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

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3
votes
2answers
51 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 ...
1
vote
0answers
67 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) + ...
1
vote
1answer
36 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 ...
0
votes
0answers
11 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 ...
1
vote
0answers
42 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 ...
0
votes
1answer
34 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
votes
1answer
22 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
votes
1answer
28 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. ...
2
votes
4answers
67 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
vote
0answers
54 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
votes
1answer
100 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 ...
0
votes
2answers
24 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
votes
0answers
69 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, ...
0
votes
0answers
20 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 ...
0
votes
1answer
31 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
vote
1answer
33 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
votes
1answer
59 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'' ...
1
vote
0answers
29 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 ...
1
vote
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' = ...
1
vote
1answer
44 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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
32 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
votes
0answers
36 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 ...
0
votes
0answers
50 views

Proof of the Lax-Wendroff theorem

The Lax-Wendroff theorem says that, if a conservative numerical scheme for a hyperbolic system of conservation laws converges, then it converges towards a weak solution. In the book "Numerical ...
4
votes
1answer
118 views

Module of the differential of a function

Given two triangles, $PQR$ and $P'Q'R'$ in $\mathbb{R}^2$, I want to find a bijection $f$ between $PQR$ and $P'Q'R'$ such that: 1) $f$ maps vertices in vertices and sides in sides (i.e. $P$ in $P'$, ...
4
votes
0answers
75 views

Simpson's Rule for Double Integrals

Simpson's Rule for double integrals: $$\int_a^b\int_c^df(x,y) dx dy$$ is given by $$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1} f(x_i,y_j) $$ where: $$W= \begin{pmatrix} ...
0
votes
0answers
19 views

How to do fixed point iteration with matrices?

I am trying to follow solution to solve $$\min[\mathbf{z},\mathbf{q+Mz}]=0$$ by fixed point iteration. If $\mathbf{M=C+B}$ then a recursive algorithm with $k$ showing the iteration can be written as ...
0
votes
1answer
22 views

Minimizing nonsmooth single variable functions?

What options is available if one wants to minimize a nonsmooth convex function of one variable? Subgradients would work, but there has to be some nice way of utilizing that we're only searching in 1d. ...
1
vote
1answer
185 views

What is the weak formulation of this problem?

Find $u\in H_D^1(\Omega)$ such that $-\nabla\cdot(a\nabla u)=0$ in $\Omega$, $\dfrac{\partial u}{\partial n}=g$ on $\Gamma_N$, $u=0$ on $\Gamma_D$. The function $a(x,y)$ is ...
1
vote
0answers
28 views

computing the area of a region using Monte Carlo integration

Suppose that I am interested in estimating the area of $\Gamma \in \mathbb{R}^2$. I do not know the exact shape of $\Gamma$ but I have a sufficiently large number of sample points $(X,Y) \in \Gamma$ ...
-2
votes
1answer
22 views

Adjust the data up curve φ(x) = α1e^(α2x) by the method of least squares

Adjust the data up curve φ(x) = α1e^(α2x) by the method of least squares: Here's what I've done so far but I think it is wrong(and sorry for the bad english) --x | 0    | 1    | 2   | 3   | 4   ...
1
vote
2answers
36 views

Newton- fixed point iteration

The formula for Newton iteration (which is a zero-finding problem) is $ x_{k+1}=x_{k}-f(x_k)/f'(x_k) $. I read in my textbook that this can be also be seen as a fixed-point iteration; where the zero ...
0
votes
2answers
37 views

Why does my answer depend on the starting values?

I'm trying to find the zeros of f(z)=c(z)-2500=0 using the secant method. I get the correct values (8.9, -2.6, 7.7, 12.3) but only if I put the starting values close. For instance this version of the ...
1
vote
0answers
22 views

What is the general idea of Nitsche's method in numerical analysis?

I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ...
0
votes
1answer
43 views

Simple Implementation of QZ-Algorithm fails in MatLab [closed]

i am still very new to numerics, but i have a question concerning a very simple Implementation of the QZ-Algorithm in Matlab. My Code Looks like: ...
0
votes
0answers
24 views

Mollifiers and Rates of Converegnce

I am interested in how quickly a.e. convergence happens to say: $|f(x) - f(x+h)|$. Originally, I thought I had proved something way too strong, but smoothed that out while typing this question up. ...
0
votes
0answers
25 views

Improper integral over product of exponentials: $\int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx$

I'm looking for a way to evaluate following integral $$ \int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx $$ f(x) resembles however a complex simulation and can therefore ...
-1
votes
8answers
303 views

Solving for n in the equation $\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$

Solving for $n$ in the equation $$\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$$ Can anyone show me a numerical method step-by-step to ...
2
votes
0answers
38 views

Separating the Complex Error Function into Real and Imaginary parts

I'm trying to do a numerical integral of the following form: $$\int_a^b (\mathbb{R}\left[\operatorname{erfi}(z)\right])^2 \, dz$$ That is, I would like to integrate the square of the real portion of ...
1
vote
0answers
24 views

Question about an boundary integral equation with a jump in the boundary

I have the following problem: $$\Delta u = 0\;in\;\Omega$$ with several boundary conditions. Applying Green's second identity the representation formula can be derived: ...
0
votes
3answers
83 views

How to find $\log{x}$ close to exact value in two digits with these methods?

I'm trying to find the result of $\log{x}$ (base 10) close to exact value in two digits with these methods: The methods below are doing by hand. I appreciate you all who already give answers for ...
1
vote
1answer
20 views

How is optimal coordinates change chosen for Chebyshev expansion?

I'm looking into SLATEC implementation of Bessel function $J_0$ computation (readable in C in GSL), namely at its part for arguments in interval $(0,4)$. There a Chebyshev expansion is used, but the ...
2
votes
2answers
43 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 ...
2
votes
1answer
13 views

Accuracy of angular difference: direct difference or difference identity

Can anyone point out if there is a difference in the accuracy of the result of calculating an angle difference when using the difference between two arctan values or when using the difference formula ...
0
votes
2answers
62 views

Simulating an orbit - numerically solving $M(E) = E + \sin(E)$

Well for a given kepler orbit (which is a ellipse) $0 \leq e < 1$. There are several functions to describe the motion of an object. $$r(\nu) = \frac{a (1 - e^2)}{1 + e \cos(\nu)}$$ Where $a$ is ...
2
votes
2answers
76 views

Fast computation of integral of Gaussian pdf

Which methods/algorithms for computation of the function $F$, where $$F(a,b) = \int_a^b e^{-t^2}dt,\quad a\leq b,$$ are the best, i. e. fast and accurate? I need to compute those integrals ...
1
vote
0answers
31 views

$Az + B\overline{z}$ as a linear operator

Given two matrices $A,B \in \mathbb{C}^{n\times n}$ with fixed $n\in\mathbb{N}^+$, let us consider the operator $$ L:\mathbb{C}^n \to \mathbb{C}^n,\\ L(z) = Az + B\overline{z}. $$ This operator is not ...
1
vote
0answers
30 views

Marking iterations on Cobweb / Staircase diagrams

In the below cobweb diagram I am interested in why the iterations $( x_1,x_2,x_3)$ are marked when the 'cobweb' intersects the line $y=x$. Image: http://i.stack.imgur.com/iiYST.png For example, why ...
0
votes
0answers
23 views

How to tell that 2 set of data are not so difference by using statistical method?

How to find stable point of these data? 2.0, -3.5, 0.0, 1.5 1.3, 6.3, 0.1, -3.4 3.3, -1.1, 3.0, 4.1 -2.5, 4.3 -1.0, 2.2 The example data is random ...