Tagged Questions

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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1
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1answer
183 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
261 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
590 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 - ...
2
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
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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
274 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
730 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
341 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
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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
200 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
103 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
425 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
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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
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3answers
634 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
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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
164 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
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0answers
121 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
465 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
477 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
176 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
117 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
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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
525 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
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
vote
2answers
391 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
votes
0answers
141 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
344 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
274 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
324 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
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
385 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
vote
1answer
374 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
vote
1answer
410 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
votes
1answer
864 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
votes
2answers
5k 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
votes
1answer
165 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
votes
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
749 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 $a,b,c$ ...