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
1answer
25 views

Partition of function into pieces for interpolation needs

I've got some experimental data obtained from my mate's research. There are two sets of (x,y) points for each curve. He asked me to interpolate function values between this points, so for each curve I ...
0
votes
0answers
10 views

using approximation to find interpolantion function?

my question is can we actually find the interpolating polynomial if we solve the approximation problem for degree m = n-1 ( where we have n data ).(i know we usually solve the approximation problem ...
0
votes
0answers
14 views

Recovering a continuous density function from its discretized version.

The probability density function is defined on [0,10], and it's discretized by taking integration over short intervals [0,0.01],[0.01,0.02], etc. Is there any kind of interpolation method ( I'm not ...
1
vote
0answers
31 views

Do nth degree polynomials derived using Least Squares Interpolation always have n+1 intersections with the function?

I have recently studied Interpolation Techniques in my College Numerical Methods class and I have this question: If we have a function $f(x)$ and we are asked to use Least Squares Interpolation(LSI) ...
1
vote
0answers
35 views

Creating FEM mesh for image region — what is the most suitable shape function?

I wish to create a FEM mesh to solve an inverse elasticity problem, for an irregular domain. This domain is given by a medical image, so it is discretised and each square on the grid has one scalar ...
0
votes
1answer
47 views

deriving an integral quadrature rule on a triangle

I'm trying to look for references on this but I've not found any. I'm probably using the wrong keywords ... Let's suppose that our domain of integration $\Omega$ is the triangle in $R^2$ with a ...
0
votes
1answer
694 views

Lagrange Interpolating Polynomials - Error Bound

Let $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$. Construct interpolation polynomials of degree at most one and at most two to approximate $f(1.4)$, and find an error bound for the ...
0
votes
1answer
30 views

Best fit in logarithmic chart

I have several variances ($\sigma^2$) which value depends on the velocity ($v$). As you can see in the graph, if increase the velocity, the variance does the same. I am studying this dependency, but ...
0
votes
1answer
42 views

Newton form of the interpolation polynomial?

Let $ f(x)=\sin(x) $ and let $ p_1 $ be the first degree polynomial that interpolates $ f $ at $0$ and $\pi/2$. Then $ p_1(x)=(2/pi)x. $ How did they get this result; how is $ p_1(x) $ found?
1
vote
1answer
52 views

Prove that the difference between a continuous function and its interpolation polynomial..?

Let $ g \in C^2 [x,y] $ and $ P $ be its interpolation linear polynomial at $ a_0 $ and $ a_1 $ in $ [x,y] $. Prove that $ \lVert g-P\rVert _\infty < 1/8(a_1-a_0)^2B $ where $ \lvert ...
0
votes
1answer
87 views

Natural Cubic Spline Unequal Spacing

I'm currently trying to create a natural cubic spline by setting up linear equations to solve for the coefficients of the spline. Where the spline follows as : ...
1
vote
0answers
25 views

Sum of $p$th powers using polynomial interpolation

It is well known that the sum of the first $n$ $p$th-powers is polynomial in $n$ and is given by: $$ \sum_{k=1}^n k^p = \frac{1}{p+1} \sum_{j=0}^p (-1)^j {p+1 \choose j} B_j n^{p+1-j} $$ where $B_i$ ...
1
vote
0answers
20 views

Interpolation of the metric tensor

I am currently facing the following problem. I have a Riemannian manifold, where the metric is only known at certain points. Are there some standard strategy to interpolate the metric in other points ...
0
votes
0answers
26 views

What equation would be used to be draw a curve like this?

I have created a little visual based programming system and am not working on the visuals, if two nodes are connected and B.x<A.x then I want a curve to be drew in the fashion above.
0
votes
0answers
29 views

Convergence of quadrature formulas and interpolating polynomials

There is a theorem of Polya (1933), which says: 1) If a interpolatory quadrature formula converges for all continuous functions on [a, b] and quadrature weights are all positive, then the formula ...
2
votes
0answers
40 views

Regarding the Lebesgue constant for interpolation

I have a question regarding Lebesgue constant $\Lambda_{n}\left(\boldsymbol{\chi}\right)$, with which the worst case error between an interpolant $p\left(\boldsymbol{x}\right)$ and the function which ...
1
vote
0answers
41 views

Find nonlinear growth formula out of a set of values

I got two sets of values: $(x=240,y=20)$ and $(x=960,y=480)$. How can I find a formula to get any values in between? Anyone knows? Also if I distribute the value of $x$ among subvalues (imagine $x$ ...
0
votes
1answer
116 views

Newton's forward-difference formula question?

