University of Florida/Egm4313/s12.team11.perez.gp/R4.1

Problem Statement
Obtain equations (2), (3), (n-2), (n-1), (n), and set up the matrix A as in (1) p.7-21 for the general case, with the matrix coefficients for rows 1, 2, 3, (n-2), (n-1), n, filled in, as obtained from equations (1), (2), (3), (n-2), (n-1), (n).

Given
As shown in p.7-21, the first equation is:

$$ 2C_2+ac_1+bc_0=d_0 \! $$ (1) p.7-21

According to p.7-20, the general form of the series is:

$$ \sum_{j=0}^{n-2}[c_{j+2}(j+2)(j+1)+ac_{j+1}+bc_j]x^j+ac_nnx^{n-1}+b[c_{n-1}x^{n-1}+c_nx^n]=\sum_{j=0}^{n}d_jx^j \! $$ (2) p. 7-20

From (2) p.7-20, we can obtain n+1 equations for n+1 unknown coefficients $$ {c_0,..., c_n} \! $$.

After referring to p.7-22, it can be determined that the matrix to be set up is of the following form:

$$ A =\begin{bmatrix} X && X && X && 0 && 0 & 0\\ 0 && X && X && 0 && 0 & 0\\ 0 && 0 && X && 0 && 0 & 0\\ 0 && 0 && 0 && X && X & X \\ 0 && 0 && 0 && 0 && X & X\\ 0 && 0 && 0 && 0 && 0 & X \end{bmatrix} \! $$

where the rows signify the coefficients $$ c_0, c_1, c_2, c_{n-2}, c_{n-1}, c_n \! $$, and the columns signify $$ d_0, d_1, d_2, d_{n-2}, d_{n-1}, d_n \! $$.

Solution
Building the coefficient matrix as shown in p.7-22 of the class notes, we can begin to solve for the coefficients as follows:

Equation associated with $$ d_0 \! $$:

j=0: $$ d_0=2C_2+ac_1+bc_0 \! $$ (1)

Equation associated with $$ d_1 \! $$:

j=1: $$ d_1=6c_3+2ac_2+bc_1 \! $$ (2)

Equation associated with $$ d_2 \! $$:

j=2: $$ d_2=12c_4+3ac_3+bc_2 \! $$ (3)

Equation associated with $$ d_{n-2} \! $$:

j=n-2: $$ d_{n-2}=[c_n(n)(n-1)+ac_{n-1}(n-1)+bc_{n-2}] \! $$ (n-2)

Equation associated with $$ d_{n-1} \! $$:

j=n-1: $$ d_{n-1}=ac_nn+bc_{n-1} \! $$ (n-1)

Equation associated with $$ d_n \! $$:

j=n: $$ d_n=bc_n \! $$ (n)

Using all of the above equations, (1), (2), (3), (n-2), (n-1), (n), we can then determine the A matrix to be:

$$                              A =\begin{bmatrix} b && a && 2 && 0 && 0 & 0\\ 0 && b && 2a && 0 && 0 & 0\\ 0 && 0 && b && 0 && 0 & 0\\ 0 && 0 && 0 && b && a(n-1) & n(n-1) \\ 0 && 0 && 0 && 0 && b & an\\ 0 && 0 && 0 && 0 && 0 & b \end{bmatrix} \! $$