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

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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
28 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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36 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
31 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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59 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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36 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
24 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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33 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 ...
2
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0answers
100 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
54 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 ...
2
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1answer
34 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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37 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
219 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
33 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
53 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
25 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
16 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
40 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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19 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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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
61 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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47 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
42 views

what is the fixed point for f with a given iteration

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
61 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
32 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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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
23 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
35 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 ...
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1answer
67 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 ...
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Is the following statement on the stability of the forward Euler method true or false?

My text asks whether the following statement is true or false: The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step ...
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Causality and viscous wave equation

I've seen several papers related to the causality condition concerning the viscous wave equation resolution but never understood how causality and stability are linked ? Conceptually, it seems hard to ...
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1answer
30 views

Numerical instabilities in the viscous wave equation

I'm currently working out the wave equation with the introduction a viscosity term, s.t. $a(x)\ddot{u}(x, t) = \nabla (b(x)\nabla . u(x, t) + c(x) \nabla . \dot{u}(x, t))$ (where $u$ is the ...
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51 views

Are there high performance computing applications for symbolic integration?

Currently there are a number of applications for numerical integration in applied mathematics and physics. Many of these are integral transforms (often Fourier or Laplace), or solving definite ...
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40 views

Avoiding substraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the substraction is blown up for small h. This I can verify ...
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0answers
20 views

Change in Singular Value Decomposition of a matrix on addition of a single row

Given that I know the svd decomposition of a matrix, is there any way to compute the svd decomposition of the matrix obtained by adding a single row to the original matrix? Is there any relation ...
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0answers
26 views

heat equation with Interface Crank Nicolson

I am currently working on solving the heat equation with an interface numerically using Crank-Nicolson. There are jump discontinuities at the interface which are dealt with using fictitious values ...
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Cubic convergence of Rayleigh quotient iteration?

Trefethen and Bau, Numerical Linear Algebra, p. 208 states that Rayleigh quotient iteration (combining Rayleigh quotient estimate for eigenvalues and inverse power iteration) converges cubically ...
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If symmetric matrix in a least-square deconvolution problem positive definite?

I want to apply Gauss-Seidel method in a least square deconvolution problem. The convolution of two vectors is written in: $h * x = z$. $$z(n) = \sum_{i=0}^{N-1}h(i)x(n-i)$$ It is a linear transform ...
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1answer
18 views

Meaning of the unique up to a normalization factor

In the following text about Gaussian quadrature by Brian Bradie I cannot understand the meaning of the author: Associated with each weight function is a special family of polynomials, unique up ...
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34 views

Quadrature formula of type Gauss.

I am preparing for a test and I have to solve a type of exercise for witch I don't have any good examples. What I have to do is to find a quadrature formula for: $$\int_{-a}^{a} w(t)f(t)dt ...
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1answer
46 views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
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1answer
43 views

Do elliptic allow for direct solvers of roots of quintic polynomials?

Galois Theory tells us that we cannot directly solve for the roots of a quintic polynomial using elementary operations and radicals. I have seen sources that use this to reason that any computer ...
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1answer
48 views

Euler's Numerical Method

Let $\eta(x;h)$ be the approximate solution furnished by Euler's method for the initial-value problem $y'=y, y(0)=1$. I proved that: $i) \eta(x;h)=(1+h)^{x/h}$; $ii) \eta(x;h)$ has the expansion ...
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1answer
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Convergence of Gauss-Seidel.

I am preparing for an exam and I have an exercise type, without any example. Give the following system : $$ \begin{bmatrix} 2 & 1\\ -1 & 2\\ \end{bmatrix} *x = ...