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

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52 views

The convergence of the fixed-point iteration for solving a cubic equation

I have a third-grade polynomial of the form $Ax^3+Bx+C$ and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this: $0=Ax^3+Bx+C$ and for a given ...
2
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45 views

Integrating sine with Monte Carlo / Metropolis algorithm

I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ...
2
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62 views

Any good approximation for this integral?

I am interested in the following integral $$ \mathcal{I}=\int_{-\infty}^\infty\mathop{dz}\left[\frac{1}{\sqrt{a+b}(z^2)^{n/4}}-\frac{1}{\sqrt{a+b\cos^2\theta}(R^2+z^2)^{n/4}}\right], $$ where $R\ll ...
2
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24 views

Is there a faster algorithm than $O(n^2)$ for calculating “cofactors” $C_k = \prod\limits_{j\neq k}(c_k - c_j)$?

Is there a faster algorithm than $O(n^2)$ for calculating "cofactors" $C_k = \prod\limits_{j\neq k}(c_k - c_j)$ ? (presumably $O(n \log_2 n)$ if one exists) In other words, if I have factors ...
2
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29 views

Computing Cholesky Factorisation by Hand

It is a common exam problem to compute the Cholesky factorisation of a small (typically 4x4) matrix. I know that this can be done by first finding the matrix $U$ in the $LU$-decomposition (e.g. by the ...
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27 views

Why does this algorithm converge?

Consider the following problem. Let $p_1, \dots, p_n \in (0,1)$ such that $\sum p_i = 1$. Let $m > 0$ such that $$ q_i := p_i + m \frac{p_i \log(p_i)}{\sum p_k \log(p_k)} < 1 $$ Suppose ...
2
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27 views

FEM for a 1D heat equation system

I want to know how to implement the (nonhomogeneous) initial boundary value problem for a heat equation; $$u_{xx}=u_t ~~~x\in (-1,1),~t\in(0,1)$$ $$u(0,x)=u_0(x)$$ $$u(t,-1)=f(t), ~u_x(t,1)=0$$ Many ...
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50 views

Are two linear system equivalent?

Let $A$ and $M$ be square matrices of size $s$ and $n$ respectively, let $k_i \in\mathbb{R^n}$ be column vectors for all $i=1,\ldots,s$. Denote $K=\left[ \begin{matrix} {{k}_{1}} \\ \vdots ...
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38 views

Numerical solution of the stationary Navier-Stokes equations

Let $d\le 3$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I'm considering an incompressible Newtonian fluid with uniform density $\rho_0$ and viscosity $\nu$. In this case, the stationary ...
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19 views

Why study the log of the absolute value of the error induced by a numerical method when solving a PDE?

Often when validating the use of a numerical PDE solution method, one considers the log of the absolute value of the error. In particular, instead of simply looking at the difference between the exact ...
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20 views

PDE Existence and Uniqueness through discretization

This is a question I have been thinking about, but I'm not sure where to look to find an answer. Have a PDE in space and time $(x,t)$. Have a time discretization of the PDE, this results in a ...
2
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29 views

Sturm-Liouville Variational Problem

I'm entirely clueless with this problem. No formal training in variational methods. Show that for function $\phi\left ( x \right )$ with $$\phi\left ( a \right )=\phi\left ( b \right )=0$$ and ...
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17 views

How to linearize this second order differential equation?

I want to linearize this second order non linear differential equation arround P($\pi/2 $,0) $\frac{d^2y}{dt}+\frac{dy}{dt}-9.8Sin(y)=5Sin(3t)$ I undestand how to linearize single variable functions ...
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40 views

Numerical intergration of a complex, oscillatory function (Bessel function, Singularities)

I am working on a university project at the moment and at some point I needed to calculate the intergral of the following function (Please refer to "Bakthiari et al. - Analysis of radiation from an ...
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55 views

Finding an error bound for Lagrange interpolation with evenly spaced nodes

I know that the error bound for Lagrange interpolation is usually $$\frac{M_{n+1}}{(n+1)!}\max_{x\in[a,b]}|(x-x_0)\cdots(x-x_n)|$$, where $M_i=\max_{x\in[a,b]}|f^{(i)}(x)|$. I'm trying to find the ...
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34 views

Approximating two-dimensional convolution

I am trying to use discrete 2d-convolution to estimate continuous double convolution. The convolution integral is $$g(x,y)=(f\ast h)(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(u,v) ...
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41 views

Transforming integrals of the form $\int \frac{f(x)}{\int g(x,y)dy}dx$ into multiple integration. Possible?

