Tagged Questions

Approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.

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Approximation of “smooth” discrete functions

Assume $f : \{1,\ldots,n\} \to \mathbb{C}$ satisfies $|D^\ell f(i)| \leq C$ for all $i \in \{1,\ldots,n-\ell\}$ and all $\ell \in \{0, \ldots, k\}$ for some $k \in \mathbb{N}$. Here, $D$ denotes ...
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How to determine the truncation error with products and quotients

If I have an equation given by $$\displaystyle Y = \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$ and I expand $a,b,c,d$ in a Taylor series, where $a$ is truncated at the $A^{th}$ order, $b$ is ...
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Newton's method for nth roots of complex numbers

Is it possible to use Newton's method to compute roots of complex numbers, say $\sqrt[n]{a+ib}$ to any desired accuracy? If yes,for what initial values will converge?
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Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
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An apparently new method to compute the $n$th root of any complex number

I found  a series of articles (in Portuguese) by a Brazilian mathematician named Ludenir Santos, where presents a series of iterative methods, he said new, to extract nth roots of any complex number ...
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The convergence of an infinite radical involving $\cos(\alpha/3)$

By using the triple angle formula for the cosine, $\cos 3\alpha$, we get the cubic equation $4x^3-3x = \cos \alpha$. Now, by expressing $x$ as $x = \frac{1}{2}\sqrt{3+\frac{\cos \alpha}{x}}$ ...
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A curious algebraic fraction that converges to $\frac{\sqrt{2}}{2}$

I have noticed that the algebraic fraction $\frac{3a+2b}{4a+3b}$ Gives better and better approximations to $\sin 45^\circ = \frac{\sqrt{2}}{2}$ For $a = b = 1$ we get $5/7 \approx 0.714$ ...
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Approximating Minkowski Sum of 3 dimensional Convex Polytopes by Sampling

Let $P_1,P_2...P_r$ be a set of convex polytopes with $n_r$ vertices in 3 dimensions. These polytopes basically represent uncertainties of '$r$' number of 3d-points respectively in space. The global ...
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Approximation of ODE [duplicate]

A problem I should solve is the following IVP: I need to find an approximation for small $\varepsilon$ of $$y'' + y + \varepsilon y^3 = 0, y(0) = 1, y'(0)=0$$ My approach so far was: I assume that I ...
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Does monomials approximate the constant function in the sense of $L^2$?

I'm trying to show that $\{x,x^2,x^3,\dots\}$ approximate the constant function in the sense of $L^2[0,1]$-convergence; i.e. that there exists a sequence of polynomials $p_n$ with $p_n(0) = 0$ such ...
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Close approximation for absolute value function

I made a very acurate approximation function for $\sqrt{n^{2}+1}$ It is $\sqrt{n^{2}+1}\approx\frac{2n(n^{2}+1)}{2n^{2}+1}+\frac{2n^{2}+1}{n(4(2n^{2}+1)^{2}+1)}$ From this I can make a very close ...
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Function Approximation by Wavelet method

If $f(x)\in{L^{2}(R)}$ , then $f(x)$ can be projected into the subspace $V_{j}$ as, $$P_{j}f(x) = \sum\limits_{k\in{Z}}c_{j,k}\phi_{j,k}(x)$$ where $k\in{Z}$. The projection equation is ...
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Algebraic approximation to differential equation

Suppose I have a first order differential equation of the form: $\frac{dx}{dt} = \frac{1}{\tau}f(x,t)$ where $f(x,t)$ is a nonlinear function. In the limit where the time constant $\tau$ is small, ...
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Runge's Approximation Theorem

This is a homework problem, so my apologies for a seeming lack of motivation. I want to understand what's going on in this problem more than I want someone to just write down the solution, so if ...
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I am rather stuck on a question from a textbook that I am practicing from which goes as follows: The forward difference formula can be expressed as $$f'(x_{0}) = \frac{1}{h}[f(x_{0} + h) - f(x_{0})] ... 0answers 21 views How to obtain an analytic approximation of this wavelet-looking function? I would like the approximating function to be infinitely differentiable. I have tried fitting (1) a polynomial and (2) a combination of sin, cos, and exp functions to approximate the function shown ... 1answer 53 views Approximating polynomials over \mathbb{C} with an entire function Given a series of polynomials p_{n} and a series of non-intersecting balls B_{n} \subset \mathbb{C} show that there exists some function f \in \mathcal{O}(\mathbb{C}) such that lim_{n \... 2answers 43 views Interpolation by splines: how to set up the equation system for finding the coefficients of the spline (in a B-spline basis) Problem. I want to interpolate a function f in some equidistant points x_0<x_1<x_2<x_3<x_4 using a quadratic spline. My attempt. I assume that we can use the interpolation points as ... 2answers 31 views How could I solve this equation:  n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1}  for x? I want to have a solution for x in this equation.$$ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1}$$Thanks ! 0answers 55 views Is the Preimage of an Open Disk a Continuous Action Let the kernel-like operator, K, be defined as K(f,R) = \text{Closure}({\{x; \lvert f(x)\rvert < \frac{1}{R}\})} where f:\mathbb{R} \to \mathbb{R} is a continuous function on the complex ... 3answers 50 views How to show that 2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2)) How can we show that: 2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2)) for large N. 0answers 14 views Finite element approximation in semi-norm |\dot|_{H^1} and L^2 Let \Omega be a polyhedral domain in \mathbb{R}^n. Consider the Poisson equation -\Delta u=f in \Omega subject to the following non-homogeneous boundary condition u= g_D on \Gamma_D; \... 0answers 37 views Integral involving double exponentials I am trying to estimate the growth in time of a certain quantity. In order to do this, I have to evaluate the following integral: \int_0^{+\infty} \exp(-b x)\exp(-c\,t \exp(-a x \log x)) dx with ... 2answers 40 views Validity of an Approximation I am considering the approximation of the following integral:$$\int_{-W/2}^{W/2}\exp[-p( x + iq)^2 ] dx \approx \int_{-\infty}^{\infty}\exp[-p( x + iq)^2 ] dx = \sqrt{\frac{\pi}{p}} For large ...
Proposition Let $\mathcal{P_e}$ be the set of functions $p_e(x) = a_o + a_2x^2 + \cdots + a_{2n}x^{2n}$, $p_e : \mathbb{R} \to \mathbb{R}$ Show that all $f:[0,1]\to\mathbb{R}$ can be ...