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:

${\displaystyle 2C_{2}+ac_{1}+bc_{0}=d_{0}\!}$ (1) p.7-21

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

${\displaystyle \sum _{j=0}^{n-2}[c_{j+2}(j+2)(j+1)+ac_{j+1}+bc_{j}]x^{j}+ac_{n}nx^{n-1}+b[c_{n-1}x^{n-1}+c_{n}x^{n}]=\sum _{j=0}^{n}d_{j}x^{j}\!}$ (2) p. 7-20

From (2) p.7-20, we can obtain n+1 equations for n+1 unknown coefficients ${\displaystyle {c_{0},...,c_{n}}\!}$.

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

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

where the rows signify the coefficients ${\displaystyle c_{0},c_{1},c_{2},c_{n-2},c_{n-1},c_{n}\!}$, and the columns signify ${\displaystyle d_{0},d_{1},d_{2},d_{n-2},d_{n-1},d_{n}\!}$.

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 ${\displaystyle d_{0}\!}$:

j=0: ${\displaystyle d_{0}=2C_{2}+ac_{1}+bc_{0}\!}$ (1)

Equation associated with ${\displaystyle d_{1}\!}$:

j=1: ${\displaystyle d_{1}=6c_{3}+2ac_{2}+bc_{1}\!}$ (2)

Equation associated with ${\displaystyle d_{2}\!}$:

j=2: ${\displaystyle d_{2}=12c_{4}+3ac_{3}+bc_{2}\!}$ (3)

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

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

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

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

Equation associated with ${\displaystyle d_{n}\!}$:

j=n: ${\displaystyle d_{n}=bc_{n}\!}$ (n)

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

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