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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Lagrange interpolation of a polynomial

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ has such property that for every distinct $x_0,x_1,...,x_n\in\mathbb{R}$ Lagrange interpolating polynomial for $f$ in these points has degree at most $n-1$. ...
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22 views

Fourier on discrete but not sequential data

I have time series data, which is discrete as it is timestamped with microsecond resolution. It is not sequential, as in not every microsecond has a value. How would I go about Fourier in such a case? ...
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Newton's Interpolation Formula: Difference between the forward and the backward formula

I was taught that the forward formula should be used when calculating the value of a point near $x_0$ and the backward one when calculating near $x_n$. However, the interpolation polynomial is unique, ...
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34 views

Subtlety about the definition of B-splines

I came across the following definition for the zero'th order B-spline $$b_0(x) = \left\{ \begin{array}{lr} 0 & |x|>1/2\\ 1 & |x|<1/2\\ 1/2& |x|=1/2. \end{array} ...
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93 views

The Puzzle of Locating Points in a Quadrilaterally-Faced Hexahedral Creature

The Disclaimer This is NOT homework... I designed the story. I thought a name MIT would be funny, while something along the line of TULSA or SU will still be decent. I know the algorithm to Q3 and 3D ...
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lagrange interpolation question here

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0, x_1=0.25 , x_2=0.5, x_3=1$ a)Find Lagrange polynomial for this function. c)Find the real approximation error. d)Find the limit of the ...
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Lagrange interpolating polynomials question?

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0$, $x_1=0.25$ , $x_2=0.5$, $x_3=1$ a)Find Lagrange polynomial for this function. So $L_3(x)=f_0(x) l_0(x)+f_1(x) l_1(x)+f_2(x) ...
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28 views

Creating formula for when to review facts before an exam

I want to create a forumla to determine when I review Anki cards before an exam. The formula should be f(today, interval, exam.date) = test.day I have the ...
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44 views

Solving interpolation between line and quadratic functions

I have two functions with known coefficients as follows: $$ f_1(x) = x^2 -2x + 2 $$ $$f_2(x) = 108 x - 206 $$ and I know that $$ (1-z) f1(x) + z f2(x) = 0 $$ $$ (1-z) f1'(x) + z f2'(x) = 0$$ ...
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279 views

Is there a cubic spline interpolation with minimal curvature?

I came across the term "cubic spline with minimal curvature". However, I am not able to find any documentations/explaination on its computation method. Can anyone help me by advising how I can go ...
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80 views

Polynomial Interpolation

My professor gave the following question as a practice for study guide. Any assistance in terms of helping me to solve this would be much appreciated. Suppose that $f$ is continuous and has ...
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19 views

Differntiability properties of kriging interpolation

What are the differentiability properties of Kriging interpolation functions? Specifically, I am interested in using it to create a random realization of a 1D function using a regularly spaced grid of ...
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62 views

How to use any function interpolation method to create two functions …

I need help desperately on this. I have been working on it for a while. Use any function interpolation method studied in the course to create two functions $x(t)$ and $y(t)$ on $0 ≤ t ≤ 1$ so ...
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65 views

Polynomial Interpolation and Error

I have numerical analysis final coming up in a few weeks and I'm trying to tackle a practice exam. Assuming $p(x)$ interpolates the function $f(x)$, find the polynomial $p(x)$ that satisfies the ...
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478 views

Cubic Spline Interpolation practice

Going over practice problems for our final exam. I'm stuck on a problem involving cubic splines. In fact, I don't even know where to begin. I need to find the natural cubic spline $S(t)$ at $t_0=0, ...
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32 views

Find a formula that describes the given day-price relation

Given data: Days Prices 0 0 1 1000 2 1000 3 1000 4 1600 5 2200 6 2800 7 3400 8 4000 9 4600 10 5200 Price is function of day. I had my formula but always put negative ...
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49 views

Calculate average speed with unknown variable accelaration

I am in the middle of a vehicle tracking project where I have to calculate the distance traveled by the vehicle in a given amount of time. Data I am getting: ...
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71 views

inequality about linear and piecewise constant interpolation?

