- 79 Views
- Updated On :

3.G rowth of Functions. 3.1 Asymptotic notation . g ( n ) is an asymptotic tight bound for f ( n ). ``=’’ abuse. The definition of required every member of be asymptotically nonnegative. Example: . In general, . asymptotic upper bound. asymptotic lower bound . Theorem 3.1.

Related searches for 3.Growth of Functions

Download Presentation
## PowerPoint Slideshow about '3.Growth of Functions' - barbra

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### 3.Growth of Functions

- The definition of required every member of be asymptotically nonnegative.

Example: asymptotically nonnegative.

- In general,

asymptotic upper bound asymptotically nonnegative.

asymptotic lower bound asymptotically nonnegative.

Theorem 3.1. asymptotically nonnegative.

- For any two functions f(n) and g(n), if and only if and

- Transitivity asymptotically nonnegative.
- Reflexivity
- Symmetry

- Transpose symmetry asymptotically nonnegative.

Trichotomy asymptotically nonnegative.

- a < b, a = b, or a > b.
- e.g., does not always holdbecause the latter oscillates between

2.2 Standard notations and common functions asymptotically nonnegative.

- Monotonicity:
- A function f is monotonically increasing if m n implies f(m) f(n).
- A function f is monotonically decreasing if m n implies f(m) f(n).
- A function f is strictly increasing if m < n implies f(m) < f(n).
- A function f is strictly decreasing if m > n implies f(m) > f(n).

Floor and ceiling asymptotically nonnegative.

For any real x

For any integer n

For any real n>=0, and

integers a, b >0

Proof of Ceiling asymptotically nonnegative.

Proof of Floor asymptotically nonnegative.

Modular arithmetic asymptotically nonnegative.

- For any integer a and any positive integer n, the value a mod n is the remainder (or residue) of the quotient a/n :
a mod n =a - a/n n.

- If(a mod n) = (b mod n). We write a b (mod n) and say that a is equivalent to b, modulo n.
- We write a ≢b (mod n) if a is not equivalent to b modulo n.

Exponential Basics asymptotically nonnegative.

Polynomials asymptotically nonnegative. v.s. Exponentials

- Polynomials:
- A function is polynomial bounded if f(n)=O(nk)

- Exponentials:
- Any positive exponential function grows faster than any polynomial.

Prove: asymptotically nonnegative.nb=o(an)

Logarithm Notations asymptotically nonnegative.

Logarithm Basics asymptotically nonnegative.

Logarithms asymptotically nonnegative.

- A function f(n) is polylogarithmically bounded if
- for any constant a > 0.
- Any positive polynomial function grows faster than any polylogarithmic function.

Prove: asymptotically nonnegative.lgbn=o(na)

Function iteration asymptotically nonnegative.

For example, if , then

The iterative logarithm function asymptotically nonnegative.

h

- Since the number of atoms in the observable universe is estimated to be about , which is much less than , we rarely encounter a value of n such that .

Fibonacci numbers estimated to be about , which is much less than , we rarely encounter a value of

Download Presentation

Connecting to Server..