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

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
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
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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'' ...
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0answers
22 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
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
36 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
29 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 ...
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0answers
34 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 ...
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0answers
32 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
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1answer
117 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 $P'$, ...
3
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0answers
57 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} ...
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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 ...
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1answer
20 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
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1answer
146 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 ...
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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 $(X,Y) \in \Gamma$ ...
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1answer
21 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   ...
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2answers
30 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 ...
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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 ...
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0answers
20 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
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1answer
40 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: ...
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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. ...
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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 ...
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8answers
302 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
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0answers
36 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 ...
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0answers
23 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: ...
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3answers
81 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
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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
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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 ...
2
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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 ...
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0answers
19 views

Please recommend a book/article for Newton-Raphson method

There are so many search results I'm a bit lost. I would like to read an article to fully understand it, including the math end, and the appliction side, please recommend one that contains also ...
0
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2answers
53 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
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2answers
73 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
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0answers
29 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
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0answers
29 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 ...
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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 ...
1
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1answer
17 views

Computing wavenumbers for discrete Fourier transform

I'm trying to implement a Fortran program to compute the derivative of a function using the FFT. To begin with, just to test my installation of fftpack, I computed the Fourier transform of ...
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0answers
10 views

Deferred Corrections vs Multigrid

I've been looking at the method of Deferred Corrections (see page 9 of this presentation) to numerically solve ODE IVPs. To me, the process looks identical to a V-cycle in a multigrid method if the ...
2
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1answer
59 views

Numerical or analytical or exisistence: Inverse Laplace Transform

Edit 1: With the hint of Ron, we can simplify the question to : $$\bar{f}(s)=\frac{1}{(s^2+1)\arctan s }$$ So what about this function's inverse Laplace Transform? Or can anyone tell me that the ...
2
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3answers
106 views

How do I get geographic coordinates of a point if I know the distance from two other points?

The keyword here is geographic. I am assuming the solution has something to do with a spherical triangle. I know that this problem has one, two, infinite or no solution at all. My specific problem ...
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1answer
23 views

turn $L\{u(t-3)(t^2)\}$ to $L\{u(t-3)[(t-3)^2+6(t-3)+9]\}$?

I was given looking at one of the examples in my textbook and it took this laplace transform $L\{u(t-3)(t^2)\}$ and turned it into this $L\{u(t-3)[(t-3)^2+6(t-3)+9]\}$ in the next step. I'm ...
1
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0answers
21 views

Upper bound, supremum norm [closed]

How to find upper limit for this: $\sup_{f \in C[a,b],f \neq 0} \frac {||p_f||_{\infty}}{||f||_{ \infty }}$ $p_f$ is interpolation polynomial of f.
5
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1answer
77 views

Reading list to master Numerical Analysis' research literature

As of lately I have been going through many research papers in my current job, and even though I have a Mathematics background at Masters level in Mathematical Finance, I sometimes struggle to follow ...
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1answer
20 views

Estimate order of convergence given error table

I looked through some other posts and didn't quite understand what was going on... Given a numerical method of calculating the solution to, say, a DE, we typically get an error table. The errors are ...
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0answers
18 views

Increase of precision in numerical Hessian computation?

In a function for numerical calculation of the Hessian [http://grizzly.la.psu.edu/~suj14/programs/Jacob.m] I saw the following 3 lines (here dh, eps, x0 and xdh are all vectors of the same size): ...
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2answers
87 views

How to determine whether a point is inside a closed region or not?

Take the following parametric equation of an implicit curve as an example: $$ \left\{\quad \begin{array}{rl} x=& 9 \sin 2 t+5 \sin 3 t \\ y=& 9 \cos 2 t-5 \cos 3 t \\ \end{array} \right. $$ ...
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0answers
34 views

Cholesky decomposition and rotation matrix inverse

I implemented three methods for inversion of a matrix, all are classic. I wanted to test for the most generalized method, while taking efficiency into account. For Cholesky decomposition, which is ...
1
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0answers
39 views

Chebyshev spectral derivation with 16 nodes for $\,f(x)=e^{\,\text{sin}^{2}\,(x)+\cos(x)}\,$ defined in $\,[0,2\,\pi].\,$

I'm making the following exercise in Matlab, and I'm having trouble expresing my result in $x\in[0,2\pi]$ not in $x\in[-1,1]$. I first done this (as shown below) in Gauss-Lobatto points, but I don't ...
0
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1answer
26 views

A problem with Maple Polynomial Curve Fitting

I've been encountering a problem while trying to use the method of least squares to fit a quadratic polynomial. Below is the question 1) Use the method of least squares to fit a quadratic polynomial ...