$\Omega\subset\mathbb{R}^3$ is a bounded, and $u(\mathbf{x},t) \in C\big(0,T,L^2(\Omega)\big)$. We divide the interval $[0,T]$ in $N$ equal subintervals with the time step $\tau$. With the notaion $$ ...
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20 views

Interpolation between curve and line

I have two functions with known coefficients as follows: $$ f_1(x) = x^2 -2x + 2 $$ $$f_2(x) = -4x + 1 $$ and I know that $$ (1-z) f1(x) + z f2(x) = 0 $$ $$ (1-z) f1'(x) + z f2'(x) = 0$$ then I ...
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404 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 ...
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65 views

Use Lagrange interpolation to prove $\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$

Suppose $f\in C^2([a,b])$ and $f(a)=f(b)=0$,use Lagrange interpolation to prove $$\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$$ I tried to use the theoretic error to prove ...
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448 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?
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111 views

How to calculate a spline for points in general position?

I want to find a curve passing through (or near) $n$ points in the plane. The catch is that the curve need not be a function. That is, a vertical line might pass through the curve in more than one ...
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48 views

Other way to write Lagrange's form (with derivative)

Prove that we can write polynomial $L_{n}\in\Pi_{n}$ which is interpolating function $f(x)$ in $n+1$ nodes $x_{0},\,\ldots,\, x_{n}$ in following form: ...
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306 views

Bilinear interpolation formula derivation

I am basically following two wikipedia articles, linear interpolation and bilinear interpolation I am having problems getting the same formulas presented for bilinear interpolation but I cannot see ...
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How to calculate interpolating splines in 3D space?

I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a ...
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99 views

Degrees of interpolating polynomials

Given a collection of $m+1$ points $\{(x_0,y_0), (x_1,y_1), ..., (x_m,y_m)\}$, we can form the interpolating Lagrange polynomial $L(x)$: $$ L(x) = \sum_{i = 0}^{m} y_i l_i(x) \\ l_i(x) = \prod_{0 \le ...
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176 views

Hermite-Birkhoff interpolation for finding a polynomial

So I have to consider $p \in\mathscr{P}_3$,where $\mathscr{P}_3$ is the set of polynomials of degree $3$, such that $p(0)=1, p'(0)=1, p(2)=1, p'(1)=2$ using the Hermite-Birkhoff interpolation with the ...
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The universal function for the class of functions defined on a finite set is computable; (Lagrange interpolation polynomials).

Theorem: A computable universal function for the class of functions of $n$ variables exists that are defined on a finite subset of $\mathbb N^n$. Attempt at proof: Each such function is completely ...
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How to calculate a trigonometric interpolation polynomial

I have the following $2 \pi$-period function f: $$ f(x) = \left \{ \begin{array}{l l l} x: & 0 < x < 2 \pi \\ \pi: & x = 0 \end{array} ...
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Wolfram Mathematica - Newton Backward Interpolation?

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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Differentiation by interpolation.

I am asked to show that the formula: $$ f'(x)\sim \sum_{i=0}^n A_i f(x_i) $$ which is derived from differentiating the interpolation polynomial is similar to that derived from checking/evaluating the ...
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DPLL Algorithm $ \rightarrow $ Resolution proof $ \rightarrow $ Craig Interpolation

I really need help here for an exam that I got tomorrow .. Let's say I got a bunch of constraints: $ c1 = { \lnot a \lor \lnot b } \\ c2 = { a \lor c } \\ c3 = { b \lor \lnot c } \\ c4 = { \lnot b ...
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$M(x)$ and $L(x)$ interpolate $f(x)$ on $n+1$ points. Show, that $f(x)$ lies between $L(x)$ and $M(x)$

We have $n+2$ points $x_0 \lt x_1 \lt x_2 ... \lt x_{n+1}$. We have two polynomials - $L$ and $M$. $L(x)$ interpolates $f(x)$ on points $x_0,...,x_n$ and $M(x)$ does so on $x_1,...,x_{n+1}$. The ...
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Intuitive proof of interpolation polynomial existence

Problem: Given a set of $n+1$ data points ($x_i, y_i$) where no two $x_i$ are the same, one is looking for a polynomial $p$ of degree at most $n$ with the property $p(x_i) = y_i$ for all $i∈ [0, n ...
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Parametric Surface Interpolation

