# 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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### Determining algebraically a point of intersection.

A student I was tutoring posed the question: "I know how to solve $$e^{-x} = \ln x$$ graphically, however how do you solve this algebraically?" I have been fiddling around with it for a while and I ...
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### Runge-Kutta Error Analysis

Could anyone explain to me how to reduce the error propagated by using Runge-Kutta of order 4? Or can anyone give me a nice reference to it.
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### How to solve the following system with parametric equation?

What is the maximum number of solutions for this system? Parameters $a_0, a_1, ..., a_{2013}$ could be any numbers. \begin{cases} y = a_0 + |x - a_1| + |x - a_2| + ... + |x - a_{2013}|\\ x^2 + y^2 = 1 ...
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### Implicit system differential equations

I came across a system of differential equations in the form: $\newcommand{\D}[1]{\frac{\mathrm{d}#1}{\mathrm{d}x}}$ \begin{align} f_1(x,y,z)\D{y}+f_2(x,y,z)\D{z}&=f_3(x,y,z),\\ f_4(x,y,z)\D{y}+...
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### Numerical methods and KKT in NLP

I am studying numerical methods and NLP. I started with gradient based methods, newton methods and KKT conditions. I found the following sentence: A local minimum is found by solving KKT conditions, ...
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### Numerical methods for ODE: Implicit, explicit, stability, stiffness

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
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### Numerical methods for ODE: Taylor vs. Interpolation approaches

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
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### Newton-Raphson method on manifolds

Has anyone explored the notion of the Newton-Raphson method on manifolds? Or to put it another way, on $\mathbb R^n$, is there a natural coordinate free way of defining an iterate of the Newton-...
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### Bounds on the size of Voronoi cells

I am working on an algorithm for which bounds on the size of voronoi cells will come in handy. Suppose that the domain $D$ is partitioned according to the Voronoi cells $D_1,\dots,D_n$ with Voronoi ...
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### Is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay

I'm studying for a test and I'd like to know is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay. I have no clue to verify the claim. Can anyone give me ...
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### LU Decomposition vs. QR Decomposition for similar problems

Suppose I want to solve the 2D Poisson equation with Neumann boundary conditions. The solution is non-unique up to an additive constant. I have previously asked a related question here for the 1D ...
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### Looking analytically if one formula is better than another one

I'm studying errors in functions on numerical methods. On my notes, I've written the Heron's Formula: Let $a\geq b\geq c$:$$A=\sqrt{p(p-a)(p-b)(p-c)}\ \ ,$$ where $$p=\frac{a+b+c}{2}.$$ This formula ...
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### Kink, Antikink, solving the wave equation numerically.

I've got a task of solving the wave equation with a potential $$u_{tt}-u_{xx}+V'(u)=0,$$ where $$V(u) = \frac{u^2(1-u)^2(1+2u)^2}{2}$$ on Python. I'm not exactly sure how to do this; my lecturer ...
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### Why use the logarithm of the relative error?

In my numerical analysis course, we had an assignment to use MATLAB to numerically solve the Poisson Equation $-\nabla\cdot\nabla u = 0$ in one dimension. We computed the numerical solution, plotted ...
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### Gaussian and binary probability random variables numerical reconstruction

in my probability class I was given this question dealing with MATLAB code the purpose of which is to create and re estimate the random variables Z1, Z2, which reads as follows: rng('default') ...
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### 'Stable' Ways To Invert A Matrix

So lets say that I need to invert a matrix that is generally dense and is poorly conditioned. What are some ways I can get an accurate inverse? Here are my candidates: SVD Inverse Inverse Via ...
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### Which error does one usually need to consider in numerical analysis?

