it in the past. But now I have run a PCA and got some questions regarding this. I already found ...
12 views

### Use QR decomposition to find LS decomposition, where S is orthonormal and L is a lower triangular matrix

Assuming that we have a black box that can find the QR decomposition of a matrix $A$, how can we use this black box to find a decomposition $A=LS$ where $L$ is a lower triangular matrix and $S$ is an ...
14 views

### Numerical Method for fitting parameters of an explicit integration to actual data

I have a heat transfer system described by, $$\{\dot{T}\} = [C^{-1}]\left([K]\{T\} + \{F\} \right)$$ where ${T}$ is a vector of the nodal temperatures of the system. From initial conditions I am able ...
25 views

32 views

### The convergence of an infinite radical involving $\cos(\alpha/3)$

By using the triple angle formula for the cosine, $\cos 3\alpha$, we get the cubic equation $4x^3-3x = \cos \alpha$. Now, by expressing $x$ as $x = \frac{1}{2}\sqrt{3+\frac{\cos \alpha}{x}}$ ...
53 views

### Check numerically the definite-positiveness on linear subspaces

I have a given matrix $W\in \mathbb{R}^n$ with known fixed entries. I would like to check the definite-positiveness of $W$ on appropriate linear subspaces. Typically I would like to show (...
37 views

### interpolation polynomial error

We have points $x_0=a \lt x_1 \lt x_2 ....x_n=b$ and $\;w_{n+1}(x)=\prod_{k=0}^{n}{(x-x_k)}$. Let $h=max_{j=0...n}|x_j-x_{j-1}|$ Let $f \in C^{n+1}[a;b]$ and $p_n\in \mathbb P_n$ be the ...
44 views

### Adams-Bashforth-Moulton two-step predictor-corrector

Can someone help me this question please, this is from past years exam.
21 views

### Preconditioning : ILU($\emptyset$) factorization and SSOR relation?

Let's suppose we have a matrix A = D - L - U , where D,L,U are diagonal , strictly lower and strictly upper triangular,respectively. Generally the preconditioning matrix, according to SSOR(Symmetric ...
20 views

### Interplanetary Optimisation using a simulator with PyGMO or SciPy

I am currently trying to use a N-body gravity simulator to model a spacecraft trajectory and using the simulator as a BlackBox to optimise the trajectory. I am thinking of using basin hopping/ ...
63 views

### Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$Y(k)=\frac{1}{k}\sum_{i=1}^k X_i$$ Notice that $Y(k)$ is ...
35 views

### Intersection of a helicoid and a line

I have a Helicoid described by the following parametric equations: $$x = u\cos(v)$$ $$y = cv$$ $$z = u\sin(v)$$ The helicoid revolves around the y-axis: Eliminating $u$ and $v$, we obtain the ...
66 views

### Error in trapezoidal rule via integral mean value theorem

During a class, I saw the following analysis of the error term in the trapezoidal rule For $f \in C^2([a,b])$, $\int_a^b f(x) \,dx - \frac{b-a}{2}[f(a)+f(b)] = -\frac{(b-a)^3}{12}f''(\eta)$ for ...
36 views

### Estimate counts with different sample sizes

Given an arbitrary time period, lets say one week, but it could be five days, one month etc.., I have a sample from a population. My sample consists of shoppers at a store. For week one my sample is ...
32 views

### Matrices over integer fields to solve complex polynomials.

Inspired by the fruitful answer to this question regarding numerically solving polynomial equations in terms of simpler fields (in that case representing real numbers as fractions of integers), I ...
42 views

### Solving differential equation with numerical method

How do I solve this differential equation numerically for $1<\beta<\inf$ knowing $\sigma$ ? $\frac{\gamma(u)}{du}=\beta\cdot\gamma(u)\cdot\gamma(\beta\cdot u)$ $\gamma(0)=\sigma$ Thank you ...
14 views

$\DeclareMathOperator{\curl}{curl}\renewcommand{\div}{\mathop{\mathrm{div}}}$ Consider the following system of equations for the unknown vector field $A$ in the unit 3d ball $B$ with boundary $S$: ... 0answers 28 views ### Newton conjugate gradient algorithm In this video, the professor describes an algorithm that can be used to find the minimum value of the cost function for linear regression. Here, the cost function is f, the gradient is g_k where ... 1answer 45 views ### Proof Bisection method [closed] Denote the successive intervals that arise in the bisection method by [a_1\,, b_1], [a_2\,,\, b_2], [a_3\,,\, b_3], and so on. Let c_n be the midpoint of [a_n, b_n]. Show that |c_n − c_{n+1}| = ... 0answers 28 views ### Second-order differential equations methods I'm looking for a method called 'Inexact Method' Idk if it goes by another name, here's what I do know: It's one of the two 2nd order differential equations methods. The other method is called '... 1answer 33 views ### To solve large systems of multivariate polynomial equations Nicolas Courtois et al. proposed the eXtended Linearization(XL) method to solve the systems of multivariate polynomial equations and analyzed the time complexity. Polynomial when the number of (... 1answer 49 views ### Iterations with matrices over simple fields approximating solutions for more complicated fields. Inspired by this question I started wondering if there exist some systematic way to construct approximation to any number one can find using matrices over a preferrably simpler field. In the question ... 2answers 49 views ### Numerical solutions of partial differential equations I'm studying mathematical physics and working on numerical solutions of partial differential equations. I am having trouble understanding the way we solve partial dif. equations, e.g., \frac{\... 2answers 23 views ### 2nd Order Runge-Kutta Method Could someone please help me with the next step of this 2nd order Runge-Kutta method. I am solving the ODE \begin{align*} x'=-\frac{x(t)}{2}, \ \ x(0)=2. \end{align*} I wish to use the second order ... 1answer 28 views ### Reconstructing a matrix Before reading on, let me acknowledge that this problem is solveable generally, however I am interested in knowing if a certain form of solution exists. If I have a square complex unitary n\times n... 0answers 36 views ### Newton's method Let a > 0. Starting from a convenient equation and using Newton's method, deduct a method to approximate 1/\sqrt a without division. How do you choose the starting value? Which is the stop ... 3answers 51 views ### For how many weeks can I group my students? I thought of an interesting question that I don't know how to solve. I imagine there are numeric results out there somewhere, but I don't know if this question has a formal name; if anyone could link ... 0answers 18 views ### What makes a geometric construction more or less stable? As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ... 2answers 70 views ### Square root of x : \sqrt{x} (Numerical Method)f(x) = \sqrt{x}$$has to be approximated by polynomial interpolation p(x_n) = f(x_n) with the positions \{x_n\} = \{1,4\}. For such problem which method is the fastest? And find p(2). My ... 1answer 21 views ### Condition number interpretation I have a (nonlinear) problem with two variables, for which I computed a relative condition number as$$K_{rel}(x_1,x_2) = \max\{1, c\},$$where I had$$\Bigg| \frac{f(x_1, x_2) - f(\tilde{x_1},\tilde{...
I want to compute the greatest $a>0$ for given $\epsilon>0$ such that $$max_{x\in [-a,a]}|f(x)-p_2(x)| < \epsilon$$ where $a$ is the distance between two grid points and the maximum is the ...
Hello I have a question in regard to moment generating functions and something I noticed but wasnt to clear on. Say that a random variable $X$ has $$MGF=M_{X}(t)=\frac{4}{4-t^2}$$ for $-2 \lt t \lt 2$...