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
63 views

Is it possible to get function from bunch of points?

I have some points which I know the position in 3D. and that points made in sequence. for example, . (x1,y1,z1) ...
2
votes
1answer
570 views

How can I find a non-negative interpolation function?

In numerical mathematics I have learnt about some interpolation methods, however today I've come across some sort of interpolation problem which I don't know how to solve or even work with: Let ...
0
votes
1answer
69 views

Legendre polynomials verification

I'm confuse on how to answer this question: Verify that the first 4 Legendre polynomials are indeed mutually orthogonal on the interval [-1,1]
7
votes
0answers
181 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The ...
0
votes
2answers
4k views

Linear Spline Interpolation

Can someone explain to me how linear splines work and what formulas are used. I can only seem to find information on cubic splines. Which I don't really understand either Specifically, if I were ...
1
vote
1answer
65 views

I need to find the name of this interpolation method.

I have found a really interesting solution for interpolating cosh. First, the solver chooses the number of nodes, then, calculates Chebyshev polynomial roots on the desired interval, then finds the ...
3
votes
1answer
99 views

Embedding of Weak Lebesgue Spaces

My question is analogous to the embedding $L^p\subset L^q(\Omega)$, for $p>q$ and for a bounded $\Omega$. In weak $L^p$ spaces, that is, $L^{p,\infty}$, does such an inclusion hold for arbitrary ...
4
votes
2answers
802 views

Why is $L^{1} \cap L^{\infty}$ dense is in $L^{p}$?

It is mentionned that using the interpolation inequality $$\Vert f \Vert_{p} \leq \Vert f \Vert^{1/p}_{1} \Vert f \Vert_{\infty}^{1-1/p}$$ one can deduce that the space $L^{1} \cap L^{\infty}$ is ...
10
votes
1answer
92 views

For a fixed degree, is there always a Lagrange polynomial below the original function?

Let $x_1<x_2< \ldots <x_n$ be $n$ real numbers, and let $y_1,y_2,\ldots,y_n$ be real values to be interpolated. Let $r\leq n$. For any $I\subseteq \lbrace 1,2,\ldots,n\rbrace$ of cardinality ...
0
votes
2answers
109 views

How can I find the gradient of this function? $f(r,t)=r^3\cos(t).$

$$f(r,t)=r^3\cos(t).$$ Is it not like this: $<3r^2,-\sin(t)>$
0
votes
1answer
76 views

Show that Chebyshev nodes cannot be covered by an equispaced points

Given Chebyshev nodes on interval [a,b], show that we cannot find set of equispaced points ${y}_j$ in [a,b] st for all i there exist some j, ${y}_i={x}_j$ where Chebyshev nodes are defined ${x}_i = ...
0
votes
1answer
137 views

smooth orientation change with quaternions

My camera orientation is looking in the $v_1$ direction. Something happens on direction $v_2$ and I want the camera to move smoothly to look at that direction. So, to find the quaternion to go from ...
0
votes
1answer
152 views

Linear interpolation in 3 points

i know it can be a stupid question , but it put a big question mark ? on me . Do you know if the linear interpolating function can be calculated for 3 points : ...
0
votes
2answers
104 views

Creating a custom exponential function

I'm trying to come up with an exponential function that starts at zero, rises quickly (in $y$) between $x = 0$ and $x = 100$ and then slowly levels off as $x$ continues into infinity. Something that ...
0
votes
1answer
52 views

polynomial interpolation

I have a function, for example $f(x)=\frac{-x^2}{2}+|x|$, which is divided on $[-1,0)$ and $[0,1]$. How do we interpolate this function with a polynomial $p$ in the maximum degree 4 with ...
0
votes
1answer
41 views

Using the TPS to interpolate between points

I have implemented the Thin Plate Spline interpolation. I successfully plotted surfaces which go through all my 3D data points. What I want to do is calculate a few points which lie between my known ...
2
votes
1answer
208 views

Convex Function to Given (Three) Data Points

Assume that a function $h(x)$ is decreasing and convex given interval $[l,u]$. I'd like to get a function which connects three points, say $(a,h(a)), (b,h(b)), (c,h(c))$, where $l\leq a<b<c\leq ...
0
votes
2answers
1k views

cubic function of the two points (2,0) (4,0)

How can I find the cubic function of two points. I have the $y$-intersect $(0,2)$ and the $y=0$ intersect with the x-axis $(4,0)$. The equation should have the form $y=x^3+2$. But when I try to ...
0
votes
1answer
462 views

