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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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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74 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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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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27 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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58 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
61 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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11 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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19 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
57 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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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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Help me to solve this question. [closed]

For a particular model of motorcycle the stopping distance at 50 miles per hour is measured at 144 feet and the stopping distance at 70 miles per hour is 296 feet. Using linear interpolation, the ...
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1answer
21 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 ...
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10 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
85 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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29 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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29 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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111 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)} ...
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2answers
81 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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Create Problems for Polynomial Interpolation [migrated]

I want to create problems concerning polynomial interpolation for students. However, I need those problems to differ in difficulty for the different students. What I could always do is solve some ...
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24 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
19 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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15 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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28 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
42 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
23 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
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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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20 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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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
27 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
65 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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35 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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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 ...
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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
40 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
34 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 ...
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How to Find the Error for Spline Interpolation Without the Original Function?

Most of the literature (e.g.: http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture11.pdf) I have consulted thus far indicates how one would determine the error of a cubic when the original ...
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7 views

Necessary and/or sufficient conditions for a set of points in the plane to be able to interpolate in them

Let $d$ be a positive integer. find necessary and/or sufficient conditions on a finite set $S$ of points in the plane such that any values on $S$ can be interpolated by a unique two variable ...
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29 views

Existence of a Blaschke product with an added boundary condition.

Suppose we have a sequence of distinct complex numbers $\{a_n\}$ such that $a_n\rightarrow 1$ and the $a_n$ satisfy the Blaschke condition $\sum (1-|a_n|)<\infty$. Does there exist a Blaschke ...
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Is a cubic Lagrange interpolation tensor product the same as bicubic interpolation?

I just implemented some interpolated texture sampling by sampling the 4x4 nearest pixels then doing Lagrange interpolation across the x axis to get four values to use Lagrange interpolation on across ...
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1answer
57 views

Alternatives / Extensions to the Thin Plate Splines method

Thin Plate Splines are a great method to find a smooth interpolating surface given scattered data. Essentially, the method involves calculating weights for a radial basis function centred around each ...
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Polynomial Interpolation When $y_i$'s are Permuted

Recall, if we have a $d$-degree polynomial $f$, evaluate it at $\textbf{x}=(x_1,\ldots,x_n)$ we would get $\textbf{y}=(y_1,\ldots,y_n)$, where $f(x_i)=y_i$ and $d+1 \leq n$. The reverse is also true, ...
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How can we calculate the tensor product of Lagrange basis polynomials?

Given data points $(x_i,y_i)$, the Lagrange basis polynomials are $$\mathcal l_j(x):=\sum_{i\ne j}\frac{x-x_i}{x_j-x_i}\;.$$ I'm reading a text targeting Smolyak's algorithm. In this text, they use ...
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2answers
55 views

function: bending the y=x line

My question has many relative questions but I didn't find anything exact to my needs. Let's take the function $f(x)=x$ with $x\in[0,100] $. I need to bend this and make it a curve. f will be a ...
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38 views

Proof that the Runge Phenomenon occurs

Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?
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34 views

Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
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64 views

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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22 views

Error estimate of polynomial quadratures missing some terms

Normally, for trapezoid rule and simpson's rule, etc, error analysis is done by using the error formula for interpolation. However, if the polynomial is restricted to some terms, for example, a ...
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1answer
41 views

Find function by 2 tangents and 2 points

I am looking for explicit function descriptions $F_1(s)$ and $F_2(s)$, following the line plotted. The line is just a description, but $F_1$ should never exceed $F_m$ and start at $s_0$ with a tangent ...
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1answer
21 views

How do I use interpolation with the Z table?

My textbook has an example of interpolation, but I am not sure how the book did it since it doesn't explain it. It says if we want $P(Z < 1.246)$ we must use interpolation and the steps given are: ...
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23 views

How does sinc interpolation work?

Every now and then I come across mention of sinc interpolation. Trying to read up on it, I have yet to get what it's about. I have done basic DSP work, have programmed stuff using FFT (using just a ...