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

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Matlab project - Jacobi method for tridiagonal matrices…

I have to do a project in Matlab to my University and I don't quite understand what I should do. I was given script that solves systems of equations with Jacobi's method with given tolerance and ...
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3answers
83 views

Prove that $x^3 -3x^2 +6 = 0$ has only one real root

I know that if I take the derivative of $$x^3 -3x^2 +6 = 0$$ and prove it is always greater than zero, I'll find that this functions is always increasing, and therefore if I find an interval where ...
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1answer
43 views

Integral over the unit ball in $\mathbb{R}^n$

Let $f(x)=|x|^r$ on $B_1(0)$ real valued function.Where $B_1(0)$ is the unit ball in $\mathbb{R}^n$. I am trying to show that if $r>1-n$ f has a weak derivative. ATTEMPT: I know from the ...
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1answer
86 views

Newton's method convergence implementation

How can I solwe this problem: Experimentaly examine convergence Newton's method for conformation: \begin{align} 2x^3-y^2-1=0 \\ xy^3-y-4=0 \end{align} for various loaded inputs with start points ...
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1answer
42 views

Large error after factoring h from fourth-order Runge–Kutta method

Consider $$\frac{dy}{dx} = 2x-y, \qquad y(0)=1$$ That has an exact solution $y(t) = 2t+3 e^{-t}-2$. But I want a numerical solution. So I decided to use forth-order Runge–Kutta method ...
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1answer
244 views

Solve a viscous Burgers' equation with a Newton-GMRes method

I implemented a preconditioner for a GMRes method. To test this preconditioner I want to solve this one dimensional viscous Burgers' equation $$\partial_t u(x,t) + u(x,t) \partial_x ...
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0answers
35 views

3 step ODE method using interpolating polynomial

I am trying to find a 3-step method of the form $y_{i+1}=y_{i-2}+b_0f_{i-2}+b_1f_{i-1}+b_2f_i$ to solve the ODE $y'=f(t,y)$ by using an interpolating polynomial and then finding $\int_0^{3h} ...
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1answer
32 views

Galerkin Method: Why Set the Residuals to Zero?

I don't understand why the Galerkin method weighs the residual by the shape functions and sets it equal to zero. I'd like to know the reason why. Any intuitive explanation would be greatly ...
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1answer
97 views

Two dimensional Numerical integration

If I am numerically integrating my function $f(x,y)$ on a two dimensional cartesian grid, say $[0,1]\times[0,1]$ with $\Delta x=\Delta y$ using the values at the center of each cell, what is the order ...
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1answer
96 views

Discretization of differential equations, solving boundary value problem

For $\Omega = (0,1)^2 \subseteq \mathbb R^2$ and $f \in C(\Omega)$ consider the boundary value problem: $- \Delta u(x,y) = f(x,y)~ \forall (x,y) \in \Omega,~ u(x,y)=0 ~ \forall (x,y) \in \partial ...
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1answer
134 views

Bifurcation Example Using Newton's Method

I am studying dynamical systems as part of a research project. I have been using Newton's Method and studying the dynamic properties. Does anyone know where I could find a relatively simple example ...
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34 views

Conjugate Gradient

Why sometime our conjugate gradient routine never reaches the stop condition even if the result is correct? As stop condition we use the following: $$\delta > \epsilon^2 \delta_0$$ To avoid ...
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0answers
54 views

extension of trigonometric functions as basis functions to higher dimensions

Trigonometric functions forms an orthonormal basis functions for $L^2[a,b]$, with corresponding normalization coefficients. I want to know if this result can be extended to higher dimensions. For ...
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58 views

Solution of inhomogenous ODE (4th order)

Hello stackexchangers, I have an inhomogenous ODE in 4th order. This ODE is the constitutive law to describe a material by using the "Wiechert model" (p. 15) which is given by $p_0\sigma + ...
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1answer
51 views

How many fixed points does a function have?

How many fixed points does $$f(x) = \sin\left(\sin\left(\dfrac{x^3}{11} + \dfrac{x^2}{7} + \dfrac{x}{3}\right)\right) $$ have in the range $[0,1]$? How does one come to solve this? I thought ...
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2answers
180 views

Finite difference method

I wanted to ask something regarding the finite difference approximation. I used the finite difference to calculate the numerical derivatives of my function. The finite difference is given by the ...
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1answer
74 views

Distributed Newton methods for large scale problems

I am keen to know about the literature landscape for distributed convex optimization methods which use second order information like the Newton step. This is as such a less evolved area compared to ...
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1answer
249 views

Relating convergence theorem for Newton-Raphson method to Newton fractal

I have created a Newton fractal (below) using the Newton-Raphson method to find the five solutions of f = (z^5-1) The convergence theorem of Newtons method say ...
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65 views

