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

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How to define boundary conditions for a sphere to run reaction-diffusion equations on its surface?

I'm in a Biology lab, and we managed to simulate reaction-diffusion equations on a torus using periodic boundary conditions for a 2D matrix. We want to try doing the same on a sphere, but I'm a ...
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24 views

prove this using lagrange and newton divided difference error!

suppose f(x) is polynomial with degree of three.prove $f[{x}_{0},{x}_{1},{x}_{2}] = \frac{1}{2}{f}^{(2)}(\frac{{x}_{0}+{x}_{1}+{x}_{2}}{3})$ and ${x}_{0},{x}_{1},{x}_{2}$ are distinct point. I ...
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17 views

A question about the condition of quadrature formula

I am reading through my numerical mathematics script and I am currently in the chapter 4 (see listing) computer arithmetic direct solution of linear systems of equations polynomial interpolation ...
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56 views

Finding $ \max_{x \in [2,4]} \left| 2 x \cos(2 x) - (x - 2)^{2} \right| $.

This is a problem taken from Burden’s and Faires’ Numerical Analysis. Define $ f: \Bbb{R} \to \Bbb{R} $ by $$ \forall x \in \Bbb{R}: \quad f(x) \stackrel{\text{df}}{=} 2 x \cos(2 x) - (x - 2)^{2}. $$ ...
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11 views

Distribution of SDE numerically from Fokker-Planck.

I'm aware of some numerical methods related to SDEs such as Euler-Maruyama, Milstein etc. However, couldn't one also simulate the equivalent Fokker-Planck equation via finite element methods? This ...
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26 views

Is there a meaningful distinction between “direct” and “iterative” methods for solving equations?

I'll motivate this question with an example. The Abel-Ruffini theorem states that there is no general "formula" for the roots of polynomials of degree greater than 4. (Specifically it states that ...
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1answer
73 views

Name of function $(1+x)^n-1$

Is there any name for this formula $$(1+x)^n-1$$ When working with floating point numbers this can be calculated with much better precision for very small $|x|<1$ values using Taylor series ...
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1answer
24 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
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1answer
46 views

Runge-Kutta force at each time-step

Consider that I am solving a second order ODE using RK2/RK4. The ODE represents simple equations of motion: Equations of motion I am trying to solve: \begin{align} \frac{dx}{dt} &= v \\[.3em] ...
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1answer
45 views

What is the justification and intuition behind Muller's method's quadratic equation?

Usually we write the quadratic formula like this: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ But Muller's is written like this: Muller's method. Why is that?
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47 views

Interpolating $n$ points by piecewise quadratic polynomial

Given $n$ data points. Is it possible to interpolate them by piecewise quadratic polynomials with knots at the given data such that the quadratic interpolant is: (a) Once continuously differentiable? ...
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26 views

Shooting method with non-robin (or derivative) boundary conditions

I am trying to solve a problem in which I have to find all the values of $\lambda$ for which the boundary value problem has just one solution for each $a,b\in\mathbb{R}$. The problem is the following: ...
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26 views

Convergence of the Midpoint (Leapfrog) method when applied to $u'(t)=\lambda u(t)$?

So, I am trying to solve this question: where example 7.7 can be found here: http://i.stack.imgur.com/PVCIC.png My approach: Forward Euler (FE) method is given by: ...
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0answers
45 views

Really confused about LU decomposition and Doolittle algorithm

I'm really confused about the Doolittle algorithm, so I need some help. At the end of the description at wikipedia, it says It is clear that in order for this algorithm to work, one needs to ...
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1answer
26 views

Second order to first order equations

I need to write $$\frac{d^2\theta}{dt^2} + 4\sqrt{k}\,\frac{d\theta}{dt}+g\sin(\theta)=0$$ as a first order equation. What I have done so far is: Let $z = \frac{d\theta}{dt}$ Then $z' = ...
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2answers
48 views

Which topic is this question from? (Root-finding)

Prove that if $x=a$ is an approximation to one root of the equation $f(x)=0$, then $x=a-\frac{f(a)}{f'(a)}$ is a closer approximation. How to solve this question? Is this asking us to prove ...
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33 views

Question on the uniqueness of LU decomposition

Let $A \in \mathbb{K}^{n \times n}$ a matrix, and suppose that we can run the Gaussian elimination on $A$ without row or column interchange, so there exist the $LU$ decomposition of $A$. We define ...
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1answer
45 views

Name of the LU decomposition algorithm

On the wikipedia page of LU decomposition there is an algorithm that produce the decomposition. It is called Doolittle algorithm. I'm really interested who is Doolittle? Or from where the name comes ...
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27 views

Newton-Raphson for Discontinuous Spring Moment Balance

I am trying to solve the following problem where there should exist an equilibrium between spring forces and moment applied to a cylinder: To do so I am solving the equality: $ M_a - \sum_{i=0}^n ...
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1answer
28 views

Numerical phase plane?

