Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver, Chehrzad Shakiban

Chapter 9 Iteration - all with Video Answers

Educators

Chapter Questions

Problem 1

Suppose $u^{(0)}=1$. Find $u^{(1)}, u^{(10)}$, and $u^{(20)}$ when (a) $u^{(k+1)}=2 u^{(k)}$,
(b) $u^{(k+1)}=-.9 u^{(k)}$,
(c) $u^{(k+1)}=\mathrm{i} u^{(k)}$,
(d) $u^{(k+1)}=(1-2 i) u^{(k)}$.
Is the system stable or unstable? If stable, is it asymptotically stable?

Victor Salazar

Numerade Educator

06:03

Problem 1

Determine the spectral radius of the following matrices:
(a) $\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$.
(b) $\left(\begin{array}{ll}\frac{1}{3} & -\frac{1}{4} \\ \frac{1}{2} & -\frac{1}{3}\end{array}\right)$.
(c) $\left(\begin{array}{rrr}0 & 1 & 0 \\ 0 & 0 & 1 \\ -2 & 1 & 2\end{array}\right)$,
(d) $\left(\begin{array}{rrr}-1 & 5 & -9 \\ 4 & 0 & -1 \\ 4 & -4 & 3\end{array}\right)$.

Nick Johnson

Numerade Educator

Problem 1

Determine if the following matrices are regular transition matrices. If so, find the associated probability eigenvector.
(a) $\left(\begin{array}{ll}\frac{1}{2} & \frac{1}{3} \\ \frac{3}{4} & \frac{2}{3}\end{array}\right)$,
(b) $\left(\begin{array}{ll}\frac{1}{4} & \frac{3}{4} \\ \frac{2}{3} & \frac{1}{3}\end{array}\right)$,
(c) $\left(\begin{array}{ll}\frac{1}{4} & \frac{2}{3} \\ \frac{3}{4} & \frac{1}{3}\end{array}\right)$,
(d)
$\left(\begin{array}{ll}0 & \frac{1}{5} \\ 1 & \frac{4}{5}\end{array}\right)$
(e) $\left(\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right)$.
(f) $\left(\begin{array}{lll}.3 & .5 & .2 \\ .3 & .2 & .5 \\ .4 & .3 & .3\end{array}\right)$,
(g) $\left(\begin{array}{rrr}.1 & .5 & .4 \\ .6 & .1 & .3 \\ .3 & 0 & .7\end{array}\right)$,
(h) $\left(\begin{array}{lll}\frac{1}{2} & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & \frac{1}{2} & \frac{1}{3}\end{array}\right)$,
(i) $\left(\begin{array}{rrrr}0 & .2 & 0 & 1 \\ .5 & 0 & .3 & 0 \\ 0 & .8 & 0 & 0 \\ .5 & 0 & .7 & 0\end{array}\right)$,
(j) $\left(\begin{array}{rrrr}.1 & .2 & .3 & .4 \\ .2 & .5 & .3 & .1 \\ .3 & .3 & .1 & .3 \\ .4 & .1 & .3 & .2\end{array}\right)$,
(k) $\left(\begin{array}{rrrr}0 & .6 & 0 & .4 \\ .5 & 0 & .3 & .1 \\ 0 & -4 & 0 & .5 \\ .5 & 0 & .7 & 0\end{array}\right)$.

Check back soon!

Problem 1

(a) Find the spectral radius of the matrix $T=\left(\begin{array}{rr}1 & 1 \\ -1 & -\frac{7}{6}\end{array}\right)$. (b) Predict the long term behavior of the iterative system $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}+\mathbf{b}$, where $\mathbf{b}=\left(\begin{array}{c}-1 \\ 2\end{array}\right)$, in as much detail as you can.

Check back soon!

