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

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Numerical Integration of a differential equation of fisrt-order with constant and variable sampling rate

Example 1 A non-homogeneous differential equation of first-order is given and the integral of it need to be calculated. The differential equation is shown in Figure 1: $ {y}'(t) = Asin(\omega t + ...
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48 views

I need help about a Jacobian Matrix for Newton's Method

Let f (x) = $x^3$, with $x \in $. Given three points $x_{i-1} <x_i <x_{i +1}$ central difference: $$f[x_{i-1}; x_{i+1}] := \dfrac{f(x_{i+1}) - f(x_{i-1})} {x_{i+1} - x_{i+1}}$$ $f_0$ ...
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52 views

Derivation of weights for estimating second derivative from Butcher's tableau of Runge-Kutta methods

Given an explicit Runge-Kutta method (following this notation): $y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i$ $k_i = f\left(t_n + c_i h, y_n + h \sum_{j = 1}^{s} a_{ij} k_j\right)$ How would one derive ...
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82 views

Order of accuracy of the following approximation?

This is a computational analysis class, the answer shouldn't be complicated. Given the ODE: $$\begin{cases}y'=f(t,y)\\y(t^0)=y^0\end{cases}$$ What would be the order of accuracy (using Taylor ...
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33 views

How to generate random matrices when it's singular values are given?

Consider matrix S as nxn diagonal matrix with singular values populated across the diagonals in non-increasing order. I want to know how to create random matrix A whose singular values with be the ...
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37 views

What is the error when approximating $L^2([0,1])$ by a finite dimensional space?

Let $X \subset L^2([0,1])$ such that $f([0,1]) \subset [-M,M]$, for some constant $M$, and any $f \in X$. By choosing a finite dimensional basis $V=\left(v_i\right)_{i=0}^n$, where each $v_i \in X$, ...
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32 views

Check for the value of x for function f is ill conditioned

I have two examples: $$ i) \ f(x) = \sqrt{1 - x^2} $$ $$ ii) \ f(x) = \sqrt{x^2 + 1} - x $$ I must check the value of x for which the calculation of the values ​​of the function $ f $ is ill ...
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101 views

Discretization of a 2nd Order Differential Equation

Good Day to ALL! I want to UNDERSTAND the method of discretizing a 2nd order differential equation. I have tried the same, but my matlab model was a bust (it computes but the result is incoherent). I ...
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33 views

Convergence of $xe^x - R$

Basing my question on one of the previous questions I have passed before Root of the function $f(x)=xe^x-R$, I was wondering why does $xe^x - R$ always converge? I was told that the function will ...
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46 views

Show convergence by Cauchy sequence

\begin{equation} x(k+1)=\arccos\bigg( -\frac{1}{2(Dr^{\frac {|\sin(2x(k)+\theta)|}{M\sin x(k)\sqrt{A+2B\cos(2x(k)+\theta)}}}+1)} \bigg) \nonumber \end{equation} $A,B,D,r,M,\theta$ are constants. ...
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36 views

Finding zero of nonlinear function

I have a function $g(x)$ and would like to numerically find all the zeros of the function for $x$ in some interval $[a ,b]$. My function $g$ is nonlinear and discontinuous. I can compute the ...
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55 views

Richardson's Extrapolation problem

The forward-difference formula can be expressed as $f'(x_0) = 1/h [f(x_0) + h) - f(x_0)] - h/2f''(x_0) - h^2/6 f'''(x_0) + O(h^3)$. Use extrapolation to derive $O(h^3)$ formula for $f'(x_0)$. I ...
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121 views

Newton's Interpolation Polynomial Question

The approximate population of the U.S. was 150.7 million in 1950, 179.3 million in 1960, 203.3 million in 1970, 226.5 million in 1980, and 249.6 million in 1990. Using Newton's interpolation ...
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123 views

Is my form of Lorentz and Gaussian equations suitable for curve fitting?

I am attempting to curve fit gas chromatography spectra. These spectra can be Lorentz or Gaussian or a combination of both. I am using the following code for these equations. ...
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18 views

Why the Hermite Polynomial $H_{2n+1}$ of $f \in C^{2n+2}[a,b]$, it's g'(t) has (2n + 2) distinct zeros in [a,b]?

