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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4answers
195 views

What are some “natural” interpolations of the sequence $\small 0,1,1+2a,1+2a+3a^2,1+2a+3a^2+4a^3,\ldots $?

(This is a spin-off of a recent question here) In fiddling with the answer to that question I came to the set of sequences $\qquad \small \begin{array} {llll} ...
1
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0answers
699 views

How to derive Hermite polynomial from a given data set?

The problem is asking to find a Hermite polynomial to predict the position of the car and its speed when t = 10s. The Hermite polynomial formula is defined as: $$H_{2n+1}(x) = f[z_0] + ...
1
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1answer
392 views

Curve fitting with upper and lower bounds for derivatives

I compute (at a great cost) upper and lower bounds $f_u(x)$ and $f_l(x)$ of an unknown function $f(x)$ at points $x$ in $[0,1]$. Now I am interested in an estimation of the derivative $f'(x)$. I ...
0
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1answer
135 views

find a function given some values

I'me trying to remember my math classes but no luck... I've got a pair of values i.e . ...
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2answers
580 views

Is there some intuition for Lagrange interpolation formula?

How do I prove the Lagrange interpolation formula is true as stated in this link? I ask this because the article isn't self contained on intuition of each step in the proof, please don't use things ...
3
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1answer
1k views

MATLAB Hermite interpolation

Anyone know where I can find the Hermite interpolation algorithm in MATLAB. Which Hermite interpolation algorithm solves this? I need to calculate a polynomial. Example: ...
1
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1answer
120 views

Polynomial interpolation on scattered points

I was wondering how I could fit a polynomial surface through a set of points in two variables. When I look up this problem in the literature, I usually see two options: Use a tensor product, but ...
0
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1answer
660 views

Is this Hermite interpolation correct?

Someone can explain this hermit interpolation algorithm with example? Thank you, ...
2
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2answers
123 views

Interpolating polynomials

So I have this question on a homework and I just can't seem to figure it out. Let $f \in C^4 [0,1]$ and let $p$ be a polynomial of degree $\le 3$ such that $p(0) = f(0)$, $p(1) = f(1)$, $p'(0) ...
0
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1answer
341 views

Fitting for piecewise function, with constraints on first/second derivative

I have the following problem. We have a set of discrete points ($x_i$,$y_i$), defined for $0 \leq x_i \leq r$, where $r$ is an arbitrary value. For values of $0 \leq x \leq r$, the y value is ...
1
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0answers
76 views

Questions about interpolating translated points from a grid

I would like to do the following transformations on a very low resolution bitmap (64x64 pixels). I am doing this transformation on a computer images, but it has nothing to do with computers, you can ...
0
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1answer
109 views

Interpolation over trajectory at set positions on path

I have the following: 2d vector for velocity 2d start coordinate gravity acceleration I need to know the coordinate of a projectile at a given distance along the trajectory. For example: ...
2
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0answers
158 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 ...
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1answer
511 views

existence and uniqueness of Hermite interpolation polynomial

What are the proofs of existence and uniqueness of Hermite interpolation polynomial? suppose $x_{0},...,x_{n}$ are distinct nodes and $i=1 , ... ,n$ and $m_{i}$ are in Natural numbers. prove exist ...
0
votes
1answer
380 views

how to interpolate a 2d function with 6 points?

I'm implementing an algorithm that uses a so called 6-point interpolation, which I never heard before. In the article I'm reading it's described like this: $\phi(p\Delta x, q\Delta y)=[q(q-1)/2] ...
2
votes
2answers
1k views

given $y = a + bx + cx^2$ fits three given points, find and solve the matrix equation for the unknowns $a,b$, and $c$

Given $y = a + bx + cx^2$ fits three given points, find and solve the matrix equation for the unknowns $a$, $b$, and $c$. the equation fits the points $(1,0), (-1, -4),$ and $(2, 11)$ I really ...
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3answers
1k views

Linear Algebra Question (Polynomial Interpolation)

Given the data for an experiment: Velocity: 0, 2, 4, 6, 8, 10 Force: 0 , 2.9, 14.8, 39.6, 74.3, 119 (One force value listed below one velocity value in a table) Find an interpolating ...
2
votes
1answer
427 views

What is a cardinal basis spline?

Wikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as "Bk". Meanwhile, cnx.org Signal Reconstruction says: The basis splines Bn are shown ... ...
2
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1answer
776 views

Comparing the maximum error between Lagrange, Hermite, and Spline Interpolation Methods

I am reading Numerical Analysis by Atkinson. I am curious how do I choose the appropriate number of data points in each method so that I can make fair error comparisons? Some background: For each of ...
3
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1answer
1k views

Interpolating point on a quad [closed]

I have a quad defined by four arbitrary points, A, B, C and D all on the same plane. I then have a known point on that quad, P. I want to find the value of 's' as shown in the diagram above, where ...
2
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0answers
297 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 ...
5
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1answer
2k views

