Data Structures

FJWUDept.

BS Software Engineering

Sparse Matrix and Polynomials

Aroosa Neelum RaffiqueSaba Arshad

Group Members

3

Sparse Matrix• A matrix is sparse if many of its elements are zero• A matrix that is not sparse is dense• The boundary is not precisely defined

•Diagonal and tridiagonal matrices are sparse•We classify triangular matrices as dense

• Two possible representations•array•linked list

SPARSE MATRIXSparse matrix of dimension 7 x 7.

COLUMNS0 1 2 3 4 5 6

0 0 0 0 -5 0 0 01 0 4 0 0 0 0 72 0 0 0 0 9 0 0

ROWS 3 0 3 0 2 0 00

4 1 0 2 0 0 0 05 0 0 0 0 0 0 06 0 0 8 0 0 0 0

SOURCE CODE:

/*Program to demonstrate addition and multiplication of Two Sparse Matrix */#include < iostream.h >#include < conio.h >#define x 25class sparce{ private: int a [ x ] [ x ], b [ x ] [ x ], c [ x ] [ x ], m, n, p, q; public: void init ( ); void input ( ); void add ( ); void mul ( ); void display ( int [25][25], int, int ); void convert( int [25][25], int, int );};void sparce :: init ( ){ int i, j; for(i = 0; i < x;i + + ) for( j = 0; j < x; j + +) c [ i ] [ j ] = 0;}

cont...

void sparce :: input(){ int i,j; cout<<"\nEnter order Of First matrix::"; cin>>m>>n; cout<<"\nEnter order Of Second matrix::"; cin>>p>>q; cout<<"\nEnter"<<m*n<<"Elements Into First Matrix\n"; for(i=0;i<m;i++) for( j = 0; j < n; j + + ) cin>> a[ i ] [ j ]; cout<<"\nEnter"<<p*q<<"Elements Into Second Matrix\n"; for(i = 0; i < p ; i + + ) for ( j = 0; j < q ; j + + ) cin>>b [ i ] [ j ];}void sparce :: add ( ){ int i, j; if( m = = p && n = = q ) { for( i = 0 ; i < m ; i + + ) for( j = 0; j < n; j + + ) c[ i ] [ j ] = a [ i ][ j ] + b [ i ] [ j ]; convert( c, m, n); } else cout<<"\nAddition Is Not Possible";}

continue...

void sparce :: mul ( ){ int i, j, k; if(n = = p) { for( i = 0; i < m; i + +) for( j = 0; j < q; j + + ) for( k = 0; k < n; k + + )

c[ I ] [ j ] + = a [ I ] [ k ] * b [ k ] [ j ]; convert(c, m, n); } else cout<<"\n Multiplecation Is Not Possible";}void sparce :: display(int c[25][25], int m, int n){ int i,j; for( i = 0 ;i < m; i + + ) { for( j = 0 ; j < n ; j + + ) cout<<c [ i ] [ j ]<<"\t"; cout<<"\n"; }}

void sparce :: convert(int c[25][25], int m, int n){ int i, j, k = 1,t = 0; int sp[25][25]; for( i = 0 ; i < m ; i + +) for( j = 0 ; j < n ; j + + ) if(c [ i ] [ j ] ! = 0 ) {sp [ k ] [ 0 ] = i;sp [ k ] [ 1 ] = j;sp [ k ] [ 2 ] = c [ i ] [ j ];k + + ;t + + ; }sp[ 0 ] [ 0 ] = m;sp[ 0 ] [ 1 ] = n;sp[ 0 ] [ 2 ] = t;display( sp, k, 3);}void main ( ){ sparce ob; clrscr ( ); ob.init ( ); ob.input ( ); cout<<"\nAddition of Two Sparce Matrix\n"; ob.add ( ); ob.init ( ); cout<<"\nMultiplecation Of Two Sparce Matrix\n"; ob.mul ( ); getch ( ); }

OUTPUT:

PolynomialsPolynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions. Here are some examples:

Polynomial:

• How to implement this?

There are different ways of implementing the polynomial ADT:

• Array (not recommended)• Linked List (preferred and recommended)

Polynomial:•Array Implementation:• p1(x) = 8x3 + 3x2 + 2x + 6• p2(x) = 23x4 + 18x - 3

6 2 3 8

0 2

Index represents exponents

-3 18 0 0 23

0 42

p1(x) p2(x)

•This is why arrays aren’t good to represent polynomials:

• p3(x) = 16x21 - 3x5 + 2x + 6

Polynomial:

6 2 0 0 -3 0 0 16…………

WASTE OF SPACE!

• 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.

Polynomial:

• Linked list Implementation:

• p1(x) = 23x9 + 18x7 + 41x6 + 163x4 + 3• p2(x) = 4x6 + 10x4 + 12x + 8

23 9 18 7 41 6 18 7 3 0

4 6 10 4 12 1 8 0

P1

P2

NODE (contains coefficient & exponent)

TAIL (contains pointer)

Polynomial:

• Advantages of using a Linked list:

• save space (don’t have to worry about sparse polynomials) and easy to maintain

• don’t need to allocate list size and can declare nodes (terms) only as needed

• Disadvantages of using a Linked list :• can’t go backwards through the list• can’t jump to the beginning of the list

from the end.

Polynomial:

SOURCE CODE:/*Program To Demonstrate Addition And Multiplication Of Two Polynomial Expression */#include < iostream.h >#include < conio.h >#define n 100class poly{ private: int a[n], b[n], add[n], mul[n], p, q, at; public: void init ( );void input ( ); void process ( ); void display ( );};void poly :: init ( ){ int i; for( i = 0; i < n; i + + ) a[ i ] = b [ i ] = add[ i ] = mul[ i ] = 0;}

void poly :: input ( ){ int i;cout<<"\nEnter Degree Of First Polynomial::"; cin>>p; cout<<"\nEnter Degree Of Second Polynomial::"; cin>>q; cout<<"\nEnter Values First Polynomial\n"; for( i = 0; i <= p; i + + ) { cout<<"\nEnter X^"<<i<<" Th Coefficient"; cin>>a[ i ]; } cout<<"\nEnter Values First Polynomial\n"; for( i = 0; i <= q; i + + ) { cout<<"\nEnter X^"<<i<<" Th Coefficient"; cin>>b[ i ]; }}

void poly :: process ( ){ int i, j; if( p > q ) at = p; else at = q; for ( i = 0; i <= at; i + +) add[ i ] = a[ i ] + b[ i ]; for( i = 0; i <= p; i + + ) for( j = 0; j <= q; j + + ) mul [ i + j ] + = a [ i ] * b [ j ];}void poly :: display ( ){ int i;cout<<"\Addition Of Two Polynomial Expressions Are\n\n";for( i = at; i >=0 ; i - -)cout<<add[i]<<"X^"<<i<<"+";cout<<"\n\nMultiplecation Of Two Polynomial Expressions Are\n\n";for( i = p + q; i > = 0; i - -)cout<<mul[i]<<"X^"<< i <<"+";}

void main(){poly ob;clrscr ( );ob.init ( );ob.input ( );ob.process ( );ob.display ( );getch ( );}

OUTPUT: