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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Interpolation of Polynomial

Let $f(x) = x^3 + 2x^2 + x + 1$. Find the polynomial of degree $2$ that interpolates the values of $f$ at $x = -1,0,1$. I was able to do the an initial part of this problem (not written), but I ...
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1answer
27 views

Polynomial Interpolation - Bound on Error

Let the function $f(x) = \ln(x)$ be approximated by an interpoation polynomial of degree of 9 with 10 nodes uniformly distributed in the interval $[1,2]$. What bound can be placed on the error? I've ...
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1answer
41 views

re-arranging polynomial (newton interpolation)

How does: $$ f(x_2)= f(x_0)+\frac{f(x_1)-f(x_0)}{x_1-x_0}(x_2-x_0)+a_2(x_2-x_0)(x_2-x_1)\tag1 $$ rearrange to: $$ a_2 = \frac{\frac{f(x_2)-f(x_1)}{x_2-x_1}-\frac{f(x_1)-f(x_0)}{x_1-x_0}}{x_2-x_0} ...
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2answers
107 views

How to obtain Lagrange interpolation formula from Vandermonde's determinant

Assume that we have An interval $[a,b]$ A function $f(x)$ that is continuous on $[a,b]$ $n+1$ distinct points $a = x_0<x_1<x_2<\cdots<x_n = b$ And $f(x_0),f(x_1),\ldots,f(x_n)$ Now we ...
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0answers
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Newton Interpolation jump in notes

I am told, The general expression for a second order polynomial that passes through 3 points $(x_1,y_1),(x_2,y_2)$, and $(x_3,y_3)$ can be written as: $$ p_2(x)=b_0+b_1x+b_2x^2 \tag1$$ which I am ...
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1answer
158 views

2D cubic B-splines

I have been looking at B-splines to interpolate points. Having 1-D B-splines makes perfect sense to me, but haven't been able to find something that explains 2-D B-splines well for me nor provide me ...
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2answers
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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 ...
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0answers
41 views

How to differentiate Lagrange Basis Polynomial?

How to differentiate Lagrange Basis Polynomial ? I don't know, if the term is correct, but the question is: If $x_0,...,x_n\in\mathbb R$ are pairwise distinct ...
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1answer
21 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?
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31 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; find the generalized ...
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0answers
39 views

How to use derivatives at points with interpolation

I am given given $n$ points with $x$ and $y$ values. I am also given the derivatives at each of these points. How can I use the derivatives to calculate or to improve my interpolation? I've been ...
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0answers
14 views

Simple multi-variate interpolation algorithm improvements

I currently have some code to perform multi-variate interpolation across arbitrary data sets with an unlimited number of variables and a sparse grid of data. However, the current algorithm ...
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0answers
12 views

Interpolating on the borders of differently-resolved images

I'm creating a three-dimensional model of the earth based on SRTM height data. The data set is pretty huge, so only a small fraction of the data is available at any given time. The height data is ...
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0answers
14 views

How to process this data simultaneously?

I have a few maps obtained at different energies (500x600 pixels). Energy distribution is not very homogeneous . 1st dim is X, 2nd dim is Y, 3rd dim is energy (different levels for blue, green, yellow ...
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1answer
177 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 $n\in\mathbb{N}$. ...
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1answer
133 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} ...
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1answer
50 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 ...
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1answer
54 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?
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0answers
32 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 ...
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1answer
30 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 ...
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1answer
175 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)} ...
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1answer
93 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 ...
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1answer
96 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 ...
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1answer
51 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 ...
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0answers
52 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 ...
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1answer
89 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 ...
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1answer
41 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 ...
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0answers
27 views

Possibility of 3D interpolation without decouple axis

I am wondering if it is possible to do 3d spline interpolation without decoupling the axis. Such as creating a spline function on x then a different one on y and another on z. Then for any given ...
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52 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) ...
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1answer
142 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 ...
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0answers
84 views

How to calculate the condition number and infinity norm of interpolating polynomial

Suppose I have a set of n+1 points: $\lbrace (x_i,y_i)\rbrace_{i=0}^n$ and the following polynomials that interpolate the above mesh. 1)Barycentric with 1st and 2nd kind of Chebyshev points 2) ...
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1answer
30 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]
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0answers
119 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 ...
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2answers
163 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 ...
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0answers
12 views

Fluctuations for uniformed knot vector in Lagrange interpolation

I am writing a simple implementation for Lagrange interpolation for 15 uniformedly sampled points in the semi circle. When using a slightly modified knots as following, the curve fluctuates too much ...
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1answer
40 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 ...
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1answer
49 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 ...
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2answers
100 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 ...
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1answer
77 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 ...
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2answers
53 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: $<3r^2,-\sin(t)>$
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1answer
48 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 = ...
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1answer
73 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 ...
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0answers
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About a theorem by Faber in interpolation theory

I am looking for a proof of this theorem: For any table of nodes there is a continuous function $f$ on an interval $[a,b]$ for which the sequence of interpolating polynomials diverges on $[a,b]$. ...
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1answer
57 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 : ...
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2answers
61 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 ...
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1answer
46 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 ...
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1answer
26 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 ...
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1answer
86 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 ...
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2answers
162 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 ...
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1answer
198 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 ...