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Recurrence Relations

Recurrence Relations. Chapter 6. Recursion. Defining an object in terms of itself. To recursively define a function:. Specify initial value(s) of function Give a rule for finding a function’s value for one input (n) from its values at previous inputs (n-1).

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Recurrence Relations

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  1. Recurrence Relations Chapter 6

  2. Recursion • Defining an object in terms of itself

  3. To recursively define a function: • Specify initial value(s) of function • Give a rule for • finding a function’s value for one input (n) • from its values at previous inputs (n-1)

  4. {1, 2, 4, 8, 16, … } an = 2n for n=0,1,2,… • The sequence may be defined by giving first term: a0 = 1 • And a rule for finding a term of a sequence from a previous one: an = 2an-1 for n=1, 2, 3, ….

  5. Recurrence Relation • The second part of recursive definition • A rule for • finding a function’s value for one input (n) • from its values at previous inputs (n-1)

  6. Recurrence Relation for sequence {an} • A formula that expresses an • in terms of • one or more previous terms • of sequence, a0, a1, … an-1 • The inductive part of a recursive definition.

  7. Recurrence Relation Solution • A sequence whose terms satisfy a recurrence relation • {a0, a1, a2, a3, … } • NOTE: • If we don't restrict values for the first term of a sequence, there may be many solutions to a given RR.

  8. Find a R.R. Solution (sequence satisfying a recurrence relation) • Let {an} be a solution that satisfies the recurrence relation an = an-1 - an-2 • Let a0 = 3, a1 = 5 • What are a2 and a3? • a2 = a1 - a0 = 5 - 3 = 2 • a3 = a2 - a1 = 2 - 5 = -3

  9. Is the sequence a valid solution for a Recurrence Relation? • Is sequence {an} = { 6, 9, 12, 15, … } OR an = 3n (for n>=2) • a solution for the following recurrence rel? • an = 2an-1 - an-2 for n=2,3,4,… • {an} = { 6, 9, 12, 15, 18, … } • an = 3n, an-1 = 3n - 3 • an-2 = 3n - 6 • an = 2an-1 - an-2 • 3n = 2(3n - 3) - (3n - 6) • = 6n – 6 – 3n + 6 = 3n YES

  10. Is the sequence a valid solution for a Recurrence Relation? • Is the sequence {an}, where an = n • a solution for the following recurr. rel? • an = 2an-1 - an-2 for n=3,4,… • {an} = {3, 4, 5, … } • an = n, an-1 = n - 1 • an-2 = n - 2 • an = 2an-1 - an-2 • an = 2(n - 1) - (n - 2) = n YES

  11. Recurrence Relation Example • Is the sequence {an} where an = 2n • a solution for the recurrence relation • an = 2an-1 - an-2 for n=0,1,2,3,4,… ? • {an} = { 1, 2, 4, 8, 16, 32, … } • an = 2n, an-1 = 2n-1 = 2n2-1 = 2n / 2 • an-2 = 2n-2 = 2n2-2 =2n / 4 • an = 2an-1 - an-2 • 2n = 2(2n / 2) - (2n / 4) = 3 / 4 * 2n NO

  12. Initial Conditions • Specify terms preceding the first term where the recurrence relation takes effect • A sequence is uniquely determined by • Initial conditions • recurrence relation • Without initial conditions, many solutions may exist for a given recurrence relation.

  13. Recurrence Relation Models • A person deposits $10K in savings at 11% compounded annually? • What will be in the account after 30 years? • P0 = 10,000 • Pn = P n-1 + 0.11P n-1 • = 1.11 P n-1

  14. Recurrence Relation Models • What will be in the account after 30 years? • P0 = 10,000 • Pn = 1.11 Pn-1 • P1 = 1.11 P0 • P2 = 1.11 P1 = 1.11* 1.11* P0 = (1.11)2 P0 • P3 = 1.11 P2 = (1.11)3 P0 • Pn = 1.11 Pn-1 = (1.11)n P0 • P30 = (1.11)30 P0 =(1.11)30 *10000 = $228,922.97

  15. Inclusion-Exclusion 6.5

  16. Inclusion-Exclusion Principle • The number of ways of doing two tasks • which may be done at the same time • (i.e., sets are not disjoint) • Add the number of ways to do the first task • To the number of ways to do the second task • Subtract number of ways of doing both tasks

  17. Inclusion-Exclusion Example • Of all CS students • 25 take C++ programming • 18 take discrete math • 6 take both • How many CS students are there? • 25 + 18 is too many • 25 + 18 - 6 = 37 CS students

  18. Inclusion-Exclusion • | A È B | = | A | + | B | - | A Ç B | C++ Discrete 19 6 12 25 18

  19. I-E Example • How many positive integers not exceeding 1000 are divisible by 7 or 11? • | A È B | = | A | + | B | - | A Ç B | • = ë1000/7û + ë1000/11û - ë1000/ (7*11) û • = 142 + 90 - 12 • = 220

  20. Formula for the # Element in the Union of Three Sets • | A È B È C | = ? A B C

  21. Formula for the # Element in the Union of Three Sets • | A È B È C | = | A | + | B | + | C | • - | A Ç B | - | A Ç C | - | B Ç C | • + | A Ç B Ç C |

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