UCB Math 128A, Spring 2009: Final Exam
May 16, 2009
Name:
SID:
GSI:
•No books, no notes, no calculators
•Justify all answers
•Do all of the 6 problems
•Exam time 3 hours
Grading
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Practice Final Exam - Numerical Analysis | MATH 128A, Exams of Mathematical Methods for Numerical Analysis and Optimization
Material Type: Exam; Class: Numerical Analysis; Subject: Mathematics; University: University of California - Berkeley; Term: Spring 2009;
Typology: Exams
Pre 2010
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UCB Math 128A, Spring 2009: Final Exam
May 16, 2009
Name:
SID:
GSI:
- No books, no notes, no calculators
- Justify all answers
- Do all of the 6 problems
- Exam time 3 hours
Grading
(a) Given a > 0, what (positive number) is the following iteration in-
tended to compute?
pn+1 =
p
3 n + 3apn
3 p^2 n + a
(b) Assuming that the sequence {pn}
∞ n=0 converges, find the order of
convergence.
Find the total number of floating point operations (Additions + Sub-
tractions + Multiplications + Divisions) for the following computations:
(a) λ = x
t Ax where A is an n × n matrix and x is an n × 1 vector.
(b) The matrix product C = AB where A is n × m and B is m × p.
(c) Backward substitution using the following MATLAB code:
x=b;
for i=n:-1:
for j=i+1:n
x(i)=x(i)-A(i,j)*x(j);
end
x(i)=x(i)/A(i,i);
end
(a) Derive the difference formula for f
′′ (x 0 ) with error term:
f
′′ (x 0 ) =
f (x 0 + h) − 2 f (x 0 ) + f (x 0 − h)
h^2
h
2
f
(4) (ξ), ξ ∈ [x 0 − h, x 0 + h].
(b) Suppose each evaluation of f gives a round-off error bounded by ε,
and that |f
(4) (x)| ≤ M. Calculate the optimal h which minimizes
the error in the computed f
′′ (x 0 ).
(a) Find a numerical integration rule of the form
− 2
f (x) dx = af (0) + bf (−1) + cf (−2)
which is exact for polynomials of the highest possible degree.
(b) Use the result of (a) to design a scheme for the IVP y
′ = f (t, y),
y(0) = y 0 of the form:
wi+1 = wi− 2 + αf (ti, wi) + βf (ti− 1 , wi− 1 ) + γf (ti− 2 , wi− 2 )
(c) Evaluate the local truncation error τ using Taylor expansions, and
find the best possible m such that τ = O(h
m ).
(d) Determine if the scheme is strongly stable, weakly stable, or unstable.