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

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1answer
19 views

Normal system of the least square method

I'm trying to show the following. $Pa$ is the approximation system of $y$. I want to show that finding the minimmum for the function $$f(a,y)=||Pa-y||_2^2$$ is equivalent to solve the normal system of ...
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0answers
14 views

Equation involving Bessel and Struve functions

I need to solve the equation $Z(\gamma) = r$ of the function $$Z(\gamma) = 1 - \frac{2}{\gamma} \left(J_1(\gamma) - i H_1(\gamma)\right),$$ where $J_1$ is the Bessel function and $H_1$ the Struve ...
2
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1answer
31 views

$LDL^t$ Factorization Algorithm to find a factorization of the form A

For $$ \begin{pmatrix} 4 & 2 & 2 \\ 2 & 6 & 2 \\ 2 & 2 & 5 \\ \end{pmatrix} $$ I found that $$ L=\begin{pmatrix} 1 & ...
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0answers
26 views

LU factorization Algorithm

How to show that the LU Factorization Algorithm requires $n^3/3-n/3$ multiplications/divisions and $n^3/3-n^2/2+n/6$ additions/subtractions?
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1answer
22 views

Numerical solution of ordinary differential equations, multistep method

I try to solve the following question, but I have no clue why we have $x'$ in the RHS: The formula $ x_{n+1} = (1-A)x_n + A{x_{n-1}} + \frac{h}{12}[(5-A)x'_{n+1}+8(1+A)x'_n + (5A-1)x'_{n-1}] $ is ...
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0answers
19 views

Minimizing the average

Let's say I have a nice-behaving function $f: \Bbb R^n\to \Bbb R$, and I would like to find its maximum. Then I can apply gradient search algorithms to look for that, and to cope with possible ...
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0answers
18 views

Solve Black scholes PDE without using any transformation

I know that one of the methods of solving the black scholes PDE given by : $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV ...
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0answers
17 views

Finding the error terms of the Legendre polynomial (Numerical Analysis)

(a)Let $x_0$,..., $x_k$ be $k+1$ distinct points. Let $P_k(x)$ be the Lagrange interpolating polynomial of a smooth function $f(x)$ using these points. Show that ...
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0answers
13 views

Prove Ladyzhenskaya- Babuska-Brezzi condition for Poisson problem with homogenoeus Dirichlet boundary condition

I'm considering the problem: \begin{equation} \label{eq:PM} \begin{cases} \mathbf{u} -\nabla p=0\quad \text{ in } \Omega\\ \mathrm{div} \mathbf{u}=-f \quad \text{ in } \Omega\\ p=0\quad \text{ in } ...
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0answers
24 views

Finite difference of radial Laplace operator doesn't give a symmetric (hermition in general) matrix

I'm using the central difference to convert the radial part of Laplace operator into a matrix. $\nabla^2 u = \frac{\partial^2 u}{\partial r^2}+$ $\frac{1}{r}$ $\frac{\partial u}{\partial r}$ which ...
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1answer
24 views

For what maximum positive $k$ is $2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$ true?

I am trying to find the maximum value of $k$ such that the inequality $$2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$$ is satisfied. I impose restrictions that $n \in \mathbb{Z}$ with $n \geq ...
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0answers
26 views

Proof that $f[x_0,x_1,…,x_n,\epsilon,\epsilon]=\frac{f^{n+2)}(\eta)}{(n+2)!}$

Up to now i have the following rule for divided differences: Assuming $x_0 \le x_1 \le...\le x_n$ then If $x_0 \lt x_n$ then ...
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0answers
8 views

Forward difference problem

How to compute $\Delta^{2}(cosx)$ ? I try using relation $\Delta =E-1$ where $E$ is shifting operator. Please need help.
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1answer
16 views

Number of iterations for Gauss-Seidel

I am having some difficulty understanding the following solved problem: Question: Shouldn't we have $||T||^k_{\infty} ||e^{0}||_{\infty} \leq 10^{-6}$ instead? Where does the $5$ come from? And ...
2
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2answers
56 views

Evaluating integral with a singularity.

I want to evaluate an integral numerically that contains one singularity. The software I use for this is Python. The actual integral I want to evaluate is quite long with a lot of other constants so I ...
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1answer
25 views

Avoiding loss of significance without series.

How could the function $$f(x)=\frac{\sin x}{(x^2+1)^{1/2}-1}$$ be computed to avoid loss of significance? I know that $$f(x)=\frac{\sin x((x^2+1)^{1/2}+1)}{x^2}$$ But $x^2$ has a problem.... How to ...
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0answers
13 views

Why is it that if a numeric method has quadratic rate of convergence then it can reach d digits of precision in logd iterations?

