Tagged Questions

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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2
votes
2answers
20 views

Lagrange interpolation for rational functions

Lagrange interpolation is very useful. I was wondering if there was an equivalent that is not using polynomials but rational functions, one polynomial divided by another. Look at this example: Say I ...
0
votes
1answer
31 views

Calculating coefficients of interpolating polynomial using Neville's algorithm

First of all, sorry for my bad math terminology as it's not my native language and I may misuse some terms in English. I've been tasked with writing an application which calculates the general ...
1
vote
1answer
39 views

I am not understanding this step

I am starting the chapter on differential equations and have this example to work through but I do not understand a few things Solve $dy=\frac{dy}{dx}=\frac{2x(y-1)}{x^2+1}$ solution: note that ...
1
vote
0answers
10 views

MATLAB implementation Spline Fitting

Check the attached problem please. I am a beginner in spline fitting and have a few questions: 1) How to find the coefficients c[n]. Is it by DTFT? 2) I understand how to find the derivative but ...
0
votes
0answers
23 views

In interpolation, why does my choice of $x_0…x_n$ matter?

This is more of a theoretical question regarding my choice of x's for my interpolation. I'm wondering if someone can explain to me why when I choose different x's for approximating a value at a point, ...
1
vote
0answers
14 views

Merging two univariate functions gracefully

Suppose I tell you that $$ f(0) = 0 $$ $$ f'(0) = 0 $$ and $$ f''(0) = a $$ for known $a>0$, whereas for large $x$ $$ f'(x) \approx \cosh^{-1}(x) $$ for $x>2$. Knowing nothing else ...
0
votes
0answers
13 views

Calculate (designate) area of ​​the largest area

The values ​​of a function of two variables z = f (x, y), where x, y, z are float. Calculate (designate) area of ​​the largest area of flat O. By 'flat area' O mean a sub area T wherein for each pair ...
0
votes
0answers
14 views

Calculate (designate) area of ​​the largest area

The values ​​of a function of two variables z = f (x, y), where x, y, z are float. Calculate (designate) area of ​​the largest area of flat O. By 'flat area' O mean a sub area T wherein for each ...
0
votes
0answers
29 views

What is it called when we interpolate a point INTO a grid…

Consider a uniform 2D grid, where each $(x,y)$ value on this grid has a corresponding value. So, if I want to find the value, $v$ (unknown) of a point that exists at some arbitrary co-ordinate $(x,y)$ ...
0
votes
0answers
14 views

What does the Stein–Weiss Interpolation Theorem say?

I was looking for the statement for the Stein–Weiss Interpolation Theorem, but I cant find it anywhere on internet.
1
vote
1answer
21 views

Existence of function with prescribed values?

Does there exist an infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ equal to $|x|$ when $x \in \mathbb{Z}$?
1
vote
0answers
16 views

Rational function interpolation?

We know that $n+1$ points is enough to completely determine a polynomial of degree $n$. Are there any techniques which says that a certain number of points is enough to completely determine a rational ...
0
votes
0answers
9 views

Polynomial Interpolation Existence and Uniqueness

The question I am attempting to solve is as follows: Let $f$ be a polynomial of degree $\le n$ and let $p_n$ be a polynomial interpolant to $f$, at the $n+1$ distinct nodes $x_0,x_1,...,x_n$. PROVE ...
1
vote
0answers
16 views

Change of basis from Chebyshev to monomial basis for polynomials

I'm not that familiar with Chebyshev polynomials, so I hope I'm not too far off. Suppose that I have three order pairs $(x_0, f(x_0))$, $(x_1, f(x_1))$, and $(x_2, f(x_2))$ where $f : \mathbb{R} \to ...
0
votes
0answers
6 views

Bounding the Lebesgue constant.

This is a homework question, so I would prefer hints/suggestions as opposed to full-out solutions. Given the Lagrange polynomials $\ell_i(x)=\displaystyle\prod_{j=0;j\neq i}^n\frac{x-x_j}{x_i-x_j}$ ...
0
votes
1answer
34 views

Determine error in Neville's Algorithm calculation

I've been mulling over this problem for a while and I don't even know how to start it. The book is hopelessly vague. The problem states Neville's Algorithm is used to approximate $f(0)$ using ...
0
votes
0answers
16 views

Lagrange interpolation using chebyshev nodes, Am I doing it right?

