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

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9 views

Choleski's Algorithm Query

In Choleski's algorithm, I wonder how can one be sure that the diagonals elements of L (except for the first one) to be all positive?
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1answer
61 views

What method are there for “numerically” computing arclengths!

I know the originals formula for arc-length is: $$\int_{a}^b \sqrt{1+{f'(x)}^2}$$ However most of the formulas don't have closed formed solutions, and are unsolvable in terms of this equation. So ...
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1answer
28 views

What numerical quadrature algoritm can be use to handle $\int_b^c K_0 (x-a)-K_1(x-a) dx$?

I am curious what numerical algorithm can be used to handle $$\int_b^c [K_0 (x-a)-K_1(x-a)] dx$$, where $a\lt b\lt c$ and K is the modified bessel function of the second kind. From plotting the ...
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2answers
37 views

Solution of $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$

I'm trying to prove that $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$,given that $A$ is symmetric positive-definite has only one minimum. I've found the derivative is $f'(x)=Ax-b$, and in order to find ...
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1answer
98 views

How to find the integer part of big number?

How to calculate the integer part $$\left \lfloor10^{10^{10^{10^{10^{-10^{10}}}}}} \right \rfloor ?$$ Does this equal $$10^{10^{10}}? $$ Both Maple and Mathematica fail with it. PS. Unmotivated votes ...
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59 views

Power iteration

If $A$ is a matrix you can calculate its largest eigenvalue $\lambda_1$. What are the exact conditions under which the power iteration converges? Power iteration Especially, I often see that we ...
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33 views

Numerical Integration Over Two Regions of an Ellipsoid

I would like to perform a numerical integration over the surface of an ellipsoid $D$. The domain must be split in two by a plane intersecting the ellipsoid (the intersection is arbitrary), so that we ...
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1answer
40 views

Two point Gaussian Quadrature rule

I want to use the two point Gaussian Quadrature rule to approximate (evaluate) $\int_0^1 \! 6x^2-2x+1 \, \mathrm{d}x $ Since, with the two point Gaussian Quadrature rule, n=2 and the integral of ...
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1answer
47 views

Korteweg–de Vries equation: why is there a substantial literature on their numerical solutions if they are analitically integrable?

Given the initial value problems for the Korteweg-de Vries equation $u_t + u_{xxx} = u u_x; \quad u(0,x) = u_0(x)$ I have read that they can be solved exactly by the inverse scattering method, but ...
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1answer
28 views

Find Iterative Method Convergence Rate

Given $f(x)\in C^2[a,b]$ s.t there is a point $x_0$ s.t $f(x_0)=0,f'(x_0)\ne 0 $, and the iterative method is defined as follows : $$ x_{k+1} = x_k - f(x_k)/g(x_k) ,\qquad g(x_k) = \frac{f(x_k ...
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1answer
37 views
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2answers
39 views

Solving $f(x) = e^{(-sin4x)} - 3/4$ with 3 digits after the decimal point correction

I needd to solve $f(x) = e^{(-sin4x)} - 3/4$ with 3 digits after the decimal point correction, but cannot find out how. I'd really appreciate it if anyone could point me to the solution. I think I ...
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4answers
67 views

Numerical integration - $\int_{-1}^{1} f(x)dx$

I'm currently studying numerical integration, and ive come across a question i'd like help answering. We are given an integration rule as follows: $I(f)=\int_{-1}^{1}f(x)dx = \frac{2}{3} ...
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1answer
28 views

Is the assumption $f \in C^4$ necessary for the composite Simpson's rule to be of order $p=4$?