I'm currently dealing with integrals of this form, which I approximate numerically. I wonder if I can manage to perform some transformation resulting in a double integral or any other kind of multiple ...
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44 views

Problem in integrating the FitzHugh-Nagumo model

I am using the FitzHugh-Nagumo model to perform some simulations concerning neuronal networks dynamics. The model equations are: $\frac{dv}{dt}=a_3^3v^3+a_2v^2+a_1v+bw+I\\ \frac{dw}{dt}=1/\tau(v-w)$ ...
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50 views

Convergence of difference equation to differential equation

Starting with the difference equation: $$f(x(t+dt),t+dt)= (1-a)f(x(t),t) + a f(x(t+dt),t)$$ where $x(0)$ is given and positive, $a\in(0,1)$, $f(0,t)=0$, and $f$ is increasing in both arguments. ...
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34 views

Solving Schrödinger's Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system

Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$ $r_1$ is the distance between the proton $1$ and the electron. $r_2$ is the distance between the proton $2$ and the electron. $R$ is the ...
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199 views

Abel-Plana formula for $\zeta(s)$, is this integral approximation correct?

I wrote a computer program to calculate values for $\zeta(s)$. I was scanning for something that would calculate complex values for $\zeta(s)$. I found the following approximation under the Integral ...
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440 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
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113 views

Contour integral mystery: why is the answer different from Maple/Matlab?

The mystery is that here is a fairly standard contour integral which can be done by the residue theorem. Yet when I tried to evaluate it using numerical softwares like Maple or Matlab, the answer is ...
2
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24 views

variable transformation in optimization

I have an optimization problem with two sets of parameters, $x_i \in [0,1]$ and $y_k \in [-\frac{\pi}{2},\frac{\pi}{2}]$ where $i,k \in \{1...n\}$ are indices. One way to solve this problem is using ...
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84 views

Please guide me books and online materials for this course

I have recently taken Course on Numerical Analysis. It is correspondence course. So i to do self study. I will be glad if someone mentions online videos and elementary books which contains following ...
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0answers
55 views

sum of reciprocal of roots

Say we are given $$\alpha \tan \sqrt x =(1+\alpha) \sqrt x$$ where $\alpha > 0$ ad are after its positive roots. In particular, I am interested in estimating the following ...
2
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0answers
53 views

two point block method for solving ODE

How to solve the ordinary differential equation $$y'(t) = -1000 y(t)+ 999 e^{-t}, \hspace{10mm} 0≤t≤5.$$ $y(t)=e^{-t}$, for $t<0$. Using two point block method $$hf_{n+1}= \frac{1}{3} (hf_{n+2} ...
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62 views

Newton's method on a surface

I am trying to use Newton's method to find the stationary solutions of the integro-differential equation of the form $$\frac{\partial u(r,t)}{\partial t} = -u(r,t) + \int_{\mathbb{R}^{2}}w(r - ...
2
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0answers
42 views

Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
2
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28 views

What is a stochastic differential equation of the form $dZ = f(Z_{prev}, X_{prev})dt + CdW_t$ called?