I am going to use the Lagrange interpolation. For that I want to use the chebyshev grid. I want to make sure I am doing it right. First, I have to make the chebyshev nodes, then, I transform them into ...
1
vote
4answers
89 views

How to find the 4th degree polynomial with given values at $0,1,2,3,4$?

Determine a fourth degree polynomial p that has $p(0), p(1), p(2), p(3), p(4)$ equal to $7, 1, 3, 1, 7$, respectively. Using my ideas, I first write out the points on the polynomial as $(0,7), (1, ...
0
votes
1answer
38 views

Polynomial approximation

Say that you have $n+1$ points on the interval $[a,b]$, let's call them $\{x_0,\dots,x_n\}$. Take any two different $y_1, y_2$, points on $[a,b]$. My goal is to show that there exists a polynomial $p$ ...
5
votes
1answer
100 views

Difference table for interpolation

For calculating divided (fraction) difference table for interpolating the points $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, $n(n-1)/2$ fraction was used. I ...
-1
votes
2answers
48 views

What is rule of this function?

I have these values.these are inputs and outputs of a function.I want to find rule of function.input is N. ...
2
votes
1answer
20 views

Find the largest value for $x_1$ in (0,1) such that $f(0.5)-P_2(0.5) = -0.25$ (interpolation)

I'm not really sure where to go with this problem and I'm hoping you can help. The problem states: Let $f(x) = \sqrt{x - x^2}$ and $P_2(x)$ be the interpolation polynomial on $x_0 = 0, x_1$, and ...
0
votes
1answer
28 views

Lagrange interpolation: Getting a bound and finding the error

I am struggling to understand this: The problem asks me to find the lagrange error of the polynomial approximation given the nodes $x_0 = 1, x_1 = 1.25, x_2 = 1.6$ with $x = 1.4$ The function I am ...
1
vote
0answers
52 views

Can you interpolate my polynomials if I give you some randomized values

Scenario (1) We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$, where $p$ is a prime number. Let $P_i$ be a polynomial such that $P_i \in R[x]$. We ...
1
vote
1answer
44 views

Very confused with interpolating polynomials

I have a problem from my homework that I completely botched, and no matter what I do I end up with the wrong answer. Here's the problem: For a given function $f(x)$ let $x_0 = 0, x_1=0.6, x_2 = ...
3
votes
0answers
75 views

Random multipliers of polynomial values at known points in $\mathbb{Z}_p$

Scenario (1) We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$, where $p$ is a prime number. Let $P_i$ be a polynomial such that $P_i \in R[x]$. We ...
0
votes
1answer
31 views

How to interpolate between sets of data

I'm probably using the wrong terminology, making it difficult to find a starting point. I have a set of motor data that looks like this: I can easily create a trend line for a given flow rate ...
8
votes
2answers
268 views

Interpolation polynomial Challenge

suppose $p(x)=x^k-x^t, k \neq t $ (k,t is a positive integer). function q(x) be a Interpolation polynomial from degree lower or equal n, to data $i=1,...,n+1, (x_i ,p(x_i))$. if ----------- then ...
-1
votes
0answers
17 views

Newton's interpolation formula and divided differences

I am having trouble verifying Corollary 3 depicted here where the divided differences are defined here. First off, I think the $1/(n!)$ should not be there. After many applications of the mean ...
0
votes
0answers
25 views

Interpolating array of integrals

I have an array of integrals, that I want to interpolate and differentiate to get result array. What kind of interpolation should I use to get a reasonably smooth output? (e.g. the output is ...
-1
votes
0answers
23 views

How to do a linear approximation with several parts

if i have partial data set of $ \langle x,y\rangle\in \mathbb{R}$ for a given function $f$, and i want to approximate it by $n$ partial linear functions how would i calculate those linear functions, ...
2
votes
1answer
22 views

Increasing Function or Polynomial with Prescribed Values

Consider $n$ points $(a_1,b_1), (a_2,b_2),\cdots, (a_n,b_n)$ in Euclidean plane with $a_1<a_2<\cdots < a_n$ and $b_1<b_2<\cdots < b_n$. It is easy to construct a polynomial of degree ...
0
votes
0answers
17 views

Roots of the Lagrange polynomials

This question follows my previous one Coefficients of Lagrange polynomials. Notations : $ n\in\mathbb{N}^*$ $[|1,n|]=\{1,2,\dots,n\}$ $A=(a_1,\dots,a_n)\in\mathbb{K}[X]^n$ all different numbers ...
3
votes
1answer
48 views

Weighted Interpolation over Triangle

Question: Is there a modification of simple component-wise barycentric-based interpolation of vertex values (such as colors) that accounts for arbitrary positive non-zero weights assigned to these ...
0
votes
2answers
42 views

A Polynomial that Passes through the following four points?