In my introductory numerics class, we wanted to integrate a function $f \in C[a,b]$ numerically. After developing the Simpson's rule, we proved that if $f \in C^4$ then the composite Simpson's rule ...
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0answers
15 views

Kernel density estimation of a divergent probability density function

I'm working with a 2D probability distribution function (pdf) that will be something like $$P\left(r,\theta\right)\approx\frac{3}{\pi^3}\frac{1}{e^{r}-1},$$ when written in polar coordinates (i.e. ...
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1answer
35 views

numerical algorithms for determining least common multiple of polynomials

I have a pair of rational polynomial fractions $\frac{A(x)}{B(x)} + \frac{C(x)}{D(x)}$ where A, B, C, and D are all polynomials in x, and I have their coefficients as an array of numbers. I would ...
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1answer
30 views

Optimal way to find derivative - numerically

Suppose we are given points $x_0,x_1,x_2$ evenly spaced points $(x_0-x_1=x_1-x_2)$, and $u(x_1),u(x_2),u(x_3)$ Where $u$ is some function. Find the best way to approximate $u''(x)$ using only the ...
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0answers
41 views

Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
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1answer
32 views

Integration Rule Exact Degree

Given the integration rule $Q(x) = \alpha_1f(0)+\alpha_2f(1)+\alpha_3f'(0)$ for interpolating the integral $\int_0^1f(x) dx$ , I need to find $\alpha_1,\alpha_2,\alpha_3$ values s.t Q has exact degree ...
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60 views

How to solve this complicated differential equation?

I need to know how to solve this complicated differential equation in $z$ either analytically or numerically : \begin{eqnarray} \frac{dx_1}{dz} &=& -ib_1x_1 - ikx_2 \\ \frac{dx_2}{dz} ...
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0answers
37 views

solving partial differentiation using finite difference method

I have been trying to solve right hand side (RHS) of the following one-dimensional partial derivative equation: $\frac {\partial p} {\partial t}=\frac {\partial} {\partial x} ({D(x)}e^{-\beta V(x)} ...
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1answer
26 views

Proof that Jacobi method will converge to the solution of a system Ax=b [closed]

Can anyone show me a statement that this works and a proof? Thanks
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2answers
68 views

Why does newtons method converge to the root of an equation?

I'm trying to understand why the Newton Raphson method converges to the root of a given equation? Can someone explain it to me theoretically. Thanks
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0answers
38 views

How to characterise this non-linear optimisation (linear objective function, non-linear constraints)

I was wondering if someone may be able to help me characterise this optimisation problem as I am struggling to find a numerical library that will solve it and I suspect it is because I am using the ...
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0answers
105 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
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2answers
27 views

What are the possible limits of the iteration?

Consider the function $f(x) = \sqrt{2 + x}$ for $x \geq -2$ and the iteration $x_{n+1} = f(x_n) ; n \geq 0$ for $x_0 = 1$. What are the possible limits of the iterations ? $\sqrt{2 + \sqrt{2 ...
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1answer
61 views

How to numerically handle a double integral with a singular endpoint on the outer integral

I am trying to numerically integrate $$\int_0^a f(x) \int_{\sqrt{x}}^\infty \frac{\exp(-u^2)}{\sqrt{u^2-x}}du dx$$ where a is some positive real number and f(x) is some well behaved function. The ...
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1answer
36 views

Can an iterative method converge for some initial approximation?

Studying iterative methods for solving(or approximating) linear equation systems, I came accross the following theorem$^1$: Let the following be an iterative method: $$x^{(0)},\qquad known\\ ...
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38 views

Solving solely continuous system of ode's with matlab

I'm working with the numerical integration of the system of differential equations, $\dot{x}=f(x)$ with the vectorfield, $f(x)$ being solely continuous. Examples of the systems which I'm working on ...
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1answer
237 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
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0answers
24 views

Inverse Iteration to Find Eigenvalues - Question about Method

So I'm doing Inverse Iteration in Excel to find the dominant eigenvalue and eigevector of a matrix. This particular method involves estimating an eigenvalue, multiplying the identity matrix by it, ...
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1answer
37 views

Linear interpolation by hand - Any quick ways to do this?

I have to calculate the roots of the equation $x^3 + x^2 -3x -3 = 0$ in the interval $[1,2]$ using linear interpolation to six decimal places, by hand. Now I know this is trivial in excel, but when ...
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0answers
17 views

Calculate the weights and the node in the integration formula

The problem is the following. Calculate the weights $w_1$ and $w_2$ and the node $x_1$ in the weighted integration formula $\int_0^1x^{\frac{3}{4}}f(x)dx\approx w_1f(x_1)+w_2f(\frac{3}{4})$ The ...
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2answers
57 views

Fastest way to obtain the parametric value t of a bezier curve, for a given set x coordinates.

