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

learn more… | top users | synonyms (2)

3
votes
1answer
21 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 ...
2
votes
1answer
48 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; $$x(0) = 0$$ ...
1
vote
2answers
32 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 + ...
0
votes
1answer
14 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 ...
5
votes
3answers
58 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 $n=0$} \\ ...
0
votes
0answers
13 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 ...
1
vote
0answers
14 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 ...
0
votes
1answer
27 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 ...
2
votes
1answer
577 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...
4
votes
1answer
28 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 ...
1
vote
0answers
19 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 ...
1
vote
2answers
45 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: ...
1
vote
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 ...
0
votes
0answers
36 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 ...
1
vote
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 ...
1
vote
0answers
29 views
+50

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

I am trying to implement a Finite Element Solution for the Stabilized Eikonal Equation : $$ |\nabla u| = 1 + \Gamma \Delta u, \quad \text{ where } \quad u = \text{ distance function }$$ $$ ...
2
votes
2answers
31 views

The effect of the CFL number in the numerical solution in this conservation law

I've been studying the very basics of numerical methods applied to conservation laws, and I'm having trouble understanding the role of the CFL number in the upwind scheme. I want to understand it (if ...
1
vote
2answers
947 views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
0
votes
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 ...
0
votes
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 ...
2
votes
3answers
1k views

Polynomial root finding

I have an univariate polynomial of some degree - how do I numerically find all of its real roots? I never thought I would ask this question - everyone knows how to find polynomial roots, right..? ...
2
votes
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
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
45 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 ...
0
votes
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 = ...
1
vote
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 ...
-1
votes
0answers
11 views
1
vote
1answer
47 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 ...
3
votes
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 ...
0
votes
1answer
53 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
votes
1answer
332 views

Solution to equations involving Plasma dispersion function

I am trying to solve an equation involving a complex argument for the plasma dispersion function as: $z = x + \iota y$, $ x = \omega / \sqrt2 k v_{Ti} $ $ y = \nu_i /\sqrt{2} k v_{Ti} $ $S[z] = ...
4
votes
1answer
66 views

How many iterations of the Newton's method are needed to achieve a given precision

There is a formula for bisection method to estimate number of iterations that are needed to achieve a given precision (desired significant figures) in the interval $[a,b]$ $$ n\ge ...
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 ...
4
votes
7answers
129 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
votes
0answers
21 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.
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
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 ...
12
votes
1answer
333 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
-1
votes
0answers
14 views

b spline, interpolation how many knots required? [closed]

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
votes
0answers
24 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 ...
2
votes
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 $x^\star$. ...
0
votes
0answers
75 views

How do I solve first order non-linear system of PDE: $\partial f^i(x,y)/\partial z = F^i(f^1,f^2,…,f^n)$?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} \frac{\partial f^i(x,y)}{\partial z} = F^i(f^1,f^2,...,f^n), \qquad i = 1,..,n \end{eqnarray} Where $z = x + iy$, ...
0
votes
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
votes
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
4
votes
1answer
502 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ ...
2
votes
1answer
36 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
2
votes
1answer
590 views

Comparison of Adams-Bashforth and Runge-Kutta methods of order 4

I have a system of ODE, that must to solve with numerical methods. I solve it by Adams_Bashforth with order4 and Runge-Kutta with order4 methods. Do you know with same length step which methods ...
1
vote
1answer
25 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 ...