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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3
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1answer
63 views

Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is ...
0
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1answer
16 views

Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
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1answer
16 views

Terminology: Spline interpolation

I have read two different definitions of Splines: A differentiable piecewise polynomial. A piecewise polynomial. If I build a piecewise polynomial using cubic polynomials, it's ...
0
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1answer
14 views

Need some help with applying specific boundary conditions to b-spline system of equations

I'm building a package for B-spline interpolation in Julia, and I've come across a boundary condition that I want to implement but can't wrap my head around how to do it (mathematically). Basically, ...
2
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4answers
46 views

Non-Piecewise interpolation over 3 points

I am trying to write an algorithm that interpolates between 3 values. The interpolation will be over the interval [0,1]. What I would like to do is: (Hopefully this makes sense) at x = 0, y = ...
0
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1answer
81 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 ...
0
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1answer
24 views

What is this graphed function with asymptotes at π/2 and -π/2?

I've come across this function with asymptotes at π/2 and -π/2, which crosses the axis at y=1. It doesn't seem to be polynomial or exponential—can anyone figure out what it is? The asymptotes and ...
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2answers
3k 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
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1answer
109 views

Lagrange interpolation: Evaluation of error in interpolation

I was given the following nodes: $x_1=0$ $x_2=1$ $x_3=8$ $x_4=27$ and was asked to interpolate the Lagrange polynomial of the function $f(x)=\sqrt[3]{x}$ (meaning, I have the values: ...
1
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1answer
19 views

Should an interpolation coincide the original function on the given data points?

Suppose having a model $f(x)=y$ where $f$ is unkown. Moreover, suppose you have some data points for this model i.e. $(x_1,y_1), (x_2,y_2), \dots , (x_n,y_n)$. If one can find an approximate of $f $ ...
0
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1answer
31 views

Fit $n$ Bézier paths to coordinates

I have a some coordinates $(X_i, Y_i)$ and I have to fit exactly $4$ cubic Bézier-paths to them (in other words, I have to find the 4 best fitting Bézier-paths, and by best fitting I mean that the ...
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0answers
23 views

Calculating B-Splines and dimension of spline space

I've got the following assignment: Let $S$ be the space of piecewise polynomials of degree $3$ on the intervall $[-1;1]$ with knots $x_i = -1+\frac{i}{2}, 0 \leq i \leq 4$. (a) Calculate a basis of ...
4
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1answer
80 views

Is there a name for this piecewise cubic interpolation kernel

I went looking for a way to do piecewise cubic interpolation, like natural cubic splines, but: expressible as a convolution of data points with a piecewise cubic kernel; and still C2-continuous ...
9
votes
1answer
349 views

Interpolating the primorial $p_{n}\#$

The primorial $p_{n}\#$ is given by the product $p_n\# = \prod_{k=1}^n p_k$ (where $p_{k}$ is the $k$th prime) -- is there a natural (a la the gamma function $\Gamma(z)$) way of interpolating it for ...
4
votes
1answer
38 views

Problem involving polynomial and arbitrary continuous function

Let $f\in C^4[0,1]$ and $p$ a polynomial of degree $3$. Suppose: $$f(0)=p(0),\quad f'(0)=p'(0),\quad f(1)=p(1),\quad f'(1)=p'(1)$$ Show that for each $x\in [0,1]$ there exists $\xi\in [0,1]$: ...
0
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1answer
33 views

Finding an Entire function with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$

I am really stuck on a homework problem, which boils down to the following: We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions ...
0
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1answer
508 views

Spline with varing tension, selection of tension factor

I need to perform a special interpolation, using that kind of basis : $$\varphi_{i,j}(x) = a_i + b_ix + c_i(\cosh(\tau\ x) - 1) + d_i(\sinh(\tau\ x) - \tau\ x)$$ where the $a_i$, $b_i$, $c_i$ and ...
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0answers
21 views

Polynomial Interpolation and Secret sharing

Hypothesis: We define all values and polynomials over the field $\mathbb{F}_p$ for a large prime $p$ (e.g. 128-bit). My question is related to the "Shamir secret sharing" scheme in computer ...
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1answer
463 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 + ...
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1answer
85 views

How is the B-Spline definition constructed?

I'm trying to understand how the B-Spline definition is constructed. That is, where did the knot vector and the basis functions and their recursive definition come from. The definition can be seen ...
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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 ...
4
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5answers
477 views

Why do we choose cubic polynomials when we make a spline?

Good morning, I want to learn more about cubic splines but unfortunately my class goes pretty quickly and we really only get the high level overview of why they're important and why they work. To me ...
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2answers
52 views

Find the natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes the values $S(-1)=13$, $S(0)=7$, and $S(1)=9$.

Find the natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes the values $S(-1)=13$, $S(0)=7$, and $S(1)=9$. I'm not sure how to go about this. Any solutions/hints are ...
0
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0answers
17 views

Root of interpolated polynomial when y-coordinates are permuted

Hypothesis: All values and polynomials are defined over a field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit) Suppose we have $n$ pairs of $(x_i,y_i)$. As we all know, given the ...
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1answer
32 views

Bernstein Interpolation 2D

I am well aware of the equations for 1D Bernstein Interpolation. But I do not understand how to extend it to 2D. I am guessing that the equations in the following image would do for 2D Bernstein ...
0
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1answer
25 views

Is interpolating well-sampled data (Nyquist-Shannon theorem) a cheat?

Suppose to sample a signal $s(t)$ with bandwidth $B$ with a sampling frequency $f_c$. Suppose also that the number of sample collected is $N$ (the duration of the signal acquisition is then $T = ...
0
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0answers
14 views

Continuous/interpolating alternative to order of magnitude?