Interpolation of Gaussian function - minimize relative error

I have been trying to interpolate the function $e^{-x^2}$ on interval [-15,15] using standard methods like Lagrange or Newton interpolation for over a month. The goal is for it to be bound by ...
0
votes
1answer
149 views

Lagrange interpolation of a polynomial

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ has such property that for every distinct $x_0,x_1,...,x_n\in\mathbb{R}$ Lagrange interpolating polynomial for $f$ in these points has degree at most $n-1$. ...
0
votes
1answer
25 views

Fourier on discrete but not sequential data

I have time series data, which is discrete as it is timestamped with microsecond resolution. It is not sequential, as in not every microsecond has a value. How would I go about Fourier in such a case? ...
8
votes
3answers
21k views

Newton's Interpolation Formula: Difference between the forward and the backward formula

I was taught that the forward formula should be used when calculating the value of a point near $x_0$ and the backward one when calculating near $x_n$. However, the interpolation polynomial is unique, ...
1
vote
2answers
40 views

Subtlety about the definition of B-splines

I came across the following definition for the zero'th order B-spline $$b_0(x) = \left\{ \begin{array}{lr} 0 & |x|>1/2\\ 1 & |x|<1/2\\ 1/2& |x|=1/2. \end{array} ...
1
vote
1answer
166 views

The Puzzle of Locating Points in a Quadrilaterally-Faced Hexahedral Creature

The Disclaimer This is NOT homework... I designed the story. I thought a name MIT would be funny, while something along the line of TULSA or SU will still be decent. I know the algorithm to Q3 and 3D ...
2
votes
0answers
212 views

lagrange interpolation question here

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0, x_1=0.25 , x_2=0.5, x_3=1$ a)Find Lagrange polynomial for this function. c)Find the real approximation error. d)Find the limit of the ...
1
vote
0answers
149 views

Lagrange interpolating polynomials question?

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0$, $x_1=0.25$ , $x_2=0.5$, $x_3=1$ a)Find Lagrange polynomial for this function. So $L_3(x)=f_0(x) l_0(x)+f_1(x) l_1(x)+f_2(x) ...
3
votes
2answers
1k views

Is there a cubic spline interpolation with minimal curvature?

I came across the term "cubic spline with minimal curvature". However, I am not able to find any documentations/explaination on its computation method. Can anyone help me by advising how I can go ...
1
vote
2answers
133 views

Polynomial Interpolation

My professor gave the following question as a practice for study guide. Any assistance in terms of helping me to solve this would be much appreciated. Suppose that $f$ is continuous and has ...
1
vote
0answers
28 views

Differntiability properties of kriging interpolation

What are the differentiability properties of Kriging interpolation functions? Specifically, I am interested in using it to create a random realization of a 1D function using a regularly spaced grid of ...
1
vote
1answer
118 views

How to use any function interpolation method to create two functions …

I need help desperately on this. I have been working on it for a while. Use any function interpolation method studied in the course to create two functions $x(t)$ and $y(t)$ on $0 ≤ t ≤ 1$ so ...
1
vote
2answers
99 views

Polynomial Interpolation and Error

I have numerical analysis final coming up in a few weeks and I'm trying to tackle a practice exam. Assuming $p(x)$ interpolates the function $f(x)$, find the polynomial $p(x)$ that satisfies the ...
0
votes
1answer
2k views

Cubic Spline Interpolation practice

Going over practice problems for our final exam. I'm stuck on a problem involving cubic splines. In fact, I don't even know where to begin. I need to find the natural cubic spline $S(t)$ at $t_0=0, ...
0
votes
2answers
35 views

Find a formula that describes the given day-price relation

Given data: Days Prices 0 0 1 1000 2 1000 3 1000 4 1600 5 2200 6 2800 7 3400 8 4000 9 4600 10 5200 Price is function of day. I had my formula but always put negative ...
0
votes
1answer
119 views

Calculate average speed with unknown variable accelaration

I am in the middle of a vehicle tracking project where I have to calculate the distance traveled by the vehicle in a given amount of time. Data I am getting: ...
1
vote
1answer
155 views

inequality about linear and piecewise constant interpolation?

