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

7
votes
3answers
440 views

Is there a generalization of the Lagrange polynomial to 3D?

What is a way to construct a smooth polynomial surface ($\mathbb{R}^2 \rightarrow \mathbb{R}$) with Lagrange-polynomial properties in every partial derivative? I want to try this for image ...
6
votes
0answers
144 views

estimating a particular analytic function on a bounded sector.

Let $f(z)$ be an analytic function on $C^+=\{\Re z>0\}$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...
1
vote
1answer
78 views

Looking for help for building a Spline's algorithm 10th order

I'm trying to code the following algorithm in C++ and need help to understand the build of Splines from a mathematical point of view (found on page 129 on this paper). $$ f(t) = \boldsymbol{t} \cdot ...
1
vote
1answer
45 views

A big contradiction in interpolating point and number of it's

For calculating divided (fraction) difference table for interpolating $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, $n(n+1)/2$ difference fraction was used. I ...
1
vote
1answer
58 views

Multivariate polynomials at bounded evens

Univariate polynomials Given $n$, is there a degree $cn^{c'}$ polynomial $p(x)\in\Bbb R[x]$ and a degree $dn^{d'}$ polynomial $q(x)\in\Bbb R[x]$ with fixed $c,c',d,d'>0$ such that $$m\in\Bbb ...
4
votes
1answer
80 views

Why does the Bezier Curve work?

Recently I've been looking at Bezier curves and trying to understand how they work. I know that a general Bezier curve is given by the equation $$ \vec{\mathbf{B}}(t) = \sum_{k=0}^n{b_{k,\ ...
2
votes
0answers
53 views

Formula for $s_n = \sum_{i = 1}^n i^3$ Newton's Forward Difference Interpolation

Use Newton's Forward Difference formula to find an expression for $$ S_n = \sum_{i = 1}^{n} i^3$$ This is from an Introductory Numerical Analysis paper. I cannot figure out the connection ...
2
votes
1answer
48 views

Formal interpolation derivation of polynomial

I have a polynomial $$F(x_1,x_2,x_3,x_4)=k(x_1+x_2)(x_3+x_4)$$ The polynomial can be described by $$F(x_1,x_2,x_3,x_4)=0\iff (x_1+x_2)=0\mbox{ or }(x_3+x_4)=0$$ Is there a way to formally derive ...
7
votes
3answers
168 views

Transform polygons into one another?

I am aware that there must be no standard way to achieve this, but I don't know what has been done so far. I feel like I'm missing keywords to investigate further. I have any two 2D polygons $a$ and ...
0
votes
0answers
83 views

Is absolute continuity enough for uniform convergence of Chebyshev interpolation

Wikipedia says For every absolutely continuous function on [−1, 1] the sequence of interpolating polynomials constructed on Chebyshev nodes converges to f(x) uniformly. $^{[\text{citation ...
2
votes
3answers
223 views

How to create a computationally cheap function passing through given points?

I am trying to develop a function which goes through the follow points. The function will be calculated on a microprocessor which has 20 mHz. List of given points: ...
0
votes
0answers
89 views

Lagrange interpolating polynomial

How can one find $$ L[x_0,x_1,..,x_n;\frac{1}{x+a}]?$$ The original problem asks for $$ L[x_0,x_1,..,x_n;\frac{x^{n+1}}{x\pm 1}]$$ I know there is a formula for $[x_0,x_1,..,x_n,\frac{f(x)}{a-x}]$, ...
1
vote
1answer
190 views

Find cubic Bézier control points given four points

What I need is to generate an SVG file while having a series of (x,y) ready. P0(x0,y0) P1(x1,y1) P2(x2,y2) P3(x3,y3) P4(x4,y4) P5(x5,y5) ... I need to make a ...
0
votes
0answers
21 views

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 ...
5
votes
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{ ...
0
votes
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 ...
1
vote
1answer
108 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 $$ ...
2
votes
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 ...
1
vote
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?
0
votes
1answer
76 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 ...
3
votes
0answers
39 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 ...
1
vote
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?
0
votes
1answer
27 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 ...
0
votes
0answers
35 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?
1
vote
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 ...
7
votes
2answers
370 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
82 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
112 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
votes
1answer
63 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
vote
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
votes
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 ...
0
votes
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
208 views

How does 2D kriging interpolation work?

I have a grid of points Example ...
0
votes
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 ...
0
votes
1answer
123 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
70 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
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 ...
0
votes
2answers
32 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
78 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
29 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
38 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
votes
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
votes
2answers
87 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
214 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
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 ...
0
votes
0answers
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 ...
2
votes
3answers
97 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 ...