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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5
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327 views

What is the maximum overshoot of interpolating splines in $d$ dimensions?

Consider cubic splines $s( x, y )$ which interpolate values $y = \{ y_0, y_1, \dots,y_n \}$, on the uniform grid $\{ 0, 1,\dots, n \}$. Fix $s''(0) = s''(n) = 0$ (natural splines). How big can ...
5
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289 views

Runge's phenomen: interpolation error using Chebyshev nodes oscillates

We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
4
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0answers
119 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The ...
4
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1k views

Computation of coefficients of Lagrange polynomials

For our homework we should write a program, that creates Lagrange base polynomials $L_k(x)$ based on a few sampling points $x_i$. Now i am eager to develop a formula to be able to compute the ...
3
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27 views

Lagrangian interpolation

I have a high-school student doing a modeling project using interpolation. (Hopefully someone understands what I mean without having to write out explicit examples, as I only have screenshots.) When ...
3
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0answers
106 views

How does 2D kriging interpolation work?

I have a grid of points Example ...
3
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82 views

Random multipliers of polynomial values at known points in $\mathbb{Z}_p$

Scenario (1) We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$, where $p$ is a prime number. Let $P_i$ be a polynomial such that $P_i \in R[x]$. We ...
3
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0answers
111 views

Interpolation method

I am researching about different interpolation methods, and their pros and cons. Please help me to understand ideas behind Gaussian interpolation method. Although, I looked at different papers for ...
2
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0answers
17 views

How to interpolate multidimensional functions?

I'm learning about interpolation and I wanted to ask if there's a "good" method to interpolate multidimensional functions (when the dimension can be even a few thousands)? Is there a theoretic limit ...
2
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0answers
28 views

Convex interpolation between two points with given derivatives

Let's say I have two real values $x_1$ and $x_2$, to each of which I associate $y_i$ and $y'_i$ satisfying $$ (y_2-y_1)(y'_2 - y'_1) \geq 0. \tag{1} $$ I would like to find a polynomial ...
2
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38 views

Interpolation of iterated logarithms

$$\text{Let }\log^2(x)=\log(\log(x)),\\ \text{ then }f(y,x)=\log^{\lfloor1+y\rfloor}\left(\log(x)/\log((1-x^{1/x}(y-\lfloor y\rfloor))+(y-\lfloor y\rfloor))\right)$$ gives an interpolation between ...
2
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0answers
33 views

Smallest set of Liner equations, which exactly fit a set of points

I have a set of 2-d points,(it can be of any arbitrary dimension n). I want to find the minimum set of straight lines(linear equations) which exactly passes through the given 2-d points (unlike ...
2
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0answers
55 views

Lagrange's interpolation to solve for 0 of y(x)

I have the data composing of 7 elements x is from 0 → 3 incrementing by 0.5 y is from 1.8241 → -1.5427 I am supposed to use Lagrange's interpolation of three nearest neighbor data points. I am ...
2
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0answers
106 views

lagrange interpolation question here

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0, x_1=0.25 , x_2=0.5, x_3=1$ a)Find Lagrange polynomial for this function. c)Find the real approximation error. d)Find the limit of the ...
2
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0answers
382 views

Interpolation between curves

I am trying to do a 2D interpolation between two curves as you can see from the attached picture. where $a$ is a varrying paramter. $f_{a_1}$ and $f_{a_2}$ are known, or at least I can perform ...
2
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0answers
43 views

Exercise 1.3.3(c) of GTM249(Classical Fourier Analysis)

Exercise 1.3.3(c) Let $0<p_0<p<p_1<\infty$ and let $T$ be an operator as in Theorem 1.3.2($\|T(f)\|_{L^{p_0,\infty}(Y)}\leq A_0\|f\|_{L^{p_0}(X)}$ for all $f\in L^{p_0}(X)$ and ...
2
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0answers
55 views

Smooth detour from one function to another

Suppose I am given two smooth functions $f$ and $g$ on the real line and real numbers $a<b$ such that $f(a)<g(b)$ and $f'(a),g'(b)\ge0$ I want to get a smooth $H:\mathbb R\rightarrow\mathbb R$ ...
2
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0answers
21 views