07:07

Problem 1

Use the Power Method to find the dominant eigenvalue and associated eigenvector of the following matrices:
(a) $\left(\begin{array}{rr}-1 & -2 \\ 3 & 4\end{array}\right)$,
(b) $\left(\begin{array}{ll}-5 & 2 \\ -3 & 0\end{array}\right)$,
(c) $\left(\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right)$.
(d) $\left(\begin{array}{rrr}-2 & 0 & 1 \\ -3 & -2 & 0 \\ -2 & 5 & 4\end{array}\right)$,
(e) $\left(\begin{array}{rrr}-1 & -2 & -2 \\ 1 & 2 & 5 \\ -1 & 4 & 0\end{array}\right)$,
(f) $\left(\begin{array}{lll}2 & 2 & 1 \\ 1 & 3 & 1 \\ 2 & 2 & 2\end{array}\right)$,
(g) $\left(\begin{array}{rrrr}2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2\end{array}\right)$,
(h) $\left(\begin{array}{llll}4 & 1 & 0 & 1 \\ 1 & 4 & 1 & 0 \\ 0 & 1 & 4 & 1 \\ 1 & 0 & 1 & 4\end{array}\right)$

Nicholas Barvinok

Numerade Educator

00:54

Problem 1

Find an orthonormal basis for the Krylov subspaces $V^{(1)}, V^{(2)}, V^{(3)}$ for the following matrices and vectors:
(a) $A=\left(\begin{array}{ll}0 & 1 \\ 3 & 1\end{array}\right), \mathbf{v}=\left(\begin{array}{r}1 \\ -1\end{array}\right)$;
(b) $A=\left(\begin{array}{rrr}2 & 2 & -1 \\ 2 & -1 & 0 \\ 2 & 1 & 3\end{array}\right), \mathbf{v}=\left(\begin{array}{r}-1 \\ 2 \\ 0\end{array}\right)$;
(c) $A=\left(\begin{array}{rrr}1 & 0 & -1 \\ 0 & 2 & -3 \\ 2 & -1 & 0\end{array}\right), \quad \mathbf{v}=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$;
(d) $A=\left(\begin{array}{rrrr}2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2\end{array}\right), \quad \mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 0\end{array}\right)$.

Victor Salazar

Numerade Educator

00:35

Problem 1

Let $f(x)=x$. (a) Determine its Haar wavelet coefficients $c_{j, k^*}$ (b) Graph the partial sums $s_r(x)$ of the Haar wavelet series (9.136) where $j$ goes from 0 to $r=2,5$, and 10. Compare your graphs with that of $f$ and discuss what you observe. Is the series converging to the function? Can you prove this? (c) What is the maximal deviation $\left\|f-s_r\right\|_{\infty}=\max \left\{\left|f(x)-s_r(x)\right| \mid 0 \leq x \leq 1\right\}$ for each of your partial sums?

Nick Johnson

Numerade Educator

03:09

Problem 2

A bank offers $3.25 \%$ interest compounded yearly. Suppose you deposit $\$ 100$. (a) Set up a linear iterative equation to represent your bank balance. (b) How much money do you have after 10 years? (c) What if the interest is compounded monthly?

Allison Knapp

Numerade Educator

02:30

Problem 2

Determine whether or not the following matrices are convergent:
(a) $\left(\begin{array}{rr}2 & -3 \\ 3 & 2\end{array}\right)$,
(b) $\left(\begin{array}{rr}.6 & -3 \\ .3 & .7\end{array}\right)$,
(c) $\frac{1}{5}\left(\begin{array}{rrr}5 & -3 & -2 \\ 1 & -2 & 1 \\ 1 & -5 & 4\end{array}\right)$,
(d) $\left(\begin{array}{lll}.8 & .3 & .2 \\ .1 & .2 & .6 \\ .1 & .5 & .2\end{array}\right)$.

Chris Trentman

Numerade Educator

02:58

Problem 2

A business executive is managing three branches, labeled $A, B$, and $C$, of a corporation. She never visits the same branch on consecutive days. If she visits branch $A$ one day, she visits branch $B$ the next day. If she visits either branch $B$ or $C$ that day, then the next day she is twice as likely to visit branch $A$ as to visit branch $B$ or $C$. Explain why the resulting transition matrix is regular. Which branch does she visit the most often in the long run?

