Week 3

Posted: August 26th, 2021

                                                                        Week 3                              

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Week 3

3.1 #8b.

            Use the error formula to find a bound for the error, and compare the bound to the actual error for the cases n = 1 and n = 2.

b.   f (x) =

Hence, if f (0) then, the approximate value would be = -1

Absolute error

Estimated value = 1, thus absolute error = 1 – (-) 1 = 2.

3.2 #2a.

Use Neville’s method to obtain the approximations for Lagrange interpolating polynomials of degree two to approximate each of the following and discuss the error:

a.   f (0.43) if f (0) = 1, f (0.25) = 1.64872, f (0.5) = 2.71828, f (0.75) = 4.48169

By applying classical continuity approach, an approximation points are selected based on proximity to the target points, that is, x0 = 0.5, x1 = 0.25, x2 =0.75 and x3 =0.0 respectively. Using Neville’s method, the following tableaux is obtained:

i	xi	Pi	Pi-1, i	Pi-2, i-1,i
0	0.5	P0 = 2.718
1	0.25	P1=1.6487	P0, 1= 4.278x+ 0.579
2	0.75	P2=4.817	P1, 2 =5.666x +0.232	P0,1,2 = 5.551x2 + 0.1151x + 1.273
3	0.00	P3=1.000	P2,3 = 4.642x + 1.000	P0,1,2, 3 = 4.095x2 + 0.1151x + 1.000

 Hence, P _0,1,2,3 = 2.912x³ + 1.1827x² +2.117x + 1.000. Subsequent approximations include P _0,1 (0.43) = 2.4188, P _0,1,2  (0.43) = 2.3489 and finally P _0,1,2,3 (0.43) = 2.361.

3.3 #5a.

Use the Newton backward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials and discuss the error.

a.   f (−1/3) if f (−0.75) = −0.07181250, f (−0.5) = −0.02475000, f (−0.25) = 0.33493750, f (0) = 1.10100000

 As such we obtain;

3.4 #4b.

 Approximate f (x), and calculate the absolute error.

b.   f (x) = x4 − x3 + x2 − x + 1; approximate f (0).

Hence, if f (0) then, the approximate value would be = -1

Absolute error

Estimated value = 1, thus absolute error = 1 – (-) 1 = 2.

Write One Paragraph To Explain Which Question Caused The Most Difficulty And Why?

The first question is the most difficult. The error formula for finding a bound for the error is complex and involves many steps to get the answer. Hence, it requires a strong understanding of the whole concept to solve it.