1
vote
1answer
79 views

linear recursion $y_n=A \cdot y_{n-1}$

Let $a,b, \in \mathbb{R}$. Let $x_0=a, x_1=b$ and $x_n=\frac{x_{n-1}+x_{n-2}}{2}$ for $n \geq 2$ (i) Write the recursion in the form $y_n=A \cdot y_{n-1}$ where $A$ is a $2 \times 2$ matrix and ...
0
votes
0answers
20 views

Amount of sub-matrices created by Laplace expansion

I have created a program that solves a matrices determinant using the Laplace expansion method, and I was wondering if there is a equation which provides how many sub-matrices are created and used in ...
2
votes
1answer
67 views

Question about chebyshev polynomial

chebyshev polynomials are defined as such: $T_n(x)=cos(n*arccos(x))$ I'm asked to show that $deg(T_j(x))=j$ and that $T_0,T_1,T_2,...,T_n$ are an orthogonal basis of $\mathbb R_n[x]$. I think I can ...
1
vote
2answers
35 views

Recursive formulae involving a linear operator

Given a basis $e_{1}$, $e_{2}$ in the plane, define the linear operator $F$ as $F(e_{1})=3e_{1}+e_{2}$ and $F(e_{2})=e_{2}$. Furthermore, define the sequence $u_{1},u_{2},\dots$ of vectors in the ...
5
votes
2answers
182 views

Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: $g(1) = \alpha;$ $g(2n+j) = 3g(n) + γn + β_j$ : j = 0,1 and n >= 1 I have assumed the closed form to ...
3
votes
2answers
321 views

Solving a mathematical recursion to find explicit function

I came across the recursive sequence $$ \begin{align} a_{n+1}&=(r-2)a_n+(r-1)b_n\;,\\ b_{n+1}&=a_n\;, \end{align} $$ and the explicit formula $$ a_n=(-1)^n(r-1)+(r-1)^n\;. $$ I saw that ...
0
votes
1answer
42 views

Taking recursivity out of an algebraic equation through manipulation

wizards with all your mental esoterica. I'm a lowly peasant who never took more than high-school precal. Here is my question: I've got an equation: margY = (imageHeight + (2*margY))*mYRatio; margY ...