Horner's method: application exercise

Horner's method is an algorithm for evaluating polynomials and finding approximate roots. It is named after the British mathematician William George Horner.
Polynomial evaluation
1. *Write the polynomial*: p(x) = a_n x^n + a_{n-1} x^{n-1} +... + a_1 x + a_0
2. *Enter the value of x*: x = c
3. *Calculate the result*: p(c) = a_n c^n + a_{n-1} c^{n-1} +... + a_1 c + a_0
Horner's algorithm
1. Initialize the result: r = a_n
2. *For i = n-1 up to 0*: r = r * c + a_i
3. *Returns r*: p(c) = r
Example
Evaluate p(x) = 3x^2 + 2x - 1 at x = 2.
Steps
1. *Initializes r = 3*
2. r = r * 2 + 2 = 3 * 2 + 2 = 8
3. r = r * 2 - 1 = 8 * 2 - 1 = 15
Result
p(2) = 15
Advantages
1. Efficient to evaluate polynomials.
2. Easy to implement.
Disadvantages
1. Only evaluate polynomials.
Applications
1. Numerical calculation.
2. Numerical analysis.
3. Engineering.
4. Sciences.
Software
1. MATLAB
2. Python (NumPy library)
3. Mathematica
4. Wolfram Alpha