Linear Recurrence Equation
Download
Report
Transcript Linear Recurrence Equation
Linear Recurrence Equation
Example. Fibonacci Sequence: fn= fn-1 +fn-2, with f0 = 0, f1 = 1
f n 1 1 f n
f 1 0 f
n 1
n 1
or
n 1
f n 1 1 f1
f 1 0 f
0
n 1
Thus, computing fn is equivalent to computing
using parallel prefix computation.
1 1
1 0
n1
,
LU Decomposition of Tri-diagonal Matrix
d1 f1
e d f
2
2
A 2
0 en
1
m
2
f n 1
dn
mn
u1
1 0
f1
u2
fn
u
LU
u 1 d1
How to solve?
Special case of
rational fraction
ei f i 1
ui di
u i 1
a i x i 1 bi
xi
c i x i 1 d i
mi= ei / ui-1 (2 < i < n)
(2 < i < n)
xi=(a, b, c, d) • xi
(a, b, c, d) (a’, b’, c’, d’) ?
ai 1 xi 2 bi 1
ci 1 xi 2 d i 1
ai 1 xi 2 bi 1
i ci 1 xi 2 d i 1
ai xi 1 bi
xi
ci xi 1 d i c
ai
bi
di
ai ai 1 xi 2 ai bi 1 bi ci 1 xi 2 bi d i 1
ci ai 1 xi 2 ci bi 1 d i ci 1 xi 2 d i d i 1
ai ai 1 bi ci 1 xi 2 ai bi 1 bi d i 1
ci ai 1 d i ci 1 xi 2 cibi 1 d i d i 1
Or,
(a, b, c, d) • (a’, b’, c’, d’)
= (aa’+bc’, ab’+bd’, ca’+dc’, cb’+dd’)
“ • ” is associative.
Thus, we can compute the operation.
ui
d i u i 1 eif i 1
u i 1