Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: - - PowerPoint PPT Presentation

Engineering Analysis ENG 3420 Fall 2009

Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00

2 Lecture 20

Lecture 20

Last time:

The inverse of a matrix Iterative methods for solving sytems of linear equations

Gauss-Siedel Jacobi

Today

Relaxation Non-linear systems Random variables, probability distributions, Matlab support for

random variables.

Next Time

Linear regression Linear least squares regression

Relaxation

To enhance convergence, an iterative program can introduce

relaxation where the value at a particular iteration is made up of a combination of the old value and the newly calculated value: where λ is a weighting factor that is assigned a value between 0 and 2.

0<λ<1: underrelaxation λ=1: no relaxation 1<λ≤2: overrelaxation

xi

new = λxi new + 1− λ

( )xi

• ld

Nonlinear Systems

Nonlinear systems can also be solved using the same

strategy as the Gauss-Seidel method - solve each system for one of the unknowns and update each unknown using information from the previous iteration.

This is called successive substitution.

Newton-Raphson

Nonlinear systems may also be solved using the Newton-Raphson

method for multiple variables.

For a two-variable system, the Taylor series approximation and

resulting Newton-Raphson equations are: f1,i+1 = f1,i + x1,i+1 − x1,i

( )

∂f1,i ∂x1 + x2,i+1 − x2,i

( )

∂f1,i ∂x2 x1,i+1 = x1,i − f1,i ∂f2,i ∂x2 − f2,i ∂f1,i ∂x2 ∂f1,i ∂x1 ∂f2,i ∂x2 − ∂f1,i ∂x2 ∂f2,i ∂x1 f2,i+1 = f2,i + x1,i+1 − x1,i

( )

∂f2,i ∂x1 + x2,i+1 − x2,i

( )

∂f2,i ∂x2 x2,i+1 = x2,i − f2,i ∂f1,i ∂x1 − f1,i ∂f2,i ∂x1 ∂f1,i ∂x1 ∂f2,i ∂x2 − ∂f1,i ∂x2 ∂f2,i ∂x1

Probability and statistics concepts

See class notes:

Probability NASA lecture