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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Methods for detecting kinks along a set of points (for interpolating)

Can anyone point to some literature on methods for detecting kinks/sharp turns with given a set of points. To give some background, I'm trying to get estimates of some function based on randomly ...
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0answers
6 views

interpolation terminology

I am trying to write a discussion on various interpolation methods, and I need a systematic terminology for interpolation, or the article will have an unnecessary digression to define terms, which ...
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0answers
21 views

Did I interpret this wiki article on spherical interpolation correctly?

In Lua pseudocode, I believe the wikipedia article here is saying that the formula is used in the following way: ...
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0answers
16 views

Interpolating a vector about an arc (Slerp)

In the following image, how can I solve for $k_0$? I know that $\mathbf v_1$ is a unit vector and $k_1 = \sin tω/\sin ω$.
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0answers
10 views

Linear interpolation on Plane (Marching Cubes)

Let's assume I have the following cube. Let's assume the isovalue = 0. I would like to draw the resulting triangles of the isosurface. I know that first I define which values are inside or outside ...
7
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2answers
272 views

Accurate floating-point linear interpolation

I want to perform a simple linear interpolation between $A$ and $B$ (which are binary floating-point values) using floating-point math with IEEE-754 round-to-nearest-or-even rounding rules, as ...
1
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0answers
23 views

Tricubic Interpolation

I am currently writing a plugin for 3D analysis software and I am working with a data grid where certain values are stored at XYZ coordinates, and I need to find an estimated value of a point that ...
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0answers
25 views

A simple Lagrange interpolation-type identity

I am unable to prove an identity that looks very much like the Lagrange interpolation identity, Problem: Given $f(x)$ is a monic, $n-1$ degree polynomial and $a_1, a_2, \cdots a_n$ distinct real ...
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1answer
26 views

Knowing two Vectors, and the distance to a 3rd, how to get the 3rd

If I know the two Vectors $v_1$ and $v_2$, which describe points in a 2D space, and I also know that a vector $v_3$ is on the line segment between $v_1$ and $v_2$, how can I get the $x$ and $y$ ...
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1answer
567 views

Spline with varing tension, selection of tension factor

I need to perform a special interpolation, using that kind of basis : $$\varphi_{i,j}(x) = a_i + b_ix + c_i(\cosh(\tau\ x) - 1) + d_i(\sinh(\tau\ x) - \tau\ x)$$ where the $a_i$, $b_i$, $c_i$ and ...
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1answer
31 views

Cubic spline interpolation results

I have a set of data points on which i am trying to do cubic spline interpolation. Below is the snapshot of the curve with the input data points marked in green color. And the red color marked point ...
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0answers
33 views

Strict order on propositions and interpolation

We can define a strict order on the set of propositions in countably many propositional letters in the following way: $$\varphi\sqsubset\psi \iff (\models \varphi\rightarrow\psi)\, \land (\not\models ...
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1answer
604 views

Hermite Polynomials Triple Product

Similar to the question Legendre Polynomials Triple Product, I would like to ask whether there are any explicit formulas for the inner product of the Hermite polynomial triple product \begin{align} ...
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0answers
17 views

Find the smallest maximized absolute error in polynomial interpolation

Given $$ f(x)=\begin{cases}1&,0\le x \le1 \\2x&,1<x\le2 \end{cases} $$ I found that the interpolating polynomial $p \in \mathbb{P}_{2} $ at $x_{0}=0,x_{1}=1,x_{2}=2$ ...
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0answers
22 views

Interpolate from a point on a sphere to a point on a another sphere?