The problem is the following: Having a bezier curve B(t) we have coordinate x from the curve, and we need to obtain the y values from it, hence we need to compute the t values. What is the fastest ...
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1answer
26 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
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1answer
17 views

Fixed-Point & Root Relation

Lets say I have found out the fixed point for a given function. I'm only given a point that is basically mapped to itself, but how do I find the root of the function and thus any solutions using this ...
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0answers
46 views

Effective computation of matrix commutator

Is there a faster way to compute the commutator of large (at least one of them sparse) matrices $[A,B]$ then to compute $AB$ ,$BA$ and subtract them?
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0answers
25 views

Implementation of Total Variation Regularization Algorithm (Lagged Diffusivity Algorithm)

I am trying to compute the derivative of an experimentally-measured quantity as a function of time. The data are fairly noisy, which causes problems. For instance, using finite differences (central ...
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0answers
28 views

Influence of preconditioning on degenerate eigenvectors

I'm using a hierarchical decomposition of a sparse matrix $A$ as suggested here. I find that the method essentially finds eigenvectors using the QR algorithm. $A$ has some eigenvalues with ...
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0answers
13 views

Best uniforme approximation of nule function in the meaning of Tchebychev

I would be interest to know , why exactly approximate a nule function and it is in the same time nule ? I would be like someone give me enough (papers, link ...) about "The best uniforme ...
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0answers
64 views

Newton's method for multidimensional functions

Can Newton's method be used to find the root of a function f : $\mathbb{R}^n\to\mathbb{R}^m$. Can anyone provide a proof for this? (I have checked the method of solving system of equations with ...
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0answers
49 views

Finding minimum of a distance function using matlab

I have a function for that I want to find the minimum. The function calculates the distance between two sets where a set is defined as matix of row vectors $ D = [ d_1, d_2, ..., d_n]$, $d_n$ is a $m ...
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2answers
45 views

what is the fixed point for f with a given iteration [closed]

let f be a continuous map from the interval [0,1] into itself and consider the iteration $$x_{n+1} = f(x_n)$$ then what is fixed point for f? $f(x) = x^2/4$ $f(x) = x^2/8$ $f(x) = x^2/16$ $f(x) = ...
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1answer
75 views

Double Integrals & Expected Value Monte Carlo Method

Tell me if I'm wrong Let $\Omega = [a,b]\times[c,d]\subseteq\mathbb{R}^2$, then $$ \iint_\Omega ...
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1answer
39 views

Is Cea's lemma sharp?

Given a problem in weak formulation $$ \begin{align} \text{find $u\in V$ s.th. for all $v\in V$} \\ a(u,v) = f(v) \end{align} $$ with bilinear form $a:V\times V\rightarrow\mathbb{R}$, bounded with ...
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0answers
14 views

Calculating h-ellipticity

How do we calculate h-ellipticity $E_{h}$ of standard five point discrete Laplacian of two dimensional partial differential equation?
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22 views

Numerical solution to a coupled differentio-algebraic system of equations

$$\frac{\mathrm{d}X_1}{\mathrm{d}t} = P \times ( \frac{I_a^n}{K_i\times exp(I_a*m) + (I_a)^n} ) \times ( 1-( \frac{A.X_2 + B}{ K_o})^z)$$ $$X_1 = X_2 -[ P' \times \frac{I_a^n}{(Ki*exp(I_a * m) + ...
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1answer
25 views

Non-convergence of Bairstow's method

I am writing a program to compute the roots of a polynomial with real coefficients. I am using Newton's method to get the real roots, and trying to use Bairstow's method for the complex ones. I am ...
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1answer
37 views

How to estimate curve length from random points along a (possibly not connected) curve

I'm working with a closed, smooth, planar curve. I don't know the curve exactly, but I do have an effectively-random collection of points along the curve. What's a good way to estimate the curve ...
1
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1answer
75 views

Using Newton's method to solve a non-linear system of equations over complex numbers

I have a function $f(\bar{z},z)$ mapping from $\mathbb{C}^n \times \mathbb{C}^n \rightarrow \mathbb{C}^n$, which I would like to find the roots of numerically. Since it is nicely formulated in terms ...