# 432hwk7 - x that satisﬁes the equation Ax =-b Show that...

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Math 432/532 Homework 15 points Due March 23, 2012 1. Let f ( x 1 , x 2 ) = 2 x 1 4 + x 2 2 - 4 x 1 x 2 + 5 x 2 . (a) Compute the ﬁrst two terms x (1) , x (2) of the Newton’s method sequence for ﬁnding a critical point of f with initial point x (0) = (0 , 0) . (b) Compute the ﬁrst two terms x (1) , x (2) of the steepest descent sequence for f with initial point x (0) = (0 , 0) . 2. (a) Compute the quadratic approximation q ( x 1 , x 2 ) for the function f ( x 1 , x 2 ) = 8 x 1 2 + 8 x 2 2 - x 1 4 - x 2 4 - 1 at the point (1 / 2 , 1 / 2). (b) Find the minimizer x * of the function q ( x ) from part(a). Then check your answer by doing one step of Newton’s method for ﬁnding a critical point of f with starting point (1 / 2 , 1 / 2). 3. Let f : R n R be deﬁned by f ( x ) = a + b · x + 1 2 x · Ax , where a R , b R n , and A is a symmetric positive deﬁnite n × n matrix. As we’ve seen in class, f has a unique minimizer
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Unformatted text preview: x * that satisﬁes the equation Ax * =-b . Show that if x (0) is chosen so that x (0)-x * is an eigenvector of A , then the steepest descent method reaches x * in one step—i.e., x (1) = x * . 4. (532 only) Let f : R n → R be a function with continuous ﬁrst-order partial derivatives on R n , and let M be a subspace of R n . (a) Suppose x * ∈ M minimizes f on M . Show that ∇ f ( x * ) ∈ M ⊥ . (Hint: Let x ∈ M and consider the function φ ( t ) = f ( x * + tx ) . ) (b) Under the assumption that f is also convex, show that any x * ∈ M such that ∇ f ( x * ) ∈ M ⊥ is a global minimizer of f on M ....
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