Math 164 (Summer 2022): TEST 02
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Math 164 (Summer 2022): TEST 02
Question 1 NEWTON’S METHOD .
Let f(X,Y) = X4 − 2X − X2 Y − Y2 .
(i) Write Newton’s method recursive formula to fnd the stationary point of the function f . (ii) Given that 
0 = (−1, 1)t find 
1 in a fraction form using part (i).
Question 2 CONJUGATE GRADIENT METHOD.
(i) Write f(X,Y) in the form f(X,Y) = 
(X,Y)t Q(X,Y) − (X,Y)t (B1 ,B2 )+ C . (ii) Use the conjugate gradient algorithm to construct a vector d1 using 
0 = (0, 0)t
(iii) Prove that d1 is Q-conjugate with rf(
0 ).
Question 3 GRADIENT METHOD.
Let n ≥ 1 be an interger and let A 2 Rn ⇥n be a symmetric matrix (non necesary positive definite) for which all its eigenvalues are non-zero. Let a 2 Rn be a given vector; consider the function f : Rn ! R, defined as f(x) = 
(x − a)t A2 (x − a) where A2 = AA . Assume x? is the global minimizer of f on
Rn .
(i) Write the updates in the steepest descent algorithm starting from a point x0 2 Rn to approximate the optimizer x? . Determine the step size ↵k in each step.
(ii) Assume we use a fixed step gradient algorithm to approximate x? . What is the maximal range for
the step size ↵ in terms of the eigenvalues of A that ensures global convergence for the algorithm?
2022-07-22