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
55 views

How to choose new points after grouping/resampling?

I'm resampling a signal (which takes values [0,1]) of N samples (blu points) to one with N/5 samples, where (for each group of 5 samples) I store in two arrays the max and the min values of the ...
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0answers
23 views

Best 2D interpolation for a set of points (x/y)

Given a set of X/Y points in a bounded area ( 0<=X<=Xmax and 0<=Y<=Ymax ), what are the best 'interpolation' ...
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2answers
43 views

Improve the order of accuracy for second derivative using central difference approximation

The formula $$D^{m}(h)=(4^mD^{m-1}(h/2)-D^{m-1}(h))/3$$ for $m=2,3,\ldots$improves the order of accuracy for first derivative using central diff approx. where ...
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2answers
44 views

The quadratic spline

I'd like to fit the data in table as blow x f(x) 3.0 2.5 4.5 1.0 7.0 2.5 9.0 0.5 when $x=5$, I want to find value of $f(x)$ by using ...
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0answers
10 views

Error Function using Chebyshev Node Interpolation

Given a set of measurements (at Chebyshev nodes), does an error function exist that allows you to roughly approximate the error (like the error function for Simpson's Rule)?
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23 views

Padua points or Chebyshev grid for more than two dimensions?

I'm looking for a good point grid in order to sample from a (polynomial) function $f:R^n\rightarrow R$ at discrete points lying in a rectangle. I don't need any weights or interpolating polynomial. I ...
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0answers
28 views

Proof of Runge's phenomenon for a concrete case

Let $f(x)=\frac{1}{1+25x^2}$ and range is $[-1,1]$. Given $n+1$ equidistant points $x_0 = -1,x_1,...,x_n = 1$ and their values $f(x_0),f(x_1),..,f(x_n)$, perform polynomial interpolation by the $n+1$ ...
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1answer
25 views

Find points $x$ such that $p_n(x)-f(x)$ does not converge to $0$ as $n\to\infty$

Let $f(x)=\frac{1}{1+25x^2}$ and range is $[-1,1]$. Given $n+1$ equidistant points $x_0 = -1,x_1,...,x_n = 1$ and their values $f(x_0),f(x_1),..,f(x_n)$, perform polynomial interpolation by the $n+1$ ...
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0answers
14 views

Does this operator produce an interpolating spline of it's argument?

I'm studying a bit of spline functions theory: In this book, chapter 6, a lot of error bounds are given when spline functions are used to approximate functions belonging to a specific function spaces. ...
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1answer
56 views

Proof that lagrange and newton's interpolation are the same

Its known that newton's interpolation and lagrange interpolation gives the same value All i need is to prove it
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1answer
46 views

Pyramidal Histogram Of Oriented Gradients - Trilinear interpolation

Hello im struggling with an implementation of this article: https://goo.gl/8mpIuq I performed bilinear interpolation over the histogram bins and the results are better with this interpolation, ...
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0answers
38 views

Use polynomial interpolation to find a formula

Any idea how to Use polynomial interpolation to find $P_{k+1}$ where $P_{k+1}(n)=\sum_{j=1}^n j^k$ for $2\le k \le 10$? Thanks!
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0answers
58 views

find the equation of sine (or relevant) function with trend to fit a curve or fill gap

I want to fill a gap of temperature (hourly) data. After plotting it appears to be the gap can be fit by using a sine function with a trend. Now i would like to find the equation. Any suggestions will ...
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1answer
41 views

Find an equation fitting a set of points

In a game that I play, there is a power value that is an overall evaluation on how powerful your account is. There is a function where you can level up a skill, and levelling it up gives a bonus to ...
1
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1answer
20 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
41 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) ...
0
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1answer
35 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 ...
0
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1answer
26 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
21 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
9 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 ...
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0answers
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
50 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 ...
0
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1answer
25 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
21 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
18 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
28 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 ...
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1answer
29 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 ...
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1answer
47 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, ...
2
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4answers
56 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 = ...
1
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1answer
21 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
36 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
73 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 ...
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1answer
36 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
44 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 $\xi\in [0,1]$: ...
0
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1answer
35 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
81 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
16 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
18 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 ...
1
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1answer
44 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
12 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
33 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 = ...
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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
41 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
44 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
26 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 ...