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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Fast way to compute barycentric lagrange interpolation

Is there any fast way to compute the barycentric Lagrange interpolation using matlab?
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3answers
248 views

Lagrange Interpolation Theorem?

The polynomials $p(x) = 5x^3 - 27x^2 + 45x - 21$ and $q(x) = x^4 - 5x^3 + 8x^2 - 5x + 3$ both interpolate the points $(1,2) , (2,1) , (3,6), (4,47)$. Even though these polynomials are of different ...
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1answer
69 views

Is there a closed form solution for slope lines of bilinear function?

Given a bilinear function $f(x,y) = a + bx + cy + dxy$, is there a closed form solution for a slope line passing through point $(x_0, y_0, f(x_0, y_0))$? It can exclude degenerate cases, e.g. $b = c = ...
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1answer
130 views

How obtain a (accurate) function from this graph with these points?

I need obtain the function from 0 to 20 from this graph: I have the even numbers in the {x, f(x)} format: {0, 0}, {2, 1.8}, {4, 2}, {6, 4}, {8,4}, {10,6}, {12,4}, {14,3.6},{16,3.4}, {18,2.8}, ...
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1answer
59 views

Is this interpolation, does it have a name?

I was waching Signle Variable Calculus MIT lectures (I stop on 9 about linear approximation) I was also learn interpolation at my university and I thought that I'll create my own equation for ...
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1answer
456 views

How to calculate the length of a cubic hermite spline between two points

I am using the following equation to create a cubic hermite spline: $$p_n(t) = a_nt^3+b_nt^2+c_nt+d_n$$ $$1\geq t\geq 0$$ $p_n(t)$ is the unit interval interpolation equation for dimension n. $t$ is ...
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2answers
174 views

using lagrange multipliers to fit a curve through a point

So this is part math/ part statistics. I have a set of data I'm fitting a 2nd order curve through using least squares method (matrix form). However, I've been given the requirement to pass the curve ...
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1answer
43 views

Transforming $[-3, -2, -1, 0, 1, 2, 3]$ to $[0.125, 0.25, 0.5, 1, 2, 4, 8]$

Given the range of negative/positive numbers $[-3, -2, -1, 0, 1, 2, 3]$, is there a transformation that gives me $[0.125, 0.25, 0.5, 1, 2, 4, 8]$?
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137 views

How can I generate a spline with a maximum second derivative without specifying first derivative for mid points?

I've done interpolation before with bezier splines and cubic splines, but I need to find a way to limit the second derivative throughout the curve so that there is a limit to how sharp the corner can ...
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155 views

NURBS surface fitting for a closed region on mesh

I'm developing a tool that allows users to select a closed boundary (a polygon) on the triangle mesh and then from this boundary, generate a NURBS surface fitting the original mesh surface. My idea ...
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2answers
88 views

Approximate formula for the series: $\sum_{k=1}^{+\infty}\dfrac{k^x}{(k!)^x}$

I found that this series: $$S(x)=\sum_{k=1}^{+\infty}\dfrac{k^x}{(k!)^x}$$ can be very well approximated in this way: $$S(x)=\dfrac{1}{\left(a+b\exp(cx)\right)^d}$$ with: $a=0.1876$, $b=-0.1895$, ...
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1answer
35 views

The cubic interpolation

I try to understand the cubic interpolation for my studies. The following website says " (1) The four equations above can be rewritten to this (2):" but how? Can anyone explain me the the necessary ...
3
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1answer
54 views

Optimal way to find derivative - numerically

Suppose we are given points $x_0,x_1,x_2$ evenly spaced points $(x_0-x_1=x_1-x_2)$, and $u(x_1),u(x_2),u(x_3)$ Where $u$ is some function. Find the best way to approximate $u''(x)$ using only the ...
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1answer
45 views

Integration Rule Exact Degree

Given the integration rule $Q(x) = \alpha_1f(0)+\alpha_2f(1)+\alpha_3f'(0)$ for interpolating the integral $\int_0^1f(x) dx$ , I need to find $\alpha_1,\alpha_2,\alpha_3$ values s.t Q has exact degree ...
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1answer
35 views

Points interpolation for tracking

I have set of points for ex. $A_0 (0,0); A_1 (1,2); A_3 (3,3);$ I need an object to travel between these points during some period of time. I was able to construct this trajectory with Bezier curve ...
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2answers
303 views

How to perform a monotonic function fitting of data points?