$\Omega\subset\mathbb{R}^3$ is a bounded, and $u(\mathbf{x},t) \in C\big(0,T,L^2(\Omega)\big)$. We divide the interval $[0,T]$ in $N$ equal subintervals with the time step $\tau$. With the notaion $$ ...
1
vote
2answers
1k views

Cubic spline interpolation - how to calculate second derivative

I ask this qeustion on stackexchange sites: stackoverflow, codereview, and signal processing and no one can help and they send me here :) So I implement cubic spilne interpolation in Java base on ...
0
votes
1answer
79 views

Use Lagrange interpolation to prove $\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$

Suppose $f\in C^2([a,b])$ and $f(a)=f(b)=0$,use Lagrange interpolation to prove $$\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$$ I tried to use the theoretic error to prove ...
1
vote
1answer
1k views

How to evaluate Newton's Divided Difference Polynomial in MatLab with an unknown degree?

I already have the code that finds the coefficients for the polynomial, but how do you find a value for the polynomial if given an x coordinate in MatLab code?
1
vote
2answers
352 views

How to calculate a spline for points in general position?

I want to find a curve passing through (or near) $n$ points in the plane. The catch is that the curve need not be a function. That is, a vertical line might pass through the curve in more than one ...
1
vote
1answer
55 views

Other way to write Lagrange's form (with derivative)

Prove that we can write polynomial $L_{n}\in\Pi_{n}$ which is interpolating function $f(x)$ in $n+1$ nodes $x_{0},\,\ldots,\, x_{n}$ in following form: ...
4
votes
2answers
10k views

How to calculate interpolating splines in 3D space?

I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a ...
0
votes
1answer
251 views

Degrees of interpolating polynomials

Given a collection of $m+1$ points $\{(x_0,y_0), (x_1,y_1), ..., (x_m,y_m)\}$, we can form the interpolating Lagrange polynomial $L(x)$: $$ L(x) = \sum_{i = 0}^{m} y_i l_i(x) \\ l_i(x) = \prod_{0 \le ...
0
votes
1answer
30 views

The universal function for the class of functions defined on a finite set is computable; (Lagrange interpolation polynomials).

Theorem: A computable universal function for the class of functions of $n$ variables exists that are defined on a finite subset of $\mathbb N^n$. Attempt at proof: Each such function is completely ...
0
votes
1answer
262 views

How to calculate a trigonometric interpolation polynomial

I have the following $2 \pi$-period function f: $$ f(x) = \left \{ \begin{array}{l l l} x: & 0 < x < 2 \pi \\ \pi: & x = 0 \end{array} ...
1
vote
1answer
519 views

Newton backward interpolation in Mathematica

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
1
vote
0answers
64 views

Differentiation by interpolation.

I am asked to show that the formula: $$ f'(x)\sim \sum_{i=0}^n A_i f(x_i) $$ which is derived from differentiating the interpolation polynomial is similar to that derived from checking/evaluating the ...
3
votes
1answer
103 views

DPLL Algorithm $ \rightarrow $ Resolution proof $ \rightarrow $ Craig Interpolation

I really need help here for an exam that I got tomorrow .. Let's say I got a bunch of constraints: $ c1 = { \lnot a \lor \lnot b } \\ c2 = { a \lor c } \\ c3 = { b \lor \lnot c } \\ c4 = { \lnot b ...
1
vote
0answers
37 views

$M(x)$ and $L(x)$ interpolate $f(x)$ on $n+1$ points. Show, that $f(x)$ lies between $L(x)$ and $M(x)$

We have $n+2$ points $x_0 \lt x_1 \lt x_2 ... \lt x_{n+1}$. We have two polynomials - $L$ and $M$. $L(x)$ interpolates $f(x)$ on points $x_0,...,x_n$ and $M(x)$ does so on $x_1,...,x_{n+1}$. The ...
0
votes
1answer
191 views

Intuitive proof of interpolation polynomial existence

Problem: Given a set of $n+1$ data points ($x_i, y_i$) where no two $x_i$ are the same, one is looking for a polynomial $p$ of degree at most $n$ with the property $p(x_i) = y_i$ for all $i∈ [0, n ...
1
vote
1answer
34 views

Show $\Delta^mp(x) = 0$ when $p(x) \in P_n$

Show that $\Delta^mp(x)= 0 $ when $p(x) \in \mathscr P_n$ and $m\ge n+1$, where $\mathscr P_n$ is the set of polynomial of degree $n$ and $\Delta^m$ is the operator for the $m$-th forward ...