Define the terms true error and relative error? Use Horner’ method to evaluate polynomial \(2x^3 – 3x^2 + 5x – 2\) at x = 3 and write down its algorithm.

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True Error is denoted by E is the difference between the true value and the approximate value.

True Error = True Value – Approximate value

Relative Error is denoted by Er and is defined as the ratio between the true error and true value.

\(Relative \enspace True \enspace Error = \frac{True \enspace Error}{True \enspace Value}\)

 

Solution:

We know that

a3 = 2, a2 = -3, a1 = 5, a0 = 2

Now, now sequence of constants can be determined by using recursive function as below:

b3 = a3 = 2

b2 = a2 + b3 * x = -3 + 2 * 3 = 3

b1 = a1 + b2 * x = 5 + 3 * 3 = 14

b0 = a0 + b1 * x = -2 + 14 * 3 = 40

Thus, p(3) = 40

Algorithm of Horner’s Method:

1. Start
2. Enter degree of polynomial, say n
3. Enter the value at which polynomial to be evaluated, x
4. Initially set bn = an
5. while n > 0
     bn-1 = an-1 + bn * x
6. End While
7. Display the value of b0, which is the value of the polynomial at x
8. End
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