In my Differential Dynamical Systems text book, I came across the following question: Sketch the local behavior you obtained in the phase plane and compare with a numerical phase plane plotter that ...
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27 views

How do I make a Maclaurin series expansion faster?

Suppose I want to approximate to e using the Maclaurin series. In this case, increased accuracy comes with at trade off of computation time. How do I make the Maclaurin series expand faster/ using a ...
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1answer
16 views

tridiagonal matrix with a corner entry from upper diagonal

I am trying a construct a matlab code such that it will solve an almost tridiagonal matrix. The input I want to put in is the main diagonal (a), the upper diagonal (b) and the lower diagonal and the ...
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42 views

Basic Corrected trapezoidal Rule for a Cubic Hermite Polynomial

The basic trapezoidal rule for approximating $$ I_f=∫^b_af(x)dx $$ is based on linear interpolation of f at x0=a and x1=b=a+h. Consider now a a cubic Hermite polynomial, interpolating both f and its ...
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9 views

Accurate numerical integration for “data times an analytical function”

The Question is as follows: I have an algorithm/data that provides me the value of a function $f(x,y,z)$ on the points of a grid. On the other hand I have an analytical function ...
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1answer
31 views

Numerical approximation of differentiation

I have the following task to solve: Let $b>x$ be defined, determine $w_0,w_1$ and $w_2$ in dependency of $b$ such that the approximation $f''(x) \approx w_0 f(x-h) + w_1 f(x) ...
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37 views

Numerical integration of a data set with uncertainties

I have a 1D data set {xi, yi} with no uncertainties in xi and with uncertainties dyi in yi. The resulting discrete function is monotonic and relatively smooth and I would like to integrate the ...
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37 views

How do I write this equation as a tridiagonal matrix to write the $n+1$ implicit formula?

I am doing a homework problem for my Applied Numerical Methods class, and I've worked the problem up to this point: $$ \large \frac{u_m^{n+1} - u_m^n}{k}=\frac{u_{m+1}^{n+1} - 2u_{m}^{n+1} + ...
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33 views

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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14 views

Simpson's rule and Hermite interpolant

For a uniform grid, $$x_n = -1 + nh$$ where $h = \frac{2}{N}$, I need to show that Simpson's rule is an $\mathcal{O}(h^5)$ integration rule. So far, I know to let $p(x)$ be the Hermite polynomial from ...
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46 views

Need help with a Crank Nicholson Method example problem.

I have an exam coming up and the professor released the sample test containing a Crank Nicolson question. I was out of town for those two lectures, so I missed the information. Even though I have ...
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1answer
20 views

efficient computation of Cholesky decomposition during tridiagonal matrix inverse

I have a symmetric, block tridiagonal matrix $A$. I am interested in computing the Cholesky decomposition of $A^{-1}$ (that is, I want to compute $R$, where $A^{-1}=RR^T$). I know how to compute the ...
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2answers
34 views

Analyzing derivative of function.

I have some function $g: [a,b] \to [a,b]$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $\forall x \in [a,b]: |g'(x)| \lt 1$. How can I find out if this is true or not? P.S. I ...
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2answers
55 views

Evaluate $-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$ in $\gamma$.

Evaluate $\gamma$ expressed, involving Lambert function, by $$-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$$ where $\gamma<1$. I doubt that it is possible to find a value for ...
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32 views

Can the following system be solved symbolically/analytically?

