Dorothy Goettler
NAE – HW 10
Problem 10.2.3b
Newton’s method was used to find the solution to the given nonlinear system. Each iteration of this
method involves determining the Jacobian matrix J(x). The vector y must then be determined such that
J(x(k-1))y = -F(x(k-1)). The new approximation x(k) is then obtained by adding y to x(k-1) . This is repeated
until the desired number of iterations have been performed. In this case, the result was
 1.772454
x
.
1.772454
The desired accuracy required six iterations.
Problem 10.3.2d
Broyden’s method was used to approximate a solution to the given system of equations. This method is a
generalization of the secant method. Once again the Jacobian matrix must be found with the initial
approximation. The inverse of this matrix is multiplied by the matrix of the given function once again at
the initial approximation. This product is subtracted from the initial approximation, giving a new
approximation. This approximation is then substituted into the system of equations. The difference of
the old values for the system subtracted from the new ones is the vector y. The vector s is defined as the
product of the Jacobian matrix by the system of equations. This value is combined with y and the inverse
of the previous Jacobian matrix as indicated in the textbook to yield the next approximation to the
solution.
Problem 10.4.4a
The method of steepest descent is used to determine an initial approximation to the given system of
equations using the method described in the textbook. This approximate solution can then be refined
using Newton’s method.