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Questions tagged [approximation]

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

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Finding optimal knots for function approximations

I would like to approximate a continuous (complex) function $f(x)$ in the interval $[a,b]$ $ (x\in\mathbb{R})$ by local polynomial functions of order $3$ (cubic Hermite spline or cubic C2 spline). Is ...
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approximating a univariate function $y=f(x)$ by roots of a bivariate polynomial

What is known about approximating a univariate monotone function $y=f(x)$ defined on $[0,1]$ or any finite domain by roots of a bivariate polynomial? For example a second order bivariate polynomial $...
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Convergence estimates for approximation with Gaussians / radial basis functions

tl;dr: Are there known convergence estimates for approximating a function with a radial basis family? Details: Let $\mathcal{G}$ be a family of radial basis functions, e.g. $\mathcal{G}=\{\exp(-\...
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Is the product of two “sample” matrices a sample matrix?

Let $f,g \colon \mathbb R^N \to \mathbb R$ be two (sufficiently regular) functions. Take a uniform, square grid of the unit cube in $\mathbb R^N$ and let $\{p_{ij}\}_{i,j=1, \ldots, L} \subset \...
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1answer
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Analytical approximate solution to a trascendental equation

I have the following equation to solve $$ z+e^{z^2}\operatorname{erfc}(z)=0 $$ being $$ \operatorname{erfc}(z)=1-\frac{2}{\sqrt{\pi}}\int_0^ze^{-s^2}ds. $$ I solved it numerically and appears to have ...
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Small-Angle Approximation for Cosine

The small-angle approximation for cosine is: $$ \cos (x) = 1 - \frac{x^2}{2} $$ Question: How can I find a range of values of $x$ for which this approximation gives correct results rounded to 2 ...
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The problem of the Chebyshev approximation in complex sense

Let there be given the set of points H of the real line,and the real functions $f(x)$ and $F(x; \lambda_1 ,\lambda_2 .... ,\lambda_n)$ of the real variable $x$, which are bounded on this set. We have ...
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Approximating $\int_1^3 \int_1^3 \int_1^3 x^{y^z} \mathrm dx\mathrm dy\mathrm dz$?

I was given a really nasty integral to approximate, which was $$\int_1^3 \int_1^3 \int_1^3 x^{y^z}\mathrm dx\mathrm dy\mathrm dz$$ I was completely clueless; and my interviewer gave me some hints ...
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How do terms like $x\sqrt{1-x^2}$ enhance the ability to approximate analytical functions?

In this paper Quantum Circuit Learning it wrote that the ability of a quantum circuit to approximate a function can be enhanced by terms like $x\sqrt{1-x^2}$ ($x\in[-1,1])$. Given inputs {x,f(x)}, ...
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How to approximate real numbers using members of $\mathbb Z (\sqrt d)$?

Real numbers can be approximated to successively better precision using the convergents of a continued fraction. Is there a similar way to find quadratic integers of fixed (positive) discriminant ...
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Approximation of integral

We consider the integral $$\int_0^1\frac{1}{x+3}\, dx$$ Using the trapezoid formula with $h=0.25$ we get the following: The formula is $$T(h)=h\cdot \left (\frac{f(a)}{2}+\sum_{i=1}^{n-1}f(a+ih)+\...
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Smoothing a function

Given a function $f(x)$ does there exist a sequence of smooth functions that $f_{n}(x) \to f(x)$ as $n \to 0$? I am currently trying to smooth out a kink for the general function $x^{1/p}$. An ...
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1answer
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Approximating when variable to infinity

In a book on algorithms I read that $n^2 (1+\log n)$ as $n$ approaches infinity is approximated to $n^2 \log n$. I am not sure if I understand reasoning in this. Is it because $1+\log n$ grows so ...
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Approximation of piecewise linear functions by constant function

Let $f(x) = \begin{cases} A_1 x + B_1 &\mbox{if } x \in [a, x_0] \\ A_2 x + B_2 & \mbox{if } x \in [x_0, b] \end{cases}$ and $f \in C[a, b]$ i.e. f has the "angle" form. Denote $E_t(f) = ||...
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In linear interpolation, what exactly is $\frac{x-x_i}{x_k-x_i}$ in geometric terms?

Thanks to this question: Explanation of Lagrange Interpolating Polynomial, I have an intuition for what $\frac{x-x_i}{x_k-x_i}$ is doing in polynomial interpolation. That is, it is a kind of "on and ...
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How to determine significant digits in tolerance of averaged measurements

Consider the scenario where you have a measurement system which records at a resolution of 0.001 but has an accuracy of +/- 0.01. Then, with the following measurement: ...
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1answer
49 views

Double-precision algorithm for inverse log gamma or log factorial?