Define $\operatorname{magnitude}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rfloor }$ and $\operatorname{magnitude'}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rceil }$ Currently I'm using this ...
0
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1answer
17 views

Find expression for $\sum_{k=0}^{n} l_k(0)x_k^{n+1}$

If the interpolation of $f(x)$ on the set of distinct points $x_0, x_1, \cdots x_n$ is given by $$\sum_{k=0}^{n} l_k(x)f(x_k).$$ Find an expression for $$\sum_{k=0}^{n} l_k(0)x_k^{n+1}.$$ I ...
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0answers
10 views

Divided difference acts on what space?

I'm currently trying to explain divided differences with a view to defining the Newton form of the interpolating polynomial. I'm using the definition: The $k$th divided difference of a function $f$ ...
0
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1answer
951 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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0answers
11 views

Monotonicity of 1D interpolation with signed errors

Suppose that we're given $\textbf{x} = (x_i,y_i)_{i = 1}^N \subset \mathbb{R}^2$ such that $x_i$ are distinct. There are a number of well known ways (see e.g. Implementation of Monotone Cubic ...
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1answer
43 views

Is there an equation for the exact line of best fit?

Is there some sort of equation/formula that can be used to find the exact values of $m$ and $b$ in $y=mx+b$ of any data points for the line of best fit? I want to be able to do this manually, not with ...
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2answers
41 views

How to find the polynomial given its factors? (A bit typical one here)

I recently saw this problem (and I should have paid more attention to my middle school maths classes). Find a 3 degree polynomial of $x$ which is $0$ when $x=1$ and $x=-2$, $4$ on $x = -1$ and $28$ ...
0
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2answers
37 views

Polynomial interpolation of $n+1$ points but ensure last coefficient is a certain number?

I have $n+1$ data points $(x,y)$, and I want to create an interpolating polynomial as described here https://en.wikipedia.org/wiki/Polynomial_interpolation. However there is a twist, I want to ...
1
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1answer
41 views

Number of points needed for linear interpolation of sine in $[0,\frac{\pi}{2}]$ with given error bound

I want to get a set of equispaced points in $[0,\pi/2]$ and use piecewise linear interpolation generated by those points to fit the sine function. And I want to determine how many points do I need to ...
4
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2answers
144 views

Interpolate 4 points by an increasing polynomial

I need to create a polynomial function that passes through the points $(0,0)$, $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ where $$ x_1 < x_2 < x_3\quad \text{ and } \quad y_1 < y_2 < y_3 $$ ...
4
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2answers
64 views

Interpolating $G(1)=\sum_{a=1}^{\infty} \frac{1}{a^{a}}$, $G(2) = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{(ab)^{ab}}$ on $\mathbb{C}$

Given that: $$ G(1) =\sum_{a=1}^{\infty} \frac{1}{a^{a}} $$ (this is just the Sophomore's dream series, but the rest are not) $$ G(2) = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{(ab)^{ab}} ...
1
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1answer
49 views

How does one derive Runge Kutta methods from polynomial interpolation?

In some numerical analysis classes, a neat way of deriving the Adams-Bashforth and Adams-Moulton methods is to approximate the function by a polynomial, and integrate the polynomial analytically over ...
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1answer
32 views

Finding Roots of a Polynomial Represented in Point-Value Form

Consider we have $n$ pairs of $(x_i,y_i)$. We all know that given the $n$ pairs we can interpolate a polynomial of degree at most $n-1$. Also, it is clear if we want to find roots of a interpolating ...
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2answers
251 views

Uniform convergence of Lagrange polynomials

There is a well-known theorem that states that on a closed interval $[a,b]$ any continuous function is the limit of a uniformly convergent sequence of polynomials. Proofs for this theorem usually ...
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1answer
994 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?
7
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1answer
160 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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1answer
26 views

Finding the equation of the line through X number of points?

I have a line graph with $8,000,000$ points. The X axis goes from $0$ to $7,999,999$ in increments of $1$ and the Y axis is either a $0$ or a $1$. There are no fractions on either axis. Is there an ...
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1answer
29 views

cubic spline interpolation - derivative known -

I at the moment trying to understand how to apply the interpolation method stated above. I have been given a start and end position, and for both position i know what their slope is. $\dot{X_a} = ...
3
votes
2answers
257 views

How to perform a monotonic function fitting of data points?

I'm seeking suggestions for general purpose function fitting of a set of data points, where, based on physical intuition, the relationship is expected to be "monotonic", i.e. the function should be ...
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1answer
55 views

Determine equation from graph

Background: I'm working on a script to read/parse a file generated by a piece of software I use to create music mixes. One aspect I'm having difficulty with is translating the volume value from it's ...
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1answer
29 views

Spline terminology

I am reading up on splines and as a beginner I have a basic question - Does it make sense to say - "I will fit a cubic b-spline to the data". As b-spline is just a representation of spline in terms ...
0
votes
2answers
1k views

How to find Chebyshev nodes

I want to use Chebyshev interpolation. But I am a little confused for finding Chebyshev nodes. I use the following figure to illustrate my problem. Consider I have a vector of numbers I depicted as a ...
3
votes
2answers
76 views

Smoothest function which passes through given points?

I am trying to interpolate/extrapolate on the basis of a known collection of (finitely many) points. I'm wondering if there is a way to formalize this intuitive notion: find a 'smoothest' function ...
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1answer
31 views

Using the polynomial of lowest order that interpolates $f(x)$ at $x_1$ and $x_2$, derive a numerical integration formula for $\int_{x_0}^{x_3}f(x)dx$.

Using the polynomial of lowest order that interpolates $f(x)$ at $x_1$ and $x_2$, derive a numerical integration formula for $\int_{x_0}^{x_3}f(x)dx$. I know that we aren't assuming uniform spacing. ...