Sample Exam Questions
Exam 1 Oct. 6
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You are responsible for subject material that covered in class
lecturres.
Could someone please send me a list of 'broken' links on our
class webpage......
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1. a)Put the system below into the form Ax = b and
solve the system using Gaussian Elimination
(without pivoting)
x + y -z = 0
2x -y +z = 3
-x +2y +2z = 5
b) Find the L and U factors of the factorization A = L*U
c) if we had performed pivoting then the factorization would
be of the form
PA = LU
Write down the resulting permutation matrix P.
d) Starting from x = 1, y = 1, z = 1 perform one iteration of
the Jacobi method. Give the values of the first iterates.
2. Use Newton's method to approximate sqrt(7). Here sqrt stands for
"square root of". Stop when |x_{n+1} - x_{n}| < 10e-4.
Give a simple formula for x_{n+1} in terms of x_{n}.
Use a starting value of x_0 = 1.
Draw a sketch of the curve that you are finding a root of.
3. a) Compute the Jacobi approximation to the solution to the system in
problem 1 starting with the initial seed x = 0 , y = 0, z = 0
Use three iterations.
b) Do the same for the Gauss Seidel method.
4. Can you think of a way to approximate pi using Newton's method.
Explain how you would do this and then do it to get an approximations.
Show your work.
.
5. Give simple geometric diagrams to illustrate how the following
root finding methods generate a new iterate:
a- method of false position
b- Newton's method (one dimension)
6. Show how the fixed point problem
x = cos(x)
can be converted to a root finding problem.
7. Compute a Jacobian matrix for the following non-linear system
x*x + y*x - y*z = 2
x*y*y - y*z*x = 3
x + y - z*z = 5
8. a) How many real roots does the non-linear system below possess?
x*x + y*y = 9
(x-2)*(x-2) + (y-2)*(y-2) = 4
b) Illustrate with a diagram.
c) Give the Jacobian matrix for the system above.
9. a)How do we know that the matrix below is non-singular?
1 2
0 1
b)Find it's inverse (recall a method from linear algebra)
c) How can you tell if an n x n upper triangular matrix
is non-singular.
10. Which are the matrices below positive definite -- give reasons.
An n x n matrix A is positive definite if
(Ax,x) > 0 when x is an n dimensional vector such that x <> 0
i) 1 0 0
0 1 0
0 0 1
ii) 2 -1 0
-1 2 -1
0 -1 2
iii) 3 -2 0
-2 3 0
1 -1 3
11. Put a Gauss-Seidel iteration method for the system below into
the matrix formulation discussed in class
2x - y = 3
-x + 2y -z = -1
y -2z = -1
12. consider the system
.0001x + y = b1
x + y = b2.
a) find formulas for the exact solution x and y.
b) let b1 = 1 and b2 = 2. Solve this system using Gaussian elim.
(without pivoting) and three digit (rounded) arithmetic.
compare with the exact solution.
13. Convert the following ordinary differential equation to a
a first order system
x'' + x'cos(x) + x = 0
x(0) = 1 , x'(0) = 0
14. Use the Euler with a time step of h = 1 to approximate
the solution to the initial value problem
x' = x*x x(0) = 1
for two time steps.
15. a)Compute the 2 norm (Frobenius) for the following vectors
i) (1,1,1,1) ii) (4,4,4,4)
b) compute the 1 norm (taxi cab) of the vectors above.
c) compute the max row sum norm for the matrix below
3 -2 1
2 -3 -1
-2 3 -2
16. For the equation
(*) u'' + xu' = sin(x) u(0) = 1, u(1) = -1
We wish to find an approximate solution by various methods.
a) is the bvp (*) linear or non-linear with respect to the
dependent variable u?
b) Suppose we want a uniform mesh
0 = x0 < x1 < x2 < ....< x_N = 1
with uniform spacing h.
What is h in terms of N?
c) Give a finite difference equation which approximates u(x) at
mesh point .
d) For this question, assume N = 8. Give a finite difference equation
associated with mesh point x_4 .
e) For the general case (N not specified). Put the system of finite
difference equations resulting from question c) into a matrix
form.
17. Suppose we are given the four points in the plane
(0,0), (0,1), (1,0), (1,1). Assume the first coordinate
is the x coordinate and the second is the y coordinate.
We discussed in class how we might interpolate the points
(0.0,2), (0,1,4), (1,0, 3), (1,1,6)
which lie above the four planar points. Discuss in as much
detail as you can how to do this. Carry out your computations
as far as you can to obtain an interpolating function.
18. Suppose we want to construct a piecewise polynomial T(x)
on [a,b] with a uniform mesh of N knots where each polynomial
is a quadratic polynomial. T(x) will interpolate f(x) at
the mesh points.
a) how many unknowns are there to be solved for on each
subinterval?
b) How many unknowns are there altogether.
c) How many interpolation constraints are there?
19. a. What is a cubic spline? (give the conditions which determine
it, give it's smoothness properties).
b. what is a natural cubic spline?
c. Define "Cubic Hermite interpolating polynomial"
20. Suppose we have two cubic polynomials -- the first p1(x)
defined on [-1,0] and the second p2(x) defined on [0,1].
where p1(x) = 3x^3 + 2x^2 + x + 5
and p2(x) = 4x^3 + 2x^2 + x + 5
Here "^" indicates exponentiation.
a) Do p1(x) and p2(x) form a cubic spline system for [-1,1]?
Justify your answer.
b) Do p1(x) and p2(x) form a cubic Hermite system for [-1,1]?
Again, justify your answer.
21.. For the 'Brain State in a Box' model answer the following questions---
i) how are images stored?
ii) what is a hypercube?
iii)how do we compute distances in a hypercube?
iv) in our neural ODE solver what is the initial value?
v) what is an attractor for an ODE?
vi) What would be the attractors in our neural system? .
vii) How are the original images recovered?