Math128A: Numerical Analysis Sample Final
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Math128A: Numerical Analysis sample Final
2. (12 points)
(a) Describe a method to evaluate l1必 
. No actual calculation is required.
(b) Evaluate
l 11 l22 (x2 + g2 + xg)dxdg.
3. (12 points)
(a) From the Taylor expansion of a function f (π), derive a irst-order approximation to f、(π).
(b) use Richardson,s extrapolation method to ind a 3 point 2nd order formula.
4. (12 points)
(a) Let A and B be n x n matrices. prove or ind a counter example: If AB = 0 then A = 0 or B = 0.
(b) Let A and B be n x n matrices. prove or ind a counter example: If AB = 0 then det(A) = 0 or det(B) = 0.
5. consider the iteration
xk+1 = 2xk — axk(2), k = 0, 1, . . . ,
where a > 0 is given. show that the iteration converges quadratically to 1/a for any initial guess x0 satisfying 0 < x0 < 2/a.
6. (12 points)
(a) For a function f and distinct points a, β, and T, deine what is meant by f [a, β, T].
(b) Find the Lagrange form of the polynomial P (x) which interpolates f (x) = 4x/(x + 1)
at 0, 1, and 3.
7. For the following linear system
x - ag = 1,
ax - g = 1,
describe for which values of a the system has an ininite number of solutions, no solutions, and exactly one solution, and ind the solution when it is unique.
8. Determine the free cubic spline that apprixmates the data f (-1) = 1, f (0) = 0 and f (1) = 1.
9. (a) Deine what is meant by the local truncation error, and the local order, for a single-step method for solving the ODE,s.
(b) Derive a speciic Runge-kutta method of local order 2. show your work.
10. Given matrix A =（ a(1) 1(a) ).
(a) Find the ∞ norm and the spectral radius of A.
(b) consider the iterative method
xk+1 = Axk + c, k = 0, 1, . . . ,
Find the conditions on a so that this iteration converges for any initial x0 .
2023-08-07