I was trying to understand why a method with quadratic convergence can get close to a good solution in $\log d$ iterations. Assume we have a method that has the property that the number of digits of ...
3
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2answers
31 views

Numerical method with convergence greater than 2

It is well known fact that, for solving algebraic equations the bisection method have linear rate of convergence, secant method have rate of convergence equal to 1.62(approx.) and Newton-Raphson ...
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0answers
21 views

Weight Function in gaussian quadrature

My question is pretty simple, although I know of the properties that the weight function must follow , such as being well defined,positive,continuos and integrable on the interval . I do not know how ...
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0answers
37 views

Method check: numerically calculate 1D integral of a 3D function

I have a function $f(r)$ where $r=\sqrt{x^{2}+y^{2}+z^{2}}$, $\forall x,y,z \geq 0$. I know the values of the function at many points, essentially I have a table of values with $r$ and the ...
0
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1answer
41 views

Intersecting three rays and a sphere of known radius

So I actually solved this problem using an iterative solver, but it annoys me because as far as I can tell it should be possible to do it directly. I have three known 3D "rays" that all start at the ...
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2answers
24 views

Solving a Linear IVP [closed]

I need help solving this linear Initial value problem: $$y'=-L(y(t)-\phi(t))+\phi'(t) \\ y(0)=y_0$$ where $\phi(t)=\cos(30t)$.
0
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1answer
16 views

Equivalents definition of linear convergence

Suppose that the sequence $\{x_n\}$ converges to $0$. I want to prove that these definitions are equivalent: a) We say that $\{x_n\}$ converges linearly to $0$, if there exists a number $q \in (0, ...
2
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1answer
58 views

How to Numerically Solve an integral equation.

First I really doon't have any background with integral equations! That said, I would like to solve the following: $$\int_a^b \frac{K(t)}{t-x} \phi(t) dt=f(x) , a<x<b$$ where$$\int_a^b \phi(t) ...
2
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5answers
68 views

Pade approximant for the function $\sqrt{1+x}$

I'm doing the followiwng exercise: The objective is to obtain an approximation for the square root of any given number using the expression $$\sqrt{1+x}=f(x)\cdot\sqrt{1+g(x)}$$ where ...
1
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2answers
21 views

What does it mean by the approximation $\int_a^bf(x)dx\approx\sum_{i=0}^nA_if(x_i)$ is exact for all polynomials of degree up to $2n+1$?

There is these notes about Gaussian Quadrature and I am trying to understand what does the sentence "is exact for all polynomials of degree up to $2n+1$" actually mean. Gaussian Quadrature - General ...
0
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1answer
17 views

Some doubts on Simpsons Rule by the Method of Undetermined Coefficients

There is this note about Quadratic Interpolation by Simpsons Rule that I don't quite understand how to get the LHS. Simpsons Rule by the Method of Undetermined Coefficients We seek an approximation ...
0
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1answer
34 views

Integrate $\int_{a}^{b}[\int_{a}^{x} (t-a)(t-b)(t-\frac{a+b}{2}) dt]dx $

The following integral arrives me while reading Atkinson's book on Numerical Analysis. It's pretty simple but I'm not sure what theorems should I be using: I have to integrate the following: ...
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3answers
34 views

Numerical partial derivative of an inverse function

We have a function whose inverse cannot be written in analytical form, such as: $$f(x)=kx^3+x$$ How to find $\frac{\partial}{\partial k}f^{-1}$ ? $f^{-1}(y)$ for a given $y$ can be easily found ...
0
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1answer
25 views

Number of continuous derivatives of a piecewise quadratic polynomial

I've been trying to reason through the following problem: Suppose that we interpolate $n+1$ data points with a piecewise quadratic polynomial. How many continuous derivatives can this interpolating ...
3
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1answer
25 views

Local Truncation Error of Implicit Euler

The LTE of an implicit Euler method is $O(h^2)$ because the method has order $O(h)$, but I'm not sure where to get started in proving this arithmetically. Any help would be appreciated. Thank you!
1
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1answer
62 views

Implicit Euler method and explicit Euler method

I wanna know what is the difference between explicit Euler's method and implicit Euler's method. And is the local truncation error for both of them is $O(h)$ and the coefficient of the $O(h)$ term is ...
1
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2answers
18 views

Numerical algorithm: Spectral function -> Continued Fraction

I am trying to code up a numerical algorithm which takes a spectral function of the form $$c(\zeta) = w_0 +\sum_{m=1}^N \frac{w_m}{\lambda_m+\zeta}$$ into a continued fraction of the form $$c(\zeta) = ...
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0answers
18 views

