Tagged Questions
0
votes
1answer
29 views
Coefficients of Newton interpolation polynomial
Given distinct $y_0,...,y_m$ in $\mathbb R$, let $N_m(x)$ be the Newton interpolation polynomial of degree $m$. That is, $N_m(x) = \sum_{n=0}^{m}a_nw_n(x)$ where $w_0 = 1$, $w_n(x) = ...
5
votes
0answers
55 views
Runge's phenomen: interpolation error using Chebyshev nodes oscillates
We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
0
votes
2answers
100 views
For n=3 Lagrange interpolation why is it equal to 1?
I'm studying Lagrange's formula for polynomial interpolation and I cannot seem understand why for $n=3$
$$L_0(x)+L_1(x)+L_2(x)+L_3(x) = 1$$
for all real x.
In my textbook it says as a hint that ...
1
vote
1answer
48 views
What is the motivation for this theorem on Polynomial Interpolation Error?
My textbook presents this theorem without any sort of introduction. It does cover using the Newton and Lagrange forms of the interpolation polynomial, so I've got that. Anyway, here's the theorem:
...
0
votes
1answer
122 views
Least-squares approximation polynomial
Consider the function
$\displaystyle f(x) = \frac{1}{\alpha (x-\beta)^2 + 1}$
in the interval $I = [-1,1]$. Set $\beta = 0$.
How do I get the expression for the least-squares polynomial, say $\tilde ...
2
votes
1answer
114 views
Automatic calculation of the intersection of discrete curves
first of all, let me apologize for a poor math-english translation, I'll try my very best.
I have the following situation:
I have over 16.000 data files which I generated from a biometric ...
1
vote
0answers
215 views
Interpolating polynomial with Chebyshev nodes
I am interested in constructing an polynomial that interpolates some known arbitrary function $f(x)$ over the domain $x \in [0,70]$. I want the polynomial to have degree 14 and so need 15 points.
...
0
votes
0answers
60 views
error of interpolating polynomial through 3 given points and given derivative in one point
What's the error of the interpolating polynomial $p$ which interpolates $f(x)$ in $(x_i,f(x_i))$ for $i=0,1,2$ and which has $p'(x_{i_0}) = f'(x_{i_0})$ for one $i_0 \in \{0,1,2\}$
We were given the ...
0
votes
1answer
85 views
Multiplying Polynomials with fewer coefficient multiplications
Not sure how this works! Apparently it can be done in 5-6 multiplications
Show how to multiply two degree 2 polynomials using fewer multiplications of coefficients than the naive algorithm.
1
vote
1answer
142 views
Using Lagrange's Interpolation Formula to show that boolean functions over finite fields are polynomials
Let $F_2$ be the set of all the functions from the finite field $GF(2^n)$ of $2^n$ elements to $GF(2)$. I am reading a textbook that proves that the elements of $F_2$ can be represented by ...
6
votes
1answer
185 views
Is there a name for these polynomials?
Given $t \in \mathbb{R}[0,1]$, consider the following set of polynomials:
$$
\left[-{\left(t - 1\right)}^{2} t, {\left(t - 1\right)} {\left(t^{2} - t
- 1\right)}, -{\left(t^{2} - t - ...
1
vote
2answers
2k views
Given four points on a cubic function curve, how can I find the curve's function?
Say I have a curve
$$y = ax^3 + bx^2 + cx + d.$$
I don't know $a$, $b$, $c$ or $d$, but I do know the $(x,y)$ values of four points on this curve.
How can the values of $a$, $b$, $c$ and $d$ be ...
1
vote
0answers
88 views
Extending Hermite polynomial interpolation
Working with the definition of Hermite polynomials
$x_0,\ldots,x_n$ are distinct in $[a, b]$, $f''(x)$ is continuous on [a, b], then
$$H_{2n+1}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} ...
1
vote
1answer
207 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:
$$L_j(x) = \prod_{{i=0}\atop{j \neq ...
1
vote
1answer
174 views
Piece-wise linear interpolating polynomials
Somebody please help me to obtain piece-wise interpolating polynomials for the function $f(x)$ defined by the below data:
$x=1$, $f(x)=3$; $x=2, f(x)=3$; $x=4, f(x)=21$; $x=8, f(x)=73$
I know the ...
1
vote
2answers
290 views
Determine the coefficients of an unknown black-box polynomial
Let $p$ be a polynomial of known degree $n$:
$$p(x) = a_0 + a_1 x + \ldots + a_n x^n$$
Suppose we have a magic black box that can evaluate the polynomial for us. How could one then determine the ...
1
vote
3answers
103 views
Polynomial interpolation $n+1$ distinct points
How would you show that $p(x)= \sum\limits_{i=0}^n b_i(x-c)^i$ is equivalent to
$p(x)=\sum\limits_{i=0}^n a_ix^i$ by expressing the $a_i$ in terms of $b_i$ and $c$?
Also we know that the polynomial ...
3
votes
1answer
135 views
Polynomial interpolation of the residues of a rational function
Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ ...
1
vote
1answer
230 views
A problem on Lagrange interpolation polynomials
Based on a previous question, I had the following conjecture and was wondering if anyone knew how to prove it or find a counterexample.
Consider the polynomial $$ ...
2
votes
1answer
374 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) + ...
3
votes
0answers
364 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 ...
0
votes
1answer
102 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
vote
1answer
103 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 ...
1
vote
3answers
374 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 ...
3
votes
4answers
817 views
Polynomial fitting where polynomial must be monotonically increasing
Given a set of monotonically increasing data points (in 2D), I want to fit a polynomial to the data which is monotonically increasing over the domain of the data. If the highest x value is 100, I ...
0
votes
1answer
129 views
Vandermonde and curve interpolation
I hesitate here because of an understanding with a calculation problems.
I want to calculate an interpolation using the Vandermonde matrix.
see: http://en.wikipedia.org/wiki/Vandermonde_matrix
My ...
0
votes
2answers
149 views
Solve a polynomial of degree d
In my Artificial Intelligence class, I encountered this equation where I have to solve for b, the effective branching factor of a tree:
...
3
votes
1answer
256 views
How to minimize this function difference
Sorry about this somewhat lengthy introduction to my question. I thought it might be useful to know what I'm trying to do.
I decided that I would like to have sequence of polynomials in $\mathbb{P}_n ...
6
votes
1answer
260 views
A Curious Binomial Sum Identity without Calculus of Finite Differences
Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$,
\begin{align}
\binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} ...
5
votes
1answer
274 views
Determining Coefficients of a Finite Degree Polynomial $f$ from the Sequence $\{f(k)\}_{k \in \mathbb{N}}$
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ ...
