Questions tagged [extrapolation]

For question on extrapolation, the process of estimating, beyond the original observation interval.

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extrapolation and Stein's interpolation

Here $S=\{z \in \mathbb{C} : 0 < \Re z < 1\}$, $\overline{S}=\{z \in \mathbb{C} : 0 \leq \Re z \leq 1\}$ are the open and the closed strip and $(\Omega, \mu)$ is a probability space. Let $(T_z)_{...
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26 views

Richardson extrapolation formula for Runge-Kutta method

We use the 4-stage Runge Kutta method and we have computed $y_{n+1}^{(h)}$ and $y_{n+2}^{(h / 2)}$, two approximations of $y\left(t_{n}+h\right).$ Develop a formula for the Richardson extrapolation by ...
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62 views

Inequality for decreasing functions

While trying to prove some extrapolation theorem for $B_{p}$ weights, I tried to prove the following: Suppose that $0<p\leq 1 $ and $w(x)$ is a non-negative and belongs to $B_{p}$, that is \begin{...
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21 views

Error term for backward differentiation formula

I am trying to derive the error term for the backward differentiation formula $f'(x) \approx \frac{3f(x)-4f(x-h)+f(x-2h)}{2h}$. By Taylor expansion $f(x-h)=f(x)-hf'(x)+\frac{h^2}{2}f''(x)-\frac{h^3}...
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39 views

Computing derivative as accurately as possible by finite difference formula

The central difference formula is $D_h(x)=\frac{f(x+h)-f(x-h)}{2h}$. In the form $(x,f(x))$ am given the data points $(0.40, 2.3987), (0.44, 2.4182), (0.48, 2.4377), (0.52, 2.4571), (0.56, 2.4764)$. ...
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23 views

Richardson Extrapolation on forward differencing formula

After deriving this with forward differencing formula & Taylor series: $$ D_1 f = f'(x_{i}) = \frac{f(x_{i+1}) - f(x_{i})}{h} - \frac{f''(x_{i})h}{2!} $$ which could perhaps also be written as: ...
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Nowcasting, what is it?

I stumbled upon this word in the context of corona virus forecasting. As far as I understand it is some kind of extrapolation method that is also used in economics and meteorolgy, but am struggling ...
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7 views

How to do a weighted extrapolation using a Burg (or similar)?

I had extrapolated data using the Burg Method (octave arburg) but in the period with no existing data the extrapolated value kept on asymptotically tending to an arbitrary value even when the data ...
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26 views

Gaussian normal distribution extrapolation methods

Some graphs of COVID-19 If I assume that active cases of COVID-19 fits to Gaussian normal distribution curve, how I can extrapolate current numerical data to draw whole curve? See the linked image ...
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Interpolating values using a fitted function

I have data for $x,y$ where $y=[0,0,0.016,0.65,0.97,0.99,1]$ and $x=[1,2,3,4,5,6,7]$. For this data I fitted a sigmoid type function as $f(x)=a/(1+exp(-b*x)+c)$. I fitted this using matlab cftool, and ...
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26 views

How to extrapolate the data?

The question demands me to find "rupture lifetime" for 300MPa load at a temperature of 649$^{\circ}$ C.How do I solve it?All that I know is to apply interpolation and extrapolation which I ...
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30 views

Fitting a surface to a set of 3d points

I am attempting to fit a surface to a set of 3d points. These points never overlap on the x,z plane, and do fit into a grid, but may be sparse. Our current implementation just uses bilinear ...
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24 views

Finding higher order method using Richardson Extrapolation with integration

Suppose that the interval of integration $[a,b]$ is divided into equal subintervals of length $h$ each such that $r = \frac{b-a}{h}$ is even. Denote by $R_1$ the result of applying the composite ...
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32 views

Extrapolate $y^2=4x^2-4x^4$ from plot

I am trying to extrapolate a function based on the distribution of eigenvalues of certain matrices I am working with. In a simple case I successfully described the data with the function $y^2=4x^2-4x^...
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27 views

Wrong coefficients when deriving Simpson's Rule using Richardson Extrapolation

Ok, so I am going with Wikipedia's definition of the Richardson Extrapolation, rather than the recursive definition that I know is floating out there. One step at a time. Given $A(h)$ is an estimate ...
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How to do prediction of function through extrapolation?