At every time step I can approximate the change in $Z$ using the following equation: $$ dZ = f(Z_{prev}, X_{prev})dt + CdW_t, \quad(1)$$ $$dW_t = r\sqrt{dt}$$ where $C$ is some constant, and $r$ is ...
2
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0answers
27 views

Reference request for finite difference method

I am trying to use finite difference method to solve the minimizing problem $$ J[u]=\min_{u\in BV(Q)}\{\|u-f\|_{L^1(Q)}+|u|_{BV(Q)}\} $$ where $Q=(0,1)\times (0,1)$ is a uint square and ...
2
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0answers
45 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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59 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
22 views

Methods for Extrapolating Data Close To Observations

I am aware of many different ways to go about interpolating between the values of known data points. However, whenever I come upon (I work in Quantitative Finance) the need to extrapolate data I find ...
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0answers
44 views

Weighting on position within a range of numbers

Firstly, hello all. I'm normally to be found on StackOverflow but felt that this forum was more appropriate for my question. Count this is a coffee break teaser, rather than a fully challenging maths ...
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0answers
56 views

Can trigonometric functions for double precision be implemented in terms of those for single precision?

In some program environments like GLSL there is support for single and double precision numbers for arithmetic and square roots computation, but only single precision trigonometric functions are ...
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79 views

Asymptotic expansion of root of $\epsilon x \tan(x)=1$

Indicate a range of roots of $\epsilon x \tan(x)=1$ for which it is impossible to get an approximation using expansions. Since $\epsilon$ is small, I think for the equation to hold, we need ...
2
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0answers
75 views

Fast algorithm to invert a large sparse matrix

I am interesting in sparse matrix that defined at here. I am looking for a fast algorithm to invert the matrix (better than Gaussian Elimimation). Could you suggest to me some methods that reduce ...
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0answers
42 views

A trajectory for shortened k-space data acquisition MRI

Given a real function $f:\mathbb{R}^n \to \mathbb{R}$, denote by $\hat{f}$ its Fourier Transform. I have shown that $\hat{f}(\vec \omega)=(\hat{f}(-\vec \omega))^*$ where $^*$ denotes complex ...
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61 views

$A(\theta)-$ stable method, region of absolute stability

We have to look for numerical methods for the numerical solution of $\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0 & \end{matrix}\right.$ that have 'great' regions of ...
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148 views

Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
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0answers
154 views

Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) ...
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0answers
44 views

Existence and uniqueness of solution of the ODE

Consider the initial value problem $(1)\left\{\begin{matrix} y'(t)=y^2 &, 0 \leq t \leq 2 \\ y(0)=1 & \end{matrix}\right.$. Verify that the following theorem: "Let $c>0$ and $f \in ...
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102 views

Solution of non-linear Fredholm(Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question but got no response. I rephrase it so that anyone who knows operator theory and integral equations would help me out.....I faced a problem in physics which is a non-linear ...
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0answers
34 views

Good method for finding roots that *usually* fall within an interval?

I've been using Brent's method to find the roots of a monotonic, nonlinear, non-differentiable function. The roots often fall within a known interval, but Brent's method fails if they occasionally ...
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47 views

Are there any symplectic integration techniques that are A-stable (work on stiff equations)?

The first and second Dahlquist Barriers show that (paraphrasing): Explicit multi-step methods cannot be A-stable and thus are not accurate for stiff equations. Implicit multi-step methods will only ...
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61 views

Solving equations in R

I have a question if a method is true and the proof of it: So, let us have a real function f which satisfies that f(a) < 0 and f(b) > 0, f is continuous in [a,b], and f'(x)>0 in [a,b]. Then we ...
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0answers
90 views

Numerical integration in 2D over a triangle - Quadrature formula

I am looking for highly (order 6 at least) accurate (for small triangle) quadrature formulas (using only values of the function, no derivatives) to calculate an integral of a continuous function (no ...
2
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0answers
42 views

Mean value theorem for sequences

This is a problem I am trying to solve. Given a sequence $x_n$ defined $x_{n+1}=F(x_n)$. Assume $\lim_{n \to \infty}x_n=x$ and $F'(x)=0$. Need to show that $$x_{n+2}-x_{n+1}=o(x_{n+1}-x_{n}).$$ ...
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0answers
131 views

Trapezoidal rule - Multivariable

If I wanted to integrate the function $f(x,y)$ over the region $[a,b]\times[c,d]$ with two segments, am I going about this the right way? $$I(f) = \int_a^b \int_c^d f(x,y)\ dy\,dx = \int_a^b g(x) \ ...