I'm seeking suggestions for general purpose function fitting of a set of data points, where, based on physical intuition, the relationship is expected to be "monotonic", i.e. the function should be ...
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0answers
50 views

Interpolation of iterated logarithms

$$\text{Let }\log^2(x)=\log(\log(x)),\\ \text{ then }f(y,x)=\log^{\lfloor1+y\rfloor}\left(\log(x)/\log((1-x^{1/x}(y-\lfloor y\rfloor))+(y-\lfloor y\rfloor))\right)$$ gives an interpolation between ...
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1answer
113 views

Bezier curves, control points & reparameterization

Given a Bezier curve $\gamma$(t) defined by 3 control points P0 = (-1,4), P1 = (0, 0), P2 = (1, 0) such that the curve lies on the parabola $\ y = (x-1)^2 $. Reparameterize to $\alpha$(t) = ...
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2answers
65 views

Extending cubic splines interpolation into n input variables. Is it possible?

I have the equation for cubic spline interpolation, and I can see how it works for a data set in the 2d Cartesian coordinates. I was wondering if there is a general form to the equation that allows ...
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1answer
148 views

Non-linear ways to interpolate some data.

I'm trying to solve a problem where I have a large set of data points. Each data "point" has 8 independent variables (input) and 1 dependent variable (the output). I got this data through ...
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2answers
384 views

How to non-linearly interpolate between 4 values

I'm looking for a non-linear way of interpolating between 4 values within a games engine. I have a unit square abcd. It has a different value for each edge ...
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1answer
37 views

Can you interpolate a function $f: \mathbb{R} \rightarrow \mathbb{R}^2$ piecewise (by two interpolations)?

I am currently trying to improve on-line handwriting recognition. On-line means in this case that I have the information how the symbols are written as a list of $n$ tuples of coordinates $(x(t_i), ...
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1answer
3k views

Spatial Interpolation for Irregular Grid

How would I interpolate to a point P if I have four points around it such that: Q1 = (x1,y1), Q2 = (x2,y2), Q3 = (x3,y3), Q4 = (x4,y4) If the coordinates formed a regular 2D grid I would use a ...
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1answer
56 views

Why does the interpolation error go to zero if we increase the number of sampling points?

This question is motivated by polynomial interpolation. We know that for $f\in C^{n+1}[a,b]$ and $a=x_0<\dots<x_n=b$ holds $$\| f - p_n \|_\infty \leq \frac{1}{(n+1)!} \| f^{(n+1)} \|_\infty ...
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3answers
45 views

How to calculate the degree of Lagrange polynomial to satisfy a given error?

I need help. I have $f(x)=sin(x)$. If I want to use Lagrange polynomial to make an approximation of $f(x)$, what should be the degree of that polynomial if I work in the interval $[0,\pi]$, and the ...
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1answer
123 views

Lagrange interpolation: Evaluation of error in interpolation

I was given the following nodes: $x_1=0$ $x_2=1$ $x_3=8$ $x_4=27$ and was asked to interpolate the Lagrange polynomial of the function $f(x)=\sqrt[3]{x}$ (meaning, I have the values: ...
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28 views

What is the derivative of $\frac{f^{(3)}(\xi(x))}{6}$ at $x=x_0$

The error of interpolating polynomial is $$ E_n(x)=\frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x)) $$ The derivative of $E_n(x)$ is $$ ...
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1answer
39 views

Can any function on naturals be interpolated to a smooth function on reals?

Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be an arbitrary function from naturals to naturals. Is it always possible to find a function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that for any $n ...
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1answer
35 views

Interpolation based on $n$ uniformly distributed points

We are given $n+1$ uniformly distributed points in the segment $[0,1]$: $x_i=\frac{i}{n}$, $i=0,1,...,n$ and a function $f(x)=e^{-x}$ $P(x)$ is the interpolation polynomial of $f(x)$ where ...
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A problem on interpolation

Given $X(0) \in \Bbb R$ and a continuous and bounded real-valued function $g(.)$ from $\Bbb R \to \Bbb R$, for $\delta > 0$ define the sequence $\{X_n^\delta\}$ by $\{X_0^\delta\} = X(0)$ and $$ ...
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47 views

Interpolation using four nodes

Suppose there are four points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ my target is to interpolate any point $x_I$ between $x_2$ and $x_3$. Is there any Interpolation method which gives linear ...
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2answers
1k views

How to find Chebyshev nodes

I want to use Chebyshev interpolation. But I am a little confused for finding Chebyshev nodes. I use the following figure to illustrate my problem. Consider I have a vector of numbers I depicted as a ...
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2answers
146 views

trigonometric interpolation of a sampled signal

Given N sampled points, using the FFT we can get the Fourier transform of those N points $X_k$. With N/2 the Nyquist frequency and $X_0$ the DC value. Using the inverse we can then get back the ...
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1answer
119 views

Sum of Lagrange polynomial

I have to calculate $\sum_{i=0}^n x^k_i*l_i(x)$ for $k=0,1,2,...,n$ . I've proved that $\sum_{i=0}^n l_i(x) = 1$, but I cannot figure out how it may help me calculating $\sum_{i=0}^n ...
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1answer
46 views

Computing the $L^2$ norm of the coefficients from a trigonometric Lagrange interpolation

Let $V(x_1, \ldots, x_n)$ be the classical Vandermonde matrix $$ V(x_1, \ldots, x_n) = \begin{bmatrix} 1 & x_1 & \ldots & x_1^{n-1} \\ \vdots & & & \vdots \\ 1 & x_n & ...
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399 views

Looking for a simple algorithm to scale/resize a matrix, or an image.