Given some pair of values, which are following the rules for many to one and into relationship (thus function may be possible) , how can we 'accurately' extrapolate and establish the underlying ...
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22 views

function approximation - identifying suitable functions to evaluate

I want to approximate best fitting function using first 3 inputs of X,Y values and then extrapolates for X=[5,20]. X=[2,20] and Y=[0,100], in the attached plot. which kind of function is likely to fit ...
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68 views

Which is better: Least Squares Regression or some Unknown Method I found my coworker using?

It's been a while since I did some good old regression, but I've been given a dataset with some simple x and y data and need to ...
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140 views

Natural Cubic Spline beyond boundary guarantees constant slope?

my understanding is that for if I have five points, I can interpolate a natural cubic spline out of any five points with increasing x, because degree of freedom is five. But when I try the python ...
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1answer
45 views

Propagating Uncertainties on Interpolated Data

I have a data set of 2000 $[x, F(x), \delta F(x)]$ triples, where $x$ is exact and $F$ is a measured value with an uncertainty $\delta F$. I can interpolate/fit the function however needed, and this ...
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42 views

Numerical Analysis - $n$-sided polygon tangential

i need help with this question..I'm not so sure how to go about the arguments. Any help would be appreciated. Consider a regular $n$-sided polygon tangential to and enclosing the unit circle to ...
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1answer
128 views

How can I derive the dense output of ode45?

I'm currently looking at the implementation of the Dormand-Prince 5(4) Runge-Kutta algorithm (also known as Dopr5, or ode45) in ...
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1answer
120 views

How to find a formula relating three values?

I haven't studied maths since high-school (20 years ago) and would like to find a formula to relate these values: ...
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59 views

Extrapolation of points by a function

I have got a positive series convergent series $w_j$, which is convergent. I can calculate a finite amount of terms $\sum_{j=0}^{j^*}w_j$ and would be interested in the estimation of the sum of the ...
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490 views

Please explain how Richardson Extrapolation is used in this example

I am having a hard time understanding what on earth Richardson Extrapolation is trying to do. Consider the example of approximating $\pi$ by inscribing regular polygons in a unit circle. The ...
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318 views

Norm of laplacian in dual space $H^{-2}$

how we can prove the estimate $\|\Delta u\|_{H^{-2}} \leq C \|u\|_{L^2(\Omega)}$, where $\Omega$ is a bounded open set of $\mathbb{R}^n$, and $H^{-2}$ is the dual space of $H^{2}\cap H_0^1$. Can we ...
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105 views

How to find more accurate numerical integration result using Richardson's Extrapolation given midpoint and trapezoidal conditions?

This question has been confusing me. Any help would be very appreciated.
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196 views

Trying to use Taylor series to find a formula for a Richardson extrapolation of order 6

I am trying to use Taylor series to find the formula for the Richardson extrapolation based on $f'(x)$ of order $O(h^6)$. I know the formula should have the form of: $$\Delta_1(h) - 20\Delta_1(h/2) + ...
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35 views

When extrapolating for projections, how do you know which function-form to use?

When you have a set of raw data, there are many models to extrapolate a projection. Linear, Quadratic, Logarithmic, Etc. How does one know which to use? I'm assuming its by the nature of the data ...
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1answer
203 views

Improving accuracy of f''(x) by Richardson extrapolation

I have been given the following Question, I literally have no Idea where to start with it, I understand that the original expression comes from central difference approximation, that makes sense. I ...
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389 views

Numerical Differentiation- Central Difference approximation & Richardson extrapolation

I have been told to derive "the central difference approximation for f''(x) accurate to O(h^4) by applying Richardson extrapolation to the central difference approximation of O(h^2). I have done this ...
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1answer
126 views

When to extrapolate?