Anthony Ramos

Numerade Educator

04:52

Problem 2

Answer Exercise 9.4.1 when
(a) $T=\left(\begin{array}{rr}1 & -\frac{1}{2} \\ -1 & \frac{3}{2}\end{array}\right), \mathbf{b}=\left(\begin{array}{l}0 \\ 1\end{array}\right)$;
(b) $T=\left(\begin{array}{ccc}\frac{1}{4} & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{4} \\ 1 & 1 & \frac{1}{7}\end{array}\right), \quad \mathbf{b}=\left(\begin{array}{r}1 \\ -1 \\ 3\end{array}\right)$;
(c) $T=\left(\begin{array}{rrr}-.05 & .15 & .15 \\ .35 & .15 & -.35 \\ -.2 & -.2 & .3\end{array}\right), \mathbf{b}=\left(\begin{array}{r}-1.5 \\ 1.6 \\ 1.7\end{array}\right)$.

Gideon Idumah

Numerade Educator

Problem 2

Use the Power Method to find the largest singular value of the following matrices:
(a) $\left(\begin{array}{rr}1 & 2 \\ -1 & 3\end{array}\right)$,
(b) $\left(\begin{array}{rrr}2 & 1 & -1 \\ -2 & 3 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rrrr}2 & 2 & 1 & -1 \\ 1 & -2 & 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}3 & 1 & -1 \\ 1 & -2 & 2 \\ 2 & -1 & 1\end{array}\right)$.

Check back soon!

02:09

Problem 2

Let $\mathbf{v}=\mathbf{x}+$ i $\mathbf{y}$ be an eigenvector corresponding to a complex, non-real eigenvalue of the real $n \times n$ matrix $A$. (a) Prove that the Krylov subspaces $V^{(k)}$ for $k \geq 2$ generated by both $\mathbf{x}$ and $\mathbf{y}$ are all two-dimensional. (b) Is the converse valid? Specifically, if $\operatorname{dim} V^{(3)}=2$, then all $V^{(k)}$ are two-dimensional for $k \geq 1$ and spanned by the real and imaginary parts of a complex eigenvector of $A$.

Runpeng Li

Numerade Educator

00:48

Problem 2

Answer Exercise 9.7.1 for the functions
(a) $x^2-x$,
(b) $\cos \pi x$,
(c) $\begin{cases}e^{-x}, & 0<x<\frac{1}{2} \\ -e^{-x}, & \frac{1}{2}<x<1 .\end{cases}$

Amrita Bhasin

Numerade Educator

02:16

Problem 3

Show that the yearly balances of an account whose interest is compounded monthly satisfy a linear iterative system. How is the effective yearly interest rate determined from the original annual interest rate?

Priyanka Sadarangani

Numerade Educator

01:04

Problem 3

Which of the listed coefficient matrices defines a linear iterative system with asymptotically stable zero solution?
(a) $\left(\begin{array}{rr}-3 & 0 \\ -4 & -1\end{array}\right)$,
(b) $\left(\begin{array}{cc}\frac{1}{2} & \frac{3}{4} \\ \frac{2}{3} & \frac{1}{3}\end{array}\right)$,
(c) $\left(\begin{array}{rr}\frac{1}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2}\end{array}\right)$,
(d) $\left(\begin{array}{rrr}-1 & 3 & 0 \\ -1 & 1 & -1 \\ 0 & -1 & -1\end{array}\right)$,
(e) $\left(\begin{array}{rrr}\frac{1}{2} & \frac{1}{4} & -\frac{1}{4} \\ \frac{1}{2} & \frac{3}{4} & -\frac{1}{2} \\ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2}\end{array}\right)$,
(f) $\left(\begin{array}{rrr}3 & 0 & -1 \\ 0 & 1 & 0 \\ 2 & 0 & 0\end{array}\right)$,
(g) $\left(\begin{array}{rrrr}1 & 0 & -3 & -2 \\ -\frac{1}{2} & \frac{1}{2} & 2 & \frac{3}{2} \\ -\frac{1}{6} & 0 & \frac{3}{2} & \frac{2}{3} \\ \frac{2}{3} & 0 & -3 & -\frac{5}{3}\end{array}\right)$.

