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

0
votes
0answers
16 views

Changing the order of the elements of the divided difference Polynomial Interpolation

Apparently this is rather trivial but I don't understand why what I've highlighted in green is correct.
0
votes
3answers
58 views

what is difference between numerical integration and interpolation?

I am studying finite element method.While studying i am confuse with numerical integration and interpolation.Is this two methods are same or different?. If they are different then is there any ...
0
votes
2answers
125 views

Intepolate from linear to step function, and one application for shading colors

I'm running after a particular function $f_\sigma : [−1,+1] \rightarrow [-1,+1]$ that could take three different forms depending on the value of its parameter $\sigma$. Could anyone help me ...
0
votes
0answers
22 views

Fast way to compute barycentric lagrange interpolation

Is there any fast way to compute the barycentric Lagrange interpolation using matlab?
1
vote
3answers
127 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 ...
0
votes
1answer
40 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 = ...
0
votes
1answer
57 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}, ...
0
votes
0answers
42 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
2
votes
1answer
53 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 ...
0
votes
1answer
99 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 ...
0
votes
2answers
54 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 ...
0
votes
0answers
26 views

How to interpolate and obtain this graph?

I was interpolating with software mathematica v9, but how can I obtain a curve similar to this?, in a easy way
-2
votes
1answer
33 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 $[0.125, 0.25, 0.5, 1, 2, 4, 8]$?
0
votes
0answers
71 views

How to calculate a bezier curve given derivative of endpoints, location of endpoints, and points on the curve?

I know how to calculate a hermite spline, which has known derivatives and locations for each point, and I know how to calculate bezier curves which go through certain points, but I need to be able to ...
0
votes
0answers
48 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 ...
0
votes
0answers
79 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 ...
2
votes
2answers
80 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: $a=0.1876$, $b=-0.1895$, ...
0
votes
0answers
54 views

Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
0
votes
0answers
41 views

How to find the tangents for a spiral geometry at each point accurately

I have been working on a progam that generates spirals from contours that have been formed by slicing a surface by various planes along its height.The contours are a collection of linear line ...
0
votes
1answer
32 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
votes
1answer
33 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 ...
1
vote
1answer
41 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 ...
0
votes
1answer
33 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 ...
2
votes
1answer
55 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 ...
2
votes
0answers
39 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 ...
0
votes
1answer
61 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) = ...
1
vote
2answers
42 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 ...
0
votes
1answer
38 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 ...
4
votes
2answers
111 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 ...
2
votes
1answer
36 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), ...
0
votes
1answer
400 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 ...
0
votes
1answer
44 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 ...
0
votes
0answers
40 views

Spectral interpolation - Rotation equivalent to translation properties of Fourier transform?

I am using a spectral code for flow simulations. My aim is to obtain flow field data from points which do not coincide with the simulation grid without using inaccurate interpolation schemes in real ...
4
votes
3answers
36 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 ...
0
votes
1answer
58 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: ...
1
vote
0answers
24 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 $$ ...
0
votes
0answers
141 views

Using Lagrange polynomial to obtain the Second Derivative Midpoint formula

The Second Derivative Midpoint/Central Formula is $$ f^{\prime\prime}(x_0)=\frac{f(x_0-h)-2f(x_0)+f(x_0+h)}{h^2}-\frac{h^2}{12}f^{(4)}(\xi) $$ I tried to get this formula using Lagrange polynomial. ...
2
votes
1answer
36 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 ...
0
votes
1answer
25 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 ...
1
vote
0answers
26 views

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 $$ ...
0
votes
0answers
38 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 ...
0
votes
2answers
98 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 ...
0
votes
0answers
14 views

Fitting Error function?

I need to fit a function of the form $$y=A+B\space \text{erf}\left(\frac{\sqrt2(x-a_0)}{w}\right),$$ where $$ \text{erf}(x)=\frac{2}{\sqrt\pi}\int_0^x e^{-t^2} ...
1
vote
2answers
69 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 ...
0
votes
1answer
62 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 ...
2
votes
1answer
39 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 & ...
0
votes
0answers
85 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 ...
0
votes
1answer
50 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 ...
0
votes
0answers
19 views

Computational interpolation?

I need to interpolate this set of equations $$ c_1\left( \frac{c_1}{b_1}-R\ln(x_1)\right)^{-1}=c_2\left( \frac{c_2}{b_2}-R\ln(x_2)\right)^{-1}=c_3\left( ...
0
votes
0answers
36 views

Approximate function algorithm using a polynomial and Boor splines

I have a defined function and a set of points with equal distance between them. The problem is that I have to approximate the graphic of that function using a polynomial function of 3rd degree and a ...