Chapter 4, Problem 54C

Explanation of Solution

a.

Simple method to compute $p\left(x\right)$:

Let us consider the “$\text{p}\left(\text{x}\right)$” polynomial of degree “n”.

The method to compute the polynomial of degree is given below:

//Define the simplePolynomial() method

public static double simplePolynomial(double a[], double n)

{

  //Declare the variable

double sum = 0;

  //Loop executes until the length of array

for(int i=0; i<a.length; i++)

  //Compute the polynomial of degree

  sum = sum + a[i] * Math.pow(n, a.length - i - 1);

//Return the polynomial of degree

return sum;

}

Running time:

In the above code, the computation of polynomial of degree is given below:

• For each and every array term,
  • a[0] is “0” operation.
  • $\text{a}\left[\text{1}\right]\text{× x}$ is “1” operation.
  • $\text{a}\left[2\right]\text{× x × x}$ is “2” operations.
  • $\text{a}\left[3\right]\text{× x × x × x}$ is “3” operations

Explanation of Solution

b.

Method to compute $p\left(x\right)$ in $O\left(n\left(logn\right)\right)$ time:

Let us consider the “$\text{p}\left(\text{x}\right)$” polynomial of degree “n”.

The method to compute the polynomial of degree is given below:

//Define the Polynomial() method

public static double Polynomial(double a[], double n)

{

  //Declare the variable

double sum = 0;

  int i=0;

  //Loop executes until the length of array

while(i<a.length)

{

  //Compute the polynomial of degree

  sum = sum + a[i] * Math.pow(n, a.length - i - 1);

  //Increment "i" by "1"

  i = i++;

  }

//Return the polynomial of degree

return sum;

}

Running time:

In the above code, the computation of polynomial of degree is given below:

• For each and every array term,
  • a[0] is “0” operation.
  • $\text{a}\left[\text{1}\right]\text{× x}$ is “1” operation

Explanation of Solution

c.

Horner method to compute $p\left(x\right)$:

Let us consider the “$\text{p}\left(\text{x}\right)$” polynomial of degree “n”.

The method to compute the polynomial of degree is given below:

//Define the HornerPolynomial() method

public static double HornerPolynomial(double a[],double n)

{

  //Declare the variable

double sum = 0;

  //Loop executes until the length of array

for(int i=1;i< a