I'm trying to do this for practice but I'm just going nowhere with it, I'd love to see some work and answers on it. Thanks :) Find a polynomial that passes through the points (-2,-1), (-1,7), ...
0
votes
2answers
41 views

Interpolating polynomial given only Y values

Can we reconstruct a polynomial with only Y values? What if the number of Y values are far more than the degree of the polynomial? Also can we obtain the root of this polynomial with this Y's value ...
1
vote
0answers
21 views

“Interpolating between estimates”?

the headline reproduces the whole problem. What is meant by saying "Interpolating between the estimates (A) and (B), we finally obtain..."? For beeing mor specific I'll give the concrete estimates ...
0
votes
0answers
21 views

Calculation of midpoint.

in a paper I am reading they use as independent variable $\theta$. They later define $b(\theta) = \dot{\theta}^2$ and $a = \ddot{\theta}$ (So a is acceleration and b squared velocity. Later, they ...
0
votes
1answer
16 views

Interpolating Polynomial

I need help with this. Find a polynomial of degree 4 of the form f(x) = ax4 + bx3 + cx2 + dx + e Plot points (1, 7),(2, 2),(3, 9),(5, 1), and (7, 5). f(x)=? Thank you
0
votes
0answers
24 views

Newton Divided Difference Table , if $f''(x)$ is given

Here is the problem: Suppose $f(0)=1$, $f(1)=f'(1)=f"(1)=0$, and $f(3)=16$. Compute the Hermite interpolation polynomial $P$. How would the divided difference table look for this, in order to ...
1
vote
0answers
17 views

On Convex Interpolation and distances

Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := ...
0
votes
0answers
18 views

lagrange interpolation of a point

Let $f(x)=\sqrt[]{x}$ be our function. Let $P_n$ be the lagrange interpolation polynom of $f$ by $n$ points and $a$ be an element in the domain of $f$. Can rational points be chosen such that ...
6
votes
2answers
205 views

Find a smooth function with prescribed moments

In several contexts I’ve encountered variants of the following problem : let $m_0,m_1,m_2$ be real numbers such that $0 < m_1 < m_0$ and $\frac{m_1^2}{m_0} <m_2 < m_1$. Then, show that ...
1
vote
0answers
34 views

Fourier Interpolation

I have this Equation, that I modeled from my measurements and simulations. $I^{exp}_{l,m} = (\mathbf{F}^{H}.\mathbf{A}.I^{true})_{l,m}$; $H$ is the Hermitian transpose and $\mathbf{F}^{H}$ is a block ...
1
vote
1answer
64 views

There is a unique polynomial interpolating $f$ and its derivatives

I have questions on a similar topic here, here, and here, but this is a different question. It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives ...
2
votes
2answers
100 views

Constructing an increasing function with prescribed values at three points

This should probably be very simple, but I'm just not very skilled in math :S. I want a function that takes one variable, x, ranging from 0-1. As the input approaches 0 so should the output. As the ...
0
votes
0answers
21 views

How do you find the gradient between two different curves that passes through an arbitrary point between the curves?

Given a graph like this, XC, and ZC is it possible to find YC, and if so, how? fB(x) and fA(x) are some known but different functions at ZB and ZA, respectively. fC(x) is NOT a known function but ...
2
votes
1answer
39 views

Given a set of sequences, compute a corresponding set of functions

Consider the following set of sequences: $ S_k(n)= \begin{cases} 1 & \text{$n \equiv0\pmod{k}$}\\ 0 & \text{$n\not\equiv0\pmod{k}$}\\ \end{cases} $ I want to compute a set of ...
0
votes
0answers
17 views

How do I derive the analytical form of a discrete wavelet transform?

I guess this is more of an "applied maths" question than pure maths, and here's to hoping this is the right forum :) I am using a fast discrete wavelet transform (DWT) of a 1D vector of 2^N numbers ...
0
votes
1answer
14 views

What is the derivative of a Radial Basis Interpolation function?

A radial basis interpolation function is described as: $ f(\textbf{x})=\sum_{k=1}^N c_k \phi(\lVert \textbf{x}-\textbf{x}_k \rVert_2), \ \textbf{x}\in\mathbb{R}^s $ where $\textbf{x}_k$ are the $N$ ...