(This guide could serve to both your individual and group projects)

Project Summary
You are tasked to find daily problems (real-world problems) which are solvable by numerical methods. Just select one problem and present a case presentation for this.

1. Problem
- the problem may range from simple to complex ones. But of course, good credit would be given for those who solves the complex problem

2. Case Presentation
- In the presentation, the solution for the given problem would be presented alongside the analysis undertaken that leads to it. It would be best to have a thorough approach such that the reason for using a numerical method would be provided. You can do this by differentiating your chosen method with the other available methods, if there are any that can also solve the selected problem. A discussion on your analysis would be necessary and an additional algorithm is required. You can provide a working program for the algorithm.

Deliverables
Submit a hardcopy of your output (group and individual works)
Have a 5-10 minutes presentation (prepare for questions afterwards)

for questions: stephen_562001 @ yahoo.com

For your finals lecture notes, you can download lectures from this site: http://numericalmethods.eng.usf.edu
That is where the resources in our class lectures were taken from.

Those items colored in red are the ones discussed/used in class, the rest can be utilized as supplemental notes.

Here is a listing of the topics:

• Regression -
  

  BACKGROUND
  	A primer on statistical terminology for regression analysis [PDF] [DOC] Introduction to regression [PDF] [DOC] Test your knowledge on background of regression analysis [HTML] [PDF] [DOC]
  PHYSICAL PROBLEM
  	Infrared spectroscopy [PDF] [DOC]
  NUMERICAL METHOD
  	LINEAR REGRESSION MODEL
  		How does Linear Regression work? [PDF] [DOC] A Power point presentation on Linear Regression [PDF] [PPT] Simulation of Linear Regression [PDF] [MFILE] Test your knowledge of Linear Regression [HTML] [PDF] [DOC]
  	NONLINEAR REGRESSION MODELS
  		How does Nonlinear Regression work? [PDF] [DOC] A Power point presentation on Nonlinear regression [PDF] [PPT] Simulation of Nonlinear Regression without Data Linearization [PDF] [MFILE] Simulation of Nonlinear Regression with Data Linearization [PDF] [MFILE] Simulation of Polynomial Regression [PDF] [MFILE] Test your knowledge of Nonlinear Regression [HTML] [PDF] [DOC]
  ANECDOTES
  	Comparing nonlinear regression models - with and without data linearization [PDF] [MFILE] On adequacy of regression models [PDF] [DOC] [PDF] [MFILE] Multivariate least squares fitting [PDF] [DOC]

  
  
• Interpolation -
  

  BACKGROUND
  	What is Interpolation? [PDF] [DOC] History of Interpolation [PDF] [DOC] Test your knowledge on background of interpolation [HTML] [PDF] [DOC]
  PHYSICAL PROBLEM
  	A physical problem of finding how much time it would take water to boil in a kettle. It requires interpolating specific heat of saturated water vs. temperature data. [PDF] [DOC]
  NUMERICAL METHOD
  	DIRECT METHOD
  		How does the direct method work? [PDF] [DOC] A Power point presentation [PDF] [PPT] Simulations of the method [PDF] [MFILE] Test your knowledge of the method [HTML] [PDF] [DOC]
  	NEWTON DIVIDED DIFFERENCE METHOD
  		How does the Newton's divided difference polynomial method work? [PDF] [DOC] A Power point presentation [PDF] [PPT] Simulations of the method [PDF] [MFILE] Test your knowledge of the method [HTML] [PDF] [DOC]
  	LAGRANGE METHOD
  		How does the Lagrange method work? [PDF] [DOC] A Power point presentation [PDF] [PPT] Simulations of the method [PDF] [MFILE] Test your knowledge of the method [HTML] [PDF] [DOC]
  	SPLINE METHOD (not part of the coverage)
  		How does the spline method work? [PDF] [DOC] A Power point presentation [PDF] [PPT] Simulations of the method [PDF] [MFILE] Test your knowledge of the method [HTML] [PDF] [DOC]
  ANECDOTES
  	The lurking dangers of extrapolation! [PDF] [DOC] [MFILE] [PDF] Why is higher order interpolation is a bad idea? [PDF] [DOC] [MFILE] [PDF] Comparison of spline and polynomial interpolation. [PDF] [DOC] [MFILE] [PDF] How choice of points of interpolation affects approximations! [PDF] [DOC] [MFILE] [PDF] How splines can help in developing a shorter path for a robot! [PDF] [DOC] [MFILE] [PDF] Runge [HTML] [PDF] [DOC] Newton [HTML] [PDF] [DOC] Lagrange [HTML] [PDF] [DOC]

  
  
• Differentiation -

  [Chapra] Chapter 4 - Numerical Differentiation part only - download

          [Chapra] Chapter 23 - Numerical Differentiation, excludes Richardson - download 

• Integration -

  [Chapra] Chapter 21 - Study Trapezoidal, Simpson's Rule and Romberg Integration - download

If you would like to select specific examples related to your field of study (chemical, mechanical, electronics, electrical, computer, etc...) you may opt to use this link: http://numericalmethods.eng.usf.edu/resource.html and from here, using the drop down select, choose the Mathematical Procedure (could be regression, interpolation...), the Math package (you may use matlab) and lastly, your engineering major then press the Go button.

