Principles of Mathematics in Operations Research

198 14 Special Transformations

y(k) = Aky 0 and A^0 = I.

When A is singular, there does not exist a unique solution y(—1) satisfying
Ay( — 1) = 2/o- When yl is non-singular,

j,(fc) = ^-^1 J/(A:+l).

Then, j/(-l) = A^yo, y(-2) = ^~^2 2/o, ••• where ^-fc = A^A-^^1 =
(^-^1 )fc, A = 2,3,... Recall that, if A = SJS'^1 then Ak = S^S'^1. Then,

y(k) = SJS-^1 y 0 , k = 0,1,...

For the non-homogeneous case,

y(fc + 1) = Aj/(fc) +/(fc).

If >1 is nonsingular,

2/(fc) = ^fc!/o+p(*0,

where p(fc + 1) = Ap(k) + f(k); p(0) = 0. This yields

-l

p(k) = -YlAk~^1 - vf{.v).

v—k

Example 14.3.1 For k = 0,1,...,

Vi(k + l) = y 2 (k) + l, yi(0) = 3,

y 2 {k + 1) = yi(k) + 1, 3/ 2 (0) = 1.

A

k

= l

l + (-l)fc l-(-l)fc

1- (-l)fc l + (-l)fc , A

ky

0 =

2+(-l)fe

2-(-l)fc

fc-i

p(*) = « £

«=0

A;-M + (-l)"(2-fc + w)

fc-u-(-l)u(2-fc + u)

W^e know,

>£<*_,,, >*G+I> and

M=0

fc-i fc-i

|(2 - *) ^C-

1

)" + | S "(-

1

)" = | - §(-!)

u=0 u=0

1 r 2fc^2 -3(-l)fc + 3

_2k^2 + 4k + 3{-l)k - 3

5

P(k) = -

y(k) = k^2 + k + (-!)* +