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:

$2C_2+a{c}_1+b{c}_0=d_0$ (1) p.7-21

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

$\sum _{j=0}^{n-2}[c_{j+2}(j+2)(j+1)+a{c}_{j+1}+b{c}_j]x^j+a{c}_{n}n{x}^{n-1}+b[c_{n-1}{x}^{n-1}+c_n{x}^n]=\sum _{j=0}^n{d}_j{x}^j$ (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:

$A=\left[\begin{array}{cccccccccc}X& & X& & X& & 0& & 0& 0\\ 0& & X& & X& & 0& & 0& 0\\ 0& & 0& & X& & 0& & 0& 0\\ 0& & 0& & 0& & X& & X& X\\ 0& & 0& & 0& & 0& & X& X\\ 0& & 0& & 0& & 0& & 0& X\end{array}\right]$

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: $d_0=2C_2+a{c}_1+b{c}_0$ (1)

Equation associated with $d_1$:

j=1: $d_1=6c_3+2a{c}_2+b{c}_1$ (2)

Equation associated with $d_2$:

j=2: $d_2=12c_4+3a{c}_3+b{c}_2$ (3)

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

j=n-2: $d_{n-2}=[c_n(n)(n-1)+a{c}_{n-1}(n-1)+b{c}_{n-2}]$ (n-2)

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

j=n-1: $d_{n-1}=a{c}_{n}n+b{c}_{n-1}$ (n-1)

Equation associated with $d_n$:

j=n: $d_n=b{c}_n$ (n)

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

                          $A=\left[\begin{array}{cccccccccc}b& & a& & 2& & 0& & 0& 0\\ 0& & b& & 2a& & 0& & 0& 0\\ 0& & 0& & b& & 0& & 0& 0\\ 0& & 0& & 0& & b& & a(n-1)& n(n-1)\\ 0& & 0& & 0& & 0& & b& an\\ 0& & 0& & 0& & 0& & 0& b\end{array}\right]$