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MATH2600 — MATH2601 Coursework Assessment

发布时间：2022-08-22

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MATH2600 — MATH2601

Coursework Assessment

Please scan your solutions to the three questions to a PDF ﬁle and upload it onto Gradescope in the area "Submit My Work", on the VLE. (In order to scan to PDF, Gradescope recommends using Evernote Scannablefor iOS and Genius Scanfor Android and iOS.)

Your work will be markedfor proper mathematical style, as well asfor correctness and completeness. Clearly state any resultfrom lecture notes, textbooks or other sources that you use. You are expected to submit well-presented and structured scripts, using proper mathematical writing.

In order to reduce round-oﬀ errors, keep as many digits as possible in your calculations, but your results in Questions 2 and 3 should be reported rounded to 4 or 5 decimal places.

Question 1 [10 marks]

The polynomial P (x), of degree n - 1, interpolates the n distinct points (xi, fi), i.e. P (xi) = fi , i = 1, . . . , n. Assuming that P (x) is known, we wish to construct, without using the Lagrange polynomials Lk(x), the polynomial of degree n, Q(x), that interpolates the new point (xn＋1, fn＋1) in addition to the n points (xi, fi), i = 1, . . . , n.

Let Q(x) = P (x) ＋ W (x), where W (x) is a polynomial to determine. First, show that the points xi , i = 1, . . . , n, are n roots of W . Then ﬁnd the degree of W and conclude that one can write

W (x) = C (x - xi), (1)

where the constant C is to be evaluated using Q(xn＋1) = fn＋1 . Then show that

Q(x) = P (x) ＋ ┌fn＋1 - P (xn＋1)、

(2)

Using Lagrange interpolation, ﬁnd and then simplify the polynomial P (x) that interpolates the data (- 1, 6), ( 1, 2) and (2, 3) . [3 marks]

Then, use equation (2) to ﬁnd the polynomial Q(x) that interpolates (3, -2) in addition to the three points given above, and simplify it. [3 marks]

Question 2 [10 marks]

Newton’s root-ﬁnding method is said to be fast in general. Explain what this means mathematically and ﬁnd an unseen example of nonlinear equation where Newton’s method converges rapidly. [2 marks] Write the Newton algorithm for your equation then perform a number of iterations and, using these data, analyse the error in the approximation to the exact root to show that the algorithm behaves as expected. [3 marks]

However, the convergence of Newton’s step becomes sometimes linear. Explain what this means mathematically and in what circumstances this happens. [2 marks]

Find another unseen example of nonlinear equation where Newton’s method now converges linearly. Again, write the Newton step, perform a number of iterations and, using these data, analyse the error in the approximation to the exact root to show that the algorithm behaves as expected. [3 marks]

Question 3 [10 marks]

The solution of initial value problems for diﬀerential equations can be approximated numerically by means of one-step methods (e.g. Euler or Runge-Kutta) or multi-step methods (e.g. Adams- Bashforth). Explain brieﬂy the diﬀerence between these approaches and why one might prefer to use one rather than the other (assuming two numerical schemes of the same order). [2 marks] A function, y(t), satisﬁes the ordinary diﬀerential equation

The variable t takes discrete values, tk, with a constant step-size h = tk＋1 - tk (for all k e 一). By ﬁnding a suitable quadrature formula for integration of equation (3) over the interval [tn, tn＋1], derive the three-step Adams-Bashforth scheme,

yn＋1 = yn ＋ [23f (tn, yn) - 16f (tn- 1, yn- 1 ) ＋5f (tn-2, yn-2 )] , (4)

where yn is an approximation to y(tn) . [4 marks]

Verify that y(t) = cos(t) is the solution of the initial value problem

= cos(t) - sin(t) - y, y(0) = 1. (5)

Use the three-step Adams-Bashforth scheme (4) to calculate an approximation to y(π) using a step-size of h = π/6 and calculate the absolute error from the exact solution at each step. The ﬁrst two steps should be computed using the forward Euler method. [4 marks]