Math 171B: Mathematical Programming
Instructor: Jennifer Erway
Spring Quarter 2007
Homework Assignment #2
Due: Friday, April 13
Our goal in this homework is to review some basic concepts from calculus of real and multivariate functions,
gain a better understanding of Newton’s method for solving nonlinear equations involving vector-valued func-
tions of several real variables, to learn about the convergence properties of Newton’s method, and to understand
how to implement the method on a computer. For our review of calculus, we wil l need to recall the definitions
of continuity and differentiation, in the cases of both real-valued and vector-valued functions of one and many
variables. Key tools throughout the course will be Taylor series and Taylor Remainder Theorem. We will do
a couple of simple implementation problems to start us off using MATLAB.
The starred exercises are those that require the use of MATLAB You MUST do the MATLAB problems
to get credit for the homework.
Exercise 2.1 Let F(x) denote a twice-continuously differentiable function of one variable. Assuming only
the mean-value theorem of integral calculus: F(b) = F(a) + Rb
aF0(t)dt, derive the following variants of the
Taylor-series expansion with integral remainder:
(a) F(x+h) = F(x) + Rx+h
xF0(t)dt.
(b) F(x+h) = F(x) + hR1
0F0(x+ηh)dη.
(c) F(x+h) = F(x) + hF 0(x) + hR1
0[F0(x+ηh)−F0(x)] dη.
(d) F(x+h) = F(x) + hF 0(x) + h2R1
0F00(x+ηh)(1 −η)dη . (Hint: Try expanding F0(x+h) using a
formula like part (b) and then differentiate with respect to husing the chain rule.)
Exercise 2.2 Find the gradient vector F(x) = ∇f(x) of the following functions, and then find the Jacobian
matrix of F(x). (The Jacobian matrix of F(x) = ∇f(x) is the same as the Hessian matrix ∇2f(x) of f(x)).
(a) f(x) = 2 x2−x2
12+ (x1−3)2.
(b) f(x) = (2x1+x2)2+ 4(x2−x3)4.
Exercise 2.3 Find f0(x), ∇f(x), and ∇2f(x) for the following functions of nvariables:
(a) f(x) = 1
2xTHx, where His an n×nconstant matrix.
(b) f(x) = bTAx −1
2xTATAx, where Ais an m×nconstant matrix and bis a constant vector.
(c) f(x) = kxk=Pn
i=1 x2
i1/2.
Exercise 2.4 Given a vector xof appropriate dimension, find the Jacobians F0(x) for the following functions.
(a)
F(x) = ex1(4x2
1+ 2x2
2+ 4x1x2+ 4x2+x1+ 5)
ex1(3x2+ 2x1+ 4)
(b)
F(x) = (A−xnI)¯x
¯xT¯x−1,
1