$g(x) = f(t) - H_{2n+1}(t) - \frac{(t-x_0)^2 \cdots (t-x_n)^2}{(x-x_0)^2 \cdots (x-x_n)^2}[f(x) - H_{2m+1}]$. Thank you.
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38 views

Approximating the speed of an object from a finite list of approximate positions

An observer watches an object and notes that, for $1 \leq i \leq n$, the object was seen approximately at point $p_i$ in the plane at time $t_i$. For the purpose of this problem, suppose that there ...
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46 views

Secant Method Convergence

Prove that if the iterates in Secant's method converge to a point $r$ for which $f'(r)\ne0$, then $f(r)=0$. Hint: For the secant method, the mean-value theorem is useful. This is the case $n=0$ in ...
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191 views

Newton's Method and Secant Method Convergence proof

Show that if the iterates in Newton's method converge to a point $r$ for which $f'(r)\ne0$, then $f(r)=0$. Also establish the same assertion for the secant method. Hint: For the secant method, the ...
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23 views

Determining that a difference scheme is “dissipative of order 2r” in practice

First off, a difference scheme is dissipative of order 2r if there is a constant $\delta > 0$ such that for all the associated frozen coefficient schemes the eigenvalues of the amplification matrix ...
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77 views

What are some examples of difficult sums?

I'm looking for sums such that evaluating $f(x)$ is easy and fast, but evaluating $$\sum_a^{a+n}{f(x)}$$ is slow and hard. To be more scientific, evaluating $f(x)$ takes time $O(n)$, but evaluating ...
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530 views

Finite difference approximation of heat equation with source term

I am using the implicit finite difference method to discretize the 1-D transient heat diffusion equation for solid spherical and cylindrical shapes. The general equation is: $$ ...
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56 views

Bisection method, involving inequalities of roots and its intervals

If the bisection method generates intervals $[a_0, b_0], [a_1, b_1]$, and so on, which of these inequalities are true for the root r that is being calculated. Give proofs or counterexamples. a. $|r ...
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47 views

Please help me about conjugate gradient method

As I know, the error function of neural network is the sum of difference between actual output and the target value. But in conjugate gradient method, they use quadratic function: $E(w) = \frac{1}{2} ...
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65 views

Numerical Error in Computation - What Are the Students' Expected to Know?

I'm going to conduct an educational research about math undergraduates' conceptions about "numerical error." So I'm making a list of items I think someone with a BSc in mathematics is expected to know ...
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110 views

If $f$ is a differentiable function and $f'$ is bounded show that there exists a unique fixed point

let $f$ be a continuously differentiable function from $\mathbb{R}$ to $\mathbb{R}$ and such that $|f'(x)| \leq 4/5$ for all $x \in \mathbb{R}$. Show that there exists a unique $x$ in $\mathbb{R}$ ...
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50 views

Approximating $x - y$ when $x$ and $y$ are really close to each other.

We have this formula $ A = \sqrt {x + \frac {2}{x}} - \sqrt{x - \frac {2}{x}}$ and also $x\gg1$ as you know in these situations we simply multiply $A$ with $ \sqrt {x + \frac {2}{x}} + \sqrt{x - ...
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17 views

Do primitive recursive languages suffice for numerical simulations?

According to the Wikipedia article, a primitive recursive programming language can't do while-loops (but it can do for-loops), but the upshot is that primitive recursive functions are not partial ...
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68 views

Solve: $2div(G(\nabla H_{u,v}\nabla H_{u,v}^\top)\nabla H_{u,v})$

From the work: Algorithms for 3D Shape Scanning with a Depth Camera. Specifically on reducing the following equation in terms of the variable $X\in\Re^{k\times t} $, to be able to program it in ...
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52 views

Numbers immediately to the right and left of $2^k$

What are machine numbers immediately to the right and left of $2^k$ ? How far is each from $2^k$? Please, help!
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131 views

FFT for Fourier Integrals of analytic Functions

So, I'm trying to implement a fourier-transform of an analytic function through DFT or FFT as I'm only interested in a certain frequency range. So far I've tested Numerical Recipes' FFT algorithm ...
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128 views

Numerical Approximation of Black-Scholes, too big for MATLAB?

