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.
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