Numerical Methods for Linear and Nonlinear Systems
MA MA 3257/CS

Spring 2013
Naive Gaussian Elimination
The following algorithms implement naive Gaussian elimination followed by back substitution to
compute the solution of Ax = b, where A is an n n matrix with ijth entry aij and b is an
nvector with ith component bi .
These are s
Numerical Methods for Linear and Nonlinear Systems
MA MA 3257/CS

Spring 2013
Gaussian Elimination with Partial Pivoting
The following algorithms implement Gaussian elimination with partial pivoting followed by back
substitution to compute the solution of Ax = b, where A is an n n matrix with ijth entry aij
and b is an nvector wit
Numerical Methods for Linear and Nonlinear Systems
MA MA 3257/CS

Spring 2013
MA 3257/CS 4032
FINAL EXAM SOLUTIONS
March 1, 2013
In working the following exam, use no books or notes except the handout provided.
Show your work on all but the rst problem. Use the backs of the pages if you need
more space.
1. (12 points) Say whether t
Numerical Methods for Linear and Nonlinear Systems
MA MA 3257/CS

Spring 2013
Final Handout
The following are our theorems for iterative methods. The numbering is that on the
Iterative Methods Handout.
Theorem 3. The iterates
x(k+1) = T x(k) + c,
k = 0, 1, . . .
converge for every x0 if and only if (T ) < 1. If (T ) < 1, then for e
Numerical Methods for Linear and Nonlinear Systems
MA MA 3257/CS

Spring 2013
Preliminary Material Practice Problems.
1. Say whether the following are true or false:
a. Machine epsilon is the smallest number (in absolute value) in the oating point
number system.
b. Floatingpoint numbers are distributed uniformly throughout their r
Numerical Methods for Linear and Nonlinear Systems
MA MA 3257/CS

Spring 2013
MA 3257/CS 4032
MIDTERM EXAM SOLUTIONS
February 7, 2013
In working the following exam, use no books or notes except the handout provided. Show your work on all
but the rst problem. Use the backs of the pages if you need more space.
1. (12 points) Say whe
Numerical Methods for Linear and Nonlinear Systems
MA MA 3257/CS

Spring 2013
Practice Problems on Bisection and Newtons Method
Bisection.
1. Consider the equation x = ex . How do you know there is a solution in [0, 1]? How
many iterations of the bisection method, starting with initial interval [a, b] = [0, 1],
are necessary to be
Numerical Methods for Linear and Nonlinear Systems
MA MA 3257/CS

Spring 2013
Practice Problems on FixedPoint Iteration and Convergence of Sequences
Fixedpoint iteration.
1. Consider the function g(x) tan x, where is a constant. For what values of
do the xedpoint iterates converge to x = 0 from a suciently good starting point?