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

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1answer
20 views

Semi-discrete time discretisation of PDEs, stability and convergence.

I'm having a lot of difficulty and I was hoping someone could help me out. I'm looking at a variety of PDEs but for the sake of this post, lets just look at the advection equation $$u_{t} = -u_{x}$$ ...
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0answers
23 views

Smallest possible number of steps

We have the following Runge Kutta Butcher tableau: $$ \begin{array}{c|ccccc} \tau_1 =0 & a_{11}=0 & a_{12} = 0\\ \tau_2 =\frac{3}{2} & a_{21} = \frac{3}{2} & a_{22} = 0\\ \hline ...
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1answer
31 views

Find value of expression

$$\begin{matrix} x'=-y \\ y'=-x' \end{matrix} \begin{matrix} x(0)=1\\ y(0)=0 \end{matrix}$$ Calculate with the explicit Euler's method the expression $A=(x_n)^2+(y_n)^2$, where $x_n, y_n$ are the ...
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2answers
46 views

Use a numerical method to approximate the roots of $x^2-1000.01 x+10=0$.

Problem: Find the roots of the following equation with calculations of four significant digits. Then use a method to find the roots of the equation with the maximum accuracy. $$x^2-1000.01 ...
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1answer
36 views

Iterative method to compute only the positive eigenvalue's and corresponding eignevectors of a very large matrix?

I have a very large dense matrix (~10000 X ~10000) which is not full rank . I want to compute only the positive eigenvalues and corresponding eigenvectors instead of computing all of them. I have ...
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1answer
28 views

How can we achieve absolute stability?

The following Runge-Kutta method is given. $$ \begin{array}{c|ccccc} \tau_1 =0 & a_{11}=0 & a_{12} = 0\\ \tau_2 =\frac{5}{2} & a_{21} = \frac{5}{2} & a_{22} = 0\\ \hline & b_1 ...
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0answers
23 views

Explicit euler method

Consider the differential equation $$\left\{\begin{matrix} y'=2x\\ y(0)=0 \end{matrix}\right.$$ Which is the value $y(1)$ ? Which is the approximation that the explicit Euler method gives us for ...
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1answer
31 views

Order of accuracy of Runge-Kutta method

The following Runge-Kutta method is given by the Butcher tableau: $$ \begin{array}{c|ccccc} \tau_1 =0 & a_{11}=0 & a_{12} = 0\\ \tau_2 =1 & a_{21} = \frac{1}{2} & a_{22} = ...
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1answer
46 views

Why use methods as Newton, ridder or secant method for root finding? [closed]

Why use methods as Newton, ridder or secant method for root finding? I am bit confused for what reason someone would use these method to determine the root of a function, as it can easily be ...
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0answers
17 views

How to turn the integro-differential equation into an ODE

I want to get the numerical solution of the integro-differential equation by Mathematica but failed. Maybe the first step should be turning that into an ODE, is there some method? {0.01+10 (0.01 ...
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2answers
36 views

What is the purpose of the weight function $w(x)$ in a Finite Element Method?

I have just started looking into finite element methods. Suppose we have an equation for the strong $$L(u) = s$$ Then the integral form of the equation is given by $$\int_0^1 L(u)w(x) dx = ...
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0answers
21 views

asymptotic smooth kernel log(|x-y|)

I am currently trying to show that the function $\log(|x-y|)$ is an asymptotic smooth kernel function, in the sense that: for $x,y \in \mathbb{R}^2$ there exist constants $C_{1},C_{2} > 0$ and an ...
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1answer
43 views

Maximum Green function

good morning does anyone know how to prove that $|G(x,t)| \le \frac{1}{2}$, where $G(x,t)=\begin{cases} \frac{(x-1)(t+1)}{2} \quad -1 \le t \le x \le 1\\ \frac{(x+1)(t-1)}{2} \quad -1 \le x \le t ...
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0answers
10 views

Nonuniform partition - euler method

Consider a nonuniform partition $a=t_0< t_1< \dots < t_{\nu}=b$ and assume that if $h_n=t^{n+1}-t^n, 0 \leq n \leq N-1 $ is the changeable step, then $\min_{n} h_n > \lambda \max_{n} h_n, ...
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0answers
18 views

NP-hardness of solving congruence equations in several variables

You are given the following equation modulo $N$ (where the $\beta_i$'s are given integers modulo $N$, and the $x_i$'s are unknown integers modulo $N$): $$\beta_1x_1 = \beta_2 x_2 = \ldots = \beta_l ...
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2answers
31 views

Is there an explicit formula for finding the number of iterations needed for an approximation for Newton's method?

