Bhushan's Corner: Recurrence Relation

Monday, 23 September 2013

Recurrence Relation

Particular Solution of Recurrence Relation:

Form of f(r)	Description of f(r)	General form of Particular solution
F₁r^t+F₂r^t-1+F₃r^t-2+…+F_t+1	Polynomial of degree t	P₁r^t+P₂r^t-1+P₃r^t-2+…+P_t+1
(F₁r^t+F₂r^t-1+F₃r^t-2+…+F_t+1)β^r	Composite (Polynomial with degree t and β is not characteristic root of recurrence relation.)	(P₁r^t+P₂r^t-1+P₃r^t-2+…+P_t+1) β^r
(F₁r^t+F₂r^t-1+F₃r^t-2+…+F_t+1)β^r	Composite (Polynomial with degree t and β is characteristic root of recurrence relation with multiplicity m.)	r^m(P₁r^t+P₂r^t-1+P₃r^t-2+..+P_t+1)β^r

Homogeneous solution of Recurrence Relation:

Description of Characteristic equation:	Homogeneous solution:
Has k distinct characteristic root	a_r^(h)₌A₁ α^r₁ + A₂ α^r₂ + A₃ α^r₃ +….+ A_k α^r_k
Has some multiple roots with multiplicity m	a_r^(h)₌(A₁r^m-1+ A₂r^m-2+ A₃r^m-3+...+ A_m) α^r₁

1 comment:

Unknown24 September 2013 at 05:45
Thank you sir !
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