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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2
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1answer
161 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
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3answers
116 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
202 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
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1answer
1k 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
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1answer
494 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
102 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
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2answers
382 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
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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
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0answers
138 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
207 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
331 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
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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
270 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
181 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
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1answer
318 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
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1answer
286 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 ...
3
votes
1answer
3k views

linear interpolation in 3 dimensions

Say that I have 2 points in 3 dimensional space specified in Euclidean coordinates $p_0(x_0,y_0,z_0)$ and $p_1(x_1,y_1,z_1)$. How would I go about finding the coordinates of an unknown point that ...
2
votes
1answer
378 views

Determining whether a function is Piecewise Polynomial

I am trying to determine whether or not a function is piecewise polynomial. The function is as below: Let $\ X$ be a continuous random variable with support on $\ \Omega_x$, and with corresponding ...
1
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1answer
358 views

Produce piecewise monotonic function with 3 points and their slope with MATLAB

I'm trying to reproduce the shape of an airfoil's camber line using the leading edge angle, the trailing edge angle, the chord and the max camber value. I cannot use a spline because it overshoots ...
1
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1answer
400 views

A problem on Lagrange interpolation polynomials

Based on a previous question, I had the following conjecture and was wondering if anyone knew how to prove it or find a counterexample. Consider the polynomial $$ ...
5
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1answer
819 views

Remainder term of Lagrange Interpolation Polynomial

Suppose $x_0,x_1,\ldots,x_n$ are $n+1$ distinct numbers in the interval $[a,b]$ and $f\in C^{n+1}[a,b]$. Then for each $x$ in $[a,b]$, there is a number $\xi$ in $(a,b)$ such that $$f(x) = P(x) + ...
2
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2answers
4k views

Finding the formula of an nonlinear function

Is there any way, if given multiple points and you wanted to find the equation of the graph where these points lie, how would you find out: First off equation is it? Line, Parabola, Hyperbola, etc? ...
3
votes
1answer
2k views

How to draw a smooth curve through given 2D points.

Let me ask about spline functions. I tried spline() function of Octave then I found it was almost I wanted , to draw a smooth curve through given 2D points. But for some points data , it plots ...
0
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1answer
164 views

Cubic spline: Help understanding Wikipedia markup

On the cubic hermite spline Wikipedia page, the formula for interpolating between $x_k$ and $x_{k+1}$ is given by ...
6
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1answer
5k views

What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline?

I am reading a book about computer graphics. It is confusing about the various splines and their algorithms. What is the difference between natural cubic spline, Hermite spline, Bézier spline and ...
2
votes
3answers
727 views

Interpolation to a power function

We have an experiment which have the variables $x$ and $y$. $x$ and $y$ can be measured into pair $(x_i,y_i)$. Now I'm finding a way to interpolate it into a power function $y=a+bx^c$. Which $a,b,c$ ...
1
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2answers
2k views

How to determine function based on input and output

I'm not very good at mathematics, so please bear with me. How can you determine / define a function based on sets of values of its input and output parameters. You have: $f(x_{1_1}, x_{1_2}, \ldots ...
1
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1answer
321 views

How to interpolate this table

I have a 3D function(al) f whose independent variables are A,C,D and E. Various tables are provided to show the function values. For each individual table, value of A is a constant. The nth table ...
2
votes
2answers
801 views

Multivariate function interpolation

I have a (nonlinear) function which takes as input 4 parameters and produces a real number as output. It is quite complex to compute the function value given a set of parameters (as it requires a very ...
4
votes
1answer
426 views

Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.

I am trying to understand Hermite Interpolation. Here is my pedagogical example. I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation. I choose the Chebyshev zeros ...
0
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1answer
4k views

Linear interpolation

The blue curve is a set of (X,Y) coordinates. Orange segment passes through two of these (X,Y) coordinates (black dots of ...
1
vote
1answer
144 views

grading accuracy of two lines [duplicate]

Possible Duplicate: Error measurement between given perfect 2D shape and freeform shape drawn by user I am programming (with vectors) an application which requires a user to draw line ...
4
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0answers
1k views

Computation of coefficients of Lagrange polynomials

For our homework we should write a program, that creates Lagrange base polynomials $L_k(x)$ based on a few sampling points $x_i$. Now i am eager to develop a formula to be able to compute the ...
4
votes
4answers
194 views

What are some “natural” interpolations of the sequence $\small 0,1,1+2a,1+2a+3a^2,1+2a+3a^2+4a^3,\ldots $?

