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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Does cubic spline interpolation preserve both monotony and convexity?

I have a question. Let's say i have a function $f(\cdot)$ such that $Dom(f) = [a,b]$. The function is at least of class $C^2$ and it is both strictly monotone and convex. My question is, does a ...
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1answer
50 views

Difference between varying a matrix or a variable over time

My question is: I have a Rotation_Matrix and I want to interpolate the rotations over time (from instant 0.0 to instant 1.0). I've done it with two approaches: For the first case, I extracted the ...
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1answer
31 views

Finding y-value in data point to determine coefficient on interpolating polynomial.

Let $P(x)$ be the interpolating polynomial for the data $(0, 0)$, $(0.5, y)$, $(1, 3)$ and $(2, 2)$. Find $y$ if the coefficient of $x^3$ in $P(x)$ is $6$. I tried finding the Lagrange interpolating ...
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1answer
20 views

One parametric family that interpolates continuously between identity and natural logarithm on (0,1]

I am looking for a family of continuous functions $f_p$, $(0,1]\to\mathbb{R}$, and $p\in [1,\infty)$ that fulfill $$ f_1 \equiv \log(x) \\ \lim_{p\to \infty} f_p \to x$$ for $x\in (0,1]$. I ...
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27 views

Gauss Markov Theorem vs Least Squares

I am learning about the Gauss Markov theorem - I know I have not understood it correctly. Assuming the conditions are met, I was aware that there exists infinite solutions for a line of best fit.I ...
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54 views

Linear, Bilinear and B-spline interpolation

I've read that the linear interpolation isn't differentiable everywhere and it would be better to model a continuous-space image using quadratic or cubic B-spline interpolation because is ...
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2answers
17 views

how to map series of coordinates onto a series of coordinates with different resolution

I have a set of target coordinates and a set of actually clicked coordinates which should be approximately the same, but not identical. The y coordinates are equal, however, the x-coordinates differ, ...
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6 views

Order of continuity of the cosine interpolant

I can't found any reference about the order of continuity of the well-known cosine interpolant: (1 - cos(t * PI) / 2; , where t is in the interval [0, 1] Anyone ...
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23 views

Fuzzy Logic Calculations for Non-Intersecting membership values

So i am not a fuzzy logic expert but wanted to see if what I am trying to do falls into something that I can solve using fuzzy logic. I have 3 categories A, B & C and for each category I have 4 ...
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1answer
24 views

Quadratic spline with the smallest sum of squares of derivatives

In quadratic spline interpolation, a formula is given and all we really need to do is calculate the values $$S_i'(t_i) = z_i$$ In quadratic spline interpolation specifically, one has a single degree ...
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1answer
31 views

Spline interpolation degrees of freedom

When using cubic spline interpolation, we have to solve $n-1$ equations with $n+2$ unknowns. What we can do is set $z_0 = z_n = 0$, which gives the natural cubic spline. But could we also set some ...
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1answer
52 views

Having trouble interpolating a polynomial using Newton's Method

Given the data: Find an interpolating polynomial using Newton's method. I am a bit confused on how to do this. I think it should be fairly simple.. Here is what I have so far: $p_0(x) = -1$ $p_1(...
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2answers
28 views

Interpolating polynomials

If I interpolate a polynomial of degree n with n+1 points, will I always get the polynomial itself back? if so, does it work for k < n+1 points?
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2answers
101 views

Cubic spline with clamped boundaries

I have a cubic spline interpolation problem to work through. I think I understand what is required of the question, but my biggest concern is the nature of clamped versus natural boundaries. All the ...
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19 views

Existence of a continously differentiable interpolator of a finite dataset

Formally, a function of several real variables f: Rm → Rn is said to be differentiable at a point x0 if there exists a linear map J: Rm → Rn such that $\lim_{\mathbf{h}\to \mathbf{0}} \frac{\|\mathbf{...
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18 views

Other proofs of uniqueness of interpolating polynomial

I think that one of the well known proofs is this one: Let $f:[a,b]\to\mathbb{R}$ be a function and $P_n:[a,b]\to\mathbb{R}$ be the interpolating polynomial for $f$ on $[a,b]$. Let the nodes of ...
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15 views

Express interpolating polynomial as a linear combination of interpolating polynomials for subintervals

Given a set of points $(x_i, u_i), \; i = 1, \dots, n$ (for my application $n = 5$) one can construct an interpolating polynomial $$ P(x) = \sum_{i=1}^n u_i \ell_i(x), \quad \operatorname{deg} P = n - ...
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229 views

Difference between ordinary kriging and simple kriging with normalization

Simple kriging assumes known mean, which it seems can be induced by normalizing the data in each dimension around a mean of 0. What is the difference between performing simple kriging in this manner ...
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83 views

Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...
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19 views

Under what condition given $(x_1, y_1\cdot r_1),…,(x_n, y_n\cdot r_n)$ we can interpolate polynomial $T$ that has specific random root?