Numerical Analysis - Richardson Method

About the Richardson Method, how should we initialize the function? I am trying to solve: $\frac{\partial u}{\partial t}-\frac{\partial^{2}u}{\partial x^{2}}=0$ where $x\in [0,2]$ and $u(x,0)=sin(2\pi ...
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0answers
19 views

Quadrature over a (smooth, compact, convex, etc.) Riemannian manifold

Problem setting Consider three points on the surface of the earth (which I want to assume to be a perfect ellipsoid here) that are pairwise sufficiently close for unique geodesics to be found between ...
2
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3answers
44 views

Numerical computation of $\sqrt{x} - \sqrt{y}$ where $x$ and $y$ are almost equal

A problem asks the following Compute $\sqrt{4321} - \sqrt{4318}$ using 4 decimal digit arithmetic. This comes out to $0.02,$ when the exact result should be $0.02282...$. It then asks the ...
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0answers
42 views

Reverse engineer numerical results to fractions of remarkable numbers?

Numerical methods output decimal numbers that oftentimes result from the division of two (or more) numbers: $1.5708... = \frac{\pi}{2}$ $0.3679... = \frac{1}{e}$ $0.7071... = \frac{\sqrt2}{2}$ ...
0
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1answer
21 views

Implicit Euler's Method

Show that two solutions $x_n$ and $y_n$ generated by implicit Euler satisfy the inequality $|x_n - y_n|$ $\leq$ $|x_0 - y_0|$. Can someone push me in the right direction by explaining to me how ...
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0answers
24 views

Where am I going wrong in my hermite polynomial

So I have tried a few different ways and I arrive at one answer P(x)=8$x^3$-11$x^2$+2x+1 Values are \begin{matrix} x & f(x) & f'(x) \\ 0 & 1 & 2 \\ 1 & ...
2
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2answers
97 views

How was the Runge-Kutta method derived?

By K values, I mean the values described here: https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#Explicit_Runge.E2.80.93Kutta_methods I know how the K values in the Runge-Kutta method can be ...
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0answers
36 views

Unconstrained minimisation problem Newton's method

min f(x) = $ x_1^4 + 2x_1^2x_2^2 + x_2^4 $ is an unconstrained min problem. The first question asks to show that $(0,0)$ is the unique minimiser. I have done the following.. Would I need to add ...
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0answers
10 views

Order of convergence is equals or greater than order of consistency

I proved that if the Numerical Method is stabile and consistent then the method is convergent . But I don't know how to prove that the order of convergence is equals or greater than order of ...
0
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1answer
45 views

Determine numerical infinity for Schrodinger equation $−\psi''(z) − (iz)^ N \psi(z) = E\psi(z)$

Consider the following one dimensional Schrodinger equation within the complex plane of $z$ $$ −ψ''(z) − (iz)^ N ψ(z) = Eψ(z). $$ where $N$ can be any real number, the boundary condition is $ψ(z) → 0$ ...
1
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1answer
25 views

Numerical solver for maxwell equations?

Just curious if someone has come across a package where I can simply solve the basic maxwell equations(just the curl equations). I'm just interested in solving it on a 2-d plate out of interest. ...
1
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0answers
15 views

Alpha Max Plus Beta Min Calculation

I read about the Alpha Max Plus Beta Min algorithm described here. Here is a screenshot from the wikipedia page: I think understand what the algorithm is supposed to do. It makes an approximation ...
0
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1answer
52 views

Perturbed differential equation

Given a differential equation $y'(t)=f(t,y(t))$, where f satisfies the condition $(u-v)(f(t,u)-f(t,v))\le0$ for all $u$ and $v$. Suppose $W$ satisfies a perturbed differential equation ...
2
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1answer
66 views

Differential equation and exact solutions

Given a differential equation $y'(t)=f(t,y(t))$, where f satisfies the condition $(u-v)(f(t,u)-f(t,v))\le0$ for all $u$ and $v$. If $U(t)$ and $V(t)$ are exact solutions, I want to show that ...
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0answers
38 views

What is the typical $\epsilon$?

In doing some self-learning in numerical methods I having come across the following a number of times. $\epsilon$ is the smallest computational unit such that, $x + \epsilon = x$. What value does ...
0
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1answer
15 views

Accuracy of $e^x$ approximation using remainder of Taylor Series Approximation.

The Problem: If we consider approximating $e^x$ on $[-1,1]$, the Taylor Theorem for $x_0 =0$ says we can represent $e^x$ using a polynomial with a (known) remainder: $e^x =\{1 + x + \frac{1}{2!}x^2 ...
0
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1answer
18 views

Find condition number for vector of roots

Consider $f(z)=z^2+az+b$ we put it's root in vector $[z_1, z_2]^T \in \mathbb{C}^2$. Find condition number in maximum norm of finding the root vector when changing variable a where $a=-2, b=3$ I know ...