RECEQUN

RECEQUN - RECURRENCE EQUATION We solve recurrence equations...

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Unformatted text preview: RECURRENCE EQUATION We solve recurrence equations often in analyzing complexity of algorithms, circuits, and such other cases. A homogeneous recurrence equation is written as: a t n + a 1 t n-1 + . . . . + a k t n-k = 0. Solution technique: Step 1 : Set up a corresponding Characteristic Equation: a x n + a 1 x (n-1) + …. + a k x (n-k) = 0, x (n-k) [a x k + a 1 x (k-1) + … +a k ] = 0, a x k + a 1 x k-1 + . . . . + a k = 0 [ for x =/= 0] Step 2 : Solve the characteristic equation as a polynomial equation. Say, the real roots are r 1 , r 2, . . . . , r k . Note, there are k solutions for k-th order polynomial equation. Step 3 : The general solution for the original recurrence equation is: t n = ∑ i=1 k c i r i n Step 4 : Using initial conditions (if available) solve for the coefficients in above equation in order to find the particular solution . Example 1 : t n – 3t n-1 – 4t n-2 = 0, for n >= 2. {Initial condition: t0=0, t1=1} Characteristic equation: x n – 3x (n-1) – 4x (n-2)...
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RECEQUN - RECURRENCE EQUATION We solve recurrence equations...

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