NumericalMethod

some algorithm and code of numerical method

Algorithm: Bisection Method

Start
Function Definition function f(x) - Define the function for which you want to find the root.

Function bisection(a, b, error) Input: a, b (interval [a, b] where the root lies), error (desired precision) Output: c (the estimated root)

3.1. Check

if( f(a) * f(b) >= 0){
        print "Error: One of the interval endpoints must be negative.";
         Return NaN (Not a Number).
}

3.3. Set c = a

3.4. While (b - a) / b >= error:

3.4.1. Set c = (a + b) / 2

3.4.2. Check if f(c) == 0.0:
       If true, break the loop.

3.4.3. Else if f(c) * f(a) < 0:
       Set b = c

3.4.4. Else:
       Set a = c

3.5. Return c as the estimated root.

Function Main()

4.1. Declare variables a, b, error, and root.

4.2. Input interval [a, b] and the desired precision (error).

4.3. Call the bisection function with arguments a, b, and error, and store the result in root.

4.4. Check if the result is a valid number (not NaN).
```
4.4.1. If true, print "Root = " + root.
4.4.2. Else, print "Error."
```
End

Horner's Rule:

Horner's rule for polynomial division is an algorithm used to simplify the process of evaluating the polynomial f(x) at a certain value X=X0 by dividing the polynomial into monomials (polynomial of 1st degree). Each monomial involves a maximum of 1 multiplication and 1 addition. The result of 1 monomial is added to the result obtained from the next monomial.

Algorithm:

Start
Enter the degree of polynomial, say n.
Enter the value at which the polynomial is to be evaluated.
Enter the coefficient of the equation.
initially, set bn = an;
while(n>0){ bn-1 = an-1 + bn*x }
print(bn);

Algorithm For Lagrange’s interpolation

start
read the no.of data, say n.
read the value at which value is needed, say y.
read the avilable data points x and f(x).
perform,

for( i=0; i<n; i++){
    for(j=0; j<n; j++){
        if(i!==j)
          L[i] = L[i] + (y - x[j]) / (x[i]-x[j])
    }
}

then,

for(i=0; i<n; i++){
    sum = sum + L[i] * f[i];
}

print interpolation value "sum" at y.

prabeshdulal/NumericalMethod

NumericalMethod

Algorithm: Bisection Method

Horner's Rule:

Algorithm:

Algorithm For Lagrange’s interpolation