#mae107

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Exam4

Numerical Differentiation:

Forward approximation

Backward approximation

Central approximation

Note: Forward difference

Numerical Integration:

Rectangle

Trapezoid

Simpson's rule (1/3) (Lagrange interpolation)

Legendre-Gauss quadrature

Numerically solving ODE:

Explicit Euler

Implicit Euler (Unconditionally stable)

Crank Nicholson (Unconditionally stable)

Runge-Kutta 2

Runge-Kutta 4

[lecatalog]

What I have been up to?

How to find roots of f(x)=0

             --> f(x)=Ax - b, if A is an invertible matrix (x=inv(A)*b):

                        --> Gauss elimination

                        --> Iterative methods : -> Jacobi (S is a diagonal matrix of A)

                                                               -> Gauss-Seidel (S is a triangular matrix of A)

             --> if non-linear equation(s):

                        --> Newton's method

                        --> Higher dimension Newton's method (know how to compute the Jacobian and how to write Matlab iteration!)

If function is not given, you are not screwed. You just have to get some values of f(x)!

           --> Estimation:  -> Least-square method (curve fitting) (remember the    pseudo-inverse!!!)

                                     -> Polynomial interpolation-->Lagrange 

                                     -> Taylor series

          --> Exact solution:  -> Taylor series

[lecatalog]

Week 2- Linear Algebra (Continued)

Exam 1 went well :)

Homework 1 was a disaster... ;(

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Homework 2 discuss finding determinant of a matrix and its characteristic polynomial. Finding eigenvalues. Another part is iteration method. After more than 6 hours on wednesday, working on Matlab trying to recursively code for a function to find the determinant of a matrix. This morning (thursday), I finally succeeded. Yeay!

However, so much more to go... To me, it's a steep learning curve as my programming skills are so rusty (background in Fortran90 about 4 years ago). I have to relearn basically everything from how to store any value or array to any loops... I can do it! It is true than during these past years, working for labs that have me run Matlab programs (image processing), it's not quite actively learning. All I had to do was to input stuff and twitch the program to show what I want but nothing like starting a program from scratch. 

I find it rather hard right now but I believe that it's a good time for practice. Especially, I will have to make programs for my signals & systems(MAE 143A) class as well.

[lecatalog]

Week 1- Linear Algebra (Continued)

Chapter 10- LU Decomposition and Matrix Inversion

Why?

less time consuming

efficient way to get matrix inverse

> attach notes here...

[lecatalog]

Welcome to Numerical Methods!

Week 1- Linear Algebra

Chapter 9-Gauss Elimination

There are few ways to solve small systems (less or equal to 3 equations) before using computer methods: 

Graphical methods

Cramer's rule (nasty)

Elimination of unknowns

The graphical method shows for systems of 2 equations, the intersection of the lines represents the solution. For 3 simultaneous equations, each represents a plane in 3-D coordinate system. The point of intersection of the 3 planes is the solution. This method is nice for visualization but for a system beyond 3 equations, it's not practical. There are 3 cases in which graphs help:

2 equations representing parallel lines --> No solution (singular system)

2 lines are coincident --> Infinite solutions (singular system)

ill-conditionned system where the lines have almost the same slope therefore difficult to see the intersection + cause sensitivity to round-off errors.

Determinants and Cramer's rule: [244]

Singular systems have zero determinants. 

Elimination of unknowns=Naive Gauss Elimination [248]

--> forward elimination (each row is normalized by dividing by pivot element)+ back-substitution. If implemented on computers, the program must avoid dividing by zero. "Naive" method because it cannot take care of this problem. To implement this on a computer, it takes too much time.

Gauss-Jordan [269]:

What are the differences?

Unknowns are eliminated from all other equations rather than just the susequent ones.

All rows are normalized by dividing them by their pivots. (Elimination steps gives an identity matrix instead of an upper triangular one. Therefore, no need for back-sustitution).

Issues of round off, scaling, and conditioning.

> Check out Chapter 12 for Case Studies for fun!