Problem Statement
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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).
As shown in p.7-21, the first equation is:
2 C 2 + a c 1 + b c 0 = d 0 {\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:
∑ 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 ] = ∑ j = 0 n d j x j {\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 c 0 , . . . , c n {\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:
A = [ 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 ] {\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 c 0 , c 1 , c 2 , c n − 2 , c n − 1 , c n {\displaystyle 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 {\displaystyle d_{0},d_{1},d_{2},d_{n-2},d_{n-1},d_{n}\!} .
Solution
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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 {\displaystyle d_{0}\!} :
j=0: d 0 = 2 C 2 + a c 1 + b c 0 {\displaystyle d_{0}=2C_{2}+ac_{1}+bc_{0}\!} (1)
Equation associated with d 1 {\displaystyle d_{1}\!} :
j=1: d 1 = 6 c 3 + 2 a c 2 + b c 1 {\displaystyle d_{1}=6c_{3}+2ac_{2}+bc_{1}\!} (2)
Equation associated with d 2 {\displaystyle d_{2}\!} :
j=2: d 2 = 12 c 4 + 3 a c 3 + b c 2 {\displaystyle d_{2}=12c_{4}+3ac_{3}+bc_{2}\!} (3)
Equation associated with d n − 2 {\displaystyle d_{n-2}\!} :
j=n-2: d n − 2 = [ c n ( n ) ( n − 1 ) + a c n − 1 ( n − 1 ) + b c n − 2 ] {\displaystyle d_{n-2}=[c_{n}(n)(n-1)+ac_{n-1}(n-1)+bc_{n-2}]\!} (n-2)
Equation associated with d n − 1 {\displaystyle d_{n-1}\!} :
j=n-1: d n − 1 = a c n n + b c n − 1 {\displaystyle d_{n-1}=ac_{n}n+bc_{n-1}\!} (n-1)
Equation associated with d n {\displaystyle d_{n}\!} :
j=n: d n = b c 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:
A = [ b a 2 0 0 0 0 b 2 a 0 0 0 0 0 b 0 0 0 0 0 0 b a ( n − 1 ) n ( n − 1 ) 0 0 0 0 b a n 0 0 0 0 0 b ] {\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}}\!}