I compute the set of states that can be achieved by a system defined as xdot=f(x(t),u(t)) with given initial states x(0)=x0, u is element of U and t element of [0,tf]. Result is a ...
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Show $\Delta^mp(x) = 0$ when $p(x) \in P_n$

Show that $\Delta^mp(x)= 0 $ when $p(x) \in \mathscr P_n$ and $m\ge n+1$, where $\mathscr P_n$ is the set of polynomial of degree $n$ and $\Delta^m$ is the operator for the $m$-th forward ...
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Proof that $f[x_{i0},…,x_{ik}] = f[x_0,…,x_k]$

i try to show, that $f[x_0,...x_k]$ is a symmetric function of $x_i$. What means, that for a permutation $x_{i0},...,x_{ik}$ of numbers $x_0, ...,x_k$ applies: $$f[x_{i0},...,x_{ik}] = ...
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200 views

How do I interpolate the derivative of a catmull-rom spline?

I am creating an implementation of a cubic hermite spline in Python. One feature I would like to add is a method to compute the slope (IE the derivative) for a given T value. Currently, I can do it ...
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Interpolation- Barycentric coefficients for nodes that are Chebyshev points of the second kind.

So I came across the following theorem: If the interpolation node are Chebyshev points of the second kind given by : $$ x_k=\cos \left( \frac{j\pi}{n}\right) \qquad ( 0 \leq j \leq 0) $$ Then the ...
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Polynomials of the form $g(x)=f[x,x_1,x_2,…,x_m]$.

Can anybody point me to some materials about polynomials of the form $g(x)=f[x,x_1,x_2,...,x_m]$, meaning that for given $x$ they will give back a leading coefficient of function $f$ interpolated at ...
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$p$-polynomial of $n$'th degree, $q(x)=p[x,x_1,x_2,…,x_k]$, prove that q has the same leading coefficient.

So I have a polynomial $p$ of $n$'th degree and q given by $q(x)=p[x,x_1,x_2,...,x_k]$, meaning that for $x$ it gives back the leading coefficient in interpolation of $p$ on points $x,x_1,...,x_k$. ...
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How stable is my quaternion interpolation?

after some experimentation to optimize slerp I found that finding the middle between quaternion is rather cheap (for $t=0.5$) in particular: (with $\theta$ the angle between $q_1$ and $q_1$) ...
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Successive parabolic interpolator for sub pixel interpolation

Is it possible to use successive parabolic interpolator for doing sub pixel interpolation. In the case of non sub pixel interpolation it is very easy to apply successive parabolic interpolation as ...
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122 views

how covert joysitck (x, y) coordinates to robot motor speed?

I am trying to formulate an equation to calculate left and right motor speeds of a robot. The robot is four wheeled drive with independent motors connected to tank-like best on each side. see this ...
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407 views

Derive the error term of Basic Corrected trapezoidal Rule for a Cubic Hermite Polynomial

The basic trapezoidal rule for approximating $I_f = \int_{a}^{b}f(x)dx$ is based on linear interpolation of $f$ at $x_0=a$ and $x_1 = b$. The Simpson rule is likewise based on quadratic polynomial ...
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38 views

Spline interpolation problem akin to Bezier spline

Given three pairwise distinct points $p_1, p_2, p_3 \in \mathbb{R}^2$, I'd like to find a function $f: \mathbb{R} \to \mathbb{R}^2$ with at least $f \in C^1$ such that $f(0) = p_1, f(1) = p_3, f'(1) ...
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27 views

Parametric Cubic Spline interpolation

I'm trying trying to interpolate a hyper-surface from a n dimensional set of points using parametric interpolation . Now i see that there's a derivative term at each interior point of the curve that ...
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18 views

Interpolation of a function given two sets

$P$ and $Q$ are sets of $k+1$ points. $P\bigcap Q$ has $k$ points. $p(x)$ in $\mathbb{P}_k$ interpolations $f(x)$ at points of $P$. $q(x)$ in $\mathbb{P}_k$ interpolations $f(x)$ at points of $Q$. Let ...
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42 views

Tile a spline with bricks

I have a series of identically sized virtual Lego bricks. I need to "tile" these bricks along the length of a Hermite spline to create a curved road. The spline is two-dimensional. E.g. the curve only ...