I am at the moment trying to come up with an solution which is capable of interpolating between a point on a sphere A to a point on a sphere b. The interpolation should both provide me with minimal ...
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1answer
12 views

find estimation of interpolation error for non differential function

Given $f(x)=|x|^{1/2}$ , $-1\le x\le 1$ , I have found the interpolating polynomial $ p(x)=x^2$ for $x_{0}=-1,x_{1}=0,x_{2}=1$. How to estimate $$\max_{-1\le x\le 1}|f(x)-p(x)|$$ now that $f$ is not ...
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1answer
22 views

interpolation error using higher derivatives

Given $x_{0},x_{1},x_{2}\in[a,b] $ each one different from the others,$f \in C^{4}[a,b]$ and $p\in\mathbb{P}_{3}$ so that $$p(x_{i})=f(x_{i}), i=0,1,2 $$ and $$p'(x_{1})=f'(x_{1})$$ prove that: ...
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0answers
27 views

Function that satisfies the given (x,y) values

I am trying to come up with a function that (approximately) satisfies these (x,y) values. (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 6), (9, 5), (10, 4), (11, 3), (12, 2), (13, 1), (14, 2), ...
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1answer
519 views

Newton backward interpolation in Mathematica

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
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1answer
8 views

Given this discrete non-linear set of values how can I get an equation for it?

I want to generate a equation in the form f(x) = {...} for this discrete data below. As X doubles Y halves but its a bit more complicated. Using an online Polynomial Interpolation calculator I got: ...
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0answers
26 views

Interpolating discrete data with completely monotone analytic functions

Suppose we have a positive integer $n$ and a finite list of real numbers $\{a_1,\,a_2,\,\dots,\,a_n\}$. We want to find a real-analytic function $f:[1,n]\to\mathbb R$ such that $f(m)=a_m$ for all ...
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1answer
19 views

Conditions under which a discretely defined function can be extended convexly

Suppose we have a set of points $u_1,\ldots, u_m \in \mathbb{R}^d$. Suppose $F$ is a function into the reals defined at each of the points $u_i$. My question is how do we know when $F$ is really ...
2
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1answer
69 views

Do nth degree polynomials derived using Least Squares Interpolation always have n+1 intersections with the function?

I have recently studied Interpolation Techniques in my College Numerical Methods class and I have this question: If we have a function $f(x)$ and we are asked to use Least Squares Interpolation(LSI) ...
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1answer
16 views

Discrete surface interpolation

I'm working on implementation of a Fast surace interpolation using hierarchial basis functions (Szeliski et al) algorithm. The idea is: given a discrete function measurements of its values (depths) ...
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1answer
21 views

How to show a piecewise quadratic interpolant is $H^1$

I am preparing for a final exam and came across this question: Suppose that $\Omega\subset\mathbb{R}^2$ is an open bounded domain with triangulation $\mathscr{T}$. Suppose that $v_h$ is a ...
2
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1answer
56 views

Cubic Spline Interpolation

My problem is to find a interpolating cubic spline to the points $$\left\{(0,0), \left(\frac{\pi}{2}, 1\right), \left(\pi,0\right), \left(\frac{3\pi}{2}, -1\right),(2\pi,0)\right\}$$ I did as ...
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0answers
17 views

Interpolation / point fitting onto a logarithmic line segment

I have figure which is logarithmic scale on both axis. There's a line on that figure, I know two points on that line and want to interpolate a third point on that line based on the two known points. ...
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0answers
21 views

Interpolation of random data

I have points $(t_1,x_1(t)),(t_2,x_2(t)), \cdots , (t_n,x_n(t))$ and I would like to estimate values of $x_k(t)$ where $1 < 2 < \cdots < k <\cdots <n$. How can I do this. I have read ...
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0answers
30 views

Least square polynomial interpolation

Given an arbitrary continuous function f(x), let Pn(x) be the polynomial of degree at most n that approximates f(x) in the least squares sense. Is it true that Pn(x) interpolates f(x) at n + 1 points? ...
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12 views

Use past and future data to predict or estimate missing values

I have a huge data set for a single variable $z$ , say WEATHER, not necessarily complete one. That is it has many holes in it(missing data) $z \hspace{3mm} is \hspace{3mm} a \hspace{3mm} 6000\times1$ ...
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0answers
28 views

Algorithm Identification

Background I'm currently working with a system that has a 4-dimensional function. Currently, an algorithm is used to speed up calculation of the final value via interpolation, and two of the ...
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1answer
47 views

