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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3
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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
1answer
265 views

interpolating the primorial $p_{n}\#$

The primorial $p_{n}\#$ is given by the product $p_n\# = \prod_{k=1}^n p_k$ (where $p_{k}$ is the $k$th prime) -- is there a natural (a la the gamma function $\Gamma(z)$) way of interpolating it for ...
2
votes
4answers
96 views

Function generation by input $y$ and $x$ values

I wonder if there are such tools, that can output function formulas that match input conditions. Lets say I will make input like that: $y=0, x=0$ $y=1, x=1$ $y=2, x=4$ and tool should ...
1
vote
1answer
151 views

Cubic spline interpolation - how to calculate second derivative

I ask this qeustion on stackexchange sites: stackoverflow, codereview, and signal processing and no one can help and they send me here :) So I implement cubic spilne interpolation in Java base on ...
1
vote
1answer
410 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 ...
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 ...
6
votes
1answer
207 views

Cauchy Integral Formula for Matrices

How do I evaluate the Cauchy Integral Formula $f(A)=\frac{1}{2\pi i}\int\limits_Cf(z)(zI-A)^{-1}dz$ for a matrix ...
5
votes
1answer
347 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 $n+1$ ...
3
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1answer
634 views

Remainder term of Lagrange Interpolation Polynomial

Suppose $x_0,x_1,\ldots,x_n$ are $n+1$ distinct numbers in the interval $[a,b]$ and $f\in C^{n+1}[a,b]$. Then for each $x$ in $[a,b]$, there is a number $\xi$ in $(a,b)$ such that $$f(x) = P(x) + ...
7
votes
2answers
332 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
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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 ...
4
votes
0answers
187 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 ...
2
votes
1answer
244 views

Interpolation inequality

Lef $u$ be at least a $C^2$ function on $\mathbb{R}^n$. Let's denote the gradient by $D$. Also, (using the multiindex notation), define the seminorm $$||D^ku|| = ...
1
vote
1answer
337 views

Thin Plate Spline interpolation of scattered $z(x,y)$ data

I am trying to understand Thin Plate Spline interpolation of scattered data. As I understand it TPS is just a special case of Radial Basis Function interpolation: $$ z(x,y) = p(x,y) + \sum_i ...
1
vote
2answers
124 views

Finding a simple spline-like interpolating function

I am looking for a continuous function $y=f(x,\alpha)$ for the interval $0\le x \le 1$ such that $0\le y \le 1$ and $y(0,\alpha)=0$ and $y(1,\alpha) = 1$ and $y(\alpha,\alpha) = 1-\alpha$ and ...
1
vote
1answer
897 views

Construct / find the simplest function based on data

Let's say I have these 7 natural numbers (all between 0 and 255): 255, 23, 45, 32, 87, 52, 146 How can I find a function F(x) that, once computed, gives me back ...
4
votes
1answer
400 views

Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.

I am trying to understand Hermite Interpolation. Here is my pedagogical example. I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation. I choose the Chebyshev zeros ...
3
votes
2answers
338 views

Spline interpolation versus polynomial interpolation

What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
3
votes
1answer
186 views

Curve through four points — simple algebra??

The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to ...
3
votes
1answer
192 views

Fritsch and Carlson is non-linear?

I was reading about this interpolation method and saw that it was mentioned that the algorithm is non-linear. What does that exactly mean? I am confused because I don't get what is "non-linearity" in ...
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 ...
2
votes
2answers
1k views

How to calculate interpolating splines in 3D space?

I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a ...
2
votes
1answer
111 views

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
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1answer
57 views

inequality about linear and piecewise constant interpolation?

$\Omega\subset\mathbb{R}^3$ is a bounded, and $u(\mathbf{x},t) \in C\big(0,T,L^2(\Omega)\big)$. We divide the interval $[0,T]$ in $N$ equal subintervals with the time step $\tau$. With the notaion $$ ...
1
vote
1answer
45 views

$p$-polynomial of $n$'th degree, $q(x)=p[x,x_1,x_2,…,x_k]$, prove that q has the same leading coefficient.

So I have a polynomial $p$ of $n$'th degree and q given by $q(x)=p[x,x_1,x_2,...,x_k]$, meaning that for $x$ it gives back the leading coefficient in interpolation of $p$ on points $x,x_1,...,x_k$. ...
1
vote
2answers
372 views

Easing function, constant velocity then decelerate to zero

I'm trying to write an interpolator for a translate animation, and I'm stuck. The animation passes a single value to the function. This value maps a value representing the elapsed fraction of an ...
1
vote
1answer
628 views

How would I create a exponential ramp function from 0,0 to 1,1 with a single value to explain curvature?

I need an exponential function that will take linear input from 0,0 to 1,1 and give me back an exponential shaped curve such that changes in X near the 0 point result in small increases in Y, but each ...
1
vote
3answers
478 views

Polynomial regression interpolation? [duplicate]

Possible Duplicate: Writing a function $f$ when $x$ and $f(x)$ are known I'm not versed in mathematics, so you'll have to speak slowly... If I want to fit a curve to the points, ...
1
vote
3answers
984 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 ...
0
votes
1answer
70 views

Degrees of interpolating polynomials

Given a collection of $m+1$ points $\{(x_0,y_0), (x_1,y_1), ..., (x_m,y_m)\}$, we can form the interpolating Lagrange polynomial $L(x)$: $$ L(x) = \sum_{i = 0}^{m} y_i l_i(x) \\ l_i(x) = \prod_{0 \le ...
0
votes
1answer
312 views

Resolve a thin plate spline function

I am trying to keep the outline of an object in a video. So I have the coordinate of the outline of the object in the image $t$ and after computing the optical flow I have the coordinate in the image ...
0
votes
1answer
225 views

Derivative of a function defined by the divided difference of another function.

Given a function $f$ of class $C$ $^{n+2}$ in an interval $[a,b]$ and $x_{0}=a<x_1<x_2 ... <x_n = b$ a subdivision of $[a,b]$ into $n+1$ points. Given another function $g$ defined in the ...
0
votes
1answer
131 views

Is it posible to interpolate convex hull in 2d space

I have $n$ points (in this example $11$) and I need to interpolate them in such a way that I have a function $f(t) \rightarrow (R, R)$ where $t \in [0; 2\pi]$. It can be parametric curve, but I need ...
0
votes
1answer
127 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
381 views

Please help to find function for given inputs and outputs [closed]

Can you help with finding the formula for these input and output values? When $n=1$: $f\left(1,1\right)= 0.0000000000$ When $n=2$: $f\left(1,2\right)= 0.0000000000$ $f\left(2,2\right)= ...