`Math 343 Fall 97`
`Homework Assignments`
```Homework 01:   due Tuesday, Sept. 2
1. Use the LU decomposition to solve the linear systems in problem
1.3.11, page 17
2. Use the LU decomposition PA=LU to solve the linear systems in problem
1.5.13```
```Homework 02:   due Tuesday, Sept. 9
1. Use the finite difference method to solve the boundary value
problem
u" = 4u' + ln(x),    x in [1,3]
u(1)=u(3)=0
2. Use the finite difference method to solve the boundary value
problem
u" = u' + 2u + cos(x),  x in [0,Pi/2]
u(0) = -0.3,   u(Pi/2) = -0.1
(The exact solution is u = -0.1*(sin(x)+3*cos(x))  )```
```Homework 03:   due Tuesday, Sept. 16
1. Use the finite difference method to solve the boundary value
problem of partial differential equation
Diff(u,x,x)+Diff(u,y,y)=4
x in [0,1],  y in [0,4/3]
u(0,y)=y^2,    u(1,y)=(y-1)^2,
u(x,0)=x^2,    u(x,4/3)=(x-4/3)^2
with h = k = 1/3,   n = 3,  m = 4
(The exact solution is u(x,y)=(x-y)^2
2. (Interpolation problem) For given data
(x[i],y[i]), i=1, 2, ..., n
there is a unique polynomial
p(x) = a[0]+a[1]*x+a[2]*x^2+a[3]*x^3+...+a[n-1]*x^(n-1)
such that
p(x[i])=y[i], i=1, 2, ..., n
Write down the general equation in matrix form that determines
a[0], a[1], a[2], a[3], ..., a[n-1]
3. For given data
x    -1.0   -0.6    -0.2      0.2      0.6
y     3.0     .664    .696    1.56     4.792
find the polynomial p(x) such that
p(-1.0)=3.0;  p(-0.6)=.664, p(-0.2)=.696
p(0.2)=1.56;  p(0.6)=4.792
Check with Maple to see if they are accurate.```
```Homework 04, due Tuesday, Sept 23
part 1: page 165, Problem 3.3.24, 3.3.25, 3.3.26
part 2: Bacteria grow at approximately a fixed rate r. The model
of the population is
N = N[0]*(1+r)^t
where  N:  the number of bacteria at time t
N[0]: the initial population
t:  the time in hours
Measurements were made and recorded in the following data:
t      1      1.5     2.0     2.5      3.0
N     117     130     138     152      163
Find the growth rate r.```
```Homework 05, due Tuesday, Sept 30
1. Find 4 Householder transformations that rotates the vector
| 1|      |*|   |*|   |*|      |*|
|-3|      |0|   |*|   |*|      |*|
| 2|  to  |0|,  |0|,  |*|, and |*| respectively.
| 1|      |0|   |0|   |0|      |*|
| 5|      |0|   |0|   |0|      |0|
2. Find the QR decomposition of
| -3   5   8  -1   0  |
|  2   4   1   0   7  |
| 11  -6   5  -9   3  |
| -2   0   3  12   5  |
|  4   3   -2  7   1  |
```
```Homework 06, due Tuesday, Oct. 7
1. Page 164, #3.3.18, answer the problem #2 with QR decomposition and
solve the least squares problem
2. Describe a method of solving an ordinary linear system Ax=b with QR
decomposition. Construct a 5x5 system to test that method.
3. (Extra credit) Design a method of solving the least squares problem
of Ax=b using the singular decomposition of A.```
```Homework 07, due Tuesday, Oct. 14
1. Construct a least squares problem Ax=b with a 5x2 matrix A and solve
it with SVD
2. Use SVD to compress (i.e., find enought singular values/vectors) the
following image
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*******
*******
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*******
*******```
```Homework 08, due Tuesday, Oct. 21
1. Page 252, #5.1.3
2. Use Maple to find the eigen-decomposition of
| 3  2  -4 |
| 2  0   5 |
|-4  5  -2 |
3. Solve the initial value problem of ordinary differential equation
using the eigen-decomposition
dx
---- = Ax,      x(0) = b
dt
where
|  1   2   3  |       |-3|       |x[1]|
A = |  0   1   0  |,  b = | 6|,  x = |x[2]|
|  2   1   2  |       |-1|       |x[3]|
```Homework 09, due Tuesday, Oct. 28
1. Transform the following quadratic forms into standard equations
and identify the conic section
(i) 8x^2 + 8y^2 -16xy + 33*sqrt(2)*x -31*sqrt(2)*y + 70 = 0
(ii) 6x^2 + 9y^2 -4xy -4*sqrt(5)*x -18*sqrt(5)*y = 5
2. Graph each of the conic sections above with Maple, following the
flow of transformation. That is,
rotation       translation
x  ----------> y  -------------> z
graph the z-equation as parametric curve and transform the graph
back to y-equation, and then, the x-equation
```
```Homework 10, due Tuesday, Nov. 4
Transform the following quadratic forms into standard equations,
identify the type of surfaces and sketch the graph:
(1) 2x^2 + 2y^2 + 4z^2 -4xy -8xz -8yz + 8x = 15
(2) 2xz - 2z - 4y -4z + 8 = 0```
```Homework 11, due Tuesday, Nov. 18
Find all critical points for the following functions and identify
local minimum, local maximum and saddle points.
(1) f(x,y) = x^3 - 6*x^2 + y^3 + 12*y^2
(2) f(x,y) = 1 - cos(x) + y^2/2
(3) f(x,y) = 8*x^3 + y^3 + 6*x*y```
```Homework 12, due Tuesday, Nov. 25
Page 273, #5.3.7 and #5.3.11```
```Homework 13, due Tuesday, Dec. 2
page 287, #5.4.8
page 288, #5.4.15```