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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1answer
23 views

Interpolation using barycentric coordinates of a triangle

If I have a triangle given by the coordinates of its three vertices, $r_1, r_2, r_3$, and another point inside that triangle $r$, I can find the barycentric coordinates of $r$: $$r = \lambda_1 r_1 + \...
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1answer
42 views

Lagrange interpolation not working?

I have been trying to use Lagrange Interpolation on some data to find the equation of a curve that fits the following points (x,y): (0,5.2) (1,3.8) (2,4.5) (3,5.5) (4,6) (5,6.25) (6,6.25) (7,6) (8,5....
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1answer
38 views

Interpolation of rounded data

I would like to interpolate (not fit) a data set whose points have been rounded. Lets say I have some observations $y_i$ of a function sampled at $x_i$. The sample locations $x_i$ may be considered ...
3
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1answer
24 views

Find $p(x)=a_1 x+a_2 x^2$ with $p(x_i)=f_i, i=0,1$

Given $x_0,x_1,x_0\neq x_1, x_i\neq 0, i=0,1$, and $f_0,f_1$, find $p(x)=a_1 x+a_2 x^2$ with $p(x_i)=f_i, i=0,1$. Can you say if the interpolating polynomial in this case is unique? Why? Can you write ...
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1answer
28 views

How can I derive the second degree equation for a curve if I know the slope at two points and the y-intercept = 0?

How can I derive the second degree equation for a curve if I know the slope at two points and the x and y-intercepts = 0?
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0answers
23 views

Temerature and Strain images comparison & interpolation: finding offsets in X,Y coordinates systems.

I have two kind of 2D images from a Strain test, which show temperature and strain distributions in X and Y cordinate systems. One is thermal image, which gives temperature T values in a 200x300 ...
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0answers
10 views

interpolation with missing gradients

I am just wondering if anyone has encountered a similar problem before. Let's consider a differentiable function in 1 dimension for now. Suppose that this function is expensive to compute. In addition,...
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28 views

Error of derivatives of polynomial interpolant and best nodes with lobatto constraint

I know when interpolating a function $f(x)$ by a degree $n$ polynomial $P(x)$, the error is $$f(x) - P(x) = \frac{f^{n+1}(\xi)}{(n+1)!}\prod_{i=0}^n(x-x_i)$$ What does the right side hand look like ...
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1answer
53 views

Is the following described $p(x), q(x)$ the same interpolation polynomial?

Suppose we are given an odd number of data points $x_i$ and the corresponding values $f_i=f(x_i),i=1,...,n+1$($n$ is even), which are symmetric about the origin, i.e for each $x_i$ there is a $j$ such ...
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0answers
44 views

How many multiplications and additions will be needed to evaluate the interpolating polynomial at a point using the Lagrange formulas?

Consider the interpolating problem with $n+1$ points. How many multiplications and additions will be needed to evaluate the interpolating polynomial at a point using the Lagrange formulas? I think it ...
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1answer
27 views

C2 continuous interpolation on a 4-dimensional dataset

I am currently coding up a project where interpolations must be performed such that C2 continuity be preserved along the length of the whole set. The end result ought to look like a line (which will ...
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0answers
25 views

Parabolic interpolation in $k$ dimensions

I know the values of a smooth function of $k$ variables at $3^k$ points on a cube in $k$ dimensions (where $k=2,3$ or $4$). The central value is known to be the largest. I want to estimate the ...
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1answer
20 views

Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
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1answer
29 views

Terminology: Spline interpolation

I have read two different definitions of Splines: A differentiable piecewise polynomial. A piecewise polynomial. If I build a piecewise polynomial using cubic polynomials, it's ...
0
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1answer
30 views

What is this graphed function with asymptotes at π/2 and -π/2?

I've come across this function with asymptotes at π/2 and -π/2, which crosses the axis at y=1. It doesn't seem to be polynomial or exponential—can anyone figure out what it is? The asymptotes and y=...
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1answer
63 views

Need some help with applying specific boundary conditions to b-spline system of equations

I'm building a package for B-spline interpolation in Julia, and I've come across a boundary condition that I want to implement but can't wrap my head around how to do it (mathematically). Basically, ...
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4answers
59 views

Non-Piecewise interpolation over 3 points

I am trying to write an algorithm that interpolates between 3 values. The interpolation will be over the interval [0,1]. What I would like to do is: (Hopefully this makes sense) at x = 0, y = [...
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1answer
22 views

Should an interpolation coincide the original function on the given data points?

