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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Roots of the Lagrange polynomials

This question follows my previous one Coefficients of Lagrange polynomials. Notations : $ n\in\mathbb{N}^*$ $[|1,n|]=\{1,2,\dots,n\}$ $A=(a_1,\dots,a_n)\in\mathbb{K}[X]^n$ all different numbers ...
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1answer
138 views

Weighted Interpolation over Triangle

Question: Is there a modification of simple component-wise barycentric-based interpolation of vertex values (such as colors) that accounts for arbitrary positive non-zero weights assigned to these ...
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2answers
67 views

A Polynomial that Passes through the following four points?

I'm trying to do this for practice but I'm just going nowhere with it, I'd love to see some work and answers on it. Thanks :) Find a polynomial that passes through the points (-2,-1), (-1,7), ...
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2answers
74 views

Interpolating polynomial given only Y values

Can we reconstruct a polynomial with only Y values? What if the number of Y values are far more than the degree of the polynomial? Also can we obtain the root of this polynomial with this Y's value ...
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“Interpolating between estimates”?

the headline reproduces the whole problem. What is meant by saying "Interpolating between the estimates (A) and (B), we finally obtain..."? For beeing mor specific I'll give the concrete estimates ...
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29 views

Calculation of midpoint.

in a paper I am reading they use as independent variable $\theta$. They later define $b(\theta) = \dot{\theta}^2$ and $a = \ddot{\theta}$ (So a is acceleration and b squared velocity. Later, they ...
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1answer
25 views

Interpolating Polynomial

I need help with this. Find a polynomial of degree 4 of the form f(x) = ax4 + bx3 + cx2 + dx + e Plot points (1, 7),(2, 2),(3, 9),(5, 1), and (7, 5). f(x)=? Thank you
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On Convex Interpolation and distances

Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := ...
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23 views

lagrange interpolation of a point

Let $f(x)=\sqrt[]{x}$ be our function. Let $P_n$ be the lagrange interpolation polynom of $f$ by $n$ points and $a$ be an element in the domain of $f$. Can rational points be chosen such that ...
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2answers
229 views

Find a smooth function with prescribed moments

In several contexts I’ve encountered variants of the following problem : let $m_0,m_1,m_2$ be real numbers such that $0 < m_1 < m_0$ and $\frac{m_1^2}{m_0} <m_2 < m_1$. Then, show that ...
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57 views

Fourier Interpolation

I have this Equation, that I modeled from my measurements and simulations. $I^{exp}_{l,m} = (\mathbf{F}^{H}.\mathbf{A}.I^{true})_{l,m}$; $H$ is the Hermitian transpose and $\mathbf{F}^{H}$ is a block ...
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1answer
105 views

There is a unique polynomial interpolating $f$ and its derivatives

I have questions on a similar topic here, here, and here, but this is a different question. It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives ...
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2answers
105 views

Constructing an increasing function with prescribed values at three points

This should probably be very simple, but I'm just not very skilled in math :S. I want a function that takes one variable, x, ranging from 0-1. As the input approaches 0 so should the output. As the ...
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52 views

How do you find the gradient between two different curves that passes through an arbitrary point between the curves?

Given a graph like this, XC, and ZC is it possible to find YC, and if so, how? fB(x) and fA(x) are some known but different functions at ZB and ZA, respectively. fC(x) is NOT a known function but ...
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1answer
45 views

Given a set of sequences, compute a corresponding set of functions

Consider the following set of sequences: $ S_k(n)= \begin{cases} 1 & \text{$n \equiv0\pmod{k}$}\\ 0 & \text{$n\not\equiv0\pmod{k}$}\\ \end{cases} $ I want to compute a set of ...
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30 views

How do I derive the analytical form of a discrete wavelet transform?

I guess this is more of an "applied maths" question than pure maths, and here's to hoping this is the right forum :) I am using a fast discrete wavelet transform (DWT) of a 1D vector of $2^N$ numbers ...
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1answer
47 views

What is the derivative of a Radial Basis Interpolation function?

