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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Flipping X, Y Known Values with Result Values; Table Data and Linear Interpolation

I am not knowledgeable in the terminology I need to be searching for to accomplish what I need in Excel. I have the following table of values which gives me the resulting RPM if I know the Pressure ...
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1answer
90 views

Polynomials with specified ranges in intervals

Say I have two finite intervals $[a,b],[c,d]\subsetneq\Bbb R$ where $a<b<c-1<c<d$ and $b-a=d-c=s<1$. I want to find a polynomial $f \in \Bbb R[x]$ such that $$\forall x\in[a,b],\mbox{ ...
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1answer
21 views

Meaning of indices for cubic hermite splines

While digging through some code about Perlin noise, I noticed, that a Cubic Hermite Interpolation polynome is used at some point. At this point, I wanted to know, which of the Hermite basis ...
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1answer
128 views

Surface fitting to a mesh grid of data points

I wonder if there is a technique for fitting a surface to a given mesh grid of data points? I've seen interpolating a polynomial to $2$D data, but not $3$D. E.g. say I was given the matrix $$ ...
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2answers
66 views

How to smooth a list of angles.

I'm not a math guy so maybe there is a super simple thing that my eyes cannot see. And sorry if my math terminology is not good at all. Please address me the right math terminology to use because ...
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3answers
52 views

Polynomial interpolation

I need to find the polynomial of degree 3 with respect to these conditions: $ p(0) = 1 $ $ p(1) = -1 $ $ p'(0) = 1 $ $ p''(0) = 0 $ How do I deal with the condition on the second derivative?
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1answer
77 views

Interpolation of polynomials

let $f(x)=2^x$ and $x_0=1$, $x_1=2$, $x_2=3$. Use divided differences to compute the interpolation polynomial $P(x)$ satisfying $P(x_i)=f(x_i)$, i=0,1,2 and $P'(x_1)=f'(x_1)$ and estimate error ...
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41 views

Lagrangian interpolation

I have a high-school student doing a modeling project using interpolation. (Hopefully someone understands what I mean without having to write out explicit examples, as I only have screenshots.) When ...
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1answer
26 views

Functions for interpolation

Do we always need to be given points to do interpolation? Or can we be given only a function? For lagrangian interpolation we require points, and does it apply for others also?
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1answer
28 views

Orders of data in Divided Differences and Lagrangian Interpolation

As we know that the order of data points i.e. x values do not matter in Divided Differences and The Lagrangian Interpolation. Why is that? What happens if we arrange them in order? better ...
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42 views

Neville's Method for Approximation

What do the numbers mean in Neville's method? Neville's method performs interpolation of numbers of the previous values, but what do the numbers mean?
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1answer
27 views

A Theorem about Interpolation Method?

I have a question about interpolation. I think that question is a theorem, but I don´t find nothing about that. Anyone can help me? Show that, if $g$ is the polynomial of degree $m<n$ that ...
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2answers
413 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 \| ...
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2answers
93 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 ...
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2answers
28 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 ...
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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 ...
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1answer
118 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 ...
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1answer
65 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.
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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 ...
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1answer
39 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 ...
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1answer
41 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 ...
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0answers
220 views

How does 2D kriging interpolation work?

I have a grid of points Example ...
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1answer
31 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 ...
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1answer
151 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. ...
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2answers
71 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 ...
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1answer
30 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 ...
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2answers
33 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 ...
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1answer
82 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 ...
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1answer
54 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 $y$ .
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31 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 ...
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1answer
40 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 ...
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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 ...
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2answers
89 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 $\{-1, 0, 1, 2\}$ ...
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4answers
250 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 ...
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1answer
54 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 ...
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23 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 ...
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3answers
98 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 ...
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2answers
70 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 ...
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1answer
168 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 ...
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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 ...
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0answers
56 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 ...
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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, ...
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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 ...
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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 $(x,y)$ ...
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1answer
34 views

Existence of function with prescribed values?

Does there exist an infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ equal to $|x|$ when $x \in \mathbb{Z}$?
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37 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 ...
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91 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 ...
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45 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 $\ell_i(x)=\displaystyle\prod_{j=0;j\neq i}^n\frac{x-x_j}{x_i-x_j}$ ...
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1answer
97 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 ...
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4answers
266 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, ...