0
votes
0answers
52 views

Best approximating linear polynomial for $|x|$ on $[-1,5]$ [on hold]

How to find the best approximation polynomial $p_1(x)\in P_1$ for $|x|$ on $[-1,5] $ w.r.t. $\|\cdot\|_{\infty}$? I want to use the Equioscillation theorem but have no clue.
0
votes
0answers
13 views

Changing the order of the elements of the divided difference Polynomial Interpolation

Apparently this is rather trivial but I don't understand why what I've highlighted in green is correct.
0
votes
1answer
31 views

Newton-Cotes Quadrature formula

Im trying to find more information about numerical integration methods. When is a Newton-Cotes Quadrature formula on n nodes exact?
0
votes
0answers
37 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
2
votes
1answer
42 views

Determine the coefficients of a polynomial knowing its roots

My prof. gave this problem as a bonus in an exam, and I couldn't figure out a solution. Some hints or a general method of solving it would be very nice. Given the following polynomial: ...
3
votes
1answer
54 views

What is the Most Efficient Way to Calculate the Internal Rate of Return?

I have built a program that prices financial assets and it does this in part by calculating the IRR. The problem is that it does not run as quickly as I would like it to. I currently use the ...
0
votes
0answers
29 views

Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
0
votes
1answer
29 views

numerical algorithms for determining least common multiple of polynomials

I have a pair of rational polynomial fractions $\frac{A(x)}{B(x)} + \frac{C(x)}{D(x)}$ where A, B, C, and D are all polynomials in x, and I have their coefficients as an array of numbers. I would ...
0
votes
1answer
39 views

Why does the interpolation error go to zero if we increase the number of sampling points?

This question is motivated by polynomial interpolation. We know that for $f\in C^{n+1}[a,b]$ and $a=x_0<\dots<x_n=b$ holds $$\| f - p_n \|_\infty \leq \frac{1}{(n+1)!} \| f^{(n+1)} \|_\infty ...
3
votes
1answer
62 views

Numerical evaluation of polynomials in Chebyshev basis

I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and ...
1
vote
3answers
106 views

How to choose the starting point in Newton's method?

How to choose the starting point in Newton's method ? If $p(x)=x^3-11x^2+32x-22$ We only learnt that the algorithm $x_{n+1}:=x_n-\frac{f(x_n)}{f'(x_n)}$ converges only in some ...
0
votes
0answers
35 views

How to differentiate Lagrange Basis Polynomial?

How to differentiate Lagrange Basis Polynomial ? I don't know, if the term is correct, but the question is: If $x_0,...,x_n\in\mathbb R$ are pairwise distinct ...
0
votes
0answers
29 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve c(u) of degree 4 at the points -1, 0, 1 as: c(-1) = 4; c'(-1) = 4; c(0) = 6; c(1) = -4; c'(1) = -6; find the generalized ...
0
votes
2answers
56 views

Legendre Polynomial Orthogonality and Size

Show $(P_i,P_j)=\begin{cases} 0& i \neq j \\ \frac{2}{2j+1} & i = j\end{cases}$ for $0 \leq i, j\leq2$ I'm just not sure exactly what I'm supposed to do. Do I plug in values of i and j and ...
0
votes
1answer
25 views

Lowest norm solution to a system of polynomial equations

I have a system of cubic equations: $$0=A_0+A_1 x+A_2 ( x \otimes x ) + A_3( x \otimes x \otimes x )$$ where $\dim A_0 = \dim x$ (so there are as many equations as unknowns). You may assume that the ...
0
votes
1answer
63 views

MATLAB: Approximate tomorrow's temperature with 2nd, 3rd and 4th polynomial using the Least Squares method.

The following is Exercise 3 of a Numerical Analysis task I have to do as part of my university course on the subject. Find an approximation of tomorrow's temperature based on the last 23 values ...
3
votes
1answer
121 views

Accuracy of the Newton-Cotes formulas for polynomials of degree $n+1$ and even $n$

Let $f$ be a polynomial of degree $n+1$. The Newton-Cotes formula is given by $$\int_{-t}^tf(x)\text{ d}x\approx\sum_{k=0}^nf(x_k)\int_{-t}^t\omega_{n+1}(x)\text{ d}x \tag{*}$$ where ...
0
votes
2answers
34 views

Find limits of value/derivatives defining a polynomial at 2 points to bound it in between

The following properties define the polynomial $p(x)$ uniquely: $\text{deg}(p(x))=7\\p(-1)=y_1,\ p'(-1)=d_{1,1},\ p''(-1)=d_{2,1},\ p'''(-1)=d_{3,1}\\ p(1)=y_2,\ \ \ \ p'(1)=d_{1,2},\ \ \ \ ...
0
votes
1answer
53 views

Rate of Convergence of Generalized Iterative Method

Consider the generalized iterative method for finding polynomial roots: $z_{k+1}=z_k +d\frac{(1/p)^{(d-1)}(z_k)}{(1/p)^{(d)}(z_k)}$ where d is a positive integer. Note that Newton's Method is a ...
1
vote
1answer
234 views

In EMI calculations, how to calculate “Rate” if EMI, Principal and Time are given

In EMI (Equated Monthly Instalments) calculations, the inputs are- Principal-P, Rate-r, and ...
1
vote
2answers
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 ...
3
votes
2answers
78 views

Bernstein polynomial looks like this: $B_i^n={{n}\choose{i}}x^i(1-x)^{n-i}$.Find it's $r$'th derivative.

