Polynomial representation using Arrays



A polynomial is a mathematical expression that contains more than two algebraic terms. It is a sum of several terms that contain different powers of the same variable. $p(x)$ is the representation of a polynomial. A polynomial expression representation consists of at least one variable, and typically includes constants and positive constants as well. It is represented as $a_nx^n+ a_{n-1}x^{n−1}+a_{n-2}x^{n−2}+.............+a_0x^0$, where $x$ is the variable, $(a_0,a_1,a_2,....................,a_n)$, are the coefficients and $n$ is the degree of the polynomial. The coefficients must be a real number.It is necessary for the polynomial expression that each expression consists of two parts:

Coefficient part
Exponent part

Example:

$p(x) = 21x^3 + 3x^2 + 14x^1 + 21x^0$.
Here, 21, 3, 14, and 21 are the coefficients, and 3,2,1 and 0 are the exponential values.

Representation of polynomial with single variable -

Consider a polynomial $−4+7x+6x^2$ then we can write as: $−4x^0+7x^1+6x^2$

For the one-dimensional array, store the coefficients in the index given by the coefficient of the polynomial.

In the array representation, the array first the first element stores the coefficient of the term with the lowest exponent and the last element contains the coefficient of the term with the highest exponent.

In general the polynomial

 $P(x)= a_0x^0+a_1x^1+\ldots+ a_{n-1}x^{n−1}+a_nx^n$
can be represented as


Polynomial Addition using Array

Consider two different polynomials 
$ P_1 = 3x^1+5x^2+7x^3$ and $P_2= 7x^0+3x^1+4x^2$. The degree of $p_1$ is 3 and the degree of $P_2$ is 2. The steps to add the polynomials are listed below.

1.First identify the highest degree polynomial. The degree of the resultant polynomial is same as the polynomial with the highest degree.
2.Store the coefficient in the index specified by the exponents of the polynomial.
3.Add the coefficients stored in one array with the corresponding index positions in the other array and store the result in the same index position in the resultant array.


C program to add two polynomials  
/*******************************************************************************/
#include <stdio.h>
// Function to add two polynomials and store the result in a third array
void addPolynomials(int poly1[], int degree1, int poly2[], int degree2, int result[]) {
    int maxDegree = (degree1 > degree2) ? degree1 : degree2;

    for (int i = 0; i <= maxDegree; i++) {
        int coef1 = (i <= degree1) ? poly1[i] : 0;
        int coef2 = (i <= degree2) ? poly2[i] : 0;

        result[i] = coef1 + coef2;
    }
}

// Function to print a polynomial
void printPolynomial(int poly[], int degree) {
    for (int i = degree; i >= 0; i--) {
        if (poly[i] != 0) {
            if (i < degree) {
                if (poly[i] > 0) {
                    printf(" + ");
                } else {
                    printf(" - ");
                }
            }
            printf("%d", (poly[i] < 0) ? -poly[i] : poly[i]);
            if (i > 0) {
                printf("x^%d", i);
            }
        }
    }
    printf("\n");
}

int main() {
    // Define the coefficients and degrees of two polynomials
    int poly1[] = {-7, 5, -2, 3}; // P(x) = -7 + 5x - 2x^2 + 3x^3
    int degree1 = 3;

    int poly2[] = {-1, 4, 2}; // Q(x) = -1 + 4x + 2x^2
    int degree2 = 2;

    // Determine the maximum degree among the two polynomials
    int maxDegree = (degree1 > degree2) ? degree1 : degree2;

    // Create an array to store the result polynomial
    int result[maxDegree + 1];

    // Add the two polynomials
    addPolynomials(poly1, degree1, poly2, degree2, result);

    // Print the polynomials and the result
    printf("Polynomial 1: ");
    printPolynomial(poly1, degree1);

    printf("Polynomial 2: ");
    printPolynomial(poly2, degree2);

    printf("Sum: ");
    printPolynomial(result, maxDegree);

    return 0;
}


Note that in this representation we assume that the polynomial is single variable and the exponents are positive.

here is the more general representation of a polynomial with array of structures.
We can represent a polynomial as a collection of terms and every term is having a coefficient and exponent. So, we can represent this as a list of terms having coefficients and exponents. So let us prepare a list of 2 things that is coefficients and exponent,

consider the polynomial
$p(x)=3x^5+2x^4+5x^2+2x+7$
Its represented like this

C program to add two polynomials  using structure representation
/*******************************************************************************/

#include <stdio.h>
#include <stdbool.h>

// Structure to represent a single term in the polynomial
typedef struct {
    int coefficient;
    int exponent;
} Term;

// Function to add two polynomials and store the result in a third array
void addPolynomials(Term poly1[], int degree1, Term poly2[], int degree2, Term result[], int *resultDegree) {
    int maxDegree = (degree1 > degree2) ? degree1 : degree2;
    *resultDegree = maxDegree;

    for (int i = 0; i <= maxDegree; i++) {
        int coef1 = 0, coef2 = 0;

        // Find the coefficients for poly1 and poly2 at exponent i
        for (int j = 0; j <= degree1; j++) {
            if (poly1[j].exponent == i) {
                coef1 = poly1[j].coefficient;
                break;
            }
        }

        for (int j = 0; j <= degree2; j++) {
            if (poly2[j].exponent == i) {
                coef2 = poly2[j].coefficient;
                break;
            }
        }

        result[i].coefficient = coef1 + coef2;
        result[i].exponent = i;
    }
}

// Function to print a polynomial
void printPolynomial(Term poly[], int degree) {
    bool firstTerm = true;

    for (int i = degree; i >= 0; i--) {
        int coef = poly[i].coefficient;
        int exponent = poly[i].exponent;

        if (coef != 0) {
            if (!firstTerm) {
                if (coef > 0) {
                    printf(" + ");
                } else {
                    printf(" - ");
                }
            }

            printf("%d", (coef < 0) ? -coef : coef);

            if (exponent != 0) {
                printf("x");

                if (exponent != 1) {
                    printf("^%d", exponent);
                }
            }

            firstTerm = false;
        }
    }

    if (firstTerm) {
        printf("0"); // Print "0" for the zero polynomial
    }

    printf("\n");
}

int main() {
    // Define the coefficients and exponents of two polynomials
    Term poly1[] = {{-7, 0}, {5, 1}, {-2, 2}, {3, 3}}; // P(x) = -7 + 5x - 2x^2 + 3x^3
    int degree1 = 3;

    Term poly2[] = {{-1, 0}, {4, 1}, {2, 2}}; // Q(x) = -1 + 4x + 2x^2
    int degree2 = 2;

    // Determine the maximum degree among the two polynomials
    int maxDegree = (degree1 > degree2) ? degree1 : degree2;

    // Create an array to store the result polynomial
    Term result[maxDegree + 1];

    int resultDegree;

    // Add the two polynomials
    addPolynomials(poly1, degree1, poly2, degree2, result, &resultDegree);

    // Print the polynomials and the result
    printf("Polynomial 1: ");
    printPolynomial(poly1, degree1);

    printf("Polynomial 2: ");
    printPolynomial(poly2, degree2);

    printf("Sum: ");
    printPolynomial(result, resultDegree);

    return 0;
}

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