We know given $(x_1, y_1),...,(x_n, y_n)$ we can interpolate a polynomial $P$ of of degree at most $n-1$. Let us define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is at most $n-1$, $\...
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32 views

Which method do I use ? Interpolation

I have table of $5$ values (i.e abscissa and ordinates are given). I have been asked to find derivative at particular point and also second derivative at that value. That value is between my given ...
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62 views

Specific Root of Interpolating Polynomial

We define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is fixed $n-1$, $\beta$ is chosen uniformly at random from the field of $p$ elements. We evaluate $P$ at some $x_i$ values. So we get $...
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1answer
88 views

Overlapping Polynomials

This question is related to this:Interpolating Polynomial & It's Root We have $P_3=P_2\cdot P_1$,for three non-zero polynomials. The degree of each polynomial is at least 1. Question: Does $...
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34 views

Probability That a Polynomial has Specific Root when we use Permutation Polynomial

To some extent similar question was asked here: Polynomial Interpolation and Security Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-2$, ...
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22 views

Probability that a Polynomial Has Specific Root When $y_i$'s are Not Random.

Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-1$, but degree of $P_2$ can be at most $n-1$. $P_1$ has root $\beta$, where $\beta \leftarrow \...
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1answer
76 views

Interpolating a Polynomial with a Subset of Interpolation Points

Consider we has a polynomial $P=(x-\beta)g(x)$, where $\beta \leftarrow \mathbb{Z}_p$, $p$ is a large prime, and $g(x)$ is a non-zero polynomial. Here degree of $P$ is fixed $n$. We evaluate $P$ at $...
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25 views

Derivative of multivariate splines using tensor products

I am trying to compute the derivative of a multivariate spline, in fact bi-variate I use a b-spline univariate to create a basis, for the first $x_1$ and second variable $x_2$, then I use the tensor ...
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1answer
86 views

Newton's and Lagrange form's of interpolating polynomial

Can someone hint me on this one? Question: Find the Lagrange and Newton forms of interpolating polynomial for these data points $(-1,0),(0,1), (2,3)$. Write both polynomials in the form $a x^2+bx+...
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24 views

How to determine a function from data $2$-dimensional graph

I assume it is possible. Let us say I have $3$ years worth of water amount of rain collected everyday. A $2$-dimensional graph has been created from the data and there is pattern every year. How do I ...
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1answer
113 views

Polynomial Interpolation and Security

Let polynomial $P$ be $P(x)=g(x).(x−β)$, where $g$ is a polynomial and $\beta \leftarrow \mathbb{F}_p$. We evaluate $P$ at some $\textbf{x}=(x_1,..,x_n)$. This gives us $\textbf{y}=(y_1,..,y_n)$. ...
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32 views

Polynomial Interpolating; When $y_i$'s are Changed

This is a comlpementry question to the one posted in: Polynomial Interpolation And polynomial Roots Given $\{(x_1,y_1),...,(x_n,y_n)\}$, we can interpolate a polynomial $P$. Assume polynomial $P$ has ...
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2answers
33 views

Given a Brownian motion $W$, and $k \in (a,b)$, I'm trying to find the distribution of $W(k)$ in terms of $W(b)$, $W(a)$, and $k$

I'm trying to perform this "interpolation" because I ultimately am trying to write a small library to simulate stochastic processes. I realized I might need to figure out what is the distribution of $...
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167 views

Radial Basis Functions Interpolation

$ \let\oldcdot\cdot \renewcommand{\cdot}{\!\oldcdot\!} \newcommand{\e}{\varepsilon} \renewcommand{\p}{\varphi} \renewcommand{\p}{\varphi} \renewcommand{\vp}{\vec{\boldsymbol\p}(x)} \newcommand{\P}{\...
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2answers
94 views

Polynomial Interpolation And polynomial Roots

Given $\{(x_1,y_1),...,(x_n,y_n)\}$, we can interpolate a polynomial $P$. Assume polynomial $P$ has some roots including an specific root $\beta$. Consider we change one of $y_i$ to $y'_i$. Given $\{(...
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55 views

Intuitive explanation for error in Newton's Divided Differences?

