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) Count the number of additions/subtractions and the nu
Trigonometry Formulas
y
T
r
y)
(x,
sin =
r
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s
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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) 0.000224729
(b) 42381900
(c) 2012.34 + 3.114
(d) 2.728 0.0036
(e
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 shift to the left.
Example 1. Consider y = 5 sin(2x +
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 not converge
for the xed point function g (x) = 2x.
2. U
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.
2, the period (i.e., the length of one
x
Lets consider
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 p|
= 1.
|pk1 p|2
Suppose |p0 p| = 106 and |p0 p| = 0.1.
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 unknown coecients a, b, c:
(a) p(x) = ax2 + bx + c.
(b)
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 to compute the determinant of A.
(d) Use the LU factor
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 approximations.
(c) What is the absolute error of the nal app
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 original one.
Interchange any two rows.
Multiply each ele