Ankit Gupta

Numerade Educator

01:16

Problem 3

A study has determined that, on average, a man's occupation depends on that of his father. If the father is a farmer, there is a $30 \%$ chance that the son will be a blue collar laborer, a $30 \%$ chance he will be a white collar professional, and a $40 \%$ chance he will also be a farmer. If the father is a laborer, there is a $30 \%$ chance that the son will also be one, a $60 \%$ chance he will be a professional, and a $10 \%$ chance he will be a farmer. If the father is a professional, there is a $70 \%$ chance that the son will also be one, a $25 \%$ chance he will be a laborer, and a $5 \%$ chance he will be a farmer. (a) What is the probability that the grandson of a farmer will also be a farmer? (b) In the long run, what proportion of the male population will be farmers?

Tyler Moulton

Numerade Educator

01:09

Problem 3

Which of the following systems have a strictly diagonally dominant coefficient matrix?
(a)
$$
5 x-y=1 \text {, }
$$
$-x+3 y=-1$
$-x+\frac{1}{2} y+\frac{1}{3} z=1$,
(e)
$$
\begin{aligned}
& \frac{1}{3} x+2 y+\frac{3}{4} z=-3 \\
& \frac{2}{3} x+\frac{1}{4} y-\frac{3}{2} z=2
\end{aligned}
$$
$$
\frac{1}{2} x+\frac{1}{3} y=1 \text {, }
$$
$$
\frac{1}{5} x+\frac{1}{4} y=6 \text {; }
$$
(c)
$$
-5 x+y=3 \text {, }
$$
$-3 x+2 y=-2$;
$$
-2 x+y+z=1 \text {, }
$$
(d) $-x+2 y-z=-2$,
$$
x-y+3 z=1 \text {; }
$$
$$
-4 x+2 y+z=2,
$$
(f)
$$
x-2 y+z=1 \text {, }
$$
(g)
$$
\begin{aligned}
& -x+3 y+z=-1 . \\
& x+4 y-6 z=3 .
\end{aligned}
$$

Mark Augustyn

Numerade Educator

35:11

Problem 3

Let $T_n$ be the tridiagonal matrix whose diagonal entries are all equal to 2 and whose sub- and super-diagonal entries all equal 1. Use the Power Method to find the dominant eigenvalue of $T_n$ for $n=10,20,50$. Do your values agree with those in Exercise 8.2.47? How many iterations do you require to obtain 4 decimal place accuracy?

Chris Trentman

Numerade Educator

03:05

Problem 3

(a) Prove that the dimension of a Krylow subspace is bounded by the degree of the minimal polynomial of the matrix $A$, as defined in Exercise 8.6.23. (b) Is there always a Krylov subspace whose dimension equals the degree of the minimal polynomial?

Nick Johnson

Numerade Educator

Problem 3

In this exercise, we investigate the compression capabilities of the Haar wavelets. Let $f(x)=\left\{\begin{array}{ll}-x, & 0 \leq x \leq \frac{1}{3} \pi, \\ x-\frac{2}{3} \pi, & \frac{1}{3} \pi \leq x \leq \frac{4}{3} \pi, \\ -x+2 \pi, & \frac{4}{3} \pi \leq x \leq 2 \pi,\end{array}\right.$ represent a signal defined on $0 \leq x \leq 1$. Let $s_r(x)$ denote the $n^{\text {th }}$ partial sum, from $j=0$ to $r$, of the Haar wavelet series (9.136). (a) How many different Haar wavelet coefficients $c_{j, k}$ appear in $s_r(x)$ ? If our criterion for compression is that $\left\|f-s_r\right\|_{\infty}<\varepsilon$, how large do you need to choose $r$ when $\varepsilon=.1$ ? $\varepsilon=.01$ ? $\varepsilon=.001$ ? (b) Compare the Haar wavelet compression with the discrete Fourier method of Exercise 5.6.10.

Check back soon!

01:51

Problem 4

Show that, as the time interval of compounding goes to zero, the bank balance after $k$ years approaches an exponential function $e^{r k} a$, where $r$ is the yearly interest rate and $a$ is the initial balance.

Taylor Shimono

Numerade Educator

Problem 4

(a) Determine the eigenvalues and spectral radius of the matrix $T=\left(\begin{array}{rrr}3 & 2 & -2 \\ -2 & 1 & 0 \\ 0 & 2 & 1\end{array}\right)$.
(b) Use part (a) to find the eigenvalues and spectral radius of $\hat{T}=\left(\begin{array}{rrr}\frac{3}{5} & \frac{2}{5} & -\frac{2}{5} \\ -\frac{2}{5} & \frac{1}{5} & 0 \\ 0 & \frac{2}{5} & \frac{1}{5}\end{array}\right)$.
(c) Write down an asymptotic formula for the solutions to $\mathbf{u}^{(k+1)}=\hat{T} \mathbf{u}^{(k)}$.