A radial basis interpolation function is described as: $ f(\textbf{x})=\sum_{k=1}^N c_k \phi(\lVert \textbf{x}-\textbf{x}_k \rVert_2), \ \textbf{x}\in\mathbb{R}^s $ where $\textbf{x}_k$ are the $N$ ...
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1answer
24 views

Compound interest problem with increasing deposits

An Investor starts with an initial investment : $A$ He earns a steady profit of 10 percent per year. But every year he adds additional amount which increases by 15 percent every year. At the end of ...
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1answer
40 views

Interpolate daily values from monthly averages

I have a list of monthly production guarantees and I want to estimate daily values. Dividing monthly totals by days/month works, but when graphed, leads to a chunky piece-wise plot. I could use a ...
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1answer
29 views

reconstruction through sinc interpolation

I have a discrete-time signal $x_k = \sum_l a_l g(kT - l(T+\Delta T))$ where $g(t) = \frac {\sin(\pi t/(T+\Delta T))}{\pi t/(T+\Delta T)}$. Since the signal $x$ has been sampled at rate $1/T>2 ...
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2answers
276 views

Bilinear interpolation of angles

Is their a solution to do a bilinear interpolation in x,y of angles in [0°-360°[ ? The elementary formula of bilinear interpolation don't work on angles due to the discontinuity at 360°-0°. ...
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2answers
106 views

Modified Hermite interpolation

I asked similar questions here and here, but I tried to formulate this one in a sharper way. Is anyone aware of some literature on polynomial interpolation where we sample the function and its ...
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1answer
473 views

Plane fitting using svd

I am trying to get a best fit plane in a 3d space of points. I am using an svd as described in http://stackoverflow.com/questions/10900141/fast-plane-fitting-to-many-points. If I use the data provided ...
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1answer
35 views

Estimate the difference between $f$ and $p$ interpolating $f$

Suppose $p$ is the unique polynomial of degree $\leq 2$ that agrees with a function $f$ at points $a_1 < a_2 < a_3$. If the third derivative $f^{(3)}$ exists, and $x\in (a_1,a_3)$, then we can ...
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1answer
107 views

A polynomial agreeing with a function and its derivatives

If we want $$p(x_i)=a_i, \qquad x_1 < \dotsb < x_{n+1},$$ then there is a unique polynomial of degree $\leq n$ that accomplishes this (Lagrange interpolation). If we want $$p(x_i)=a_i, \qquad ...
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0answers
28 views

Is there any interactive spline fitting software?

I'd like to know if there's any software (freeware) for interactive data interpolation. What I want is to be able to visualize my data on an XY plot and drag the points to see how it affects the ...
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1answer
90 views

Natural cubic spline interpolation - check and suggest better way

I was given the following interpolation nodes: $(0,10),(\frac{1}{2},8),(1,5),(2,2),(3,1)$ and I was asked to find the natural cubic spline interpolation between every 2 points. I want to show you ...
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2answers
112 views

Why do we interpolate - no guarantee of success

this is somewhat of a general question about interpolation, I don't fully understand how can we be confident that our approximation is good, even if we know a lot of points. An example would be: ...
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1answer
27 views

Are there some scattered point configurations that would yield bad interpolation results using Radial Basis Function (RBF) interpolation?

Is Radial Basis Function interpolation sensible to the scattered point configuration? I seem to be having problems for scattered points $(x_i,y_i)$ that are illustrated below: The values $f(x,y)$ ...
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Taylor's remainder in a compact

My impression is that each function enough regular ($C^\infty$ ) in a compact is equivalent to a polynomial. Is this true? Is there a way to prove it? The expression of the Taylor's remainder just ...
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Convex interpolation between two points with given derivatives

Let's say I have two real values $x_1$ and $x_2$, to each of which I associate $y_i$ and $y'_i$ satisfying $$ (y_2-y_1)(y'_2 - y'_1) \geq 0. \tag{1} $$ I would like to find a polynomial ...
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Are there other names for multilayer perceptrons or multidimensional interpolants based on Kolmogorov's approximation work?