I was asked to find the approximate number of iterations using Newton's Method and I was curious if there were an explicit formula as there is for the bisection method which is ...
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0answers
12 views

Absolute stability Euler method

I am looking at the following exercise. We suppose that the explicit Euler method is applied at the differential equation of second order $\left\{\begin{matrix} x''(t)+(\lambda+1)x'(t)+ \lambda ...
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2answers
23 views

An example of a BVP for a second order ODE: $y''+p(x)y'+q(x)y=f(x)$ (where $\,0\leq x\leq L\,$ and $\,y(0)=\alpha\,$ $\,y(L)=\beta$)

I'm looking for an explicit example of a BVP for a second order ODE: $y''+p(x)y'+q(x)y=f(x)$ (where $\,0\leq x\leq L\,$ and $\,y(0)=\alpha\,$ $\,y(L)=\beta$). If you also have the exact ...
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1answer
31 views

Tetrahedron subdivision

What are all the possible subdivisions of the P3 tetrahedron (i.e. for each face, 3 vertices plus two points per edge, located at 1/3 and 2/3, and the centroïd of the face, so a total of 20 points for ...
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0answers
23 views

Ideas for expressing the inverse of matrix quadratic form $CAC^T$

I want to find an expression for the inverse of the matrix system $Z=CAC^T$, where $A \in \mathbb{C}^{n \times n}$ is block diagonal with dense blocks, and $C \in \{-1,0,1\}$ with dimension $m \times ...
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1answer
14 views

Taylor expansion of $f$ in stability analysis of 2-step Adams-Bashforth method

Given the two-step Adams-Bashforth method $$ u_{n+1} = u_n + \tfrac{h}{2}(3f_n - f_{n-1}) $$ find its order. Some notation: $t_n = t_0 + nh$ is the $n$-th node and $y_n = y(t_n)$; $f_n$ ...
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0answers
10 views

what are stability analysis of numerical methods

which of the stability in numerical methods (A, A(0), A(alpha), A nut, stiffly stable, L or L nut) is stringent
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0answers
5 views

how should a numerical solver handled hybrid points either as a single scheme or block method

how should I handle the hybrid point in a hybrid scheme for solving IVP problems when treating the numerical examples (both for direct method and block method)
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2answers
32 views

How to approximately guess the roots of a function

My question is : How to approximately guess the root of a function... By root i mean is the starting point guess when used in case of Newton's method or any other root formulating methods. (Without ...
2
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1answer
35 views

Solving a system of polynomials in $N$ variables

Suppose I am given some non-negative constants $(C_p)_{p=1, ..., l}$ and I would like to find an integer $N$ and vector $v \in R^N$ such that $$ \sum_{i=1}^N (v_i)^p = C_p $$ for $p=1, ..., l$. Can ...
3
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2answers
42 views

Significance of Sobolev spaces for numerical analysis & PDEs?

I never had an option to take a Functional Analysis module. I am tied up with other work for the next two months so I won't get a chance to self-study it until September. So one thing I was wondering ...
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0answers
34 views

Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output.

I've been trying to solve the following Schrödinger equation numerically, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + ...
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1answer
32 views

error bound in function approximation algorithm

Suppose we have the set of floating point number with "m" bit mantissa and "e" bits for exponent. Suppose more over we want to approximate a function "f". From the theory we know that usually a ...
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0answers
9 views

lattice variation, cylindrical discretisation of PDE

Given an energy functional $ E=\int_{0}^{\infty} \,dr.r\left[\frac{1}{2}\left(\frac{d \phi}{dr}\right)^2 - S.\phi\right] $, I am told that discretizing on a lattice $ r_{i}=ih $ (h=lattice size, i is ...
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0answers
32 views

Simplifying the Generalized Eigenvalue Problem

Let $\Sigma_1$, $\Sigma_2$ be symmetric positive-definite real $n\times n$ matrices. We want to solve the generalized eigenvalue problem $$ \Sigma_1V=\Lambda\Sigma_2V, $$ where $\Lambda$ is the ...
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1answer
28 views

Use the Forward Difference method to approximate the solution to the following PDE?

Use the Forward Difference method to approximate the solution to the following PDE: $$ u^3\frac{\partial u}{\partial t}-x^2u\frac{\partial^2u}{\partial x^2}=2x^8t^7+6x^6t^5+4x^4t^3 $$ for $0\le ...
0
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1answer
21 views

Determine the local truncation error of the following method

Consider the ordinary differential equation $$y'(t)=f(t,y(t))$$ Let $y_n$ be an approximate to $y(t_n)$, where $t_n = nh$ and h is constant step size. Determine the local truncation error of the ...
0
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1answer
26 views

Numerically solving equations with expectations

I have a equation $\mathbb{E}_\theta f(x,\theta)=a$, where $\theta$ is a vector real random variable with a known distribution, $a$ is a real constant, $x$ is a real (can be vector valued) variable. ...
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4answers
58 views

