Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.

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Catmull-Rom: spline and filter

On this website, the author gives this definition for Catmull-Rom splines (slide 10): $$catmullRom(t) = \frac{1}{2}\left\{\begin{array}{ll} t^3 + 5t^2 + 8t + 4 & \text{if } -2 \le t \lt -1\\ -3t^...
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Estimate the error of interpolation/extrapolation

I'd like to know how we can estimate the error of interpolation. For example, let's consider Lagrange interpolation. We don't know anything about the real function (it could be an algebraic or a ...
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Dynamic learning for efficient interpolation (combinatorics)

Please regard this as soft question and reference request. Suppose I have to test out a combination of certain candidates. For example a combination of two signals at side A and side B. Assume we ...
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55 views

Newton 3d grade polynomial , simpson 3/8

Hello guys and sorry for my bad English , i have the following homework i should composite Newtons polynomial interpolation 3d grade , Simpsons 3/8 method with matlab ! But i have some trouble, i ...
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34 views

Interpolation with logaritmic function

I want to interpolate with the function $$f(x) = a\ln(x+b)+c$$ That is, I assume some sort of logarithmic relationship, but there might be an offset. I assume that I need 3 datapoints, as there are ...
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42 views

Which interpolation method for complicated, smooth curves?

Which interpolation method should I use for complicated "smooth" curves such as $\frac{sin(x)}{x}$ for $x>0$.
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55 views

Number of points needed for linear interpolation of sine in $[0,\frac{\pi}{2}]$ with given error bound

I want to get a set of equispaced points in $[0,\pi/2]$ and use piecewise linear interpolation generated by those points to fit the sine function. And I want to determine how many points do I need to ...
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37 views

Finding Roots of a Polynomial Represented in Point-Value Form

Consider we have $n$ pairs of $(x_i,y_i)$. We all know that given the $n$ pairs we can interpolate a polynomial of degree at most $n-1$. Also, it is clear if we want to find roots of a interpolating ...
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38 views

Interpolation and divergence of $n$-th derivative

i have a curiosity. Let's say i have a function $f \in C^{\infty}[a,b]$ such that there's a $x_0 in [a,b]$ that makes and $f(x_0) = 0$ and $a_n = f^{(n)}(x_0)$ diverges (i.e. $|a_n|=\infty$) could ...
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70 views

Signal processing : future values prediction

Let $f : \mathbb{R}^+ \rightarrow \mathbb{R} $ be a continuous function. Do you have some references (books or online resource) about techniques that allow to predict $f(x_{n+1})$, knowing $f(x_0), .....
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162 views

Approximating on a line

Say I have sampled some points in $[0,1]^2$ and evaluate a function $f(x,y)$ for them. I am interested in the behavior of $f$ along a single dimension. If the points were like $(x_1,y_1),(x_1,y_2),\...
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31 views

Is it possible to interpolate $e^n$ in more than one way?

The most basic definition of exponentiation is repeated multiplication, $$e^n = e \cdot e \cdot \cdot \cdot \cdot e$$ $n$ times However, if $n$ is a rational number such as $2.4$, this ...
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Composite 3 point Gauss integration of f(x) over an interval

I'm attempting to express the composite 3-point Gauss formula for integrating f(x) over an interval [a,b]. $\int_{a}^b f(x) dx = \sum_{j=0}^n f(xj)*cj$ where $cj = \int_{a}^b lj(x)dx$ $lj(x)$ are ...
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Calculating Newton Form of Interpolating Polynomial

For the function $f(x)=\frac{1}{x}$, I am trying to calculate the Newton form of the interpolating polynomial for the points $x_0=2$, $x_1=3$ and $x_2=4$. I understand that the Newton form of the ...
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9 views

Taylor Remainder over an interval for polynomial interpolation

When attempting to find how big n should be so that $|e^x - p(x)| < 10^{-4}$ over the interval $[-1,1]$ using Taylor Remainder, what value should I be using for $x$ in $(x - x0)^{n+1}$? I'm using 1,...
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Convergence theorems of periodic and natural cubic Splines

I have this question on the topic convergence theorems of cubic splines, for cubic splines which theire first derivatives at the start and end points are equal to first derivatife of the $f$ function ...
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1answer
36 views

Using the polynomial of lowest order that interpolates $f(x)$ at $x_1$ and $x_2$, derive a numerical integration formula for $\int_{x_0}^{x_3}f(x)dx$.

