Homework #4
1. (a) Write out each step of Gaussian elimination on the augmented matrix for the
linear system
2x 3y + z = 1
x+yz = 2
4x + 4z = 1.
(b) Use back substitution to solve for x, y, z .
2. (a
Trigonometry Formulas
y
T
r
y)
(x,
sin =
r
.
.
.
.
.
.
s
.
y
r
cos =
x
r
tan =
y
x
Ex
sec =
1
cos
csc =
1
sin
tan =
sin
cos
cot =
cos
sin
C
.
.
.
.
.
. .
. .
. .
. .
. .
.
.
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Homework #0
1. Suppose we have a computer that performs 3-digit rounding. Find the absolute and
relative errors for the results of each of the following numbers computed in the computer.
(a) 0.0002247
Examples of Graphs of y = a sin(bx + c)
Consider y = a sin(bx + c), with b > 0.
Amplitude: A = |a|
Period: T =
2
b
Phase shift: s =
c
b
Note that s > 0 means a shift to the right and s < 0 means a
Homework #2
1. (a) Give a graphical description showing how xed point iterations converge for the
xed point function g (x) = x/2.
(b) Give a graphical description showing how xed point iterations do n
Graph of y = a sin(bx + c)
The graph of a basic sine curve y = sin x is shown below.
y
1
0
2
3
2
2
x
1
y = sin x
For a basic sine curve, one cycle corresponds to 0
cycle) is 2, and the amplitude is 1.
Homework #1
1. Let p be the exact solution and cfw_pk be a sequence of approximations satisfying
k=0
1
|pk p|
=.
|pk1 p|
10
Also let cfw_pk be another sequence of approximations satisfying
k=0
|pk
Homework #6
1. For each part, nd the interpolating polynomial for the data points
(1, 2), (1, 3), (2, 2).
by writing down a linear system involving p(1) = 2, p(1) = 3, p(2) = 2 and solving
it for the
Homework #5
1. (a) Give an example of a 2 2 matrix that doesnt have an LU factorization.
(b) Find the LU factorization of the matrix
2 0
1 1
1
2
0
1
.
A=
4
1 2 4
0
0
2
0
(c) Use the LU factorization
Homework #3
1. (a) Starting with the initial guesses of 1 and 2, draw how two approximations of the
root of f (x) = x2 2 are calculated under secant method.
(b) Find the values of the rst three approx
Gauss-Jordan Elimination Method
The following row operations on the augmented matrix of a system produce the augmented matrix
of an equivalent system, i.e., a system with the same solution as the orig