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!

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)