Natural cubic splines vs. Piecewise Hermite Splines

Recently, I was reading about a "Natural Piecewise Hermite Spline" in Game Programming Gems 5 (under the Spline-Based Time Control for Animation). This particular spline is used for generating a C2 ...
2
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1answer
174 views

Short argument/reference for uniform continuity of piecewise linear interpolation

I have a piecewise linear interpolation: $$ B(t) = \frac{t_{l+1}-t}{t_{l+1}-{t_{l}}} B_l + \frac{t-t_l}{t_{l+1}-{t_{l}}} B_{l+1} \quad \text{ if $t \in (t_l, t_{l+1})$;}$$ $B(t_l)=B_{l}$ and $B(t) = ...
9
votes
4answers
2k views

Polynomial fitting where polynomial must be monotonically increasing

Given a set of monotonically increasing data points (in 2D), I want to fit a polynomial to the data which is monotonically increasing over the domain of the data. If the highest x value is 100, I ...
0
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2answers
52 views

Interpolating a period-two sequence ($f(x) = f(x-2)$): why $(-1)^x$?

Wolfram|Alpha gives the recurrence equation solution $f(x) = c_2(-1)^x + c_1$. Why is the interpolating function $(-1)^x$? Other functions like $\cos(\pi x)$ (the real part) and even ...
0
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1answer
148 views

Vandermonde and curve interpolation

I hesitate here because of an understanding with a calculation problems. I want to calculate an interpolation using the Vandermonde matrix. see: http://en.wikipedia.org/wiki/Vandermonde_matrix My ...
0
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2answers
162 views

Solve a polynomial of degree d

In my Artificial Intelligence class, I encountered this equation where I have to solve for b, the effective branching factor of a tree: ...
2
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1answer
416 views

How to interpolate a sequence?

I have an infinite sequence (see the graphic) which I want to interpolate with an analytic function. Polynomial interpolation fails due to Runge phenomenon. What else can I do?
5
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1answer
1k views

Lyapunov's Inequality for Weak-Lp Spaces

Let $(X,\mu)$ be a measure space. Suppose that $0 < p_{0} < p < p_{1} < \infty$ and $\frac{1}{p} = \frac{1-\theta}{p_{0}} + \frac{\theta}{p_{1}}$ for some $\theta \in (0,1)$. If $f \in ...
1
vote
1answer
46 views

Equally distributing cards on a table. (interpolation)

this problem is originally from a programming task i am on. there is a number of playing cards n. the width of each card is w. now the cards should be placed next to each other on a table with equal ...
2
votes
3answers
242 views

creating smooth curves with f(0) = 0 and f(1) = 1

I would like to create smooth curves, which have f(0) = 0 and f(1) = 1. What I would like to create are curves similar to the gamma curves known from CRT monitors. I don't know any better way to ...
3
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1answer
282 views

How to minimize this function difference

Sorry about this somewhat lengthy introduction to my question. I thought it might be useful to know what I'm trying to do. I decided that I would like to have sequence of polynomials in $\mathbb{P}_n ...
1
vote
1answer
71 views

A function for forming a closing curve on all points on a 2D image

First of all, this is for image processing. The small circles above is on an image and are my list of points. The points are always in order from p1, p2, p3 to pn. And I am to create a curve that ...
3
votes
1answer
2k views

2D array downsampling and upsampling using bilinear interpolation

I am trying to understand how exactly the upsampling and downsampling of a 2D image I have, would happen using Bilinear interpolation. Now I am aware of how bilinear interpolation works using a 2x2 ...
3
votes
1answer
2k views

What is Hermite data?

Using fairly simple language, what is Hermite data? I encountered it here, http://www.frankpetterson.com/publications/dualcontour/dualcontour.pdf and could not get an answer on standard StackExchange, ...
1
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1answer
162 views

Decomposition of any point in the unit hypercube as a positive linear combination of polynomial number of vertices

I have function values at each of the vertices of the hyper cube. What would be a natural interpolation of the function to each point on and inside the cube that can be written as a positive linear ...
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3answers
1k views

Smooth transition between two lines (2d)

I have function that is defined as $$ Y = \frac{1}{15} x \longrightarrow {\rm if}\qquad 0 \leq x \leq 30 $$ $$ Y = \frac{1}{70} x + \frac{11}{7} \longrightarrow {\rm if}\qquad x > 30 $$ The ...
3
votes
2answers
3k views

Implementation of Monotone Cubic Interpolation

I'm in need to implement Monotone Cubic Interpolation for interpolate a sequence of points. The information I have about the points are x,y and timestamp. I'm much more an IT guy rather than a ...
7
votes
2answers
359 views

A Curious Binomial Sum Identity without Calculus of Finite Differences

Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$, \begin{align} \binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} ...
5
votes
1answer
369 views

Determining Coefficients of a Finite Degree Polynomial $f$ from the Sequence $\{f(k)\}_{k \in \mathbb{N}}$

Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ ...
6
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4answers
295 views

Are there smooth analogs to polynomial splines

Is possible to construct infinitely differentiable functions that interpolate through arbitrary points, the way polynomial splines do? If so, do they have a name and is there an algorithm for ...