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

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Alternatives / Extensions to the Thin Plate Splines method

Thin Plate Splines are a great method to find a smooth interpolating surface given scattered data. Essentially, the method involves calculating weights for a radial basis function centred around each ...
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1answer
17 views

Numerical analysis-secant method

I am trying to solve the secant problem here but i do not know how to derive the f(x). Question: Use secant method to approximate ln(2) to 3 decimal place, x_0= 0.6, X_1= 0.7 I need help.
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2answers
33 views

Find $x_i$ and weights such that the following integration rule is exact for all polynomials of degree $\leq 5$

I'm going over an exam I failed. I was told that I can't use the method I used to solve the following question, and I don't know why. Can you please explain and suggest a correct solution? Question ...
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0answers
11 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
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1answer
34 views

Gauss Hermite Integration of 1/(1+x^2)

I'm trying to learn Gauss Hermite Integration and was manually try to calculate the value of integral of $\frac{1}{1+x^2}$ from $-\infty$ to $+\infty$ The exact answer is simply $\pi$ ($\approx$ ...
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18 views

Time advance in Adaptive Mesh Refinement method

I am working on solving complex system of 2D PDEs governing the behaviour of plasma in a gas lamp during discharge. Recent tests have shown that because of steep gradients in temperature field and ...
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0answers
20 views

How can we calculate the tensor product of Lagrange basis polynomials?

Given data points $(x_i,y_i)$, the Lagrange basis polynomials are $$\mathcal l_j(x):=\sum_{i\ne j}\frac{x-x_i}{x_j-x_i}\;.$$ I'm reading a text targeting Smolyak's algorithm. In this text, they use ...
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1answer
29 views

Looking for finite difference approximations past the fourth derivative

I scanned the internet and could not find further representations of the central difference approximations past the fourth derivative. Are there published results past the fourth derivative? Ideally ...
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46 views

Newton's method on a surface

I am trying to use Newton's method to find the stationary solutions of the integro-differential equation of the form $$\frac{\partial u(r,t)}{\partial t} = -u(r,t) + \int_{\mathbb{R}^{2}}w(r - ...
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1answer
36 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
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18 views

Runge-Kutta methods that satisfy row condition produce same solutions for equivalent autonomous problems

Given a IVP $y'(x) = f(x,y)$ $y(a)=\eta$ in $[a,b]$, it can be written as an autonomus IVP by increasing the space dimension: $$ \tag{*} \bar y(x) = \begin{pmatrix} x\\ y(x)\end{pmatrix},\quad \bar ...
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2answers
36 views

Find an algorithm to calculate the following function

I'm solving questions from an exam I failed, and I would love some help with the following question: Question We want to calculate the following function in Matlab: $$ f(x) = \frac{e^{x^2} - (1 + ...
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3answers
72 views

Showing that the sequence $x_n=\frac{1}{3}x_{n-1}(4+x_{n-1}^3)$ where $x_0=-0.5$ quadratically converges

I am stuck at a point in solving this problem: Show that the sequence defined by: For all $$n\in\mathbb{N}, x_n = \begin{cases} -\frac{1}{2}, & \text{if $n=0$} \\ ...
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1answer
17 views

Product rule in discrete derivative in finite difference scheme.

Suppose we are on real line and I want to discretize the usual derivative operator. Take a smooth function $u$ and step size $h$. Then I could define $$ \Delta_+u(i) = \frac{u(i+1)-u(i)}{h} $$ as the ...
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0answers
15 views

For what $p$ is the condition number of a given matrix $A$, using the $p$ norm on matrices, minimal? [duplicate]

For what p is the condition number of a given matrix $A$, using the p norm on matrices, minimal? The condition number on $A$ is given as: $$K(A,p) = \|A\|_p \times \|A^{-1}\|_p$$ I tried ...
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0answers
22 views

Error estimate of polynomial quadratures missing some terms

Normally, for trapezoid rule and simpson's rule, etc, error analysis is done by using the error formula for interpolation. However, if the polynomial is restricted to some terms, for example, a ...
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1answer
32 views

Numerically evaluate Gauss' hypergeometric function ${}_{2}F_{1}(a,b;c;x) $ for large $|a|$ or $|b|$ and $x\ll 0$ or $ x \approx 1$?

I need to compute Gauss' hypergeometric function $${}_{2}F_{1}(a,b;c;x)$$ for the case where one of $|a|$ or $|b|$ is large and $x\ll 0$ or $ x \approx 1$. By employing some linear transformations, I ...
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0answers
23 views

Calculating $n$-th $q:P(q)=p \in \Bbb P$

Let $P(x)$ denote the number of ways of writing an integer $x$ as a sum of positive integers (where permutation of the array of integers in the sum doesn't count). Ex: $P(1)=1, P(2)=2,P(4)=5$. Let ...
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2answers
61 views

How to solve this equation numerically???

The equation is given by $$ \sum_{n=1}^N \min(\gamma, \beta a_n)=N$$ where $\beta$ is the variable with $\beta\in[0,\sqrt\gamma/\min(a_n\mid a_n>0)]$, $ \gamma $ is a constant with ...
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2answers
50 views

Maple not able to calculate Bernstein polynomial

Hope you can help me on this one. Please look at this simple Maple code: Obviously $B(1)=g(1)=4 \neq 0$. Why is Maple not able to compute this right? Am I doing something wrong? Kind regards PS: ...
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1answer
24 views

Gauß-Newton Example with one variable

$$T=f(t):=2 \alpha + \sqrt{\alpha^2+t^2}$$ To estimate $\alpha$ we got the measured values $T_i$ for $t_i$. Formulate the curve fitting problem and show each step in the Gauss-Newton algorithm. My ...
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1answer
29 views

Characterization of contraction mapping

Let $T$ be a mapping from $\mathbb{R}^n \to \mathbb{R}^n$. Fix $x^\star \in \mathbb{R}^n$, and suppose that the Jacobian matrix of $T(x) $ at $x = x^\star$is symmetric. Then, I know that if all the ...
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0answers
22 views

Arbitrary Lagrangian-Eulerian methods

hopefully this is an ok place to be posting about this, since it's not exactly a math question. I am working on a project that involves modeling fluid dynamics with matlab. Here is my problem: I don't ...
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1answer
63 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; $$x(0) = 0$$ ...
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0answers
37 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
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1answer
22 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
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1answer
16 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 ...
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2answers
46 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 ...
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Any one can give me solution of relationship between forward and backward interpolation operator? [closed]

Any one can give me solution of relationship between forward and backward interpolation operator ?
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1answer
14 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
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1answer
43 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 ...
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0answers
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+50

Jacobian of Stabilized Eikonal Equation $| \nabla u| = 1$

I am trying to implement a Finite Element Solution for the Stabilized Eikonal Equation : $$ |\nabla u| = 1 + \Gamma \Delta u, \quad \text{ where } \quad u = \text{ distance function }$$ $$ ...
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1answer
36 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 = ...
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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 ...
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1answer
22 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 ...
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1answer
56 views

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

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 ...
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0answers
22 views

Obtain roots of the polynomial using sturm sequences [closed]

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.
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1answer
28 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 ...
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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. ...
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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
27 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 ...
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7answers
131 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
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
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0answers
19 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 ...
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0answers
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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 ...
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0answers
34 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 $x^\star$. ...
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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 ...
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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 ...
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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 ...