Recurrence section 6.2 of unit 6 explained
- 1. CSE 2813 Discrete Structures Solving Recurrence Relations Section 6.2
- 2. CSE 2813 Discrete Structures Degree of a Recurrence Relation • The degree of a recurrence relation is k if the sequence {an} is expressed in terms of the previous k terms: an c1an-1 + c2an-2 + … + ckan-k where c1, c2, …, ck are real numbers and ck 0 • What is the degree of an 2an-1 + an-2 ? • What is the degree of an an-2 + 3an-3 ? • What is the degree of an 3an-4 ?
- 3. CSE 2813 Discrete Structures Linear Recurrence Relations • A recurrence relation is linear when an is a sum of multiples of the previous terms in the sequence • Is an an-1 + an-2 linear ? • Is an an-1 + a2 n-2 linear ?
- 4. CSE 2813 Discrete Structures Homogeneous Recurrence Relations • A recurrence relation is homogeneous when an depends only on multiples of previous terms. • Is an an-1 + an-2 homogeneous ? • Is Pn (1.11)Pn-1 homogeneous ? • Is Hn 2Hn-1 + 1 homogeneous ?
- 5. CSE 2813 Discrete Structures Solving Recurrence Relations • Solving 1st Order Linear Homogeneous Recurrence Relations with Constant Coefficients (LHRRCC) – Derive the first few terms of the sequence using iteration – Notice the general pattern involved in the iteration step – Derive the general formula – Now test the general formula on some previously calculated (by iteration) terms
- 6. CSE 2813 Discrete Structures Solving 2nd Order LHRRCC • Form: an c1an-1 + c2an-2 with some constant values for a0 and a1 • Assume that the solution is an rn, where r is a constant and r 0
- 7. CSE 2813 Discrete Structures Step 1 • Solve the characteristic quadratic equation r2 – c1r – c2 = 0 to find the characteristic roots r1 and r2 2 4 2 2 1 1 2 , 1 c c c r
- 8. CSE 2813 Discrete Structures Step 2 • Case I: The roots are not equal an = 1r1 n + 2r2 n • Case II: The roots are equal (r1=r2=r0) an = 1r0 n + 2nr0 n
- 9. CSE 2813 Discrete Structures Step 3 • Apply the initial conditions to the equations derived in the previous step. – Case I: The roots are not equal a0 = 1r1 0 + 2r2 0 = 1 + 2 a1 = 1r1 1 + 2r2 1 = 1r1 + 2r2 – Case II: The roots are equal a0 = 1r0 0 + 20r0 0 = 1 a1 = 1r0 1 + 21r0 1 = (1+2)r0
- 10. CSE 2813 Discrete Structures Step 4 • Solve the appropriate pair of equations for 1 and 2.
- 11. CSE 2813 Discrete Structures Step 5 • Substitute the values of 1, 2, and the root(s) into the appropriate equation in step 2 to find the explicit formula for an.
- 12. CSE 2813 Discrete Structures Example • Solve the recurrence relation: an 4an-1 4an-2 where a0 a1 1 • Solve the recurrence relation: an an-1 + 2an-2 where a0 2 and a1 7
- 13. CSE 2813 Discrete Structures Exercises • 1, 3