Analogue of Helly’s theorem for non-exact interpolation

Let $\overrightarrow{x}=(x_1,x_2, \ldots ,x_n),\overrightarrow{a}=(a_1,a_2, \ldots ,a_n)$ and $\overrightarrow{b}=(b_1,b_2, \ldots ,b_n)$ be vectors in ${\mathbb R}^n$, with $a_k \leq b_k$ for every ...
2
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0answers
138 views

monotonic smoothing fit to be implemented (in python or other language)

In a post that already exists, implementation-of-monotone-cubic-interpolation, there is a good method for fitting data which necessarily includes all of the given points. But, what if I need to ...
2
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0answers
89 views

Minimal surface representation from a 3D contour

I have a set of 3D points defining a 3D contour, as shown below. The points in this contour lie in their best-fit plane and I want to obtain a 3D triangular mesh representation of the surface inside ...
2
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0answers
123 views

Ellipse radius interpolation with different radiuses

I am writing a library for graphical LCDs and I want to incorporate a function to draw a circle on the screen. I have already succeeded in drawing simple circles, however, I want to be able to pass a ...
2
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0answers
171 views

B-Spline Definition

I'm currently working on my master's project. For this, I rely on one PhD-thesis in which I found a statement I do not understand. Unfortunately, the author hasn't answered to my mails yet, so I have ...
2
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0answers
92 views

variogramas kriging

In order to find the altitude of a surface, we use the following method. My question is: What is the name of this method? We have $$ A=\begin{pmatrix} a_{1,1} & .. & .. & ...
2
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0answers
521 views

Interpolating polynomial with Chebyshev nodes

I am interested in constructing an polynomial that interpolates some known arbitrary function $f(x)$ over the domain $x \in [0,70]$. I want the polynomial to have degree 14 and so need 15 points. ...
2
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0answers
329 views

Polynomial Interpolation and Error Bound

Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the ...
2
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0answers
199 views

Runge function error second factor

I'm currently learning about the Runge function. On Wikipedia, I read the following: Consider the function: $ \dfrac{1}{1+25x^2}$ Runge found that if this function is interpolated at ...
2
votes
0answers
263 views

linear interpolation error estimate for non-smooth function

Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be ...
2
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0answers
103 views

explicit error bounds for Multivariate interpolation

I want to interpolate a function of $d$ variables over a Cartesian grid, using multivariate interpolation, while characterizing interpolation error in terms of bounds on partial derivatives of the ...
2
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0answers
157 views

(Experimental) Can it be shown that this extension of the secant-interpolation has quadratic convergence?

Background: I needed some efficient but simple interpolation-methods aside of Newton's iteration because I want to have it in contexts, where the derivative of a function is not always known. So an ...
2
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0answers
281 views

Is there software that interpolates/extrapolates data using a discrete Fourier?

I've read various methods of Fourier interpolation and extrapolation detailed in articles such as Interpolation and Extrapolation Using a High-Resolution Discrete Fourier Transform—so what I'm ...
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0answers
26 views

How to smooth a list of angles.

I'm not a math guy so maybe there is a super simple thing that my eyes cannot see. And sorry if my math terminology is not good at all. Please address me the right math terminology to use because ...
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0answers
14 views

Estimate accuracy of inaccurate fast function having exact values of slow one

Let’s say we have functions $F$ and $H$ to calculate a series $S$ of integers and that: $S_{i} = H(x_{i}) = F(x_{i}) + e_{i}$ Being $e_{i}$ the error of $F(x_{i})$ to estimate $S_{i}$ The problem ...
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0answers
25 views

Interpolation between two points

I am looking for an interpolation between two points $P$ and $Q$. I need the curve to have derivative (direction) $\vec{v_1}$ at point P and $\vec{v_2}$ at point Q. In addition, there is a maximum ...
1
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0answers
32 views

Cubic polynomial interpolation

Let $f(x) = x^2\cdot (x-1)^2 \cdot (x-2)^2 \cdot (x-3)^2$. What is the piecewise cubic Hermite interpolant of $f$ on the grid $x_0 = 0$, $x_1 = 1$, $x_2 = 2$, $x_3 = 3$. Let $g(x) = ax^3 + bx^2 + cx ...
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0answers
18 views