Bernstein polynomials are defined like this $B_i^n={{n}\choose{i}}x^i(1-x)^{n-i}$.I need to prove that $r$'th derivative of it is equal to: ...
1
vote
2answers
56 views

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$ I tried to to solve it from the right side: ...
1
vote
2answers
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 ...
1
vote
1answer
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: ...
0
votes
1answer
154 views

Graeffe's root finding method

What are the practical applications of Graeffe's root finding method?I searched a lot but couldn't find.I found that it is used in aerodynamics and electric circuit analysis.But don't know much about ...
1
vote
0answers
24 views

how to show that ortogornal polynomials are a basis of $P_n$ and how to prove that $deg\left(\phi_n\right) = n$

Let $\{\phi_1,\phi_2,.......\}$ be a row orthogornal polynomials which are not zero. With $\phi_i \in P_i$ and inner product $<.,.>$ on $C\left([a,b]\right)$. show that ...
1
vote
1answer
97 views

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 ...
0
votes
0answers
26 views

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 ...
1
vote
1answer
49 views

$p$-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 $x,x_1,...,x_k$. ...
1
vote
1answer
402 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 ...
1
vote
2answers
99 views

Simple Polynomial Interpolation Problem

Simple polynomial interpolation in two dimensions is not always possible. For example, suppose that the following data are to be represented by a polynomial of first degree in $x$ and $y$, ...
1
vote
0answers
74 views

Analytical solution(root) for a tenth order polynomial?

is it possible to develop an analytical solution (root) for such a polynomial: $f(x)=\left(x^{10}-c_1^2\right)*\left(c_2-x\right)^2-0.2*\left(x^2-1\right)*c_1^2$ with $c_1$ and $c_2 >0$. Numerical ...
2
votes
1answer
59 views

Chebyshev polynomials with non-negative constants

Please let help me solve the following problem that I encountered while engaging in my research. I'm dealing with a class of functions, in which each function has a unique series representation of ...
3
votes
2answers
119 views

Root of the function $f(x)=xe^x-R$

How can we find the root of the function $f(x)=xe^x - R$ for a general R where $R>=-1/e.$ I don't have any idea as to how to even approach this. Came across this problem during my self-study in ...
0
votes
1answer
53 views

Condition Number of Polynomial (Condition Number = 0)

I'm calculating the condition number of a polynomial equation $$ y = (x-2)^{9} $$ for this equation, the Jacobian is equal to ...
2
votes
3answers
112 views

If I know that a polynomial (of order $k \gt 2$) has at most $1$ positive real root - can I find that easily?

[update 2] Urgghh - the time-consumption really stems only from the construction of the h-order polynomial. The time for finding the roots (only 10 to 20 times Newton-iteration because of my nice ...
0
votes
1answer
90 views

Bounding a bicubic polynomial

My actual situation is working with bicubic polynomials, (that is, polynomials of the form $\sum_{i=0}^3 \sum_{j=0}^3 a_{i,j} x^iy^j$) defined on the unit square $[0,1]^2$ (actually these are ...
0
votes
0answers
37 views

What are the coeff. of this polynomial?

My matlab generates this answer for the problem: p = -20.2090 17.3368 272.9057 -0.7528 Is it correct? It seems I've gotten different results in another ...
1
vote
0answers
56 views

Fast way to switch between lagrange and newtonian representation of polynomials?

I just wanted to know whether there is a fast algorithm to switch between these two representation methods of polynomials $\in \mathbb{R}[x]$? By Lagrange I am refering to: enter link description ...
3
votes
1answer
146 views

affine transformation of a polynomial

If I have a set of polynomials defined on six points of a triangle, such that $\phi_i(p_j) = \delta_{ij}$, how do I use an affine transformation to get new polynomials so that $\bar{\phi_i}(\bar{p}_j) ...
3
votes
2answers
289 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 $L_1,L_2,L_3,K,V_1,V_2,V_3,C$ and $Z_n$ ...
5
votes
0answers
228 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
83 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
49 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 ...
22
votes
2answers
515 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 ...
4
votes
1answer
661 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) ...
2
votes
1answer
936 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
452 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
336 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 ...