Use Newton's forward-difference formula to construct interpolating polynomials of degree two, and three for the following data. Approximate the specified value using each of the polynomials. I ...
1
vote
1answer
67 views

Interpolation with a constrained range between given control points

I am trying to create an algorithm that creates smooth color gradient functions, given control points in the red, green, and blue components. Mathematically, each curve would have a domain [0, 1] ...
0
votes
1answer
50 views

Lagrange polynomials question

Construct the Lagrange interpolating polynomials for the following functions, and find a bound for the absolute error on the interval $[x_0, x_n]$. $f(x) = e^{2x}\cos3x, ~~~~~~~x_0=0, ~~~~~~~x_1=0.3, ...
2
votes
1answer
120 views

Possible generalizations of Hadamard's three line lemma

Let $f$ be an analytic function on a sector $$ S=\left\{re^{i\theta}:0<r<\infty,\; 0<\theta<\gamma<\frac{\pi}{2}\right\} $$ with opening angle $\gamma$ at the origin. Suppose $f$ is ...
0
votes
0answers
23 views

Linear interpolation vs polynomial interpolation

Why linear interpolation is better than polynomial interpolation when we want to approximate $f(0.25)=e^{0.25}$? I can't formulate a concrete explanation. I thought that maybe it has a link with the ...
1
vote
0answers
50 views

Estimate the error of the Lagrange Interpolating polynomial

consider the function, f(x) = 1/(1+x^2) Estimate the error of the Lagrange Interpolating polynomial of 5 equally spaced points in the interval [-5,5]. Is there a proper way to find this logically ? ...
0
votes
1answer
312 views

Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
0
votes
1answer
68 views

Reverse spline interpolation

Say I have a number of sets $(x, y)$ for $x \in \{0, 1, \dots, 255\}$. I want to find the least number of points to reproduce the set with a certain accuracy using linear interpolation. What is the ...
0
votes
0answers
34 views

Finding the period of the function

My question is as follows:- Find the trigonometric interpolant $ \bar{f}(x)$ for $f(x)= \frac{\pi}{x+3\pi}$ and $n=1$. Thant is, find coefficients $c_{-1}, c_0 ,c_1$ such that ...
1
vote
1answer
73 views

General name for interpolation and extrapolation

I would like to know if there is a technical term to cover both interpolation and extrapolation. The reason why I am asking is that I am writing a computer program to do interpolation and ...
0
votes
1answer
37 views

Getting error of interpolating polynomial by subtraction.

$f(x)= \frac{1}{1+x^2}$ and when I computed the interpolating polynomial of 5 equally spaced points in [-5,5] I got $ p(x)= 0.0053x^4 -0.1711x^2 +1$ Now I need to estimate the error in the ...
0
votes
0answers
25 views

numerical errors near at the borders

I use some kind of partitioning on my data and then I do some interpolation and some other mathematical operations using chebyshev points. I have noticed that in the borders of each partition, It ...
0
votes
0answers
30 views

Help on Lagrange Error Calculation [duplicate]

Here is an example in Burden's Numerical Analysis book. My problem is in bold In example 2 we found the second Lagrange polynomial for $f(x)=1/x$ on $[2,4]$ using the nodes $x_0=2$, $x_1=2.75$, ...
1
vote
1answer
51 views

computing interpolating polynomials of 5 equally spaced points in a given interval

The question I faced is as follows:- Consider Runge's function. $f(x)= \frac{1}{1+x^2}$ Compute and graph the interpolating polynomials (atop a graph of Runge's function itself) of with 5 equally ...
1
vote
1answer
124 views

3D Interpolation for Irregular Grid

I have a cloud of data points in three dimensions (x,y,z) that carry a value of some sort (lets say temperature T). I was wondering what the best options were for interpolating the temperature to a ...
0
votes
0answers
62 views

Trouble doing simple polynomial interpolation [duplicate]

I need to do a polynomial interpolation of a set of $N$ experimental points; the functional form I have to use to interpolate is this: $$ f(x) = a + bx^2 + cx^4$$ as you can see the coefficient that ...
1
vote
1answer
48 views

Trouble doing polynomial interpolation

I need to do a polynomial interpolation of a set $N$ of experimental points; the functional form I have to use to interpolate is this: $$ f(x) = a + bx^2 + cx^4,$$ as you can see the coefficient that ...
7
votes
3answers
470 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
80 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
82 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
58 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
50 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
175 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
85 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
226 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
90 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 $[x_0,x_1,..,x_n,\frac{f(x)}{a-x}]$, ...
2
votes
1answer
224 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 ...