Check back soon!

02:08

Problem 4

The population of an island is divided into city and country residents. Each year, $5 \%$ of the residents of the city move to the country and $15 \%$ of the residents of the country move to the city. In $2003,35,000$ people live in the city and 25,000 in the country. Assuming no growth in the population, how many people will live in the city and how many will live in the country between the years 2004 and 2008 ? What is the eventual population distribution of the island?

Nick Johnson

Numerade Educator

02:20

Problem 4

For the strictly diagonally dominant systems in Exercise 9.4.3, starting with the initial guess $x=y=z=0$, compute the solution to 2 decimal places using the Jacobi Method. Check your answer by solving the system directly by Gaussian Elimination.

Joseph Palsic

Numerade Educator

View

Problem 4

Prove that, for the iterative method (9.82), $\left\|A \mathbf{u}^{(k)}\right\| \rightarrow\left|\lambda_1\right|$. Assuming $\lambda_1$ is real, explain how to deduce its sign.

Victor Salazar

Numerade Educator

00:26

Problem 4

True or false: A Krylov subspace is an invariant subspace for the matrix $A$.

Taylor Shimono

Numerade Educator

08:45

Problem 4

(a) Explain why the wavelet expansion (9.136) defines a linear transformation on $\mathbb{R}^n$ that takes a wavelet coefficient vector $\mathbf{c}=\left(c_0, c_1, \ldots, c_{n-1}\right)^T$ to the corresponding sample vector $\mathbf{f}=\left(f_0, f_1, \ldots, f_{n-1}\right)^T$.
(b) According to Theorem 7.5, the wavelet map must be given by matrix multiplication $\mathbf{f}=W_n$ c by a $2 \times 2^n$ matrix $W=W_n$. Construct $W_2, W_3$ and $W_4$. (c) Prove that the columns of $W_n$ are obtained as the values of the wavelet basis functions on the $2^n$ sample intervals. (d) Prove that the columns of $W_n$ are orthogonal. (e) Is $W_n$ an orthogonal matrix? Find a formula for $W_n^{-1}$. (f) Explain why the wavelet transform is given by the linear map, $\mathrm{c}=W_n^{-1} \mathbf{f}$.

Foster Wisusik

Numerade Educator

01:58

Problem 5

For which values of $\lambda$ does the scalar iterative system (9.2) have a periodic solution, meaning that $u^{(k+m)}=u^{(k)}$ for some $m$ ?

Raj Bala

Numerade Educator

Problem 5

(a) Show that the spectral radius of $T=\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right)$ is $\rho(T)=1$.
(b) Show that most iterates $\mathbf{u}^{(k)}=T^k \mathbf{u}^{(0)}$ become unbounded as $k \rightarrow \infty$.
(c) Discuss why the inequality $\left\|\mathbf{u}^{(k)}\right\| \leq C \rho(T)^k$ does not hold when the coefficient matrix is incomplete. $(d)$ Can you prove that $(9.28)$ holds in this example?

Check back soon!

01:42

Problem 5

A certain plant species has either red, pink, or white flowers, depending on its genotype. If you cross a pink plant with any other plant, the probability distribution of the offspring is prescribed by the transition matrix $T=\left(\begin{array}{ccc}5 & .25 & 0 \\ .5 & .5 & .5 \\ 0 & .25 & .5\end{array}\right)$. On average, if you continue crossing with only pink plants, what percentage of the three types of flowers would you expect to see in your garden?

Joshua Eastwood

Numerade Educator

01:13

Problem 5

(a) Do any of the non-strictly diagonally dominant systems in Exercise 9.4 .3 lead to convergent Jacobi algorithms? Hint: Check the spectral radius of the Jacobi matrix.
(b) For the convergent systems in Exercise 9.4.3, starting with the initial guess $x=y=z=$ 0 , compute the solution to 2 decimal places by using the Jacobi Method, and check your answer by solving the system directly by Gaussian Elimination.

