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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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 ...
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1answer
554 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
358 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
237 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
212 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
128 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
401 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
524 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
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1answer
317 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
122 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 ...
2
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1answer
398 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
394 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
220 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
298 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
709 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
391 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
870 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
509 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
128 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
157 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
217 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
116 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
506 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
733 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
185 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 ...
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1answer
80 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) = ...
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0answers
132 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
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1answer
533 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
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1answer
553 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 ...
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2answers
2k 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
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1answer
243 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 ...
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3answers
136 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
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1answer
217 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 $I_g(z)$ ...
2
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1answer
3k 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 ...
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1answer
686 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
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0answers
113 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 ...
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2answers
472 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 ...