Using the polynomial of lowest order that interpolates $f(x)$ at $x_1$ and $x_2$, derive a numerical integration formula for $\int_{x_0}^{x_3}f(x)dx$. I know that we aren't assuming uniform spacing. ...
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second order interpolation with positive weights

I have a function evaluated at predefined grid points $x_i=\frac{i}{N},\ i=0,...,N,\ f(x_i)=y_i$ I am looking for an interpolation scheme satisfying $\|I(f)-f\|_{L^\infty} = \mathcal{O}((1/N)^3)\...
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45 views

Real vs Complex Interpolation

The two major classical interpolation theorems in analysis are Riesz-Thorin Theorem (complex method) and Marcinkiewicz Theorem (real method). One can see the statements of the theorems and realize ...
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Difference quotient's degree

I am trying to prove following theorem. It's given without a proof in my notes, and I can't understand what does it really mean. It appears that it means that just like when we take a derivative of $w(...
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44 views

Partial Derivative of Newton's Divided Difference

Let $x_0<x_1<...<x_n$, and let $f$ be continuously differentiable. Show that $$ \frac{\partial}{\partial_{x_i}} f[x_0,x_1,...,x_n]=f[x_0,x_1,...,x_i,x_i ,x_{i+1},...,x_n] $$. I have the ...
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Prove the Taylor polynomial $p(x)=\sum_{j=0}^{k-1}{1\over j!}f^{(j)}(x_0)(x-x_0)^j$ interpolates $f$ at $x_0, x_0,…, x_0$ ($k$ repetitions).

Prove the Taylor polynomial $p(x)=\sum_{j=0}^{k-1}{1\over j!}f^{(j)}(x_0)(x-x_0)^j$ interpolates $f$ at $x_0, x_0,..., x_0$ ($k$ repetitions). I'm not sure how to approach this. Any solutions or ...
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Computing splines using Hermite interpolants

The form for a cubic Hermite interpolant has the form $p_i(x)=a_i+b_i(x-x_i)+c_i(x-x_i)^2+d_i(x-x_i)^3$ according to the following conditions: $p_i(x_i) = y_i$ $p_i(x_{i+1}) = y_{i+1}$ $p_i'(x_i)=...
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Spline interpolation explanation

I'm trying to learn about spline interpolation, and I'm struggling to understand what h_i^3 is in the second and third equation. I don't understand how they derived that equation. I'm trying to code ...
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33 views

Newton method for interpolation of polynomials

I have a function $f(x) = \frac1{1+25x^2}$, and 7 equally spaced nodes on the interval $[-1, 1]$ My points are $(-1, 1/27), (-0.6, 11), (-0.3, 4.25), (0, 2), (0.3, 4.25), (0.6, 11), (1, 1/27)$. I'm ...
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Find a cubic hermite interpolation of the function

Let be $I=[0,2]\subset\mathbb{R}$ and $f:I\to\mathbb{R}$ with $$f(x):=\frac{x^2-5}{-x^3+x^2-4}.$$ Define a polynomial $p$ using the cubic Hermite interpolation method with the grid points $x_0=0,x_1=...
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Can I make this numerical integration continuously differentiable?

Suppose I have the discrete values $f(x_i)$ for every $x_i$ greater than some $\varepsilon$, and I want to numerically calculate the following integral: \begin{equation} n = \int_\varepsilon^\infty f(...
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69 views

Derivative of Lagrange interpolating polynomial

I'm using a textbook titled "Numerical Analysis" by Richard L. Burden, 9th edition. I'm having a problem with a particular derivation The Lagrange interpolating polynomial is given by $$f(x) = \sum_{...
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Error of Lagrange Interpolating polynomial is zero for polynomials of low degree

Let $f$ be a function defined on the interval $[x_0 −h, x_0, x_0 +h]$, and $f \in C^3[x_0 −h, x_0 +h]$. Let $h$ be the Lagrange interpolation polynomial of $f$ at the nodes $x = x_0 − h$, $x_0$, $x_0 ...
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Degree of Lagrange polynomial to satisfy a given error

Consider the function $f(x)=e^{x-1}$ on the interval $0\leq x\leq1$. For all $N=1,2,...$ we have the uniform partition $x_i=ih$, $i=0,1,...,N$, $h=1/N$. We find the Lagrange polynomial for $x_i$. If ...
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43 views

Interpolation polynomials

Let $p_k$ be the polynomial of degree $\leq k$ such that $p_k(x_i)=y_i$ for $0\leq i \leq k$. Prove that $p_k=p_{k-1}$ if and only if $p_{k-1}(x_k)=y_k$. I'm a first year PhD student and I ...
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50 views

Let $f(x)=1/x$ and prove that $f[x_0,x_1,…,x_n]=\prod_{i=0}^nx_i^{-1}$. [closed]

Let $f(x)=1/x$ and prove that $f[x_0,x_1,...,x_n]=\prod_{i=0}^nx_i^{-1}$. I'm sure how to approach this or even how/why we need $f(x)=1/x$. Any solutions or hints are greatly appreciated.
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Error formula when using a polynomial interpolation