Multistep Method: Gear's Formula Interpolation

Please explain how to do this. How can we use Lagrange Interpolation to derive this formula? Thanks in advance.
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1answer
15 views

build function passing for specific points

I have to solve a problem very similar to this how-to-create-a-function-passing-through-given-points I need a function that draw a curve like the blue one in the picture here thus passing as ...
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0answers
19 views

Interpolate with smoothing parameter

I need to implement in C++ interpolation with smoothing parameter. To the non-familiar with this function: The smoothing parameter gets a value from 0 to 1. 0 brings absoulte linear interpolation ...
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1answer
181 views

Upper bound for the error magnitude

For the function $f(x) = \mathrm{e}^x$ on the interval $[0,1]$, by using polynomial interpolation with $x_0 = 0$, $x_1 = 1/2$, and $x_2 = 1$, find the upper bound for the magnitude $$ \max_{0 ...
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0answers
28 views

Given two entire functions $f_1,f_2$ without common zeros, prove that one can find some entire functions $g_1,g_2$ such that $f_1g_1+f_2g_2=1$ [duplicate]

The question Let $f_1,f_2$ be some entire functions without zeros in common, so for every $z∈ℂ$ we have $|f_1(z)|^2+|f_2(z)|^2≠0$. Prove that there exist two entire functions $g_1,g_2$ such that: $$ ...
2
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3answers
64 views

Polynomial interpolation

I need to find the polynomial of degree $3$ with respect to these conditions: $$\begin{cases} p(0) = 1\\ p(1) = -1\\ p'(0) = 1\\ p''(0) = 0 \end{cases}$$ How do I deal with the condition on ...
2
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1answer
32 views

Approximating Fresnel integrals with standard functions

I would like to approximate the Fresnel S and Fresnel C with standard functions. I've started with the $ S(x) $ function: $$ approxS(x) = sgn(x) * \left ( sgn(x)* \left ( \frac{ \sin( \frac{x^2}{2} ...
2
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1answer
24 views

Fourier series on incomplete data [closed]

Given a periodic function that's only partly specified, e.g.: $$f(\theta)=\begin{cases}1 & \text{if } \cos(\theta)>a\\ -1 & \text{if } \cos(\theta)<-a\end{cases}$$ Obviously the ...
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1answer
1k views

Plane fitting using svd

I am trying to get a best fit plane in a 3d space of points. I am using an svd as described in http://stackoverflow.com/questions/10900141/fast-plane-fitting-to-many-points. If I use the data provided ...
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3answers
50 views

SmoothStep: Looking for a continuous family of interpolation functions

Background: SmoothStep is a simple sigmoid-like function defined as S(x) = 3x^2 - 2x^3. It is monotonically increasing from (0, 0) to (1, 1), is rotationally symmetric over that interval, and has ...
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0answers
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Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
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2answers
2k views

Spline interpolation versus polynomial interpolation

What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
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0answers
12 views

Comparing smoothness among approximations

We are interpolating a missing fragment of a 2D curve given a set of sample points. Our method generates several candidates of curve pieces to fill the missing part, but we want to select the solution ...
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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
7 views

Roll forward a payment

If I earned $100 per month from Jan 1, 2016 to April 30, 2016 how do I determine my projected 2016 salary if I am assuming an annual trend rate of 7.8 % starting May 1st? I would think it would be ...
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0answers
11 views

interpolation preserving boundedness property

I'm trying to construct interpolation for a function $m$ such that \begin{equation*} 0\leq m(x)\leq 1,\quad\forall x\in\Omega\subset \mathbb{R}^1. \end{equation*} I tried to use ...
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1answer
76 views

Equivalent condition for interpolation polynomial

Let $(x_1,y_1),...,(x_n,y_n)\in \mathbb{R}^2 $, where $x_i\neq x_j$ if $i\neq j$. Let $p$ be a polynomial such that $$\det\begin{pmatrix} p(x)& 1 & x & x^2 &\dots & x^n \\ ...
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1answer
28 views

Second Order Accurate Interpolation

On a grid I am having the values of a physical quantity say for example Temperature, at the E,W,N,S and P node all of them being calculated using a second order discretization scheme. I want a second ...
2
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0answers
73 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...