Inverse Chebyshev Recurrence

The Chebyshev polynomials (of the first kind) are a sequence of polynomials defined recursively by $$ \begin{cases} T_{0}(x) = 1 \\ T_{1}(x) = x \\ T_{n}(x) = 2xT_{n-1}(x) - T_{n-2}(x) \end{cases} $$ ...
1
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0answers
52 views

complex rank-one update

I'm trying to find the eigendecomposition of a rank-one update to a complex matrix $D + uv^T$. The matrix $D$ is diagonal, but not the identity. It has unique imaginary entries along the diagonal. ...
3
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0answers
91 views

$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t$ with $x_t$ Ornstein Uhlenbeck process - what to do? [UNRESOLVED]

I consider the following equation: $$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t, \tag{1}$$ where $a=$ constant and where $x_t$ follows an Ornstein Uhlenbeck process (see here under Alternative ...
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2answers
22 views

If the iteration $x^{k+1}=x^k-t_kH_k^{-1}\nabla f(x^k)$ converges superlinearly to a stationary point $x^*\ne x^k$, then $t_k\to 1$

Let $f\in C^2(\mathbb{R}^n)$ $(H_k)_{k\in\mathbb{N}_0}\subseteq\text{GL}_n(\mathbb{R})$ $x^0\in\mathbb{R}^n$ and $$x^{k+1}:=x^k+t_k d^k\;\;\;\text{for }k\in\mathbb{N}_0\tag{1}$$ with ...
3
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0answers
65 views

Problem using the Fourier transform and convolution to compute an integral

I'm trying to write a subroutine (in Fortran) to compute integrals of the form $$I=\int_{-L}^{L} f(x)g(y-x) \:\mathrm{d}x, $$ using the convolution theorem and fast Fourier transforms. In my routine, ...
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0answers
10 views

Order of error Verlet integration

I have a simulation of moving particles. The integration method I'm using is Velocity Verlet. Wikipedia states that the order of the error of this method is $2$. However, if I calculate the order of ...
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1answer
18 views

Prove that the sum of the Lagrange (interpolation) coefficients is equal to 1

Prove that the sum of the Lagrange (interpolation) coefficients is equal to 1. Please suggest me a book-reference or give a solution for me. Thanks a lot in advance. If $f = ...
1
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1answer
32 views

In an ODE dynamic system, is there a convient way or algorithms for estimating the parameters which make the ODE solution satisfing some constraint?

I have construct a ODE dynamic system like this $$molA(t)==sa$$ $$molB'(t)=sb-db\;molB(t)+\frac{kab\;molA(t)\;molB(t)}{molB(t)+Jab}-\frac{kgb\;molG(t)\;molB(t)}{molB(t)+Jgb} $$ $ molC'(t)=sc-dc\ ...
4
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1answer
56 views

What is so good about the $L^2$-norm of the second derivative being small?

One of the main properties of cubic splines is the minimality property which basically means that if $s$ (cubic spline) and $g$ (some other function) interpolate $f$ in a certain way then $$\Vert s'' ...
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0answers
16 views

periodic boundary conditions and the FEM

I am trying to set up the mass matrix for a 1D system which I want to solve using finite elements. So the mass matrix is defined as $$ M = \int{NN^T}dL, $$ where $N$ is the finite element linear ...
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0answers
58 views

How do I solve a Second order differential equation that is a variation of the Sine Gordon Equation?

$$0.1 \frac{d^2 \varphi}{d\tau^2}+\frac{d\varphi}{d\tau}+\sin\varphi-1.1=0$$ Im not quite sure how to reduce this equation. The inclusion of $\sin \varphi $ throws me off some what. If it helps $\tau ...
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0answers
29 views

define the origin of rotation from groups of points.

So I have a group of points $K = [X_{1},X_{2},...,X_{n}]$ in $R^{3}$ that have been rotated by some unknown angles around an unknown point $p$ giving a new list of points $K' = ...
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0answers
28 views

Confidence Interval Algorithm

I am trying to write a C++ program for parameter estimation(with Confidence Interval information) of an Exponentially distributed data set. I understand that $\lambda \bar{X} \sim \Gamma(n, n)$. To ...
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2answers
34 views

Integration of a function that is numerical solution of differential equation

I've obtained a numerical solution of a differential equation in a form of a vector (i.e., M(170,1)) by using ode45 (MATLAB) and ...
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2answers
26 views

Show that a zero of $f$ is a fixed point of $g$

I want to show that a solution of the equation $x^2+cos(x)-10x=0$ is a fixed point of $g(x)=(x^2+cos(x))/10$. I tried using the quadratic equation but my solution doesn't simplify nicely in $g$. I'm ...
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0answers
24 views

Numerically solving the diffusion-reaction equation with boundary values

I want to solve a nonlinear PDE (steady-state diffusion reaction): $\Delta u = f(u)$ That has the following boundary conditions: $u_y(x,0) = 0$ $u(x,h) = m$ I am trying to solve it via newton's ...
0
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0answers
31 views

Approximation of the coefficients of the Fourier Series via the FFT

Is there literature on the approximation of the coefficients of the Fourier Series via the FFT? The approach I'm interested is merely numerical, consisting of computing the integrals with the ...