ME 352 Fall 2008

Demonstration of Euler's method. Quick review of trouble-spots in the course

Overview of ODE Solvers, Euler's method

Discuss the meaning of Equations (8.37) and (8.38). Sample solution by inspection of diagonal systems. Solution of triangular systems by backward or forward substitution. Return to discussion of the role of condition number.

Solving diagonal systems, triangle systems and conceptual overview of Gaussian Elimination. Review formal solution. Define the residual. Introduce the condition number and discuss meaning of ill-conditioning. Review of how to compare two vectors: use norm(x-y) not norm(x)-norm(y). Discuss meaning of Equation (8.37)

Conclude the sample problem by showing the MATLAB implementation. Discuss Example 8.2 from the textbook -- a more challenging problem where the unknowns are not so obvious. Return to a discussion of the need for linear independence of the columns of A, and of consistency of the system of equations.

Review of the example problem: develop a two-dimensional interpolant h = f(p,T)

We began with a consideration of the matrix-vector product. I showed that the matrix-vector product y = A*x can be interpreted as a linear combination of the columns of matrix A. From there we quickly looked at the implications for solving linear systems of equations. If A*x = b has a solution, then b must lie in the column space of A. Next I passed out a single sheet of paper with a homework problem from last year. The goal of the assignment was to formulate a two-dimensional interpolant for the enthalpy of steam as a function of pressure and temperature.

More vector operations: linear combinations, inner product, contrast with array operators.

Vector operations: addition, scalar multiplication, linear combinations

I discussed the design problem for column buckling and showed how it could be implemented in MATLAB using the fzero function. Handouts showed code and a map of input and output variables between the code segments. Near the end of lecture, I reviewed MATLAB vectors and matrices. I defined vectors and matrices and discussed why array operators were needed.

I spent more time than I had planned answering questions and reviewing material. We reviewed Chapter 5 material in some detail.

Forgot to turn on recorder for first half of lecture, which dealt with a wrap-up of bisection, and then Newton's method. In this second half, I showed slides to describe the secant method and the reverse-quadratic interpolation in fzero. I then did a demonstration of how to use fzero to solve a simple problem.

Started with review of m-files. Introduction to root-finding: formulation, graphical interpretation, and what is a bracket.

Using MATLAB to plot f(x) and identify location of roots. Introduction to brackPlot

Unavoidable errors: roundoff and truncation errors. Limits imposed by using a finite number of digits. Practical implications: convergence tests. Machine precision.

Data types: numeric, string, logical. Demonstrate fprintf with a progressive sequence of small m-files that provide more control over the output of a table of numbers.

MATLAB programming: conditional execution, use of automated stopping criteria in the implementations of sine series

MATLAB programming: review loops over elements of a vector, implementations of sine series

MATLAB programming: while loops

MATLAB programming: MATLAB is not Excel; Sequential execution, for loops

Demonstration of using MATLAB for data reduction for a capillary viscometer

Overview of MATLAB: Launching, evaluating simple expressions, calling built-in functions

Introduction and Class Logistics