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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Eigen library: spline interpolation vs spline smoothing

The C++ library Eigen provides an "unsupported" splines module which is giving me troubles. The task is typical: a data-acquisition device provides me with a time series that is sort-of regular and ...
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1answer
33 views

a mapping function is requested

I have $n$ data points in $d$-dimensional space, i.e $x_1,x_2,...x_n$ where $x_i\in R^d$. I want an interpolation function $f:R^d-> R$ on these data points to map them to $y_1,y_2,...,y_n$, where ...
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1answer
403 views

Using Lagrange's Interpolation Formula to show that boolean functions over finite fields are polynomials

Let $F_2$ be the set of all the functions from the finite field $GF(2^n)$ of $2^n$ elements to $GF(2)$. I am reading a textbook that proves that the elements of $F_2$ can be represented by ...
2
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1answer
294 views

Interpolation inequality

Lef $u$ be at least a $C^2$ function on $\mathbb{R}^n$. Let's denote the gradient by $D$. Also, (using the multiindex notation), define the seminorm $$||D^ku|| = ...
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1answer
62 views

how to make 2d shape deformations?

I'm trying to make an animation transition between 2 shapes, example: transform a square into a circle... or transform a triangle into 2 squares pretty much like this: ...
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1answer
45 views

Unique solution on subspaces whose union is dense implies unique solution globally?

Let $V$ denote the space of all $f : [0,1] \to {\mathbb R}$ such that the second derivative $f''$ is continuous except on a finite set, equipped with the norm $N(f)=|f(0)|+|f’(0)|+||f''||_{\infty}$ ...
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1answer
655 views

Natural Cubic Spline S on [0,2]

A Natural Cubic Spline S on $[0,2]$ is defined by: S(x)= $$S_0(x)=1+2x-x^3 \to 0 \leq x < 1 $$ $$S_1(x)=2+b(x-1)+c(x-1)^2+d(x-1)^3 \to 1 \leq x \leq 2$$ Find b,c and d This question ...
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0answers
1k views

Finding error bounds for hermite interpolation

I am unsure how to find the error bounds for Hermite interpolation. I have some kind of idea but I have a feeling that I am going wrong somewhere. $f(x)=3xe^x-e^{2x}$ with my x-values being 1 and ...
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2answers
2k views

Why is Lagrange interpolation numerically unstable?

Here is my understanding of the polynomial interpolation problem: Interpolating by inverting the Vandermonde matrix is unstable because the Vandermonde matrix is ill-conditioned, so "difficult" to ...
3
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1answer
784 views

Akima spline interpolation

I want to use Akima interpolation on series of points. I have those points in 3D [x, y, z]. But in all resources, I found, there is only f(x) and x (so [x,y]). In Natrual Cubic Spline I am using this ...
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0answers
435 views

Polynomial Interpolation and Error Bound

Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the ...
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1answer
168 views

Linear Interpolation?

Just want to ask if this scenario is still possible in linear interpolation. If not, what interpolation would I need to get the missing value: X1 = 50 Y1 = 35 Z1 = 60 X2 = 70 Y2 = 40 Z3 = 70 X3 = ...
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0answers
35 views

Constructing a sequence [duplicate]

Possible Duplicate: Terms of a Sequence Construct a sequence of interpolating values $Y_n \text{ to }f(1 + \sqrt{10})$, where $f(x) = (1 + x^2)^{-1}$ for $-5 \leq x \leq 5$, as follows: ...
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2answers
1k views

What is a simple formula to interpolate 2 unknowns between 2 knowns?

What is a simple formula to find 2 intermediate values between 2 known values? f(1)=a, f(2)=?, f(3)=?, f(4)=b If there would be only 1 unknown, then it would be ...
3
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1answer
564 views

Determining partial derivatives and cross products for bicubic interpolation using function values only?

I'm trying to implement a bicubic interpolation algorithm. In order to calculate the interpolated values, I need to calculate sixteen coefficients used in the calculation process - and that's where ...
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1answer
81 views

How to find the equation from data?