I know extrapolating is probably dependent on the question being asked. But are there some guidelines on when extrapolation is "okay"? If you can't be as certain that a set of data will maintain a ...
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234 views

Extrapolate a sum using partial sums at powers of two

In an online textbook for MIT OCW 18.013a, Calculus with Applications, the author uses residue calculus to derive the well-known formula $$\sum_{n>0} n^{-2} = \frac{\pi^2}{6}$$ (See Some Special ...
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307 views

Application of Richardson Extrapolation

Apply Richardson extrapolation to $ S_{2h} = \frac{2h}{3}[f(x_0) + 4f(x_2) + f(x_4)]$ $S_h =\frac{h}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)] $ to derive a Newton–Cotes formula based on the ...
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46 views

extrapolation to $\infty $ without knowing the exponent

I obtained a curve numerically (see below). Apparently, the curve will converge to some value at $\infty $. How can I extract this limit value? The problem is that I do not know how this curve ...
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87 views

How to extrapolate smoothly, based on discrete values

I am trying to figure out how to value a property (piece of land) based on the original purchase price, and shifted proportional to irregular government valuations. This is for the purpose of a legal ...
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1answer
1k views

Finding correct step-size for the Euler method

I don't understand how to find the correct step-size $h$ for the Euler method. My script says the following: One method consists in computing the numerical solution for an arbitrary $h$ and then $...
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1answer
423 views

Linear Interpolation/Extrapolation

I have the following grid of data. The first row represents a value called $m$ that ranges between 0.8 and 1.2, and the first column represents days that can be anywhere between 0 and $\infty$ (but ...
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1answer
578 views

Extrapolation methods for ODE

While preparing for an exam I revisited extrapolation methods for ODEs and I noticed that I don't understand it properly. Although I have an exercise-related question, there is a soft question: why do ...
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2answers
16k views

1e+0.8= What? What does E mean? [duplicate]

Hello I came across a math equation and I was wondering what did the "e" stand for? the equation is : y=47931x-1E+0.8 Can someone please help me by showing what the E stands for and the answer for my ...
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41 views

How to test hypothesis where population size is unknown?

Assume that somebody claims that driving a car or being an airplane pilot is just like riding a bike and anybody can do it with just a little practice and therefore it is unnecessary to have a license....
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37 views

extrapolate terms of sequence, knowing sums

I'm not familiar with filters etc, so I'm asking here if anyone knows an existing method to do the following. I have a sequence of observations $S_n$ and I know that they satisfy $S_n=\sum_{i=1}^m a(n,...
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212 views

Composite rule of Newton cotes? Simpson + Trapzoid

What is the best combination of Trapezoidal rule and Simpson's rule to solve this problem? I used: Trapz from 1 to 2 1/3 Simpson's from 2 to 4.5 Trapz from 4.5 to 6 3/8 Simpson's from 6 to 9 1/3 ...
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323 views

Proof that using Richardson extrapolation once yields Simpson's rule

Given that $\int_{a}^{b}f(x)\,dx \approx\frac{h}{2}\left[f(b) + f(a) \right]$, it must be shown that $\int_{a}^{b}f(x) \, dx \approx \frac{h}{6}\left[f(b) + 4f\left[\frac{a+b}{2} \right ] + f(a) \...
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176 views

Generalized application of Young's inequality

Let $a,b\geq 0$ and let $(p,q)$ be conjugate exponents, which means $p,q>0$ and $p^{-1}+q^{-1}=1$. Then, we have the so-called Young's inequality: $$ab\le \frac{a^{p}}{p}+\frac{b^{q}}{q}$$ Now, ...
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759 views

Non-linear Extrapolation formula

Given a set of at least 180 data-points (dates as X and associated non-negative values as Y) - how would one go about predicting/forecasting future data-points? I'm not even sure whether what I'm ...
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1k views

How to extrapolate data using the second order rate equation?

I would like to model the second order rate equation using the following criteria: \begin{align} A_\text{initial}&=8\,500\,000,\\ A_\text{final} &=1\,200\,000,\\ t &=38. \...
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29 views

Name of the field of study that details extrapolation of a series based on subset sample data

Apologies if my notation and/or terminology is way off - I'm not well versed in mathematics. I'm looking for the name of the field of mathematics that might help me solve my problem. Here's my problem:...
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1answer
48 views

Extrapolation of $c^{-3/2}$. Optimal data points?

I have a computer code that does some complicated calculations and results in a real number. At the end I had to perform a infinite sum, $$ S=\sum_i^\infty x_i \;, $$ which I approximate by a finite ...
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62 views

Extrapolating the sequence of derivatives of an analytic function

Suppose that $f(x)$ is an analytic complex or real function on the real domain and we only know $f(x_0)$ and the first $n$ derivatives of $f(x)$ at $x=x_0$. Is there an algorithm whose accuracy ...