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

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1answer
59 views

Reasons for different answers when finding area using Simpsons rule and numerical integration? [closed]

I have a function $\sqrt{x^4(x+4)}$ to be integrated from 0 up to -4. Using Simpson's will give me 19.02 but using normal numerical methods giving me -19.5 ! What's the reason behind this difference ...
0
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1answer
32 views

Automatic way to have a good initial guess for the iterative methods ( newton method) and for high dimensional nonlinear problems

I am solving a variety class of nonlinear systems where I need to reduce in optimal manner the number of iterations of either the newton or modified newton method. For this end I am trying to figure ...
0
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1answer
16 views

Higher accuracy of numerical derivative in 2D case

Recently, I face a problem about solving a PDE (2D in spatial direction) and I stuck on the discretization of the 1st order derivative. My stencil is as follow There are five points in my stencil. ...
1
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1answer
21 views

Numerical method for nonlinear wave equation

I need to solve the following nonlinear wave equation numerically $U_{tt}=(1+\epsilon U_{x}^2)U_{xx}$ with Initial conditions. what is the best method for solving it? I tried the finite elements ...
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0answers
37 views

How to study numerical analysis?

As the title says, I'm curious about what methods can be used when trying to study numerical analysis (or numerical methods ). I have no problem studying abstract algebra or real analysis, since that ...
4
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7answers
134 views

Evaluating numerically $\int_0^{\infty}e^{-t^2 /100} \sin \pi t $

What is an appropriate method to approximate $$I=\int_0^\infty e^{-t^2 /100} \sin \pi t \ dt?$$ This is for a Physics problem, but in fact I need this in general, as my professor and book taught us ...
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0answers
19 views

Numerical Solutions of Fredholm Integral Equations of the First Kind

Can anyone recommend me some papers about numerical solutions of Fredholm integral equations of the first kind? Thanks in advance
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0answers
33 views

How to calculate $det(X^T X)$ efficiently, update one column of X each time

$X_{1} = (A, b)$, where $X_{1}$ is a $n\times p$ matrix, $A$ is a $n\times (p-1)$ and $b$ is $n\times1$. First calculate $\det(X_{1}^T X_{1})$, then update $b$ with $c$, st. $X_{2} = (A, c)$ and ...
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1answer
20 views

What does the following iteration formula do?

The question is: What does the following interation formula do?: $x_{k+1}=2x_k-cx_{k}^2$. I already tried to identify this with newtons method. I.e. I tried to bring that into the form ...
2
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0answers
35 views

Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
0
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0answers
35 views

Final year dissertation/project ideas for numerical methods

In my final year, I have to submit a project/dissertation on Numerical Methods. I have done a course on it, which included some proofs and programming. Just eager to get ideas that I can look at. PS ...
1
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1answer
26 views

Geometric interpretation of the derivative of a Bezier curve

For a given set of control points $b_0, b_1, \ldots, b_n$, the Bezier curve is defined as $$b^n(t) := \sum_{j=0}^n b_j B_j^n(t),$$ where $B_j^n(t):=\binom{n}{j}t^i(1-t)^{n-i}$ are Bernstein ...
0
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0answers
33 views

For which starting values the iteration convergences?

Given: $g(x)=\frac{1}{2}(x+\frac{a}{x})$ for $a\in \mathbb R_{>0}$ Question: For which starting values $x_0>0$ does the iteration $x_{k+1}=g(x_k)$ converges? My thoughts: Should I find an ...
1
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0answers
27 views

Stability in partial differential equations

I have the following PDE, with parameters $a$ and $b$: $$ \frac{\partial c}{\partial t} = \frac{\partial}{\partial z} \left( a c + b \frac{\partial c}{\partial z}\right) $$ with, for now, just one ...
0
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2answers
22 views

How to represent non-linear operators computationally?

I have a finite dimensional vector space V, and want to compute a non-linear operator $R: V \rightarrow V$. I want to have a "general" form of this operator R. I think of the following series ...
2
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2answers
52 views

Calculating values of $1 - \cos(x)$ for $x$ near zero using computer arithmetic

Explain why calculating values of $1 - \cos(x)$ where $x$ near zero using the trigonometric identity $1 - \cos(x) = 2\sin^2\big(\frac{x}{2}\big)$ will result in more accurate results. Is it because ...
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0answers
14 views

Retail Inventory Prediction - More scientific approach?