When interpolating a smooth function $f$ using $n+1$ points, the error in the interpolation is bounded by $e(x) \leq$ $f[x_0,\ldots,x_n,x] \cdot \prod_{i=0}^n(x-x_i)$. This seems kind of interesting ...
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1answer
27 views

Polynomial Fitting of Circular Data Object

This is a very odd question. I have a one dimensional data set that is graphed on a histogram. I am trying to curve fit this data set (using the class midpoints as the x values, and the frequencies as ...
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25 views

When do I use a specific interpolation method?

I am having a course on Numerical Analysis and I was wondering if I can use any interpolation method to interpolate any data, or one method has some specific advantages over another. Here are some of ...
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1answer
91 views

How is the B-Spline definition constructed?

I'm trying to understand how the B-Spline definition is constructed. That is, where did the knot vector and the basis functions and their recursive definition come from. The definition can be seen ...
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2answers
59 views

How does Newton Interpolation work?

How does the Newton Interpolation work? The definition can be found here: http://www.nptel.ac.in/courses/122104018/node109.html Not how it's defined since that's mathematically clear, but I'm trying ...
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1answer
30 views

Confused with an interpolation problem using Lagrange.

I'm really confused about the following interpolating problem.Not sure if this is the right method. For $n =3$, explain why $$ x_0^jL_o(x) + x_1^jL_1(x) + x_2^jL_2(x) + x_3^jL_3(x) = x^j, \ \ j \leq ...
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1answer
18 views

Existence of an interpolator of a finite data

I am wondering if the following problem has an affirmative answer: Does any finite dataset can be interpolated by a continuous and smooth function. Formally, let $E\subset \mathbb{R}^{n}$ be a finite ...
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40 views

Interpolating Distances (Values) on a Square or Grid

Let's say that I have values on a grid, where each intersection on the grid represents the distance from that point on the grid to the nearest point on the edge of some shape. A negative value means ...
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68 views

Maximum Error Bound on a Cubic Spline, Chebyshev Polynomials

Most notations for the error resulting from interpolation of cubic splines (e.g. http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture12.pdf) requires knowledge of the original function to ...
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1answer
70 views

Is this algorithm for 3D spherical interpolation correct?

I am attempting to write a spherical interpolation algorithm for for the application of smooth 3D animation in a game. The scripting language that the game engine uses is Lua. It is often easier for ...
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1answer
93 views

Finding the closest function describing a “magnetic line” (given magnetic readings)

I'm collecting data from a smartphone magnetometer while I move a magnet along a straight line (a slider). I am collecting the values of the magnetic field strength along the three axes. I would like ...
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59 views

Least squares have null determinant

I want fitting my data using bicubic interpolation: $$f(x,y)=\sum_{i=0}^{3}\sum_{j=0}^{3}a_{ij}x^iy^j$$ Let known $$f(0, 0)=1; f(2, 0)=1;f(1, 1)=0;f(0, 2) = 1; f(2, 2)=1$$ I used least squares method, ...
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41 views

Interpolation of random processes

Let $\left(\Omega, \mathcal{F}, \left\{\mathcal{F}_{t}\right\}_{t\geq 0}, P \right)$ be a complete probability space with a nondecreasing family of right continuous sub-$\sigma$-algebras $\mathcal{F}$...
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17 views

Taking FFT of an interpolated function

As part of a research project relating to metrological uncertainties in CT reconstruction I am attempting to calculate, from a reconstructed volume, the MTF of the imaging system. I am following the ...
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1answer
50 views

Strange behavior with coordinate transformation of square and quadrilateral

I am trying to map coordinates from a quadrilateral to a square. The coordinates are square: $(500,900)(599,900)(599,999)(500,999)$ quad: $(454,945)(558,951)(598,999)(499,999)$ where the $i^{th}$ ...
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1answer
42 views

Explaining Non-Uniquuness of an Interpolation Polynomial

I am stucked at this problem: If $f\in C^1[a,b]$ and $x_0,...,x_n$ are $n+1$ distinct points in $[a,b]$, Then there exist unique polynomial $H_{2n+1}$ of degree at most $2n+1$ that satisfies the ...