Tagged Questions
3
votes
2answers
54 views
Solving 3 simultaneous cubic equations
I have three equations of the form:
$$ i_1^3L_1 + i_1K +V_1 + (i_2+i_3+C)Z_n = 0 $$
$$ i_2^3L_2 + i_2K +V_2 + (i_1+i_3+C)Z_n = 0 $$
$$ i_3^3L_3 + i_3K +V_3 + (i_1+i_2+C)Z_n = 0 $$
where $ ...
5
votes
0answers
57 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 ...
2
votes
1answer
57 views
Reference request: Newton-Kantorovich hypothesis for polynomials of integral coefficients
Kantorovich's theorem states that the Newton method for finding the
roots of a nonlinear function is guaranteed to converge if a
parameter $h$, determined by the values of the function and its
...
0
votes
0answers
26 views
robust computation of Groebner basis
I am trying to solve numerically polynomial systems of equations, over the reals. I am coming across the following phenomenon: let's say that i have a system of 7 equations with 7 unknowns. I am using ...
20
votes
2answers
381 views
How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?
How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible.
I proceeded by using the ...
2
votes
1answer
250 views
Runge-Kutta 4 - solving system of 6 differential equations (BVP)
I'm facing a tricky problem. I need to solve a system of 6 differential equations numerically, but I don't have 6 IVP (initial value problem) conditions, instead I have 6 BVP (boundary valye problem) ...
1
vote
1answer
168 views
Polynomial root finding
I have an univariate polynomial of some degree - how do I numerically find all of its real roots?
I never thought I would ask this question - everyone knows how to find polynomial roots, right..? ...
6
votes
2answers
222 views
Convergence of fixed point iteration for polynomial equations
I'm looking for the solution $x$ of
$$x^n+nx-n=0.$$
Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, 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
312 views
Lagrange basis function
Let $x_0,...,x_n$ be distinct real numbers and $l_k(x)$ be the
Lagrange's basis function. $\delta_n = \prod^n_{k=0}(x-x_k)$.
Prove that:
a. - $\sum^n_{k=0}(x_k-x)^jl_k(x)\equiv 0$, for ...
15
votes
2answers
423 views
Wiggly polynomials
I'd like to be able to construct polynomials $p$ whose graphs look like this:
We can assume that the interval of interest is $[-1, 1]$. The requirements on $p$ are:
(1) Equi-oscillation (or ...
0
votes
1answer
126 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
79 views
Method for finding roots of real trigonmetric polynomial
Given a real valued trigonometric polynomial,
$$ f(x) = \sum_{k=0}^{n} a_k \cos(k x + \phi_k) $$
what is the current fastest numerical method to find the roots of the polynomial for a given error? I ...
2
votes
1answer
131 views
Proof of a lower bound of the norm of an arbitrary monic polynomial
In my course I have come across the following problem:
The Chebyshev polynomial of degree $n$, $T_n(x)$, is defined on $[-1,1]$ by $T_n(x)=\cos n\theta$.
Let $q_{n+1}(x)$ be any monic ...
1
vote
0answers
217 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
61 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
0answers
56 views
PrimeFactorUsingSpreadsheet-Positive integer maximum of 15 digits (from 1 to 999,999,999,999,999)
I need your help or suggestion to my project "Prime Factor By Saccuan's Lab"
Prime Factor Using Spreadsheet-Positive integer maximum of 15 digits (from 1 to 999,999,999,999,999) and if you can to ...
5
votes
1answer
258 views
Finding all roots of polynomial system (numerically)
I want to numerically find all the roots of a system of polynomials (n equations in n variables). Since I can compute the Jacobian for the system (analytically or otherwise), I can use the Newton ...
4
votes
1answer
451 views
What are the best methods for solving cubic and quartic equations by computer programs?
We know that there are closed form formulas for real roots of degree 4 and 3 polynomials, but people sometimes advise to use numerical (e.g. Newton) methods anyway. They claim that closed formulas ...
3
votes
2answers
325 views
Computing roots of high degree polynomial numerically.
Here is my problem ; for my research, I believe that the complex numbers I am looking at are precisely the (very large) set of roots of some high degree polynomial, of degree $\sim 2^n$ where $1 \le n ...