Question in a nutshell: Can anyone point me to an algorithm for computing to double-precision floating-point (roughly 16 digits) the inverse of either log gamma or log factorial? In other words, if ...
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A variant of Kronecker's approximation theorem?

Let $\tau,\sigma\in(0,\infty)$ with $\frac{\tau}{\sigma}\notin\mathbb Q$. By Kronecker's approximation theorem, we know: (1) For each $x\in \mathbb R$ and $\epsilon>0$, there are $m,n\in\mathbb ...
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Any good approximation for $\Phi^{-1}(1-\Phi(a))$?

Let $\Phi$ be the standard Gaussian CDF and $a > 0$. Question Is there any good approximation for $\Phi^{-1}(1-\Phi(a))$ ?
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1answer
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Approximation of eigenvalues of matrix

We have the matrix \begin{equation*}A=\begin{pmatrix}1/2 & 1/5 & 1/10 & 1/17 \\ 1/5 & 1/2 & 1/5 & 1/10 \\ 1/10 & 1/5 & 1/2 & 1/5 \\ 1/17 & 1/10 & 1/5 & ...
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Least Squares : Approximation of cubic polynomial

I want to determine an approximation of a cubic polynomial that has at the points $$x_0=-2, \ x_1=-1, \ x_2=0 , \ x_3=3, \ x_4=3.5$$ the values $$y_0=-33, \ y_1=-20, \ y_2=-20.1, \ y_3=-4.3 , \ y_4=32....
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Can I generalise $x_{n+1}$ in this case?

I have the following algorithm for producing rational approximations of $\sqrt x$. We take an initial $x_0$ and apply the following steps: $$x_n=\sqrt{\mu_n}+\sqrt{x+\mu_n}\tag1$$ $$x_{n+1}=\sqrt{x+\...
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Error upper bound of minimax polynomial approximation over $[a,b] \cup [c,d]$

I found that Jackson's inequality in approximation theory provides us a nice upper bound on the (infinite-norm) error of minimax polynomial over an interval $[a,b]$ as noted in [1, p. 16]. Can it be ...
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Kernel evaluations of and order approximations of 2nd order Volterra integral equation

The integral equation $u:[a,b]\to \mathbb{R}$ $$u(t) = f(t) + \int\limits_a^t K(t,s)u(s)ds$$ defined on the interval $[a,b]$, with $f:[a,b]\to \mathbb{R}$ and $K: [a,b]^2 \to \mathbb{R}$ some known ...
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1answer
50 views

Proving error bound for Simpson's rule

The Simpson's rule can be stated as follows: $$\int\limits_{x_0}^{x_2}f(x)dx\approx \frac{h}3\left[f(x_0)+4f(x_1)+f(x_2)\right]$$ The way I'm trying to find the error bound for the Simpson's rule is ...
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Solve $(2xy+e^x) dx + (x^2 y -\sin{y}) dy=0$

Solve: $(2xy+e^x) dx + (x^2 y -\sin{y}) dy=0$ Clearly, the differential equation is not exact. After applying various methods to no avail, (including power series and Fourier expansion) I cannot see ...
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Show vector is approximately an eigenvector of matrix, thus find eigenvalue

Say we have matrix $\mathbf{A}$ $$ \mathbf{A}=\begin{pmatrix} -3&2&0\\ 4&-6&2\\ 0&1&-1 \end{pmatrix} $$ We now must show that $\mathbf{v}=\begin{pmatrix}-1.34&-0.8&1\...
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Finite difference approximation for iteration matrix?

Taking $A=D+L+U$, I am able to find that $(\lambda D + L+U)v=0$ If $A$ and $D_i$ are defined as Why is the finite difference approximation: $-u_{j,k-1} - u_{j,k+1} - \lambda u_{j+1,k} - \lambda u_{...
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Approximating values while calculating percentage changes

At times, in certain types of data interpretation questions that usually get asked in aptitude examinations, some techniques are employed to cut time on calculation and get a near perfect answer. One ...
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What is the cheapest known finite dimensional approximation of Lipschitz functions.

Let $Lip_{1}([0,1]^{d})$ the set of all Lipschitz functions on $[0,1]^{d}$ with Lipschitz constant less or equal to $1$. I would like to approximate the set with respect to the uniform topology by a ...
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Negative order of accuracy

Suppose we analyze the order of accuracy of a finite difference approximation of a derivative, $$f'(x)=\frac{1}{2h} \left[f(x-2h) -4f(x-h) +3f(x)\right]$$ and we conclude that the order of accuracy ...
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41 views

Approximating $\log(X-Y)$

Is there a way to approximate $\log(X-Y)$ as $f(X)+f(Y)$?
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How to find a function's approximation?