A-stability of Runge-Kutta methods

I am studying Runge-Kutta methods, but I can't understand why explicit Runge-Kutta methods are not A-stable. Someone can explain it to me?
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1answer
93 views

Higher order numerical PDE schemes near boundaries, implementation in MATLAB

Followup to my previous question. The first order scheme proved unstable for my pde: $$f_t + A y f_x - B x f_y =0$$ So I'm looking to implement a higher order scheme (using these tables). I was ...
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0answers
21 views

Estimate accuracy of inaccurate fast function having exact values of slow one

Let’s say we have functions $F$ and $H$ to calculate a series $S$ of integers and that: $S_{i} = H(x_{i}) = F(x_{i}) + e_{i}$ Being $e_{i}$ the error of $F(x_{i})$ to estimate $S_{i}$ The problem ...
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2answers
161 views

Looking for numerical methods for finding roots of convex vector function ${\bf f}({\bf x})={\bf 0}$

Consider the function ${\bf f}:\mathbb{R}^n\to\mathbb{R}^m$ defined as ${\bf f} = (f_1,f_2,\ldots,f_m)$ where each $f_i:\mathbb{R}^n\to\mathbb{R}$ is twice-continuously differentiable convex in ${\bf ...
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1answer
106 views

Understanding what exactly an upper bound on an error is in numerical analysis

I think the hardest part of numerical analysis for me is understand what constitutes an "upper bound", and this has caused me alot of strife because often times my answer differs from the book, but ...
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27 views

Best way to fit an equation for the given graph

I have 450 pair $(x,y)$ of data. The plot is like this: I need to fit an equation: $y=f(x)$ for the given data, and to find out values of $y$ when $x=500$. Now, my question is: What kind of ...
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2answers
46 views

Algorithm to solve the system $\sum_{i=1}^nx_i^k = k!c_k$, $k=1,2,\ldots,n$ efficiently

$$ x_1 + x_2 + \cdots +x_n = c_1 $$ $$ \frac{x_1^2}{2} + \frac{x_2^2}{2} + \cdots +\frac{x_n^2}{2} = c_2 $$ $$ \vdots $$ $$ \frac{x_1^n}{n!} + \frac{x_2^n}{n!} + \cdots +\frac{x_n^n}{n!} = c_n $$ ...
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1answer
62 views

A question on a error bound with trigonometric functions

I have a link to a paper on a solution below http://math.berkeley.edu/~zworski/128/psol07.pdf This is related to my other question on the same problem. For problem 7, the author achieves a second ...
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2answers
505 views

Notation laplace operator squared $\Delta^2$

I have the following expression (in a numerical context) $$\Delta_h u(x) = \Delta u(x) + \frac{h^2}{12} \Delta^2 u(x) + O(h^4)$$ The $\Delta$ is the Laplace operator so $\Delta u = u_{xx}+u_{yy}$. ...
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1answer
42 views

Confusion about trigonometric error bounds in numerical analysis

I have a link to a paper on a solution below http://math.berkeley.edu/~zworski/128/psol07.pdf For problem 7, the author of the paper does something like so: $$f''(\xi) = -5e^{2\xi}sin3\xi + ...
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71 views

Magnus series expansion

In the theory of the Magnus series expansion, it can be found that $$ \Omega(t) = \int_0^t A(\tau)d\tau - \frac{1}{2}\int_0^t \left[ \int_0^\tau A(\sigma)d\sigma, A(\tau) \right]d\tau + ...
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1answer
33 views

Composition of two functions in normed spaces

Let $\Omega_1, \Omega_2 \subset \mathbb{R^n}$ be bounded. The mapping $ F: \Omega_1 \rightarrow \Omega_2 $ shall be bijective, continuously differentiable and such that $||DF(x)||$ and ...
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1answer
65 views

Quadrature obtained from Simpson's rule, and its order of error

Express $Q$ as a weighted combination of the five function values $f(a)$ through $f(e)$ and establish that its order is six. (See section 6.2.) This is from Numerical Methods by Moler, ...
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2answers
324 views

Help with Runge-Kutta method for solving systems of differential equations

I am currently doing an investigation with SIR model for predicting the progress of an infectious disease. However, I am not very much familiar with systems of differential equations,so I would need ...
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1answer
56 views

Help find error bound of trapezoidal quadrature

I'm having trouble finding the error bound of this function. My professor says it's "trivial" and thus, he refused to offer me any help beyond a simple hint I kind of knew anyway. I am given this ...
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1answer
184 views

MATLAB, 1st order 2d hyperbolic equation, problem with convergence.