I know things about linear/quadratic fittings etc. I'm just wondering, if i know a set of data for value e.g. z=[-2.563 -0.1932 -0.1502 -0.1102 -0.836 -0.5234] and l=[1 2 3 4 5 6] m=[6 5 4 3 2 1] I ...
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1answer
372 views

Interpolation error for the exponential function

I'm studiyng for my exam of scientific computing, specifically to the subject of interpolation techniques, I'm stuck with this problem: How many equally spaced nodes must be taken to interpolate the ...
3
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1answer
245 views

Fitting a surface to 2D measurements

I am looking for a way to fit a surface given a set of measured data $(x, y) \mapsto z$. A typical example would consist of anywhere between $10$ and $30$ measurements spread evenly over a disc. ...
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1answer
216 views

Physical meaning of spline interpolation

I remember that when I took my Numerical Analysis class, the professor said the spline interpolation take its name from a kind of wood sticks used to draw curved lines. Also Wikipedia say that the ...
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1answer
129 views

How to determine the function factors in order to fit the curve?

I am trying to calculate innovation and imitation factor by ovserving the usage of specific service among the population. After going through an overview of the paper I wrote the fuction in matlab: ...
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2answers
185 views

Generalization of $\frac{x^n - y^n}{x - y} = x^{n - 1} + yx^{n - 2} + \ldots + y^{n - 1}$

I thought about a generalization for the formula $$\frac{x^n - y^n}{x - y} = x^{n - 1} + yx^{n - 2} + \ldots + y^{n - 1}$$ It can be written as $$\frac{x^n - y^n}{x - y} = x^{n - 1} + yx^{n - 2} + ...
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3answers
406 views

How can I find out 2 unknowns in a cubic equation?

I need to give a bit of a background first, so please bare with me. I have a set of values that represent servo motor position values. By default I end up with a large set of values and I'd like to ...
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2answers
43 views

found a function from equations and inequality?

I'm Software engineer and I'm having little issue solving this problem let's called H. Well I'm looking for the mathematical expression of the function f(x) based on 3 equations and one inequality. ...
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1answer
536 views

Function to represent curve between two 2D points

I need a (simplest) function that interpolates values in range from predefined point $A$ to $B$ with rules: it must be smooth curve direction near $B$ must be the same as predefined $D$ vector ...
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1answer
319 views

Reconstruct Control points in a Bézier Curve?

I have a curve that I know is a (non-periodic) Cubic Bézier Curve (because I constructed it as such). I stored each ordered pair in the curve, but not the control points. Is it mathematically ...
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3answers
123 views

Which Interpolation should I use to create a curve?

I'm pretty weak in the field of mathematics, but a strong programmer. I am looking for a mathematical solution that, given two points on a line will give me a curve between them, including those two ...
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0answers
267 views

Runge function error second factor

I'm currently learning about the Runge function. On Wikipedia, I read the following: Consider the function: $ \dfrac{1}{1+25x^2}$ Runge found that if this function is interpolated at ...
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1answer
401 views

Finding an interpolating polynomial and natural cubic spline for a given accuracy

I'm trying to make an exercise but I don't know how to start. Is there somebody that can give me a hint so that I can start with the exercise. The exercise is: Consider the function $f(x) = \sin(x)$ ...
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1answer
403 views

Resolve a thin plate spline function

I am trying to keep the outline of an object in a video. So I have the coordinate of the outline of the object in the image $t$ and after computing the optical flow I have the coordinate in the image ...
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1answer
5k views

How to perform simple linear interpolation on a data set

With the following data set, what is the best way to interpolate the data for each time. ...
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1answer
223 views

How to write this in sigma notation?

Newton's formula for interpolation is $$P(x)=c_1+c_2(x-x_1)+c_3(x-x_1)(x-x_2)+c_4(x-x_1)(x-x_2)(x-x_3)+\cdots$$ I prefer sigma notation, when it is possible. Can this be written in sigma notation?
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1answer
200 views

Is there a name for these polynomials?

Given $t \in \mathbb{R}[0,1]$, consider the following set of polynomials: $$ \left[-{\left(t - 1\right)}^{2} t, {\left(t - 1\right)} {\left(t^{2} - t - 1\right)}, -{\left(t^{2} - t - ...
3
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0answers
302 views

linear interpolation error estimate for non-smooth function

Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be ...
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1answer
720 views

Numerically find the intersection point between two sets of data.

