(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!

If you can't see the slideshow(above), just download the file.
Lecture04 - Gaussian Elimination, LU Decomposition and Gauss-Seidel Method - [Download]

If you can't see the slideshow(above), just download the file.
Lecture03 - Linear Algebraic Equations - [Download]

Aside from the class lecture slides provided here, lectures notes (in PDF format) are also made available here.

1. Approximation and Errors [download]

2. Systems of Linear Equations [download]

3. Root Finding [download]

*These are accompanying notes for the Numerical Methods for Engineers, 4/e, Chapra, et. al. by Ronald D. Ziemian, et. al.

(the 4th and 5th editions of the NME book have only slight differences)

Here are some book resources that you can download. Due to our school's internet firewall, you may not be able to download using the school's network.

Numerical Methods Books

[1]. Numerical Methods For Engineers and Scientists Second Edition by Joe D.Hoffman

[2]. Numerical Methods in Engineering with MATLAB by J. Kiusalaas

[3]. Fundamental Numerical Methods and Data Analysis by G. W. Collins

[4]. Numerical Recipes in C by W. Press, et. al.

Scilab Books/Notes

[1] Modeling and Simulation in Scilab/Scicos by S. Campbell, et. al.

[2] Scilab Primer

[3] Tutorials on Scilab

Disclaimer: These ebooks are not hosted on this blog. Only links to file sharing site/s is/are provided. This blog author is not responsible for the files.