(This is a spin-off of a recent question here) In fiddling with the answer to that question I came to the set of sequences $\qquad \small \begin{array} {llll} ...
1
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0answers
641 views

How to derive Hermite polynomial from a given data set?

The problem is asking to find a Hermite polynomial to predict the position of the car and its speed when t = 10s. The Hermite polynomial formula is defined as: $$H_{2n+1}(x) = f[z_0] + ...
1
vote
1answer
356 views

Curve fitting with upper and lower bounds for derivatives

I compute (at a great cost) upper and lower bounds $f_u(x)$ and $f_l(x)$ of an unknown function $f(x)$ at points $x$ in $[0,1]$. Now I am interested in an estimation of the derivative $f'(x)$. I ...
0
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1answer
133 views

find a function given some values

I'me trying to remember my math classes but no luck... I've got a pair of values i.e . ...
1
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2answers
539 views

Is there some intuition for Lagrange interpolation formula?

How do I prove the Lagrange interpolation formula is true as stated in this link? I ask this because the article isn't self contained on intuition of each step in the proof, please don't use things ...
3
votes
1answer
1k views

MATLAB Hermite interpolation

Anyone know where I can find the Hermite interpolation algorithm in MATLAB. Which Hermite interpolation algorithm solves this? I need to calculate a polynomial. Example: ...
1
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1answer
119 views

Polynomial interpolation on scattered points

I was wondering how I could fit a polynomial surface through a set of points in two variables. When I look up this problem in the literature, I usually see two options: Use a tensor product, but ...
0
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1answer
551 views

Is this Hermite interpolation correct?

Someone can explain this hermit interpolation algorithm with example? Thank you, ...
2
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2answers
120 views

Interpolating polynomials

So I have this question on a homework and I just can't seem to figure it out. Let $f \in C^4 [0,1]$ and let $p$ be a polynomial of degree $\le 3$ such that $p(0) = f(0)$, $p(1) = f(1)$, $p'(0) ...
0
votes
1answer
320 views

Fitting for piecewise function, with constraints on first/second derivative

I have the following problem. We have a set of discrete points ($x_i$,$y_i$), defined for $0 \leq x_i \leq r$, where $r$ is an arbitrary value. For values of $0 \leq x \leq r$, the y value is ...
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0answers
75 views

Questions about interpolating translated points from a grid

I would like to do the following transformations on a very low resolution bitmap (64x64 pixels). I am doing this transformation on a computer images, but it has nothing to do with computers, you can ...
0
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1answer
108 views

Interpolation over trajectory at set positions on path

I have the following: 2d vector for velocity 2d start coordinate gravity acceleration I need to know the coordinate of a projectile at a given distance along the trajectory. For example: ...
2
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0answers
152 views

(Experimental) Can it be shown that this extension of the secant-interpolation has quadratic convergence?

Background: I needed some efficient but simple interpolation-methods aside of Newton's iteration because I want to have it in contexts, where the derivative of a function is not always known. So an ...
1
vote
1answer
461 views

existence and uniqueness of Hermite interpolation polynomial

What are the proofs of existence and uniqueness of Hermite interpolation polynomial? suppose $x_{0},...,x_{n}$ are distinct nodes and $i=1 , ... ,n$ and $m_{i}$ are in Natural numbers. prove exist ...
0
votes
1answer
362 views

how to interpolate a 2d function with 6 points?

I'm implementing an algorithm that uses a so called 6-point interpolation, which I never heard before. In the article I'm reading it's described like this: $\phi(p\Delta x, q\Delta y)=[q(q-1)/2] ...
2
votes
2answers
978 views

given $y = a + bx + cx^2$ fits three given points, find and solve the matrix equation for the unknowns $a,b$, and $c$

Given $y = a + bx + cx^2$ fits three given points, find and solve the matrix equation for the unknowns $a$, $b$, and $c$. the equation fits the points $(1,0), (-1, -4),$ and $(2, 11)$ I really ...
1
vote
3answers
1k views

Linear Algebra Question (Polynomial Interpolation)

Given the data for an experiment: Velocity: 0, 2, 4, 6, 8, 10 Force: 0 , 2.9, 14.8, 39.6, 74.3, 119 (One force value listed below one velocity value in a table) Find an interpolating ...