If you are reading this post on a Saturday (Feb 21), that means, the exam will be tomorrow at the COE Amphitheater, 2-5PM.

midterm quiz - 2

midterm quiz solutions

ES 84 SAMPLE PROBLEM-midterms - Get more Business Documents

Here are the powerpoint slides used during our lectures (in the zip file):
[download]

Take note of the Exam Coverage:
1. Bracketing Methods
2. Open Methods
3. Multiroot
4. Bairstow Methods (Root of Polynomials)
5. Optimization

Determine the real roots of f(x) = -1 + 5.5x - 4x^2 + 0.5x^3
(a) graphically and
(b) using the Newton-Raphson method to within Es = 0.01%. (use initial guesses x=0, x=2 and x=6)

(5.13)
The velocity v of a falling parachutist is given by

v = (gm/c)( 1-e^[-(c/m)t])

[* read as (gm over c) times the quantity of 1 minus e raise to the negative (c over m) times t]
where g = 9.8 m/s^2. For a parachutist with a drag coefficient c = 15 kg/s, compute the mass m so that the velocity is v = 35m/s at t =9 s. Use false-position method to determine m to a level of Es = 0.1%

(5.14)
Since 5.14 involves an ES 64 word problem on Moments, just solve the problem above (5.13) using the Bisection Method.

1pm-3pm Class:
#1. Use zero-through third-order Taylor Series expansions to predict f(3) for
f(x) = 25x^3-6x^2+7x-88
Using a base point at x=1. Compute the true percent relative error (Et) for each approximation.

Note: 25x^3 is read as 25 times (x raised to 3).  ^ signifies that the character on its left is raised to the no. on its right side.

Soln.


#2 Use Gauss-Jordan Elimination to solve:

2x1 + x2 - x3 = 1

5x1 + 2x2 - 2x3 = -4

3x1 + x2 + x3 = 5

Do not employ partial pivoting.

Soln.


#3.  The following system of equations is designed to determine concentrations (the c's in g/m^3) in a series of coupled reactors as a function of the amount of mass input to each reactor(the right-hand sides in g/day)

15c1 - 3c2 - c3 = 3800

-3c1 + 18c2 - 6c3 = 1200

-4c1 - c2 - 12c3 = 2350

(a) Determine the matrix inverse

(b) Use the inverse to determine the solution

Answers:
You can use the numerical methods discussed in the class such as LU to compute the inverse.

(a)

Inverse:
0.0658949 0.0103888 - 0.0106857
0.0035619 0.0546156 - 0.0276046
- 0.0222618 - 0.0080142 - 0.0774711
(b)

if we have transform the system of equations to its matrix form Ax = B,

multiplying the result from (a) [inverse of matrix] with vector B, you can then find the solution
x1= 237.75601
x2= 14.203028
x3= - 276.26892


3-5pm Class

#1 Use zero-through fourth-order Taylor Series expansions to predict f(2.5) for f(x) = ln x using a base point at x =1. Compute the true relative error Et for each approximations.

Answer:

 zero-order:
f(2.5) = 0; Et = 100%

first-order:
f(2.5) = 1.5; Et = 63.704%

second-order:
f(2.5) = .375; Et = 59.074%

third-order:
f(2.5) = 1.5; Et = 63.704%

fourth-order:
f(2.5) = .234375; Et = 74.421%

#2. Given the system of equations:

-3x2 + 7x3 = 2

x1 + 2x2 - x3 = 3

5x1 - 2x2 = 2

(a) Compute the determinant

(c) Use Gauss Elimination with partial pivoting to solve for the x's.

Answers

(a) determinant = -69
(b) x1 = 0.9130
x2 = 1.4638
x3 = 0.9855

#3. Solve the following system of Equations using LU decomposition with partial pivoting:

2x1 - 6x2 - x3 = -38

-3x1 - x2 + 7x3 = -34

-8x1 + x2 - 2x3 = -20

Answer
x1= 4
x2 = 8
x3 = -2

I will not be giving the homework which I said in class that I will give you this Christmas break. Instead, prepare for a quiz this Wednesday, Jan. 07.

Happy new year!