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CSC 427 Final Exam
创建日期：2024-04-23 17:05:19
标签： CSC 427
Final Exam

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MA/CSC 427 Final Exam 

How to submit your exam:
1. To answer the questions in this final, you may refer to notes, books, or other resources. Because of
this accessibility, your answers should be as complete as possible. However, the work must be done
by individuals. Do not exchange ideas or engage discussions with others.
2. Final answers must be typed up and submitted as one single PDF file. Any handwritten notes will not
be graded.

4. Mark the Subject line of your email by ”MA427 Final Your Name”.
5. Submission must be received by 2:30 PM, Wednesday, December 8, 2021, which is strictly
enforced by the time stamp of your email. Early submission is always welcome.
1. (20 pts; 5 pts each) Explain in words or demonstrate by examples the following terms:
(a) Round-off error of a computer.
(b) Truncation error of a numerical scheme.
(c) Numerical stability of an algorithm.
(d) Conditioning of a mathematical problem.
2. (10 pts) Tabulate all positive numbers in fractional form that can be exactly represented in the system
F (2, 2, 2, 1), where base β = 2, mantissa length t = 2 with normalization, exponent length q = 2, and
exponent shift by 2. Note that with normalization, the first bit in the mantissa is hidden.
3. (15 pts; 5 pts each) Given a set of points {(xi, fi)}ni=0 with distinct nodes {xi}ni=0,
(a) There are at least three ways to compute the interpolating polynomial. Describe what these
methods are.
(b) What are the two error formulas for polynomial interpolation?
(c) Construct by hand the polynomial that interpolates the points (−1, 3), (0,−2), (2, 1), (5,−1).
Show the steps.
4. (20 pts; 4 pts each) Concerning the numerical integration:
(a) Describe the explicit formulation of the composite Simpson’s rule.
(b) Carefully elaborate why a composite rule can do a better job for numerical integration.
(c) What is the fundamental difference between the Newton-Cotes quadrature and the Gaussian
quadrature?
(d) The derivation of the Gaussian quadrature is based on three fundamental principles by which the
theory of orthogonal polynomials becomes an critical tool. Describe without proof these basic
theorems.
(e) Given a standard Gaussian quadrature,
∫ 1
−1 g(t) dt ≈
∑n
i=1 wig(ξi), with ξi ∈ [−1, 1]., how an
integral over a general interval
∫ b
a
f(x) dx can be handled numerically?
5. (10 pts; 5 pts each) Concerning the numerical differentiation:
(a) How to estimate f ′′(a) by the central difference formula for?
(b) Use this central difference formula for f ′′(a) to explain the limitation on the step size h when
doing numerical differentiation.
6. (15 pts; 5 pts each) Concerning the Numerical ODE methods:
(a) Describe the format of a general R-stage Runge-Kutta method.
(b) Describe the format of a general linear (p+ 1)-step method.
(c) Name three fundamental differences, other than just their different appearance, between the RK
method and the multi-step method.
7. (10 pts; 5 pts each) Suppose that the 2-stage Runge-Kutta method
yn+1 = yn + hf
(
xn +
h
2
, yn +
h
2
f(xn, yn)
)
is applied to the test problem y′ = λy.
(a) Show that the numerical solution satisfies
yn =
(
1 + h¯+
h¯2
2
)n
y0,
where h¯ = λh.
(b) Suppose λ = −1, what is the maximal step size h allowable to ensure the stable?
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