I'm looking for an efficient way to determine if two paths (sets of x,y coordinates) intersect at a point. Input - (x,y) from a Mercator Projection (longitude,latitude) coordinates Output - ...
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3answers
2k views

Deriving an equation that satisfies many points

Say I have a collection of points, for example the following: (1, 167), (2, 11), (3, 255), etc Is it possible to construct an equation that satisfies all of ...
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3answers
14k views

Given four points on a cubic function curve, how can I find the curve's function?

Say I have a curve $$y = ax^3 + bx^2 + cx + d.$$ I don't know $a$, $b$, $c$ or $d$, but I do know the $(x,y)$ values of four points on this curve. How can the values of $a$, $b$, $c$ and $d$ be ...
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1answer
39 views

Finding gradual values

I'm writing some code for a pressure level sensor for propane tanks. The manual provides me with the following table with the caption: "Best accuracy will be obtained using the calibration data in ...
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1answer
400 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 ...
3
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2answers
875 views

Need a formula for a quadratic spline

I'm trying to reproduce some results from a paper and I need an explicit formula for a specific quadratic spline to do so. The problem is, I've only got a plot of it. The quadratic spline is from ...
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1answer
525 views

Derivative of a function defined by the divided difference of another function.

Given a function $f$ of class $C$ $^{n+2}$ in an interval $[a,b]$ and $x_{0}=a<x_1<x_2 ... <x_n = b$ a subdivision of $[a,b]$ into $n+1$ points. Given another function $g$ defined in the ...
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3answers
129 views

Get equation for a curve which intersects x at seemingly randomly distributed points?

Is there any type of function that when graphed would show a curve which intersects the x axis multiple times, with each point being an arbitrary distance from the last? I mean, not like a trig ...
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1answer
158 views

Why is this a linear interpolation?

Let $J_{k,n}$ be the dyadic partition of $[0,1]$, i.e. $n\in \mathbb{N}_0,k=1,\dots,2^n$, $J_{k,n}:=((k-1)2^{-n},k2^{-n}]$ and we denote with $\phi_{n,k}$ the Schauder functions over $J_{k,n}$, i.e. ...
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1answer
153 views

Interpolation of vectors with quadratic polynomial

I have following points (-|b-a|,a), (0,b), (|c-b|,c) with a, b and c as two-dimensional vectors. These should be interpolated component-by-component with a second-degree polynomial p. My problem now ...
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1answer
225 views

Sagar an Payne stress-strain relationships and Boussinesq aproximation in Matlab

I have to do 2 problems in Matlab, and the Math course is not my favourite one. However, I have tried to resolve the first problem, based on another exercise, but I'm pretty sure it's wrong. Can you ...
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1answer
117 views

Fixing end derivatives up to the second order when interpolating points

I would like to interpolate a set of points in the real plane $(x_i,y_i), \ 1\leq i \leq n$ with specified end derivatives up to the second order. That is finding $f \in ...
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2answers
514 views

What mathematical function would do this: if $x = 0$ then $y = 0$ but if $x > 0$ then $y = 1$?

$x = 0$, $f(x) = 0$ $x = 1$, $f(x) = 1$ $x = 2$, $f(x) = 1$ $x = 3$, $f(x) = 1$ ... There have been so many times I could have used this at different programming problems but I always resorted to ...
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1answer
2k views

Need to understand question about not-a-knot spline

I am having some trouble understanding what the question below is asking. What does the given polynomial $P(x)$ have to do with deriving the not-a-knot spline interpolant for $S(x)$? Also, since ...
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3answers
740 views

Polynomial regression interpolation? [duplicate]

Possible Duplicate: Writing a function $f$ when $x$ and $f(x)$ are known I'm not versed in mathematics, so you'll have to speak slowly... If I want to fit a curve to the points, ...
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1answer
67 views

Proof that infinite functions can fit a table of numerical values

Suppose while conducting experiments, I measure a finite number of variables with some constants like temperature, etc. We get a table of finite number measurements (numerical values to some decimal ...
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1answer
189 views

Is it posible to interpolate convex hull in 2d space

I have $n$ points (in this example $11$) and I need to interpolate them in such a way that I have a function $f(t) \rightarrow (R, R)$ where $t \in [0; 2\pi]$. It can be parametric curve, but I need ...