This question concerns planning inventory for a retailer. Suppose you have access to all inventory and sales metrics for all styles within the department. You are planning an e-commerce order for ...
1
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1answer
36 views

Chaikin's Algorithm: Proof of Convergence

Chaikin's algorithm is, in some sense, similar to de Casteljau algorithm in that (in the limit) it produces a curve from a set of control points. There are claims all over the internet that Chaikin's ...
0
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0answers
23 views

Multivariable gradient descent with approximation of gradinet

This is not a statistics problem I have a vector $$X=[x_1,...,x_{10}]$$ and a cost function $$y=F(X)$$ and my aim in to find the best $X$ to minimize the cost function. It is impossible to ...
0
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1answer
61 views

Cubic function: Cardano's method

(Wikipedia link) So I am writing an essay on different ways to solve cubics. But I get stuck in the Cardano's method... Mainly is the part with Cardano's method's condition $\frac{q^2}{4} + ...
1
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1answer
57 views

How to numerically solve the Poisson equation given Neumann boundary conditions?

I want to solve the Poisson equation on a 2D domain given Neumann-type boundary conditions: The PDE: \begin{equation} \nabla^2 \; u(r,\theta) \;=\; f(r,\theta) \end{equation} The boundary ...
0
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0answers
36 views

What is the minimizer of the matrix norm and it's significance?

For $M_{n\times n}$ a p.s.d real matrix, if we minimize $||M^{\frac{1}{2}}x||_2$ over $x$ under a linear constraint on $x$ as in $Ax=b$, where $b$ is non-zero. what is the significance of this $x$? ...
2
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2answers
18 views

Lipschitz-like behaviour of quartic polynomials

I have observed the following phenomenon: Let the biquadratic $q(x)=x^4-Ax^2+B$ have four real roots and perturb it by a linear factor $p(x)=q(x)+mx$, so that $m$ not too large with respect to ...
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0answers
25 views

Issues with finite-difference implicit solution of Advection-Diffusion-Reaction eqn

Objective: I am trying to numerically solve $C(x,y,t)$ from the following advection-diffusion-reaction equation in 2D space (x,y) and time. I will be testing my numerical solution with an approximate ...
1
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1answer
65 views

Expected Utility Method and a Repeated Game Solution

I am trying to replicate Bruce B. de Mesquita's (BDM) results on political game theory for prediction. Based on where actors stand on issues, their capabilities, salience, BDM's method attempts to ...
0
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1answer
75 views

Numerically solving a system of partial integro-differential equations in Matlab [closed]

Given the following system of partial integro-differential equations - $\frac{dS(t)}{dt}=\Lambda-\mu S(t)-\beta S(t)F(t),\\ \frac{\partial I(t,\omega)}{\partial t}+\frac{\partial ...
0
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0answers
82 views

How can I resolve this improper integral?

I would like to resolve this integral numerically . However, I'm not sure about the best way to do it because it is an improper integral: $$ ...
0
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2answers
25 views

Estimating line paths in vector fields.

Assume I have a vector field sampled in discrete points. For simplicity let us assume it is sampled regularly on a Cartesian grid. I want to estimate flow lines through various points in this vector ...
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0answers
23 views

Monte Carlo (or other) Approximation to Infinite Summations

This question is sort of paired with a similar post I wrote in Stack Overflow, but not a repeat because I am asking about using another mathematical method. StackOverflow I am trying to approximate a ...
0
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0answers
26 views

Approximate solutions to differential equations

If one has a differential equation for $y(x)$. If this differential equation has two solutions one for $x\ll a$ and the other for $a\ll x$, where $a$ is constant real value. My question is at what ...
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0answers
13 views

Method of Modified Equations: algebra problem

I am having trouble following my professor's lecture notes on the method of modified equations and arriving at his final solution. Starting with the recurrence defining the numerical method: ...
0
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1answer
37 views

Assumption on Runge-Kutta-Method

I think I missed a crucial step concerning Runge-Kutta-methods. If a RK-method is given by its Butcher-table, is it necessary to have $\sum_{j = 1}^s a_{ij} = c_i$? In class we discussed, that a ...
0
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0answers
21 views

Easiest way to calculate Runge-Kutta stability function

For a Runge–Kutta method with Runge–Kutta matrix $A$ and weights $b_1, \ldots, b_s$ the stability function is defined as: $$R(\zeta) = 1 + \zeta \begin{pmatrix} b_1, \ldots, b_s \end{pmatrix}(I-\zeta ...
0
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1answer
29 views