Follow up to my previous question: MATLAB: solving 1st order hyperbolic equation in 2 spacial dimensions The equation I'm solving has the form: $$f_t + A y f_x - B x f_y =0$$ I wrote the following ...
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2answers
61 views

Numeric methods?

I would please like to receive advice on the following: Prove that the given equation has a real solution, then find it, numerically; $2^x-x^2=20$ I don't know how to do this...any help?
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45 views

Does Runge Kutta need future state of system?

In order to use the RK methods, you need to know the state of the system at future time-steps which can be expensive to compute (e.g., in physics simulations). As a simple example I'll use RK-2: In ...
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1answer
30 views

Properties of carry in base $b$ multiplication

Consider $n$ bit numbers $A$ and $B$. Let they be represented in base $b$. When you multiply $A$ and $B$ using school multiplication: $(1)$ how many carry propagations can one expect? $(2)$ what ...
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1answer
171 views

Determine $p$, $q$ and $r$ so that the order of the fixed point iteration for computing $a^{1/3}$ becomes as high as possible

So I'm given the following equation for computing $a^{1/3}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this equation converges to $a^{1/3}$ as ...
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1answer
46 views

Weed out numerical artifacts from matrix inversion

I am working with the inverses to a set of large sparse matrices (in Matlab). A key indicator for my application is the number of non-zero entries in each row, and I recently discovered that I was ...
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0answers
31 views

linear differential operator 2d, order of error h^4?

I have to show that following discretization of a linear differential operator satisfies the equation $\Delta_h u(x) = \Delta u(x) + \frac{h^2}{12} \Delta^2 u(x) + O(h^4)$ $$\Delta_h u = ...
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1answer
42 views

Introduction to Newtons method

I'm supposed to come up with two ways to introduce Newtons method for the approximation of zeros for highschool students. (That is the method using tangents and with the formula $ x_{n+1} = x_{n} - ...
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1answer
57 views

Nearest-neighbor interpolation

I read in a book that the nearest-neighbor interpolation results in a function whose derivative is either zero or undefined. Can anyone explain what does it mean when the derivative of a function is ...
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1answer
16 views

Approximating the value of the limit of a sequence defined recurrently

Suppose I have a sequence defined by recurrence, i.e. $x_{n+1}=f(x_n)$ for some $f:\mathbb{R}\to \mathbb{R}$, and $x_0\in \mathbb{R}$. Suppose that $f$ is $K$-Lipschitz for some $K<1$. Then $f$ has ...
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0answers
32 views

Transformation for two different boundary functions in Stefan problem

Peace be upon on all of you, I have one-dimensional Stefan problem. Let say we have two boundary conditions of $u(t,s_{1}(t))=g_{1}(t)$ and $u(t,s_{2}(t))=g_{2}(t)$, where $u$ is temperature, $t$ is ...
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56 views

Gaussian Quadrature - Construction

Suppose you have $w(x)= 1/\sqrt{x}$ as your weight function, and the integration of the form $\int_0^1 f(x) w(x) dx$. I am tasked with creating a quadrature of exactness 3. So I know I need a ...
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1answer
54 views

Numerical Analysis approximating equilibrium points

Given the differential equations: $\frac{dx_1(t)}{dt}=x_1(t)(4-0.33x_1(t)-0.42x_2(t))$ $\frac{dx_2(t)}{dt}=x_2(t)(2-0.25x_1(t)-0.12x_2(t))$ I need to approximate the equilibrium points accurately ...
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1answer
37 views

Intermediate Forms Between Parabolic and Hyperbolic PDE (numerically)

Greetings MSE community, I have recently conducted some rudimentary experiments in matlab coding of PDE's. I have explicit and implicit numerical solutions to both the heat and the wave equation, for ...
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2answers
39 views

Numerical analysis - Showing fixed point exists

Let $g(x) = \frac{1}{2}(e^{−x})\cos x$. Prove that $g(x)$ converges to a fixed point. The answer provided by my lecturer is: $g(x)$ is continuous on $[0,1]$, and we can easily verify that $0 ≤ ...
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39 views

Pairing Two Point Clouds

So I have two point clouds $X$ and $Y$ each with $N$ points in the familiar $\mathbb{R}^3$ euclidian 3D space. I then have an inter-point distance $d(\vec x_i,\vec y_j)$ which is zero if $\vec x_i$ is ...
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38 views

How to solve an inverse of derivative ode

How can I solve $$(\phi'(y))^{-1}=y^{-c_1}+y^{-c_2},$$ where $c_1,c_2$ are constants and $(\cdot)^{-1}$ is inverse? Since I have inverse of derivative and it's nonlinear I think it has to be done ...