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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cubic spline interpolation - derivative known -

I at the moment trying to understand how to apply the interpolation method stated above. I have been given a start and end position, and for both position i know what their slope is. $\dot{X_a} = ...
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1answer
65 views

Determine equation from graph

Background: I'm working on a script to read/parse a file generated by a piece of software I use to create music mixes. One aspect I'm having difficulty with is translating the volume value from it's ...
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1answer
45 views

Spline terminology

I am reading up on splines and as a beginner I have a basic question - Does it make sense to say - "I will fit a cubic b-spline to the data". As b-spline is just a representation of spline in terms ...
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18 views

Linear interpolation from perspective-correct interpolators

This question is trying to approach this problem from a mathematical perspective. I have some value $u$ that I want to interpolate linearly, as $(1-a)u_0+a u_1$. However, I can only use ...
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2answers
96 views

Smoothest function which passes through given points?

I am trying to interpolate/extrapolate on the basis of a known collection of (finitely many) points. I'm wondering if there is a way to formalize this intuitive notion: find a 'smoothest' function ...
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29 views

error bound for polynomial interpolation with derivative matching

We all know the following formula for the maximum error (evenly spaced) polynomial interpolation: $|f(x) - p_n(x)| \leq \frac{h^{n+1}}{4(n+1)} \max_{x\in [a,b]} f^{(n+1)}(x)$ where $p_n(x)$ is the ...
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3answers
380 views

Linear interpolation with two points

Question: $p(x)$ is the linear function that interpolates $\sin(x)$ at $0$ and $\frac{\pi}{2}$. And I need to show that $\ |p(x) - \sin(x)|\le\ \frac{1}{2}(\frac{\pi}{4})^2$ My attempt: $\ ...
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37 views

How to use piecewise quadratic interpolation?

I'm attempting to get the hang of quadratic interpolation, in MatLab specifically, and I'm having trouble approaching the process of actually creating the spline equations. For example, I have 9 ...
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46 views

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\\ ...
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26 views

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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14 views

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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1answer
49 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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1answer
33 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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2answers
41 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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1answer
53 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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1answer
36 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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1answer
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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1answer
62 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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1answer
160 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 ...
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1answer
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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20 views

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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25 views

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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8 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 ...
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24 views

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
34 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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45 views

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} = ...
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1answer
36 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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13 views

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 ...
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1answer
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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1answer
15 views

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: $p_i(x_i) = y_i$ $p_i(x_{i+1}) = y_{i+1}$ ...
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30 views

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 ...
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1answer
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 ...
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63 views

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 ...
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1answer
63 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) = ...
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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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1answer
42 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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1answer
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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2answers
105 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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1answer
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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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31 views

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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4answers
724 views

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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1answer
20 views

$\bar x$ 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: $f(2) = 10$ $f(4) = 13$ ...
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50 views

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. $$m(a,b,x)=\frac {f(b)-f(a)}{b-a}(x-a)+f(a)$$ ...
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30 views

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\\ ...
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1answer
27 views

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 ...