Victor Salazar

Numerade Educator

06:28

Problem 5

Prove that the invertibility of the coefficient matrix $S^T A S$ in (9.104) depends only on the subspace $V$ and not on the choice of basis thereof.

Jimmy Yao

Numerade Educator

Problem 5

Test the noise removal features of the Haar wavelets by adding random noise to one of the functions in Exercises 9.7.1 and 9.7.2, computing the wavelet series, and then setting the high "frequency" modes to zero. What do you observe? Is this a reasonable denoising algorithm when compared with a Fourier method?

Check back soon!

02:55

Problem 6

Consider the iterative systems $u^{(k+1)}=\lambda u^{(k)}$ and $v^{(k+1)}=\mu v^{(k)}$, where $|\lambda|>|\mu|$. Prove that, for all nonzero initial data $u^{(0)}=a \neq 0, v^{(0)}=b \neq 0$, the solution to the first is eventually larger (in modulus) than that of the second: $\left|u^{(k)}\right|>\left|v^{(k)}\right|$, for $k \gg 0$.

Jacquelyn Trost

Numerade Educator

00:47

Problem 6

Given a linear iterative system with non-convergent matrix, which solutions, if any, will converge to $\boldsymbol{0}$ ?

Vishal Parmar

Numerade Educator

View

Problem 6

A genetic model describing inbreeding, in which mating takes place only between individuals of the same genotype, is given by the Markov process $\mathbf{u}^{(n+1)}=T \mathbf{u}^{(n)}$, where $T=\left(\begin{array}{ccc}1 & \frac{1}{4} & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{4} & 1\end{array}\right)$ is the transition matrix and $\mathbf{u}^{(n)}=\left(\begin{array}{c}p_n \\ q_n \\ v_n\end{array}\right)$, whose entries are, respectively, the proportion of populations of genotype $\mathrm{AA}$, $\mathrm{Aa}$, a in the $n^{\text {th }}$ generation. Find the solution to this Markov process and analyze your result.

Victor Salazar

Numerade Educator

08:28

Problem 6

The following linear systems have positive definite coefficient matrices. Use the Jacobi Method starting with $\mathbf{u}^{(0)}=\mathbf{0}$ to find the solution to 4 decimal place accuracy.
(a) $\left(\begin{array}{rr}3 & -1 \\ -1 & 5\end{array}\right) \mathbf{u}=\left(\begin{array}{l}2 \\ 1\end{array}\right)$,
(b) $\left(\begin{array}{ll}2 & 1 \\ 1 & 1\end{array}\right) \mathbf{u}=\left(\begin{array}{r}-3 \\ 1\end{array}\right)$,
(c) $\left(\begin{array}{rrr}6 & -1 & -3 \\ -1 & 7 & 4 \\ -3 & 4 & 9\end{array}\right) \mathbf{u}=\left(\begin{array}{r}-1 \\ -2 \\ 7\end{array}\right)$,
(d) $\left(\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & 1 \\ 0 & 1 & 5\end{array}\right) \mathbf{u}=\left(\begin{array}{r}1 \\ -5 \\ 0\end{array}\right)+(e)\left(\begin{array}{llll}5 & 1 & 1 & 1 \\ 1 & 5 & 1 & 1 \\ 1 & 1 & 5 & 1 \\ 1 & 1 & 1 & 5\end{array}\right) \mathbf{u}=\left(\begin{array}{l}4 \\ 0 \\ 0 \\ 0\end{array}\right)$
(f) $\left(\begin{array}{rrrr}3 & 1 & 0 & -1 \\ 1 & 3 & 1 & 0 \\ 0 & 1 & 3 & 1 \\ -1 & 0 & 1 & 3\end{array}\right) \mathbf{u}=\left(\begin{array}{r}1 \\ 2 \\ 0 \\ -1\end{array}\right)$

Nicholas Barvinok

Numerade Educator

Problem 6

The Inverse Power Method. Let $A$ be a nonsingular matrix. (a) Show that the eigenvalues of $A^{-1}$ are the reciprocals $1 / \lambda$ of the eigenvalues of $A$. How are the eigenvectors related? (b) Show how to use the Power Method on $A^{-1}$ to produce the smallest (in modulus) eigenvalue of $A$. (c) What is the rate of convergence of the algorithm?
(d) Design a practical iterative algorithm based on the (permuted) $L U$ decomposition of $A$.

