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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7
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2answers
332 views

Hardy–Littlewood-Sobolev inequality without Marcinkiewicz interpolation?

Here is the statement of the Hardy–Littlewood–Sobolev theorem. Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$ \left \| ...
1
vote
2answers
72 views

Using several curves in 3D to create a surface

I have a set of several closed curves in 3d (like image below is showing my set of curves from 3 views). To clarify my idea, i ask my questions in two different ways showed by diction 1 and diction ...
0
votes
2answers
27 views

First Derivative The slope at the sample points

I am trying to implement an interpolation function in C# and one of the parameter is an array of 4 elements, which should contains first derivative of the slope at the sample 4 points. I am not a ...
1
vote
0answers
19 views

Estimate accuracy of inaccurate fast function having exact values of slow one

Let’s say we have functions $F$ and $H$ to calculate a series $S$ of integers and that: $S_{i} = H(x_{i}) = F(x_{i}) + e_{i}$ Being $e_{i}$ the error of $F(x_{i})$ to estimate $S_{i}$ The problem ...
0
votes
1answer
108 views

How to calculate the amount of time spent interpolating from one tempo value to another

I am writing a music creation program where the user is allowed to change the tempo throughout the track. If the user had a set tempo or only changed the tempo at discrete intervals I could easily ...
0
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1answer
58 views

Quadratic spline and quadratic interpolation

I am trying to understand what is the difference between quadratic spline and quadratic interpolation. Thank you for any help and advice.
1
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2answers
28 views

How to determine the smallest interpolation degree required?

Given a set of $n$ points $(x_k, y_k)\ (k\in\{1,...,n\})$, of course a polynomial of degree $n$ can fit all points. However, in some cases the coefficient of the higher degrees actually vanish and one ...
0
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1answer
37 views

Nearest-neighbor interpolation

I read in a book that the nearest-neighbor interpolation results in a function whose derivative is either zero or undefined. Can anyone explain what does it mean when the derivative of a function is ...
0
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1answer
40 views

Developing a function of two variables from given data

Cross listed with Mathematica SE: http://mathematica.stackexchange.com/questions/66086/developing-a-function-of-two-variables-from-given-data I have been stuck on the following problem. Consider a ...
3
votes
0answers
188 views

How does 2D kriging interpolation work?

I have a grid of points Example ...
0
votes
1answer
29 views

Interpolation in quadtrees/octrees

I'm looking for an interpolation algorithm for quadtrees and octrees that is derived from bi(tri)linear or bi(tri)cubic interpolation. I'm mostly interested in the case where: the interpolant is ...
0
votes
1answer
117 views

Application for interpolating periodic B-spline

I need to draw a cubic C^2 continous, closed (periodic boundary conditions) B-spline which should interpolate a set of control points. If possible it would be great if I could specify the knot vector. ...
0
votes
2answers
66 views

Second Degree Polynomial Interpolation, error related

We want to create a table of the exponential integral function $$E_{1}(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dt, x>0$$ over the interval $x \in [1,10]$ with stepsize $h$. How large can $h$ be if a ...
0
votes
1answer
29 views

Newton polynomial interpolation degree

8It is asked to find the polynomial of adequated degree to estimate $\sqrt{1.035}$. The following table is given: We know that 1.03 and 1.04 need to be used.Calculation the divided differences ...
0
votes
2answers
30 views

Finding an algorithm to mark a lens barrel

I have a zoom lens that only has a handful of focal lengths marked on the zoom ring. I want to make some intermediate marks, but I don't know the math required. I do have the approximate angles of the ...
1
vote
1answer
72 views

Interpolation between two points

I am looking for an interpolation between two points $P$ and $Q$. I need the curve to have derivative (direction) $\vec{v_1}$ at point P and $\vec{v_2}$ at point Q. In addition, there is a maximum ...
0
votes
1answer
53 views

Finding the value of y using Lagrange Formula

Let $p_2(x)$ be the interpolating polynomial for the data $(0 , 0) , (0.5 , y) , (1,3)$ from Lagrange formula. The coefficient of $x^2$ in $p_2(x)$ is $-2$ , Find the value of $y$ .
0
votes
0answers
26 views

Interpolating between many inputs and two outputs

We have a piece of computer software that we need to estimate the minimum requirements for. The requirements will be parametrized by certain usage factors, and expressed in terms of CPU and memory ...
2
votes
1answer
34 views

How to interpolate multidimensional functions?

I'm learning about interpolation and I wanted to ask if there's a "good" method to interpolate multidimensional functions (when the dimension can be even a few thousands)? Is there a theoretic limit ...
0
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1answer
39 views

Interpolation polynomial types

I was wondering if both the Maclaurin and Taylor series are two types of interpolation polynomials? I was under the impression that they were not because they only go though one point in an interval ...
0
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2answers
83 views

Cubic spline solving equation

$$S(x)=\begin{cases} x^3 +4x^2 -2x +7 & \text{ if } -1\leq x\leq 0, \\ x^3 - 2x^2 +4x +5& \text{ if } 1\leq x\leq 2, \end{cases}$$ is a cubic spline with knots $\{-1, 0, 1, 2\}$ ...
2
votes
3answers
181 views

Why do we choose cubic polynomials when we make a spline?

Good morning, I want to learn more about cubic splines but unfortunately my class goes pretty quickly and we really only get the high level overview of why they're important and why they work. To me ...
2
votes
1answer
53 views

Cubic polynomial interpolation

Let $f(x) = x^2\cdot (x-1)^2 \cdot (x-2)^2 \cdot (x-3)^2$. What is the piecewise cubic Hermite interpolant of $f$ on the grid $x_0 = 0$, $x_1 = 1$, $x_2 = 2$, $x_3 = 3$. Let $g(x) = ax^3 + bx^2 + cx ...
0
votes
0answers
21 views

Error of linear Interpolation with intermediate points obtained from an explicit RKM

For the initial value problem $y'(t)=f(t,y(t))$ $(f\in C^\infty(\mathbb{R^2}))$ with $t\in [a,b]$ and $y(a)=y_0$ let $u_k, k=0,...,n$ be the approximation of $y(t_k)$ obtained from an explicit ...
2
votes
3answers
93 views

Does a sequence of moments determine the function?

Related questions and answers: Find a smooth function with prescribed moments When do equations represent the same curve? Consider a real valued integrable function $f(x)$ at the interval $a \le x ...
2
votes
2answers
63 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
149 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
48 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
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0answers
49 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
36 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
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0answers
18 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
1answer
40 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 $(x,y)$ ...
1
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1answer
33 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
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0answers
33 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 ...
1
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0answers
88 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 ...
1
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0answers
42 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
87 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 ...
1
vote
4answers
231 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
53 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 $p$ ...
2
votes
1answer
115 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 ...
0
votes
2answers
58 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
30 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 ...
1
vote
1answer
182 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
56 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
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1answer
63 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
88 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
104 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 ...
1
vote
2answers
295 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 ...
0
votes
0answers
33 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 ...
2
votes
1answer
29 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 ...