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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2
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2answers
69 views

Lagrange interpolation, syntax help

I am told, the basic interpolation problem can be formulated as: Given a set of nodes, $ \{x_i, i=0, ..., n\} $ and corresponding data values$\{y_i, i=0, ..., n\}$, find the polynomial $p(x)$ of ...
0
votes
1answer
25 views

how to show a data set satisfying a equation?

I have a set of data points like $(1,6)$, $(4,9)$, etc., and I am given a specific linear equation with two variable like $y = a/b + b x$. How can I show that the data points fit the curve?
0
votes
0answers
63 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve $c(u)$ of degree $4$ at the points $-1$, $0$, $1$ as: $c(-1) = 4$; $c'(-1) = 4$; $c(0) = 6$; $c(1) = -4$; $c'(1) = -6$; ...
4
votes
1answer
260 views

Integer valued polynomial through some known points

I have 2 questions, but I'll put both of them here since they are closely related: An integer valued polynomials $P(x)$ is a polynomial whose value $P(n)\in\mathbb{N}$ for every $n\in\mathbb{N}$. 1-...
2
votes
1answer
619 views

Hermite Polynomials Triple Product

Similar to the question Legendre Polynomials Triple Product, I would like to ask whether there are any explicit formulas for the inner product of the Hermite polynomial triple product \begin{align} \...
0
votes
1answer
143 views

Solving sub-triangles for Barycentric Interpolation (Triangle Geometry)

I'm trying to solve this triangle, so I can implement a barycentric interpolation, but I'm having trouble solving everything. I have all the base values for each of triangular sections and with a ...
0
votes
1answer
257 views

Reverse range of numbers, scaling

I have a float that goes from 1 to 0 .Im trying to make it so that the order is reversed and scaled so it goes from 0 to -80 Just wondering if there is a straight forward way to do this?
2
votes
0answers
36 views

Smallest set of Liner equations, which exactly fit a set of points

I have a set of 2-d points,(it can be of any arbitrary dimension n). I want to find the minimum set of straight lines(linear equations) which exactly passes through the given 2-d points (unlike ...
0
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1answer
53 views

interpolation inequality- how we use it to get into this inequality

I was working to get some inequality, and the author use the following inequality and call it "interpolation inequality" $$\|u\|_{L^2} \leq c\|u\|_{H^{-1}}^{\frac{1}{2}}\|\nabla u\|_{L^2}^{\frac{1}{2}...
6
votes
1answer
838 views

Interpolation inequality on Holder space

Let $0< \beta < \gamma <1$. Show that the interpolation inequality holds. $$||U||_{C^{0,\gamma}(U)} \le ||U||^{\frac{1-\gamma}{1-\beta}}_{C^{0,\beta}(U)} ||U||^{\frac{\gamma-\beta}{1-\beta}}...
2
votes
1answer
273 views

Bézier curves and optimization

I have a very peculiar problem. Assuming that you know how B-Splines or Bézier Curves work, you may also know that if we assume the result of the function, let's say tri-dimmensional, as a position in ...
3
votes
1answer
270 views

Sigmoid function with separate control of derivative at 0 and sharpness of bend.

For a physical relationship, $f(x)$, I'm trying to model, i have a fairly good determination of some of the boundaries, such that $ f(0) = 0$ $f'(0) = B$ $ \lim_{x\to\infty} f = A $ So far, what I ...
1
vote
1answer
90 views

Lagrange Interpolation definition doubt

Based on some exercise which explains Lagrange Interpolation itself, I got some doubts: It introduces function $$f(x)=\frac{1}{x}$$ and given points $x_0=2$, $x_1=2.75$ and $x_2=4$ so the following:...
2
votes
0answers
92 views

Lagrange's interpolation to solve for 0 of y(x)

I have the data composing of 7 elements x is from 0 → 3 incrementing by 0.5 y is from 1.8241 → -1.5427 I am supposed to use Lagrange's interpolation of three nearest neighbor data points. I am ...
1
vote
1answer
261 views

concept of trilinear interpolation.

I have a big confusing about trilinear interpolation. first, Interpolation's concept is estimating between two points(I'll call it as start point and end point), right? and this is depiction of ...
0
votes
1answer
245 views

Interpolating Z-Values when given complete and incomplete XYZ pairs

I am building an application that works with PolyLineZ (ESRI Shapefile) data and rewrites outlying Z values. The minimum and maximum Z-values are defined by the user through the interface Let's take ...
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
601 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 $(x_i,...
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
182 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 Euler-...
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
66 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
101 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
947 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
142 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
158 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 : $$(x_{0},y_{0}),(x_{1},y_{1}),(x_{2},...
0
votes
2answers
110 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 $p'(x_0)=f'(...
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
213 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
472 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
22k 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
171 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
217 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) l_2(x)+f_3(...
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
134 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
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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
120 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
100 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
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1answer
121 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: ...