MATLAB implementation Spline Fitting

Check the attached problem please. I am a beginner in spline fitting and have a few questions: 1) How to find the coefficients c[n]. Is it by DTFT? 2) I understand how to find the derivative but ...
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0answers
17 views

Merging two univariate functions gracefully

Suppose I tell you that $$ f(0) = 0 $$ $$ f'(0) = 0 $$ and $$ f''(0) = a $$ for known $a>0$, whereas for large $x$ $$ f'(x) \approx \cosh^{-1}(x) $$ for $x>2$. Knowing nothing else ...
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0answers
18 views

Rational function interpolation?

We know that $n+1$ points is enough to completely determine a polynomial of degree $n$. Are there any techniques which says that a certain number of points is enough to completely determine a rational ...
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0answers
36 views

Change of basis from Chebyshev to monomial basis for polynomials

I'm not that familiar with Chebyshev polynomials, so I hope I'm not too far off. Suppose that I have three order pairs $(x_0, f(x_0))$, $(x_1, f(x_1))$, and $(x_2, f(x_2))$ where $f : \mathbb{R} \to ...
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0answers
16 views

Bounding the Lebesgue constant.

This is a homework question, so I would prefer hints/suggestions as opposed to full-out solutions. Given the Lagrange polynomials $\ell_i(x)=\displaystyle\prod_{j=0;j\neq i}^n\frac{x-x_j}{x_i-x_j}$ ...
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0answers
55 views

Can you interpolate my polynomials if I give you some randomized values

Scenario (1) We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$, where $p$ is a prime number. Let $P_i$ be a polynomial such that $P_i \in R[x]$. We ...
1
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0answers
21 views

“Interpolating between estimates”?

the headline reproduces the whole problem. What is meant by saying "Interpolating between the estimates (A) and (B), we finally obtain..."? For beeing mor specific I'll give the concrete estimates ...
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0answers
19 views

On Convex Interpolation and distances

Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := ...
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37 views

Fourier Interpolation

I have this Equation, that I modeled from my measurements and simulations. $I^{exp}_{l,m} = (\mathbf{F}^{H}.\mathbf{A}.I^{true})_{l,m}$; $H$ is the Hermitian transpose and $\mathbf{F}^{H}$ is a block ...
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0answers
21 views

Is there any interactive spline fitting software?

I'd like to know if there's any software (freeware) for interactive data interpolation. What I want is to be able to visualize my data on an XY plot and drag the points to see how it affects the ...
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0answers
23 views

Taylor's remainder in a compact

My impression is that each function enough regular ($C^\infty$ ) in a compact is equivalent to a polynomial. Is this true? Is there a way to prove it? The expression of the Taylor's remainder just ...
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0answers
26 views

Blended surface

Partially blended surfaces are extensively used in the literature for shape preserving interpolation. Most of these shape preserving partially blended surface interpolation is based on the result that ...
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0answers
24 views

What is the derivative of $\frac{f^{(3)}(\xi(x))}{6}$ at $x=x_0$

The error of interpolating polynomial is $$ E_n(x)=\frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x)) $$ The derivative of $E_n(x)$ is $$ ...
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0answers
26 views

A problem on interpolation

Given $X(0) \in \Bbb R$ and a continuous and bounded real-valued function $g(.)$ from $\Bbb R \to \Bbb R$, for $\delta > 0$ define the sequence $\{X_n^\delta\}$ by $\{X_0^\delta\} = X(0)$ and $$ ...
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0answers
20 views

Newton Interpolation jump in notes

I am told, The general expression for a second order polynomial that passes through 3 points $(x_1,y_1),(x_2,y_2)$, and $(x_3,y_3)$ can be written as: $$ p_2(x)=b_0+b_1x+b_2x^2 \tag1$$ which I am ...
1
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0answers
124 views

Lagrange interpolating polynomials question?

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0$, $x_1=0.25$ , $x_2=0.5$, $x_3=1$ a)Find Lagrange polynomial for this function. So $L_3(x)=f_0(x) l_0(x)+f_1(x) l_1(x)+f_2(x) ...