I am having problems with the following question: Use the linear approximation $(1+x)^k\approx 1+kx$ to find an approximation for the function $f(x)$ for values of $x$ near zero $$f(x)=\sqrt[3]{\...
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result of multiplication having zeroes after the decimal

Given the multiplication $3.25 \times 0.4$, the primary school students learn that we multiply the digit 4 to the number 325 which result in 1300 we count and then add the number of decimal place ...
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1answer
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Bilinear Interpolation - Alternative Calculation

Problem description: Given four points $P_i$ with coordinates $(x_i, y_i, z_i)$ find the $z$-value at point $C$ with known $(x_c, y_c)$ that lies within the quadrilateral formed by the $P_i$s. I am ...
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Approximating $\pi$ with arctangent

Use the fact that $\frac{\pi}{4} = \text{arctangent}(\frac{1}{2}) + \text{arctangent}(\frac{1}{3})$ to determine the number of terms summed to ensure an approximation to $\pi$ less than $10^{-3}$. So ...
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Proving that a given function $f^*$ is the best least square approximation

In De Boor (1972) it is stated that Let be $\$ $ a finite dimensional linear space of functions defined on the interval $[a,b]$. We are searching for the best approximation from $\$$ to $g$. ...
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The best approximation method to recover original polygon outline before rasterization procedure

I have a polygon, originally created as a Bézier Curve (black outline on the picture), and then saved as a polygon with enough points to call it smooth (at this scale). Then this polygon was ...
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comparison of wavelet coefficients

Consider a function the following spaces: $$ \{f : \|(i\omega)k \hat f(\omega)\|_p ≤ 1, k ∈ N ∪ 0, p ∈ (1, ∞)\}. $$ Denote by $ \psi^m_D$ an orthonormal Daubechies wavelet of order m. One can find ...
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Solve given equation to a approximate value

Can some explain to me how the relation shown below gives $\log k$ on approximation? $$x = \sum_{i=0}^{k-1} \frac{1 }{k-i}$$
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Approximation of an $L^1-$ function of two variables by trigonometric polynomials.

We know as a theorem that the trigonometric polynomials are dense in $L^1([0,1))$ For instance for a Lebesgue integrable function we use the Fejer kernel $$F(x)=\sum_{n=-N}^N(1-\frac{|n|}{N+1})e^{2\...
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Approximate C0-Funktion with C1-Funktions

Let $I=[a,b]$ and $f\in C(I).$ I want to show that there exists a $g\in C^1(I),$ such that for any $\varepsilon > 0$ $|f(x)-g(x)|<\varepsilon$ for all $x \in I.$ By Stone-Weierstrass ...
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First order approximation of $\zeta$(s) at s=1

I was playing around with Wolfram Alpha. I found one interesting thing when I asked it to evaluate this particular summation.$$\Sigma_{n=1}^\infty\frac{1}{n^{1+10^{- 10}}}$$ It returned this$$ \...
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2answers
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Picard's method of succesive approximations

Construct first $3$ succesive approximations $x_0,x_1,x_2$ for the following Cauchy problems: $$x'=-x+t^2$$ $$x(0)=2$$ I have no idea how to start this... any ideas?
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Using the complex exponential function to calculate the $\sin(30^{\circ})$ [closed]

I need help with the calculation of the following function $$\sin(\omega t)=\frac{(e^{iωt}-e^{-iωt})}{2j}\implies \sin(30^{\circ})= \sin\left(\frac \pi 6 \right) = \frac{e^{\frac \pi 6 i}-e^{-\frac \...
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1answer
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Approximation of tan(f(n))

I have the function $f(n)= \pi (n+\frac{1}{2})+\epsilon \frac{2\pi(n+1/2)-a}{2\pi(n+1/2)}$ when $\epsilon<<1$, I can't understand what identity I need to use to prove that: $\tan(f(n))\approx \...
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1answer
24 views

Higher Dimensional Random Walks

I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in an arbitrary direction (uniformely on the unit sphere $S^1$), the expected distance ...
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Nonlinear approximation by piecewise constants and the expectation of the error

Suppose $\Omega=[0,1]$ and $f$ is continuous, monotonic, and of bounded variation on $\Omega$ with $M:= \text{Var}_\Omega(f).$ Let $\| \cdot\|:=\| \cdot\|_{L_\infty(\Omega)}.$ Let $T=\{0=t_0,...,t_n=...
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1answer
41 views

Approximate function by stacking building blocks

I need some help with a 'generalised Lego problem': Given a function $f(x)\geq 0$ on an interval $[a,b]$, and a 'building block' function $p(x)\geq 0$ with compact support. The maximum of f shall be ...
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Some multivalued problems - Yosida approximation

I study multi-valued problems on the basis of M. Chipot's book and I have a seemingly simple problem. Set for a.e. $x \in \Omega$, $\lambda > 0$, $\forall t \in \mathbb{R}$: $$ J_\lambda (x,t) = ...