Second derivative numerical estimate - stability and approach

I would like to know how to estimate second derivatives of a function sampled discretely with constant spacing. Let there be a function $f(x)$. I sample its values $\{f(x_i)\}$ at points $\{x_i\}$ ...
2
votes
1answer
74 views

Calculating the x-intercept of the line that passes through the points $ (x_0,y_0)$ and $(x_1,y_1)$

I got this problem from the book Numerical Analysis 8-th Edition (Burden): Suppose two points $(x_0,y_0)$ and $(x_1,y_1)$ are on a straight line with $y_1\neq y_0$, Two formulas are available to ...
0
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0answers
53 views

Matlab Coding finding zeros without using fzero or roots function

So i am a completely new at Matlab. I'm basically suppose to develop a function in Matlab that finds the zeros of a cubic polynomial. real and complex. I'm pasting below what I have so far. I started ...
1
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1answer
31 views

Prove the solution $u_i$ of equation $\lambda r_i+\frac{1}{\theta}(u_i-v_i)+\beta \left (\sum_{i=1}^Nu_i-1\right )$

I have an cost function such as $$E(U)=\lambda \sum_{i=1}^{N} \int_{\Omega}r_iu_idx+\frac{1}{2\theta}\sum_{i=1}^{N} \int_{\Omega}(v_i-u_i)^2dx+ \frac{\beta}{2} \int_{\Omega}\left ( ...
1
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1answer
50 views

What does the code do and what ODE is that?

This exercise I came across asks what kind of method of solving ODE is that: ...
1
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1answer
54 views

Why are Runge Kutta's method and Euler's so different?

I am solving a system of linear equations $\underline {\dot A}=\underline A\cdot \underline x$ numerically. I have done this in the popular methods of of Euler and Runge Kutta. I have noticed quite a ...
1
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0answers
28 views

Numerical method for wave equation with nonlinear forcing in 1+1 D

I am looking for numerical method to solve the equation $$ \square \phi = \frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial t^2} = \lambda \phi^3 \, $$ for real $\phi = ...
5
votes
2answers
233 views

Numerical methods (for ODE/PDE) that could take approximate solutions/good initial guesses, and further refine it to an certain accuracy

I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping; thus there is no convergence issues caused by time-stepping because of ...
1
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2answers
87 views

Numerical mathematics, Lagrange interpolation..

I am trying to solve this problem, but I don't have any idea. Maybe it doesn't look at first sight that Lagrange interpolation can be used, but I found this problem in that chapter of Numerical ...
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0answers
21 views

Polynomial interpolation- numerical analysis

Using polynomial interpolation show that the polynomials $p(x) = (x-1)(x-2)(x+1) $and $f(x) = x^3 - 2x^2 - x + 2$ are one and the same......its absurdly simple...... I tried to show by Newtonian ...
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0answers
28 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
0
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1answer
50 views

Is there an analytic or at least a numerical solution to an eqaution of the form $\sqrt{k_1\sqrt{x}+k_2}\;\Big(k_3x+k_4\sqrt{x}+k_5\Big)+k_6=0$?

So the problem comes from cosmology and I want to solve for the unknown function $a(t)$, which is the scale factor for the universe. So I have an integral involving $a$: $$ ...
2
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1answer
33 views

Use of ergodic theory in numerical simulations

Is ergodic theory used in numerical simulations? The kind of application I have in mind is: for $\alpha$ irrational, $( n\alpha \mod 1)_{n \geq 0}$ is equi-distributed on $[0,1]$, and I imagine that ...
1
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1answer
32 views

Distance between points

I am wondering how can I solve following problem. ...
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2answers
82 views

Estimating the value of an improper integral numerically

My question is how can I estimate the value of an improper integral from $[0,\infty)$ if I only have a programming routine that gives me the function evaluated at 100 data points, or 100 values of ...
0
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1answer
30 views

Gradient descent with linear perturbation

Given a convex, differentiable function $f$ (from a Hilbert space to $\mathbb{R}$) with a minimum (say $x^*$), I know you can find $x^*$ using gradient descent. Suppose now that you apply gradient ...
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0answers
55 views

3Dimensional runge kutta and Euler method ( help to verify the idea and proposition)

I been discussing this idea with a tutor for sometime. However it turn out that the proof is not comprehensible.Can someone please help to verify the the proof for 3D Euler method and runge kutta ...