PhD Qualify Exam in Numerical Analysis October 18, 2017
1. (2010 Fall, Easy) Let{xn}∞n=0be a sequence generated by xn+1 = g(xn), where g(x) is a function defined on a given interval [a, b] ⊂ R.
(a) (5%) Give a sufficient condition for g(x) so that it has a fixed point in [a, b].
(b) (5%) Give a sufficient condition for g(x) so that{xn}∞n=0 is conver- gent of order k, where k is a positive integer.
(c) (5%) Show that the Newton’s iteration is locally quadratically con- vergent provided the iteration converges to a simple root.
2. (2017 Spring, Average)(10%) Suppose A∈ Rn×nis nonsingular and that we have solutions to linear systems Ax = b and Ay = g where b, g ∈ Rn are given. Show how to solve the system
[ A g hT α
] [ x µ
]
= [ b
β ]
in O(n) flops, where α, β ∈ R and h ∈ Rn are given and the enlarged matrix
[ A g hT α
]
is nonsingular.
3. (2006 Fall, Easy)10%Give an example to illustrate that (a+b)c̸= ac+bc may happen in a real calculator.
4. (Average) Consider the 2× 2 matrix A =
[ 1 ρ
−ρ 1 ]
(a) (7%) Under what conditions will Gauss-Seidal iteration converge with this matrix?
(b) (8%) For what range of ϖ will the SOR method converge?
(c) (5%) What is the optimal choice for the parameter ϖ?
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5. (Average) We learn in calculus that e = lim
h→0(1 + h)1h.
(a) (10%)Assume that there are constants K1, K2, . . . such that e = (1 + h)1h + K1h + K2h2 + K3h3+· · · .
Use extrapolation on the approximations to produce an O(h3) approximation to e when h is sufficient small.
(b) (5%)Do you think that the assumption in part 6.(a) is reasonable?
6. (2010 Fall, Easy)(15%)Is it possible to use af (x + h) + bf (x) + cf (x−h) with suitably chosen coefficients a, b, c to approximate f′′′(x)? How many function values at least are required to approximate f′′′(x)?
7. (2011 Spring, Challenged) (15%)Consider the initial-value problem y′ = f (t, y), for a≤ t ≤ b with y(a) = α
Let
w0 = α
wi+1 = wi+ hϕ(ti, wi, h) for i > 0,
and we0 = α
e
wi+1 =wei+ h eϕ(ti,wei, h) for i > 0,
give two one-step methods for approximating solution of y(t) with local truncation error τi+1(h) = O(hn) and eτi+1(h) = O(hn+1), respectively, i.e.,
y(ti+1) = y(ti) + hϕ(ti, y(ti), h) + O(hn+1) and
y(ti+1) = y(ti) + h eϕ(ti, y(ti), h) + O(hn+2).
Please use these two methods to construct an adaptive step-size control method for the considering initial-value problem.
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