Note, this is a homework problem. Okay, so the problem is to approximate the Black-Scholes model using a semi-discretization with forward Euler in time and the usual spatial finite difference for ...
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68 views

Small symbols behind parantheses

I am currently reading the following paper where the author uses constantly small symbols after parantheses, but I do not know what this means. I am particularly interested in equation (23), so you ...
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48 views

Comparison of upper bounds of operator norms

I am working with a $C^{1}$ function $F: D \rightarrow \mathbb{R}^{n}$ with Frechet derivative $F'$, and assume the following condition (1) $||F'(z)^{-1}\left(F'(y) - F'(x))(y-x)\right)|| \leq ...
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187 views

Solve Master Equation (Differential Equation) Numerically

any ideas on how to solve the following master equation numerically? $$ \frac {\partial P(x,y,t)} {\partial t} = rP(x-1,y,t) + (1-2r)P(x,y-1,t) + rP(x+1,y-2,t)+\gamma P(x,y+1,t) - (1 + \gamma)P(x,y,t) ...
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42 views

Appropriate numerical scheme for eigen value problem

I am looking for a numerical scheme which can easily handle the following eigen value problem I already had the analytical results of this problem, now I want to know how to treat this problem ...
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98 views

Numerical differentiation with non-uniform step sizes?

Most of text I've read so far about numerical differentiation with higher-order methods has a fixed step size, including the one described on Wikipedia. I have a set of data from a simulation from ...
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25 views

Level set of 2D gaussians

I'm looking for a numerical method to rapidly find the level set $\{x,y\} | f(x,y)=h$, where $f(x,y)$ is a sum of a 2d gaussians (forming a mixture model), and $h$ a constant. Here's an example ...
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37 views

What are the coeff. of this polynomial?

My matlab generates this answer for the problem: p = -20.2090 17.3368 272.9057 -0.7528 Is it correct? It seems I've gotten different results in another ...
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115 views

Numerically Integrating Function Over Sphere Surface Using Irregular Differentials

I'm trying to (numerically) integrate a scalar-valued function over the surface of a unit sphere with irregularly spaced differentials. I take mathematical standard spherical ...
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45 views

Counting the number of additions/subtractions for Gauss-Jordan Elimination

We have a matrix of the form $$\Big[\;\;\;n\times n\;\;\;\Big|\;\;\;n\times m\;\;\;\Big].$$ Where the set up is basically solving $m$ linear systems of equations all with the same coefficient ...
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47 views

Estimate parameters in $y=y_{0}(1-\frac{t}{\tau})e^{-\alpha t/\tau}$

Given the function $y(t)$ with two independent parameters $\tau$ and $\alpha$ $$ y=y_{0}\left(1-\frac{t}{\tau}\right)e^{-\alpha t/\tau}, $$ We have two data points (experimental data) $ ...
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67 views

Testing the stability of a numerical method..

I have been working on this exam question on Numerical Methods for Ordinary Differential Equations: I have completed the first part already - with some help: Show that the method is of order 3 if ...
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935 views

Trapezoidal integration rule for double integrals with non-equally spaced points

I've found this very useful link about 2D Trapezoidal Rule (or Composite Trapezoidal Rule) that works when the points are equally spaced. I want to implement an analogous rule for non-equally spaced ...
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133 views

Bluestein Algorithm for Fast Fourier Transform

Can anyone demonstrate the full algorithm of Fast Fourier Transform? Because from Wikipeida and other internet sources, I saw that there are different ways of padding. So can anyone tell me when the ...
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33 views

Linking a finite difference operator with with derivatives

Suppose differential operator is defined as $$L^N_\epsilon=\epsilon D^-+I$$ where $D^-V_i=(V_i-V_{i-1})/h_i$ and $h_i=x_i-x_{i-1}$. If I have the the exact solution of an ODE as ...
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64 views

Divergence preservation and change of variable

I'm currently studying a numerical scheme which preserves the divergence through time, i.e $div(A^{n+1}) = div(A^{n})$ where $A$ any vector field in $\mathbb{R}^n$ and $n, n+1$ time stations. This is ...
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86 views

How to calculate the error of $x-(x^3-1)^{1/3}$ substituting $x$=35

The equalation $f=\ln({x-\sqrt[3]{x^3-1}})$ is calculated for $x=35$. The root is calculated by taking 8 significiant digits (after the decimal point). What is the error in calculating $f$? ...
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43 views

Reflection lines generation with Matlab

Given a tensor-product surface in Matlab how can I generate reflection lines over this surface with Matlab? Thanks.
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40 views

Estimate a level set of the form $A \equiv \{\mathbf{x} \mid f(\mathbf{x})=\alpha \}$

Suppose I have a continuous function $f(\mathbf{x}):\mathbb{R}^d \mapsto\mathbb{R}$. I am interested in the level set $A \equiv \{\mathbf{x} \mid f(\mathbf{x})=\alpha \}$. Suppose the lebesgue measure ...
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223 views

Newton iteration- estimate the error

I was wondering whether there are equations available to estimate the a priori and a posteriori error for newton's method? My idea was to use that it is a fixed point iteration and therefore one can ...