Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.

learn more… | top users | synonyms

1
vote
1answer
30 views

How to find the intersection of a line and a plane with interpolation ( given two points in the opposite side of the plane)

I have two points in the opposite side of a plane (P1,P2) in 3D space, and i know their signed distances to the plane(D1,D2). how can i use interpolation to calculate the point that is the ...
2
votes
1answer
26 views

How to resample translated grid to ensure consistent interpolation?

I have a grid of values, sampled at certain locations. I'd like to translate grid by some offset and resample it. The questions is: what should be the resampling and interpolation formulas such that ...
0
votes
1answer
70 views

When fitting a polynomial to data points, how to determine the reasonable degree to use?

I have wondered the following: Suppose that there is a set of data points $(x_i,y_i)$. Then I would like to know if it is more reasonable to assume if there is a polynomial relation of degree $m$ ...
1
vote
1answer
81 views

How does one derive Runge Kutta methods from polynomial interpolation?

In some numerical analysis classes, a neat way of deriving the Adams-Bashforth and Adams-Moulton methods is to approximate the function by a polynomial, and integrate the polynomial analytically over ...
1
vote
1answer
107 views

Find the approximation for the interpolation of $f(x)$ by a polynomial of second degree

Assume that $f(x)$ has a minimum in the interval $x_{n-1}\leq x\leq x_{n+1}$ where $x_k=x_0+kh$, $k$ being an integer. Show that the interpolation of $f(x)$ by a polynomial of second degree yields the ...
0
votes
1answer
23 views

Should I use interpolation when finding median, and quartiles?

I am a S1 maths (Edexcel) AS student in the UK. My question: Say we have a stem-and-leaf diagram with 26 values. We want to find the lower quartile. To get the marks for our specification, we need to ...
2
votes
1answer
54 views

Why polynomial interpolation is considered as better than others?

Why polynomial interpolation is considered as better than others? In case of interpolation, the function $\phi(x)$ to approximate the unknown function $f(x)$ may be polynomial, exponential, ...
1
vote
3answers
65 views

Given the following data pairs, find the interpolating polynomial of degree 3 and estimate the value of y corresponding to x = 1.5.

So i was given this question Given the following data pairs, find the interpolating polynomial of degree 3 and estimate the value of y corresponding to x = 1.5. a) $(0, 1), (1, 2), (2, 5), (3, 10)$ ...
0
votes
2answers
35 views

Does there exist smooth or $C^2$ function for some infinite given points $a(n)$?

I know that there exist some smooth function (polynomial) for finite numbers of values. The question is if there exists function (not necessary unique) which is twice differentiable and $f(n)=a(n)$?
2
votes
0answers
71 views

Finding an error bound for Lagrange interpolation with evenly spaced nodes

I know that the error bound for Lagrange interpolation is usually $$\frac{M_{n+1}}{(n+1)!}\max_{x\in[a,b]}|(x-x_0)\cdots(x-x_n)|$$, where $M_i=\max_{x\in[a,b]}|f^{(i)}(x)|$. I'm trying to find the ...
1
vote
0answers
43 views

Can monotone cubic interpolation be implemented explicitly in B-spline form?

I have been interpolating cubic splines to some data, but it is now clear that I need my curves to be monotonic. Wikipedia and StackExchange sources describe how to impose the monotonicity condition ...
0
votes
0answers
30 views

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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
1answer
22 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 ...
0
votes
0answers
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 ...
0
votes
0answers
58 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 ...
0
votes
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, ...
0
votes
0answers
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 ...
0
votes
0answers
24 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 ...
1
vote
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 ...
2
votes
1answer
34 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 ...
0
votes
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(...
0
votes
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?
1
vote
2answers
103 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 ...
0
votes
0answers
20 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{...
0
votes
0answers
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 ...
0
votes
0answers
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 - ...
0
votes
0answers
236 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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 $n-1$, $\...
0
votes
0answers
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 ...
0
votes
0answers
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 $...
0
votes
1answer
91 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 $...
0
votes
0answers
35 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 $n-2$, ...
0
votes
0answers
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 \...
2
votes
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 $...
0
votes
0answers
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 ...
1
vote
1answer
88 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+...
0
votes
0answers
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 ...
0
votes
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 $\textbf{y}=(y_1,..,y_n)$. ...
0
votes
0answers
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 ...
1
vote
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 $...
7
votes
0answers
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}{\...
0
votes
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 $\{(...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
0answers
26 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 ...
1
vote
1answer
92 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 ...
0
votes
2answers
60 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 ...