4
votes
2answers
150 views
Refraction equation, quartic equation
Given two points $P$ and $Q$, a line ($A$, $B$ - orthogonal projection of $P$, $Q$ onto the line) and a coefficient $n$, I want to find out such point $C$ that $\frac{\sin{a}}{\sin{b}}=n$ (in fact, ...
1
vote
1answer
160 views
Linear interpolation for finding root of $f(x)$
When using linear interpolation, with similar triangles, to find the root of a function you narrow down the interval the root is in.
If $f(1) < 0$ and $f(2) > 0$ then the root is in $[1, 2]$
...
1
vote
3answers
298 views
Interval bisection to find a root of f(x)
I'm attempting to understand Interval bisection. I'm given a simple question in my textbook, and I can do the process easily, I just don't know when to stop. The question is "Use Interval bisection to ...
1
vote
1answer
210 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:
$$L_j(x) = \prod_{{i=0}\atop{j \neq ...
1
vote
1answer
178 views
Piece-wise linear interpolating polynomials
Somebody please help me to obtain piece-wise interpolating polynomials for the function $f(x)$ defined by the below data:
$x=1$, $f(x)=3$; $x=2, f(x)=3$; $x=4, f(x)=21$; $x=8, f(x)=73$
I know the ...
2
votes
0answers
182 views
What is a more modern way of getting numerical solution/roots of polynomial in base 10 than this?
I know Graeffe's method to approximate roots is great, but people always find troubles when they convert the root in n-th-root form into base 10 form by hand, for example, a large number in ...
1
vote
2answers
292 views
Determine the coefficients of an unknown black-box polynomial
Let $p$ be a polynomial of known degree $n$:
$$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
1answer
42 views
How to construct a polynomial with minimum deviation from zero on the complex region?
I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
7
votes
3answers
251 views
Simplifying $1 - x + x^2 - x^3 + … + x^{98} - x^{99}$ to an equivalent expression.
I am doing an exercise to see the error when solving this polynomial for $x = 1.00001$ using nested multiplication.
I believe the correct way to achieve this simplification (based on a lecture) is to ...
3
votes
0answers
367 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 ...
2
votes
0answers
77 views
What important problems require one to solve large systems of polynomial equations?
What is an extremely important problem that requires one to solve large systems of polynomial equations? I've heard of a number of "general areas" where the problems crop up (robotics, coding theory, ...
3
votes
2answers
570 views
Fast and robust root of a cubic polynomial with constraints
I'm looking for a fast and robust method for finding a root of a cubic polynomial
$x^3 + px^2 + qx + r$
To make the search more robust and faster, I'd like to leverage these properties:
The ...
3
votes
1answer
216 views
How do iterative methods applied to the companion matrix of a polynomial $p(\lambda)$ relate to $p$ itself?
A few days ago, I had a vague question in my mind about "matrix methods" for finding roots of a polynomial. Now I can ask at least a semi-precise question, thanks to the post
How to calculate complex ...
3
votes
2answers
394 views
Finding the real roots of a polynomial
Recent posts on polynomials have got me thinking.
I want to find the real roots of a polynomial with real coefficients in one real variable $x$. I know I can use a Sturm Sequence to find the number ...
2
votes
2answers
254 views
Representing affine transform of Legendre polynomials
I have a function defined as a set of weighted Legendre polynomials:
$f(x)=\alpha_0 P_0(x) + \alpha_1 P_1(x) + \alpha_2 P_2(x) +\ldots$.
I have another function similarly defined with Legendre basis ...
0
votes
1answer
310 views
Solving Polynomials in Computer Algebra Systems
Apart from low degree polynomials (2, 3, and 4) and factoring to lowest degrees, what are the method(s) used to find all the roots of a high-degree polynomial equations having only complex roots, and ...
7
votes
1answer
205 views
Krylov-like method for solving systems of polynomials?
To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
4
votes
3answers
728 views
Eigenvalues of matrix
I want to find all the roots of a polynomial and decided to compute the eigenvalues of its companion matrix.
How do I do that?
For example, if I have this polynomial: $4x^3 - 3x^2 + 9x - 1$, I ...
