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

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2
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1answer
22 views

Is there a numerical solution for a system of three 1st order nonlinear ODE?

How would I go about solving the following system of non-linear ODEs for $x(t), y(t), z(t)$ $$x' = y $$ $$y'=\sin(x)+z$$ $$z'=y-z$$ I have the following initial conditions; $$x(0) = 0$$ ...
0
votes
0answers
22 views

Solving a boundary-value problem where the function is not differentiable at the boundary?

Let us say we have a initial-boundary value problem $$ \frac{\partial u}{\partial t} = Lu $$ on $(0, T]\times [0, \infty)$ with initial condition $u(0, x)=h(x)$. I don't specify $L$ here in the hope ...
2
votes
1answer
17 views

Will numerical routines for the Exponential Integral function E_n work when n is continuous?

So I am a mathematical biologist of sorts. I rely heavily on Mathematica which often provides analytic results couched in terms of special functions which I then try to go and learn about. Right now ...
2
votes
1answer
15 views

Implicit finite differences: Sufficient conditions for non-negativity

Given the finite difference approximation for black scholes with zero interest rate, $$ \frac{V_n^{m+1}-V_n^m}{\Delta t} + \frac{1}{2}\sigma^2S^2 \frac{V_{n+1}^{m}-2V_n^m+V_{n-1}^{m}}{\Delta ...
2
votes
2answers
42 views

Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
-1
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0answers
11 views

Any one can give me solution of relationship between forward and backward interpolation operator? [on hold]

Any one can give me solution of relationship between forward and backward interpolation operator ?
1
vote
1answer
13 views

What's the difference between these two spaces?

In the finite element method, $Q1$ element is defined by $\textrm{span} \{1, x, y, xy\}$. And $\textit{rotated } Q1$ element is defined by $\textrm{span}\{1, x, y, x^2-y^2\}$. Please tell me what ...
3
votes
1answer
42 views

Runge Kutta stability

I am facing a problem solving a ODE with a Runge-Kutta 4th order method: The expression in order to solve is : \begin{equation} Ay^{''}+By^{'}+Cy= Cu \end{equation} \begin{equation} y =OUTPUT ...
0
votes
1answer
34 views

Error estimate for Midpoint rule of ratio of integrals

Let's say that I partition an interval $[a,b]$ such that $x_{0} = a$, $x_{k} = a + k\Delta$, until $x_{K} = b$ $\Delta$ is the length of the subinterval. I assume equal length, and thus $\Delta = ...
0
votes
0answers
12 views

Lipschitz method writing the unique solution.

So the problem gives $f(t,y) = y \cos t$ with $t$ between or equal to $0$ and $2$. I already know the lipschitz method holds with $L=1$. But I'm not sure how to find the unique solution which turned ...
0
votes
1answer
21 views

Numerical differentiation (approximation with three supporting points )

Given the supporting points $x-2h,x-h,x+2h$. Determine the difference quotient Du(x) in the form $$Du(x)=au(x-2h)+bu(x-h)+cu(x+2h)$$ for the numerical approximation of $u'(x)$ of order $2$. What ...
1
vote
1answer
47 views

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

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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0answers
19 views

Obtain roots of the polynomial using sturm sequences [on hold]

Obtain the number of roots of the polynomial $x^4 – 3x^3+20x^2+44x + 54 = 0$ in the interval $[0,4]$ using Sturm sequences.
0
votes
1answer
27 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
votes
1answer
15 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
vote
1answer
18 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 ...
-1
votes
0answers
13 views

b spline, interpolation how many knots required? [on hold]

Hi I would like to get help with these questions. How many control points $d_i$ are involved when evaluating a cubic B-spline at a single points. The point are deboor. How many knots are necassary ...
0
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0answers
23 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
votes
7answers
128 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 ...
0
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0answers
16 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
0
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0answers
18 views

How to perform the following integration using dblquad in MATLAB

I am trying to perform the following integration in MATLAB \begin{equation} \begin{split} F &= @(x,y)(e^{(-0.5([x - \mu_1 \hspace{5pt}y-\mu_2])\Sigma^{-1}([x - \mu_1 ...
0
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0answers
29 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 ...
0
votes
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
votes
0answers
32 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
29 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
vote
1answer
23 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
vote
0answers
24 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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0answers
31 views

Roots of System of Logarithmic Equations [closed]

What is the general equation for the system of logarithmic equation?
0
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0answers
30 views

roots of system of logarithmic equation using newton raphson method [closed]

Can I apply Newton-Raphson Method in getting the roots of system of logarithmic equations? For Example $\log x+\log y$ and $\log x−\log y=31$. What is its basis? I mean, what is its general equation?
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
votes
2answers
42 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 ...
0
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0answers
13 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
vote
1answer
35 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
votes
0answers
21 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
votes
1answer
55 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
vote
1answer
37 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
votes
0answers
32 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
votes
2answers
16 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 ...
0
votes
0answers
28 views

Is this idea correct?

Given a curve passing through point $(p_0, v_0)$ and defined in a standard way as $k(p, v) = k(p_0, v_0)$, i can find the 1st term,a 2nd term b and so on by expand k in taylor series and consider ...
0
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0answers
22 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
vote
1answer
57 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
votes
1answer
47 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
votes
2answers
24 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 ...
0
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0answers
21 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
votes
0answers
25 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 ...
0
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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
votes
1answer
24 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
20 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 ...