Show that if H0 = I the Davidon–Fletcher–Powell method is the conjugate gradient method. What...

Show that if H0 = I the
Davidon–Fletcher–Powell method is the conjugate gradient method. What similar
statement can be made when H0 is an arbitrary symmetric positive definite
matrix?

In the text it is shown that for the Davidon–Fletcher–Powell
method Hk+1 is positive definite if Hk is. The proof assumed that αk is
chosen to exactly minimize f (xk + αdk ). Show that any αk > 0
which leads to pT k qk > 0 will guarantee the positive definiteness of Hk+1.
Show that for a quadratic any αk = 0 leads to a positive definite Hk+1.