Download presentation

Presentation is loading. Please wait.

Published byLauryn Lavell Modified over 3 years ago

1
The Polynomial ADT By Karim Kaddoura For CS146-6

2
Polynomial ADT What is it? ( Recall it from mathematics ) An example of a single variable polynomial: 4x 6 + 10x 4 - 5x + 3 Remark: the order of this polynomial is 6 (look for highest exponent)

3
Polynomial ADT (continued) Why call it an Abstract Data Type (ADT)? A single variable polynomial can be generalized as:

4
Polynomial ADT (continued) …This sum can be expanded to: a n x n + a (n-1) x (n-1) + … + a 1 x 1 + a 0 Notice the two visible data sets namely: (C and E), where C is the coefficient object [Real #]. and E is the exponent object [Integer #].

5
Polynomial ADT (continued) Now what? By definition of a data types: A set of values and a set of allowable operations on those values. We can now operate on this polynomial the way we like…

6
What kinds of operations? Here are the most common operations on a polynomial: Add & Subtract Multiply Differentiate Integrate etc… Polynomial ADT (continued)

7
Why implement this? Calculating polynomial operations by hand can be very cumbersome. Take differentiation as an example: d(23x 9 + 18x 7 + 41x 6 + 163x 4 + 5x + 3)/dx = (23*9)x (9-1) + (18*7)x (7-1) + (41*6)x (6-1) + …

8
Polynomial ADT (continued) How to implement this? There are different ways of implementing the polynomial ADT: Array (not recommended) Double Array (inefficient) Linked List (preferred and recommended)

9
Polynomial ADT (continued) Array Implementation: p1(x) = 8x 3 + 3x 2 + 2x + 6 p2(x) = 23x 4 + 18x - 3 6238 0 2 Index represents exponents -3180023 0 42 p1(x)p2(x)

10
This is why arrays arent good to represent polynomials: p3(x) = 16x 21 - 3x 5 + 2x + 6 Polynomial ADT (continued) 6200-30 016 ………… WASTE OF SPACE!

11
Polynomial ADT (continued) Advantages of using an Array: only good for non-sparse polynomials. ease of storage and retrieval. Disadvantages of using an Array: have to allocate array size ahead of time. huge array size required for sparse polynomials. Waste of space and runtime.

12
Polynomial ADT (continued) Double Array Implementation: Say you want to represent the following two polynomials: p1(x) = 23x 9 + 18x 7 - 41x 6 + 163x 4 - 5x + 3 p2(x) = 4x 6 + 10x 4 + 12x + 8

13
p1(x) = 23x 9 + 18x 7 - 41x 6 + 163x 4 - 5x + 3 p2(x) = 4x 6 + 10x 4 + 12x + 8 Polynomial ADT (continued) 2318-41163-53410128 9764106410 Coefficient Exponent Start p1(x)Start p2(x) End p1(x)End p2(x) 0 28

14
Polynomial ADT (continued) Advantages of using a double Array: save space (compact) Disadvantages of using a double Array: difficult to maintain have to allocate array size ahead of time more code required for misc. operations.

15
Polynomial ADT (continued) Linked list Implementation: p1(x) = 23x 9 + 18x 7 + 41x 6 + 163x 4 + 3 p2(x) = 4x 6 + 10x 4 + 12x + 8 23 918741618730 4 610412180 P1 P2 NODE (contains coefficient & exponent) TAIL (contains pointer)

16
Polynomial ADT (continued) Advantages of using a Linked list: save space (dont have to worry about sparse polynomials) and easy to maintain dont need to allocate list size and can declare nodes (terms) only as needed Disadvantages of using a Linked list : cant go backwards through the list cant jump to the beginning of the list from the end.

17
Polynomial ADT (continued) Adding polynomials using a Linked list representation: (storing the result in p3) To do this, we have to break the process down to cases: Case 1: exponent of p1 > exponent of p2 Copy node of p1 to end of p3. [go to next node] Case 2: exponent of p1 < exponent of p2 Copy node of p2 to end of p3. [go to next node]

18
Polynomial ADT (continued) Case 3: exponent of p1 = exponent of p2 Create a new node in p3 with the same exponent and with the sum of the coefficients of p1 and p2.

19
Polynomial ADT (continued) Introducing Horners rule: -Suppose for simplicity we use an array to represent the following non-sparse polynomial: 4x 3 + 10x 2 + 5x + 3 - Place it in an array, call it a[i], and compute it…

20
Polynomial ADT (continued) 4x 3 + 10x 2 + 5x + 3 A general (and inefficient) algorithm: int Poly = 0; int Multiply; for (int i=0; i < a.Size; i++) { Multiply =1; for (int j=0; j*
{
"@context": "http://schema.org",
"@type": "ImageObject",
"contentUrl": "http://images.slideplayer.com/6/1607588/slides/slide_20.jpg",
"name": "Polynomial ADT (continued) 4x 3 + 10x 2 + 5x + 3 A general (and inefficient) algorithm: int Poly = 0; int Multiply; for (int i=0; i < a.Size; i++) { Multiply =1; for (int j=0; j
*

21
Polynomial ADT (continued) 4x 3 + 10x 2 + 5x + 3 Now using Horners rule algorithm: int Poly = 0; for (int i = (a.Size-1); i >= 0 ; i++) { Poly = x * Poly + a[i]; } iPoly 34 24x + 10 14x2 + 10x + 5 04x3 +10x2 + 5x +3 Time Complexity O(n) MUCH BETTER!

22
Polynomial ADT (continued) So what is Horners rule doing to our polynomial? instead of: ax 2 + bx + c Horners simplification: x(x(a)+b)+c

23
Polynomial ADT (continued) So in general a n x n + a (n-1) x (n-1) + … + a 1 x 1 + a 0 EQUALS: ( ( (a n + a (n-1) )x + a (n-2) ) x + … + a 1 ) x + a 0

Similar presentations

Presentation is loading. Please wait....

OK

Review Pseudo Code Basic elements of Pseudo code

Review Pseudo Code Basic elements of Pseudo code

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on product design and development Ppt on artificial intelligence techniques Latest ppt on global warming Ppt on forward rate agreements Ppt on mathematics programmed instruction Ppt on c language fundamentals grade Ppt on network switching methods Ppt on no plastic bags Ppt on rural entrepreneurship in india Ppt on taking notes