I have the following system of equations with variables $a,m$, and I'm wondering—can this system be solved symbolically/analytically? \begin{align} m &= 100 + \frac{ \left( 200 ...
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1answer
24 views

Having trouble with discretization and boundry value problems

I have the following homework question: Consider the boundary value problem $y''(x) + 5y'(x) − (2 + x)y(x) = e^x$ on $x ∈ (0, 2)$ with boundary conditions $3y(0) + y'(0) = 5$ and $y'(2) = 7$. ...
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1answer
50 views

The Gherkin (an egg shaped building) - equation for the curve in order to calculate the surface area of revolution

I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that ...
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1answer
37 views

Finding a root by bisection method in Excel

Working on a maths assignment and we're trying to use Excel for a bisection method. $$\frac12 e^{x/2}+\frac{1}{2x}-\frac32=0$$ Here is a pic, I can't get the formula to work with the exponent. This ...
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1answer
19 views

What is the physical meaning of 2 nodes being same while fitting an interpolating polynomial?

When we are trying to find out constants for Newton's interpolating polynomial, we use divided difference method to find the constants. Then we have Hermite-Genocchi formula to find those constants ...
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47 views

Asymptotic expansion of root of $\epsilon x \tan(x)=1$

Indicate a range of roots of $\epsilon x \tan(x)=1$ for which it is impossible to get an approximation using expansions. Since $\epsilon$ is small, I think for the equation to hold, we need ...
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2answers
51 views

A further question on asymptotic expansions of all real roots of xtan(x)=ϵ

I have asked a related question here How to find asymptotic expansions of all real roots of $x \tan(x)=\epsilon?$, however, when I discussed with my adviser today, he argued the solution is flawed. ...
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1answer
34 views

How do I write the generic finite difference approx of f'(x) using Lagrange interpolating polynomial approximation?

I have the following homework problem: (10 points) Differentiation Formulas by Lagrange Interpolating Polynomials. (a) Write the generic finite difference approximation to f'(x) using the Lagrange ...
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1answer
64 views

questions on polynomial Lagrange Interpolation of order $n$?

I ran in One Ex in my book when I‌ prepare for final exam on numerical method. how can help me how we solve such a problem? if $P(x)$ and $Q(x)$ be two polynomial Lagrange Interpolation of order $n$ ...
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1answer
42 views

How to integrate this using tan(x/2) substitution?

How do I integrate cos(x)/(sqrt(5)+cos(x)) ? I have been advised to use t = tan(x/2) substitution but ended up with a polynomial of degree 4 over one of degree 6 to integrate, which did not have an ...
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2answers
25 views

About minimal curvature of splines

I am given a the following problem set: Let $s$ be a natural cubic spline that interpolates a function $f \in \mathcal{C}^2 ([a,b])$ at points $a = x_0 < x_1 < \ldots < x_n =b$ with ...
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1answer
50 views

Quick question that I can't find anywhere online about Runge-Kutta

I'm writing a presentation on modelling fluid flow. We used Runge-Kutta second order to describe the flow as a numerical method. I just want verify that Runge-Kutta fourth order would be of a higher ...
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1answer
33 views

y''+y=cos(t) what is the smallest possible value of t for which |y(t)|>10?

Not sure if this is correct, but I was able to find a general solution of the form: y= c1cos(t)+c2sin(t)+(1/2)tsin(t) I'm not sure how I would go about finding the smallest possible value to make the ...
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20 views

Local Truncation error of Gaussian Quadrature

We have error estimate formula for Gaussian quadrature is: $$ \frac{(b-a)^{2n+1}(n!)^4}{(2n+1)[(2n)!]^3}f^{(2n)}(\xi) \; \; \; a < \xi < b$$ Suppose that we have 10 Gaussian points, so how can ...
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18 views

Solving System of Boundary Value problem

The boundary value problem: $$y'' + Q(t)y = f(t)$$ satisfying $$Ay(a) +By(b) = g$$ where A, B and Q are the matrices of order n. After calculation, we can get the form of solution will be $$y(x) = ...
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1answer
24 views

How to determine if an equation represents a cubic spline?

Given the equation $$ f(x) = \left\{ \begin{array}{lr} 2x^3+x^2+4x+5 & : 0 \le x \le 1\\ (x-1)^3 + 7(x-1)^2 + 12(x-1)+12 & : 1 \le x \le 2 \end{array} \right. ...
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62 views

How do I apply this PDE as an image filter?

I'm trying to preprocess a height map image with a helmholtz-type equation as described in this paper. The equation is: $$ddx(h') + ddy(h') + y(h'-h) = 0$$ I solved for h and got: ...