Suppose having a model $f(x)=y$ where $f$ is unkown. Moreover, suppose you have some data points for this model i.e. $(x_1,y_1), (x_2,y_2), \dots , (x_n,y_n)$. If one can find an approximate of $f $ ...
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0answers
41 views

Calculating B-Splines and dimension of spline space

I've got the following assignment: Let $S$ be the space of piecewise polynomials of degree $3$ on the intervall $[-1;1]$ with knots $x_i = -1+\frac{i}{2}, 0 \leq i \leq 4$. (a) Calculate a basis of ...
3
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1answer
74 views

Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is defined....
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1answer
37 views

Finding an Entire function with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$

I am really stuck on a homework problem, which boils down to the following: We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions ...
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1answer
45 views

Problem involving polynomial and arbitrary continuous function

Let $f\in C^4[0,1]$ and $p$ a polynomial of degree $3$. Suppose: $$f(0)=p(0),\quad f'(0)=p'(0),\quad f(1)=p(1),\quad f'(1)=p'(1)$$ Show that for each $x\in [0,1]$ there exists $\xi\in [0,1]$: $$...
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1answer
39 views

Fit $n$ Bézier paths to coordinates

I have a some coordinates $(X_i, Y_i)$ and I have to fit exactly $4$ cubic Bézier-paths to them (in other words, I have to find the 4 best fitting Bézier-paths, and by best fitting I mean that the ...
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0answers
21 views

Polynomial Interpolation and Secret sharing

Hypothesis: We define all values and polynomials over the field $\mathbb{F}_p$ for a large prime $p$ (e.g. 128-bit). My question is related to the "Shamir secret sharing" scheme in computer ...
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0answers
17 views

Root of interpolated polynomial when y-coordinates are permuted

Hypothesis: All values and polynomials are defined over a field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit) Suppose we have $n$ pairs of $(x_i,y_i)$. As we all know, given the ...
4
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1answer
83 views

Is there a name for this piecewise cubic interpolation kernel

I went looking for a way to do piecewise cubic interpolation, like natural cubic splines, but: expressible as a convolution of data points with a piecewise cubic kernel; and still C2-continuous ...
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0answers
18 views

Continuous/interpolating alternative to order of magnitude?

Define $\operatorname{magnitude}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rfloor }$ and $\operatorname{magnitude'}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rceil }$ Currently I'm using this $\...
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1answer
19 views

Find expression for $\sum_{k=0}^{n} l_k(0)x_k^{n+1}$

If the interpolation of $f(x)$ on the set of distinct points $x_0, x_1, \cdots x_n$ is given by $$\sum_{k=0}^{n} l_k(x)f(x_k).$$ Find an expression for $$\sum_{k=0}^{n} l_k(0)x_k^{n+1}.$$ I don't ...
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1answer
49 views

Bernstein Interpolation 2D

I am well aware of the equations for 1D Bernstein Interpolation. But I do not understand how to extend it to 2D. I am guessing that the equations in the following image would do for 2D Bernstein ...
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0answers
13 views

Divided difference acts on what space?

I'm currently trying to explain divided differences with a view to defining the Newton form of the interpolating polynomial. I'm using the definition: The $k$th divided difference of a function $f$ ...
0
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1answer
41 views

Is interpolating well-sampled data (Nyquist-Shannon theorem) a cheat?

Suppose to sample a signal $s(t)$ with bandwidth $B$ with a sampling frequency $f_c$. Suppose also that the number of sample collected is $N$ (the duration of the signal acquisition is then $T = \frac{...
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0answers
11 views

Monotonicity of 1D interpolation with signed errors

Suppose that we're given $\textbf{x} = (x_i,y_i)_{i = 1}^N \subset \mathbb{R}^2$ such that $x_i$ are distinct. There are a number of well known ways (see e.g. Implementation of Monotone Cubic ...
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2answers
42 views

How to find the polynomial given its factors? (A bit typical one here)

I recently saw this problem (and I should have paid more attention to my middle school maths classes). Find a 3 degree polynomial of $x$ which is $0$ when $x=1$ and $x=-2$, $4$ on $x = -1$ and $28$ ...
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2answers
38 views

Polynomial interpolation of $n+1$ points but ensure last coefficient is a certain number?

I have $n+1$ data points $(x,y)$, and I want to create an interpolating polynomial as described here https://en.wikipedia.org/wiki/Polynomial_interpolation. However there is a twist, I want to ...
1
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1answer
45 views

Is there an equation for the exact line of best fit?

Is there some sort of equation/formula that can be used to find the exact values of $m$ and $b$ in $y=mx+b$ of any data points for the line of best fit? I want to be able to do this manually, not with ...
1
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1answer
28 views

Finding the equation of the line through X number of points?

I have a line graph with $8,000,000$ points. The X axis goes from $0$ to $7,999,999$ in increments of $1$ and the Y axis is either a $0$ or a $1$. There are no fractions on either axis. Is there an ...
1
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1answer
34 views

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
73 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 ...
1
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1answer
49 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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0answers
20 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 perspective-...
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2answers
101 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 ...
0
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1answer
32 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 ...
3
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3answers
385 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: $\ |f(x)-p(x)...
2
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0answers
38 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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0answers
51 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\\ -3t^...
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0answers
31 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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0answers
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 ...
2
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1answer
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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1answer
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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2answers
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 $x>0$.