You have a function f that has a continous $(n+1)th$ derivative. And you use a polynomial $p_n$ to interpolate the function at points $(x_0,f_0), (x_1,f_1) ... (x_n,f_n)$. Then the error for the ...
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Interpolate inside cuboid / plane that intersects a point

I am trying to implement a trilinear interpolation algorithm for cuboids. Please excuse my lack of math jargon, this is not my area! The cuboids can be rotated in any dimension in 3D space. No two ...
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112 views

Find a polynomial of lowest possible degree

Find a polynomial $f$ of lowest possible degree such that $f(x_{1})=a_{1}$, $f(x_{2})=a_{2}$, $f'(x_{1})=b_{1}$, $f'(x_{2})=b_{2}$ where $x_{1} \neq x_{2}$ and $a_{1}, a_{2}, b_{1}, b_{2}$ are given ...
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Computing Two-Norm for interpolation of functions

$$ f(x) = x^3, p(x) = (3/2)x^2 − (1/2)x $$ The two-norm of f(x) - p(x) is: $$( \int_0^1 (f(x) - p(x))^2 dx )^{1/2} $$ p(x) interpolates f(x) at $$x=0, x=1/2, x=1$$ The result of the two-norm ...
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Determine Hermite interpolation polynomial and evaluate the appropriate error.

Determine Hermite interpolation polynomial and evaluate the appropriate error y:=f(x) ...
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Newton form vs. Lagrange form for interpolating polynomials

I'm just wondering, what are the advantages of using either the Newton form of polynomial interpolation or the Lagrange form over the other? It seems to me, that the computational cost of the two are ...
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$\bar x$ value for $f(\bar x) = 4.5$ with tabled values

The exercise asks for the $\bar x$ value as $f(\bar x) = 4.5$ using linear interpolation. I couldn't reproduce a table with MathJax, so I'll put the tabled with data as: $f(2) = 10$ $f(4) = 13$ $f(...
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Linear interpolation optimization

I want to interpolate any function $f(x)$ using only linear interpolation. So far I have found that the following equations do the trick pretty well. $$m(a,b,x)=\frac {f(b)-f(a)}{b-a}(x-a)+f(a)$$ $$L(...
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Proof of Neville's algorithm

Let be $n\in\mathbb{N}$ and $(x_0,...,x_n)$ data points with the belonging functions $(f_0,...,f_n)$. Let be for $i=0,...,n$ and $i<j\leq n$ $$\begin{align*} p_i(x)=p_{i,i}(x) &= f_i\\ p_{i,j}...
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How to find the intersection of a line and a plane with interpolation ( given two points in the opposite side of the plane)

I have two points in the opposite side of a plane (P1,P2) in 3D space, and i know their signed distances to the plane(D1,D2). how can i use interpolation to calculate the point that is the ...
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How to resample translated grid to ensure consistent interpolation?

I have a grid of values, sampled at certain locations. I'd like to translate grid by some offset and resample it. The questions is: what should be the resampling and interpolation formulas such that ...
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70 views

When fitting a polynomial to data points, how to determine the reasonable degree to use?

I have wondered the following: Suppose that there is a set of data points $(x_i,y_i)$. Then I would like to know if it is more reasonable to assume if there is a polynomial relation of degree $m$ ...
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78 views

How does one derive Runge Kutta methods from polynomial interpolation?

In some numerical analysis classes, a neat way of deriving the Adams-Bashforth and Adams-Moulton methods is to approximate the function by a polynomial, and integrate the polynomial analytically over ...
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106 views

Find the approximation for the interpolation of $f(x)$ by a polynomial of second degree

Assume that $f(x)$ has a minimum in the interval $x_{n-1}\leq x\leq x_{n+1}$ where $x_k=x_0+kh$, $k$ being an integer. Show that the interpolation of $f(x)$ by a polynomial of second degree yields the ...
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23 views

Should I use interpolation when finding median, and quartiles?

I am a S1 maths (Edexcel) AS student in the UK. My question: Say we have a stem-and-leaf diagram with 26 values. We want to find the lower quartile. To get the marks for our specification, we need to ...
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1answer
54 views

Why polynomial interpolation is considered as better than others?

Why polynomial interpolation is considered as better than others? In case of interpolation, the function $\phi(x)$ to approximate the unknown function $f(x)$ may be polynomial, exponential, ...
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3answers
64 views

Given the following data pairs, find the interpolating polynomial of degree 3 and estimate the value of y corresponding to x = 1.5.

So i was given this question Given the following data pairs, find the interpolating polynomial of degree 3 and estimate the value of y corresponding to x = 1.5. a) $(0, 1), (1, 2), (2, 5), (3, 10)$ ...
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Does there exist smooth or $C^2$ function for some infinite given points $a(n)$?

I know that there exist some smooth function (polynomial) for finite numbers of values. The question is if there exists function (not necessary unique) which is twice differentiable and $f(n)=a(n)$?