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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12
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2answers
181 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} + ...
2
votes
3answers
329 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 ...
1
vote
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. ...
0
votes
1answer
451 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 ...
0
votes
1answer
286 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 ...
0
votes
3answers
111 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 ...
2
votes
0answers
199 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 ...
1
vote
1answer
346 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 $f(x) = \sin(x)$ ...
0
votes
1answer
367 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 ...
0
votes
1answer
4k 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. ...
1
vote
1answer
186 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?
6
votes
1answer
196 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 - ...
2
votes
0answers
263 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 ...
0
votes
1answer
598 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 - ...
3
votes
3answers
1k 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 ...
2
votes
3answers
10k 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 ...
1
vote
1answer
36 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 ...
0
votes
1answer
292 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
votes
2answers
746 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 ...
0
votes
1answer
364 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 ...
1
vote
1answer
149 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. ...
1
vote
1answer
151 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 ...
0
votes
1answer
202 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 ...
0
votes
1answer
104 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 ...
1
vote
2answers
431 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 ...
0
votes
1answer
1k 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 ...
1
vote
3answers
656 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, ...
1
vote
1answer
63 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 ...
0
votes
1answer
165 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 ...
1
vote
1answer
79 views

Lagrange form and differences

For a function f and distinct points $\alpha$, $\beta$, $\gamma$; what is meant by $f[\alpha,\beta,\gamma]$? Find the Lagrange form for the polynomial $P(x)$ that interpolates $f(x) = ...
1
vote
0answers
123 views

Extending Hermite polynomial interpolation

Working with the definition of Hermite polynomials $x_0,\ldots,x_n$ are distinct in $[a, b]$, $f''(x)$ is continuous on [a, b], then $$H_{2n+1}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} ...
1
vote
1answer
482 views

Lagrange Coefficients in Maple

I'm trying to compute Lagrange coefficients in Maple. Having found the $n$ roots of a Lagrange polynomial, I want to calculate the $j$-th coefficient: $$L_j(x) = \prod_{{i=0}\atop{j \neq ...
1
vote
1answer
493 views

Piece-wise linear interpolating polynomials

Somebody please help me to obtain piece-wise interpolating polynomials for the function $f(x)$ defined by the below data: $x=1$, $f(x)=3$; $x=2, f(x)=3$; $x=4, f(x)=21$; $x=8, f(x)=73$ I know the ...
2
votes
2answers
1k views

Determine the coefficients of an unknown black-box polynomial

Let $p$ be a polynomial of known degree $n$: $$p(x) = a_0 + a_1 x + \ldots + a_n x^n$$ Suppose we have a magic black box that can evaluate the polynomial for us. How could one then determine the ...
2
votes
1answer
186 views

Divided difference coefficient of product of two functions

For any function $f$ and distinct reals $x_1,\ldots,x_n$, denote by $f[x_0,\ldots,x_n]$ the coefficient of $x^n$ of the minimal polynomial interpolating $f$ at $x_0,\ldots,x_n$. Let $f$ and $g$ be ...
1
vote
3answers
132 views

Polynomial interpolation $n+1$ distinct points

How would you show that $p(x)= \sum\limits_{i=0}^n b_i(x-c)^i$ is equivalent to $p(x)=\sum\limits_{i=0}^n a_ix^i$ by expressing the $a_i$ in terms of $b_i$ and $c$? Also we know that the polynomial ...
3
votes
1answer
203 views

Polynomial interpolation of the residues of a rational function

Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ ...
1
vote
1answer
2k views

Construct / find the simplest function based on data

Let's say I have these 7 natural numbers (all between 0 and 255): 255, 23, 45, 32, 87, 52, 146 How can I find a function F(x) that, once computed, gives me back ...
1
vote
1answer
548 views

Interpolation error

Working with a homework problem where I'm to derive an estimation of the interpolation error, and compare it with the actual error. This part is ok and I'm done with it. But while working with this in ...
2
votes
0answers
103 views

explicit error bounds for Multivariate interpolation

I want to interpolate a function of $d$ variables over a Cartesian grid, using multivariate interpolation, while characterizing interpolation error in terms of bounds on partial derivatives of the ...
1
vote
2answers
403 views

How to find the best interpolating function if we know $y(x_i)$ and $dy(x_i)/dx$

Imagine you are given a set of data points $\{x_i,y_i\}$, supplemented by a list of known first derivatives $\{y'_i\}$. How would you construct an interpolating function $y(x)$ (which satisfies ...
1
vote
0answers
76 views

How to approximate a trigonometric curve by Bezier curves?

Let me ask how to approximate a trigonometric curve by Bezier curves? Is there any known algorithm? Thank you in advance.
0
votes
0answers
142 views

How do I create a shape from a square corners' values?

I'm working on a 3D algorithm, so my problem applies to cubes, not squares. But for convenience, I'll stick to 2D. Each corner of a square can contain up to 100 units, depending of the values at each ...
4
votes
1answer
211 views

Polynomial interpolation

Let $P=[a,b]\times (c,d)$. Assume that we have given $n$ points $(x_1,y_1),...,(x_n,y_n)\in P$, such that $x_i\neq x_j$ for $i\neq j$; $i,j=1,...,n$. Does there exist a polynomial $f$ such that ...
0
votes
0answers
350 views

How to use the cspline formula?

I learned here that cspline is possibly suitable for my problem. Using the formula bellow for a paticular curve I have a problem and let me ask it. $p(t)=(2t^3-3t^2+1)\cdot p(0)+(t^3-2t^2+t)\cdot ...
1
vote
0answers
22 views

Need help showing piecewise cubic $Z(x,y)$ is $C^1$

We work in $\mathbb{R}^2$. Given a non-degenerate triangle $\triangle ABC$ and an interior point $P$, we specify the value of a function, value of its gradient at the $A,B,C$, and we specify the ...
1
vote
1answer
277 views

How to draw a smooth curve between 2 points given the 2 tangents at them?

Let me ask a question , given 2 points on the XY plane and given the 2 tangents at them, how to compute an arbitrary chosen smooth curve passing the 2 given points. For details, traveling along the ...
7
votes
2answers
182 views

Negative value of $\sqrt[3]{20}$

Given $f(x)=\sqrt[3] x$, find an approximation of $\sqrt[3]{20}$ using Lagrange interpolation method. $x_0=0$, $x_1=1$, $x_2=8$, $x_3=27$ and $x_4=64$ $f(x_0)=0$, $f(x_1)=1$, $f(x_2)=2$, ...
0
votes
1answer
329 views

4 point quadratic curve

I can define a curve that passes through 3 points using a quadratic equation: ax2 + bx + c = 0 I would like to know is it possible to define a curve that passes ...
9
votes
1answer
288 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 ...