Are there other names for multilayer perceptrons that are used outside of the neural net community? At its core, multilayer perceptrons form a multidimensional interpolant of the form $$ ...
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1answer
22 views

Different forms of a quadrature

I am solving the following problem: Find the quadrature of the following form: $Q(f) = Af(−1) + Bf(0) + > Cf(1)$, which has the highest degree and interpolates the integral: $\int_{-3}^{3} ...
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1answer
112 views

Accurate floating-point linear interpolation

I want to perform a simple linear interpolation between $A$ and $B$ (which are binary floating-point values) using floating-point math with IEEE-754 rounding rules, as accurately as possible. Please ...
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1answer
25 views

Maximum of error in equidistant interpolation

Interpolating a function, to estimate the error, knowledge of the function $$\omega(x)=\prod_{i=0}^k (x-x_i)$$ with $x_i$ being the sampling points, is required. In the equidistant case, this would ...
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1answer
39 views

Finding polynomial optimal in terms of least squares approximation

Find polynomial $w$ of degree at most $2$ optimal in terms least squares approximation for a function $f(x)=x^3$ in the norm $\|g\|=\sqrt{(g,g)}$, given that: $$ (f,g) = \int\limits^1_0 f(x)g(x)dx. ...
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1answer
39 views

Collective term for interpolation and extrapolation

Is there a collective term for both interpolation and extrapolation? If there is such a term, what is it?
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47 views

How can missing data be organised or classified (Interpolation vs Approximation)?

I'm looking for a way to distinguish between the various types of missing data techniques? Can someone help to clarify or organize these categories in sub-sections or indicate similarities or ...
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1answer
48 views

How to find a graph's equation from its points

I have a set of data that constitutes the graph on the picture. What I want to know is how would I find the equation equivalent to that kind of graph? The X are on the interval $[1,10]$.
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Relationship between Lagrange interpolation and Taylor expansion

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h_{-1}$ and $x_{1} = x_0 + h_1$ with $h_1, h_{-1}$ > 0. Given a smooth function f, and an approximation to $f'(x_0)$ given by the ...
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29 views

Blended surface

Partially blended surfaces are extensively used in the literature for shape preserving interpolation. Most of these shape preserving partially blended surface interpolation is based on the result that ...
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2answers
246 views

General method to “naturally interpolate” to a complex map?

Given a region of the complex plane and a map $z \to f(z)$, is there a general way to "naturally interpolate" the point $z$ to $f(z)$ in such a way that the movement follows a "natural" smooth path ...
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1answer
79 views

Calculating a cubic spline goes wrong

I am trying to solve a old exam and really stuck at the cubic splines. We have the function $f(x) = \cos^2(\frac{x}{2})$ and the points $x_0 = \frac{\pi}{2}$, $x_1=0$ and $x_2 = \frac{\pi}{2}$. ...
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estimate an upper bound for the error of an interpolation polynom

The task is to estimate the error of an interpolation polynom $p(x)$ to an function $f(x)$. The sampling points are $x_0 = -1,\ x_1= 0,\ x_2=1,\ x_3=3$ So i already calculated the polynom which does ...
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1answer
64 views

Product of Chebyshev polynomials of the second kind?

So Wikipedia has this formula for a product of two Chebyshev polynomials of the second kind evaluated at a fixed $x$ with different indices: $$ U_n(x)U_m(x)=\sum_{k=o}^{n}U_{m-n+2k}(x) $$ Which would ...
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1answer
45 views

Formula to link two exponential values together - doesn't quite work

Basically, I've done a script, and I'm stuck on a formula for it. After I run the code on a cube, based on two different inputs (detail level and vertex average iterations), the resulting size will be ...
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24 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.
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3answers
151 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 ...
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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 ...
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31 views

Fast way to compute barycentric lagrange interpolation

Is there any fast way to compute the barycentric Lagrange interpolation using matlab?