Test 2

Test 2 for MTH 335 covers a bit of chapter 3, chapter 4 and a bit of chapter 5.

Question 1:

A method for finding a zero of a function $v=v(x)$ can be written as a fixed point problem through:

$$~ F(x) = x - \frac{v}{v'} + \frac{v v''}{2(v')^3} ~$$

That is, if $x_0$ is given and $x_n = F(x_{n-1})$ converges to $s$, then $v(s) = 0$.

Assume $F$ has the following 3 derivatives:

$$~ F' = \frac{-v^2 v'''}{2(v')^3} + \frac{3v^2(v'')^2}{2(v')^4}, ~$$ $$~ F'' = \frac{v}{v'^2} \cdot ( \frac{v \cdot v''''}{2v'} + \frac{9v v'' v'''}{2(c')^2} - \frac{6v(v'')^3}{(v')^3} - v'''' + \frac{3(v'')^2}{v'}), ~$$ $$~ F''' = \frac{1}{v'} \cdot ( -\frac{-v^2 v'''''}{2(v')^2} + \frac{6v^2 v'' v''''}{(v')^3} + \frac{9v^2(v''')^2}{2(v')^3} - \frac{36v^2 (v'')^2v'''}{(v')^4} + \frac{30v^2 (v'')^4}{(v')^5} - \frac{2v v''''}{v'} + \frac{17 v v'' v'''}{(v')^2} - \frac{21v(v'')^3}{(v')^3} - v''' \frac{3(v'')^2}{v'} ). ~$$

Based on the above, what is the order of convergence of $F$ as $F(x_n) \rightarrow s$? Be specific as to how you know and if you needed any assumptions beyond $v$ having sufficient derivatives.

Question 2:

For what values of $a$ will the following matrix be symmetric and postive definite? Write down the values of $a$ you can prove are true along with a proof.


Question 3:

By hand, find the Cholesky decomposition of the symmetric positive definite matrix

3×3 Array{Int64,2}:
 2  1  1
 1  2  1
 1  1  2

Use this decomposition to solve $Ax=b$ with $b=[1, 3, 1]$. (Assume [a,b,c] means a column vector.)

Question 4:

Use scaled row pivoting and Gaussian elimination to find $P$, $L$, and $U$ so that $PA=LU$ when $A$ is given by

3×3 Array{Int64,2}:
 -9  1  17
  3  2  -1
  6  8   1

Question 5:

Fix $n$, and let $\delta_{ij}$ be the $n\times n$ matrix with entries that are $0$ except row $i$ column $j$ and then it is $1$. We can express our two elementary row operations then as:

For what values of $i,j,k,l$ is $\delta_{ij} \cdot \delta_{kl}$ non zero? When it is non-zero, what is the answer in terms of $\delta$?

Using this fact, show that $P_{ij}P_{ij}$ is the identity.

Using this fact, suppose $i \leq j,k,l$, what conditions on $i,j,k,l$ will ensure $R_{ij}\cdot P_{k,l} = P_{k,l} \cdot R_{ij}$?

Question 6:

Consider $\Pi_n$ the vector space of all polynomials of degree $n$ or less. Define a "dot" product as $dot(p,q) = \int_{-1}^1 p(x) q(x) dx$. Define the $\| p \| = \sqrt{dot(p,p)}$.

Question 7:

Given $A$, $b$ and a $Q$ with $\| I - Q^{-1}A \| <1$, the iterative algorithm

$$~ Q x^{n} = (Q-A)x^{n-1} + b ~$$

will converge to $x$, a solution of $Ax=b$.

Let A be given by

2×2 Array{Int64,2}:
 1  2
 3  5

let $Q$ be the diagonal matrix of $A$, and $b=[1, 1]$.

If $x^0 = [0, 0]$ compute $x^1$ and $x^2$ (show your work). What do these values converge to? Is it a solution to $Ax =b$? (Why or why not?)

Question 8:

We saw in class that if $A$ is $n \times n$ and has $n$ distinct eigenvalues, with $\lambda_1$ the largest in absolute value, then $A^kx \approx \lambda_1^k a_1 x_1$. We can use this to converge on the largest eigenvalue through matrix multiplication with this algorithm:

for k in 1:M
  y = A*x
  r = y[1]/x[1]
  x = y

Start with $A$ given by

2×2 Array{Int64,2}:
 1  2
 3  5

and $x=[1,1]$. Perform 3 steps ($M=3$) to compute values for $r_1$, $r_2$, and $r_3$. The eigenvalues of $A$ are $-0.16...$ and $6.16...$. Does it appear that $r_n$ is converging to the largest one? Show all your work.

Question 9:

To solve $Ax=b$ with a full $n\times n$ matrix, we saw in class that it takes basically $n^3/3$ ops to find $U$ and $n^2/2$ ops to perform the back substitution. For a tri-diagonal matrix (where no more than $p=3$ entries in a row are non-zero), it takes $np(2p+1)$ steps to find $U$ and $n(p+1)$ steps to do the back substitution.

For $n=10,000$. Compare the number of ops to solve $Ax=b$ when $A$ is a full matrix to the number of ops to solve $Ax=b$ when $A$ is tri-diagonal ($p=3$). Compute both, and then find their ratio.

In a simulation, it took $3.88 \cdot 10^{-4}$ seconds to solve $Ax=b$ when $A$ was tri-diagonal and $12.45\cdot 10^0$ seconds when $A$ was a full matrix. Is the ratio of times more, less our basically the same as you expect from your calculation above?

Question 10:

Let $\kappa$ be the condition number, $\kappa(A) = \|A\| \cdot \|A^{-1}\|$, for some matrix norm.

If $A$ and $B$ are two non-singular square matrices of the same size, prove that: $$ \kappa(AB) \leq \kappa(A) \kappa(B). $$