PhD Qualify Exam in Numerical Analysis, March 21, 2018

PhD Qualify Exam in Numerical Analysis, March 21, 2018

1. (Average) Consider the initial value problem

(I.V.P.)

 _dy

dt = f (t, y) for t ∈ (a, b), y(a) = α.

Show that the difference method w₀ = α,

w_i+1 = w_i + a₁f (t_i, w_i) + a₂f (t_i+ δ, w_i+ βf (t_i, w_i)),

for each i = 0, 1, . . . , N − 1, cannot have local truncation error O(h³) for any choice of a1, a2, δ and β. ^(10%)

2. (Easy) The iteration equation for the secant method can be written in the simpler form

xn = f (xn−1)xn−2− f (x_n−2)xn−1

f (x_n−1) − f (x_n−2) .

Explain why, in general, this iteration equation is likely to be less accurate than the one given by

xn = xn−1− f (x_n−1)(x_n−1− x_n−2) f (x_n−1) − f (x_n−2) .

(10%)

3. (Average) Show that the numerical quadrature formula

Q(P ) =

n

X

i=1

c_iP (x_i)

can not accomplish an exact computation, provided that polynomial P (x) is of degree greater than 2n − 1, regardless of the choice of c₁, c₂, . . . , c_n and x₁, x₂, . . . , x_n. ^(10%) 4. (Average) A sequence {p_n} is said to be superlinearly convergent to p if

n→∞lim

|p_n+1− p|

|pn− p| = 0.

(a) ^(7%) Show that if p_n→ p of order α for α > 1, then {p_n} is superlinearly convergent to p.

(b) ^(8%) Show that p_n = _n¹n is superlinearly convergent to 0 but does not converge to 0 of order α for any α > 1.

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5. Let x^(k)= T x^(k−1)+ c, k = 1, 2, . . . with a given x⁽⁰⁾ be an iterative method for solving the linear system Ax = b, where A ∈ R^n×n and b ∈ Rⁿ.

(a) (Average) Show that

kx^(k)− xk ≤ kT k^kkx⁽⁰⁾− xk and

kx^(k)− xk ≤ kT k^k

1 − kT kkx⁽¹⁾− x⁽⁰⁾k,

where x is the fixed point of the iteration x^(k) = T x^(k−1) + c, provided that kT k < 1. (10%)

(b) (Easy) Show that the Jacobi iteration converges if A is strictly diagonally domi- nant. ^(5%)

(c) (Average) Show that the Gauss-Seidel iteration converges to a solution of Ax = b if A is strictly diagonally dominant or A is symmetric positive definite. ^(10%) 6. (Average) Suppose that A ∈ R^n×n is nonsingular and that for specific given right hand

side vectors b, g ∈ Rⁿ, solutions to linear systems Ay = b and Az = g, respectively, are already known. Show how to solve the system

 A g h^T α

  x µ



=

 b β



in O(n) flops where α, β ∈ R and h ∈ Rⁿ are given and the matrix A₊ =

 A g h^T α

 is nonsingular. ^(10%)

7. (Average) Apply Householder reflection transformation to verify that det(I_n+ xy^T) = 1 + x^Ty, where x, y ∈ Rⁿ. ^(10%)

8. (Easy) Show that if B is singular, then 1

κ(A) ≤ kA − Bk kAk , where κ(A) = kAkkA⁻¹k. ^(10%)

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