When analyzing performance of a numerical method, I have considered and plotted $$\| x_n - x^* \|$$ where $x_n$ is the $n$-th iteration and $x^*$ is the true value (in our toy examples, it wasn't hard ...
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### Method to Linearise PDE

I have a Monge-Ampere-type PDE I wish to solve using a finite difference method: $$(1-u_{xx})(1-u_{yy}) -u_{xy}^2 = f(x,y).$$ Is there generally a preferred method for linearising the system after ...
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### A proof of the $QR$ algorithm when $A$ is symmetric and tridiagonal that doesn't involve shifting

Consider the following special case of the $QR$ algorithm: Let $A$ be symmetric and tridiagonal. Define the following sequence of matrices: $A_1 :=A$, and for $n \ge 1$ decompose $A_n=Q_n R_n$ ...
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### Proof of Runge's phenomenon for a concrete case

Let $f(x)=\frac{1}{1+25x^2}$ and range is $[-1,1]$. Given $n+1$ equidistant points $x_0 = -1,x_1,...,x_n = 1$ and their values $f(x_0),f(x_1),..,f(x_n)$, perform polynomial interpolation by the $n+1$ ...
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### Iterative methods: What happens when the spectral radius of a matrix is exactly 1?

I know that an iterative method (I'm using Jacobi and Gauss-Seidel in this case) will converge iff the spectral radius (max absolute value of eigenvalues) of its iterative matrix is strictly less than ...
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### How to find values of x where $a_i x$ are nearly integers? $a_i \in \Bbb R$

I have a set $\{a_i \in \Bbb R | \ i <=7 \}$, and I'm looking for a way to find values of $x$ where given $\epsilon > 0$, $$\forall i \ \exists n_i \in \Bbb{Z} \ \ |a_i x - n_i| < \epsilon$$ ...
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### make the Exponential interpolation vanish and show it has a unique solution

I have several questions concerning different parts of the question: a) Is it sufficient to show that ${1, e^x,...,e^{nx}}$ are linearly independent over the vector space of differentiable functions ...
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### Deriving a New Iteration Method by Solving a Quadratic Equation

My Question: Derive a new iteration method for solving $f(x)=0$ by solving the quadratic equation $$f(x_k)+f'(x_k)(x-x_k)+\frac{1}{2}f''(x_k)(x-x_k)^2=0$$ Complete your algorithm by specifying ...
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### Gauss-Green cubature in 2d

Hello friends of maths, I've given an arbitrary polygonal cross section (in cartesian coordinates $y$ and $z$). On this cross section, there acts an arbitrary stress-field $\sigma = f(y,z)$ as ...
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For the standard 4th order Runge Kutta: where the system is assumed to be smooth (so that the RHS has no discontinuous points) $\mathbf{y'} = \mathbf{F}(t,\mathbf{y})$ $\mathbf{y(t_0)} = \... 0answers 85 views ### The convergence of the fixed-point iteration for solving a cubic equation I have a third-grade polynomial of the form$Ax^3+Bx+C$and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this:$0=Ax^3+Bx+C$and for a given ... 0answers 42 views ### Numerical solution of$k\nabla^2 p(\vec{x}) = \nabla(\vec{f}(\vec{x})p(\vec{x}))$I have the following equation: $$k\nabla^2 p(\vec{x}) = \nabla(\vec{f}(\vec{x})p(\vec{x}))$$ where the constant$k>0$and vector field$\vec{f}(\vec{x})$are both known. I wish to numerically ... 0answers 58 views ### Integrating sine with Monte Carlo / Metropolis algorithm I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ... 0answers 43 views ### Newton's method of finding roots of an equation. Consider Newton's method on finding the roots of$x^3-x=0$, how to show that$x_n$converges to$1$for any$x_0>1/\sqrt{3}$? My attempt: The Newton's method says$x_{k+1}=x_k-\frac{x_k^3-x_k}{...
I am interested in the following integral $$\mathcal{I}=\int_{-\infty}^\infty\mathop{dz}\left[\frac{1}{\sqrt{a+b}(z^2)^{n/4}}-\frac{1}{\sqrt{a+b\cos^2\theta}(R^2+z^2)^{n/4}}\right],$$ where $R\ll 1$,...