I am looking for a simple algorithm to scale a matrix of any size. Given matrix A of dimensions [w1,h1]. Given a scaling factor (or resizing factor) SF, which is a real number (not necessarily ...
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1answer
73 views

properties of third order derivative

I have a question in my assignment (about interpolation) which has the condition that the third-order derivative is continuous at $x_2$ and $x_{N-1}$. That is, $S'''(x_2)=S'''(x_{N-2})$. The question ...
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1answer
57 views

Is the Lagrange polynomial integer-valued for points with consecutive integer x-values?

What I'm really wondering is, does Lagrange polynomial interpolation have an answer for every question of "what's the next integer in this sequence"? Does it define an infinite integer sequence to ...
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1answer
54 views

Numeric Analysis Interpolation of $f(x) , f'(x) $

There is a problem i'm finding quite difficult to solve, i'd be grateful if anyone could point me to the solution : We want to interpolate the function $f(x)$ and it's derivative $f'(x)$ s.t ...
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1answer
1k views

Find the linear-to-linear function whose graph passes through the given three points

Find the linear-to-linear function whose graph passes through the points $(1, 1)$, $(4, 2)$ and $(30, 3)$. So by using the $$f(x)=\frac{ax +b}{x+d}$$ I got my final answer to be ...
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1answer
84 views

Maximum of $w(x)=\prod\limits_{i=0}^8(x-x_i)$

What is the maximum of $w(x)=\prod\limits_{i=0}^8(x-x_i)$ on the interval $[-1,1]$, with $\bullet$ equidistant nodes $x_i$, $(x_0=-1,x_8=1).$ $\bullet$Chebyshev nodes, $\displaystyle ...
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3answers
140 views

Stuck with integral

Having this: $\int x\sqrt{1-x^2}dx$ Substitution: $t = 1-x^2$ $dt = -2xdx => dx=\frac{-2x}{dt}$ So: $$\int x\sqrt{1-x^2}dx = -\int x t^\frac{1}{2}\frac{2x}{dt} = -\int \frac{2x^2 ...
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1answer
83 views

Inverse distance weighting where neighbouring values are zero

I am trying to interpolate a set of rainfall data in order to find the rainfall at an unknown point. I have been using the inverse distance weighting interpolation method (details given here: ...
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1answer
109 views

Defined matrix in Catmull Spline Curve

I am trying to use Catmull spline curve in my program , I am trying to understand it but why we only use below given Matrix , because the examples I saw I only found the below one In Catmull spline ...
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2answers
104 views

Polynomial interpolation using derivatives at some points

Given $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y_5)$, we can interpolate a polynomial of degree 4 using Lagrange method. But, when we are given $(x_1, y_1), (x_2, y_2), (x_3, y_3), ...
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1answer
78 views

nonlinear interpolation question

Hi all, here is an screenshot from book called rotation transforms for computer graphics by John Vince. I think here is an error ,if it doesn't please explain how do m+n equals 1 Thank you
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1answer
291 views

Use the zeroes of T3 to construct an interpolating polynomial

Use the zeroes of T3 to construct an interpolating polynomial of degree two for the function x^3 on the interval [-1,1] Okay, so I have been looking at Finding the zeroes using Chebyshev polynomials ...
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like Gauss-Chebyshev integration formula using Lagrange polynomials

Suppose that $L_k(x)$ is Lagrange Interpolation Polynomial for points $x=1,0,-1$. How to show that: $$\int_{-1}^{1}\frac{f(x)}{\sqrt{1-x^2}}dx=\sum_{k=-1}^1C_kf(k)+E$$ where ...
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1answer
413 views

chebyshev nodes on a 2D grid

I want to use chebyshev nodes for interpolation using lagrange formula. My grid is two dimensional and i do not know how to determine the nodes of chebyshev in a 2D grid point?
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1answer
42 views

Writing a Hermite Interpolation script

Tasked with writing a MATLAB script that computes the Hermite interpolation of a function. Specifically, it asks to find: $$p(x)\in \Pi_{2n+1} \text{, such that } p(x_0)=f(x_0), p'(x_0)=f(x_0)... ...