Check back soon!

Problem 6

Apply the Inverse Power Method of Exercise 9.5.7 to the find the smallest eigenvalue of the matrices in Exercise 9.5.1.

Check back soon!

02:09

Problem 6

Prove that $(9.92,93,94)$ give the same Arnoldi vectors $\mathbf{u}_k$ and the same coefficients $h_{j k}$ when computed exactly.

Abdul Vahid M

Numerade Educator

Problem 6

Write the Haar scaling function and mother wavelet as linear combinations of step functions.

Check back soon!

03:40

Problem 7

Let $u(t)$ denote the solution to the linear ordinary differential equation $\dot{u}=\beta u$, $u(0)=a$. Let $h>0$. Show that the sample values $u^{(k)}=u(k h)$ satisfy a linear iterative system. What is the coefficient $\lambda$ ? Compare the stability properties of the differential equation and the corresponding iterative system.

Ajay Singhal

Numerade Educator

View

Problem 7

Suppose $T$ is a complete matrix. (a) Prove that every solution to the corresponding linear iterative system is bounded if and only if $\rho(T) \leq 1$. (b) Can you generalize this result to incomplete matrices? Hint: Look at Exercise 9.1.40.

Victor Salazar

Numerade Educator

03:48

Problem 7

A student has the habit that if she doesn't study one night, she is $70 \%$ certain of studying the next night. Furthermore, the probability that she studies two nights in a row is $50 \%$. How often does she study in the long run?

Sohyun Lee

Numerade Educator

View

Problem 7

Let $A$ be the $n \times n$ tridiagonal matrix with all its diagonal entries equal to $c$ and all 1 's on the sub-and super-diagonals. (a) For which values of $c$ is $A$ strictly diagonally dominant? (b) For which values of $c$ does the Jacobi iteration for $A \mathbf{u}=\mathbf{b}$ converge to the solution? What is the rate of convergence? Hint: Use Exercise 8.2.48. (c) Set $c=2$ and use the Jacobi Method to solve the linear systems $K \mathbf{u}=\mathbf{e}_1$, for $n=5,10$, and 20. Starting with an initial guess of $\mathbf{0}$, how many Jacobi iterations does it take to obtain 3 decimal place accuracy? Does the convergence rate agree with what you computed in part (c)?

Nick Johnson

Numerade Educator

01:46

Problem 7

The Shifted Inverse Power Method. Suppose that $\mu$ is not an eigenvalue of $A$.
(a) Show that the iterative system $\mathbf{u}^{(k+1)}=(A-\mu \mathrm{I})^{-1} \mathbf{u}^{(k)}$ converges to the eigenvector of $A$ corresponding to the eigenvalue $\lambda^*$ that is closest to $\mu$. Explain how to find the eigenvalue $\lambda^*$. (b) What is the rate of convergence of the algorithm? (c) What happens if $\mu$ is an eigenvalue?

James Chok

Numerade Educator

05:25

Problem 7

Solve the following linear systems by the Conjugate Gradient Method, keeping track of the residual vectors and solution approximations as you iterate.
(a)
$$
\begin{gathered}
\left(\begin{array}{rr}
3 & -1 \\
-1 & 5
\end{array}\right) \mathbf{u}=\left(\begin{array}{l}
2 \\
1
\end{array}\right), \quad(b)\left(\begin{array}{rrr}
6 & 2 & 1 \\
2 & 3 & -1 \\
1 & -1 & 2
\end{array}\right) \mathbf{u}=\left(\begin{array}{r}
1 \\
0 \\
-2
\end{array}\right), \quad(c)\left(\begin{array}{rrr}
6 & -1 & -3 \\
-1 & 7 & 4 \\
-3 & 4 & 9
\end{array}\right) \mathbf{u}=\left(\begin{array}{r}
-1 \\
-2 \\
7
\end{array}\right), \\
\text { (d) }\left(\begin{array}{rrrr}
6 & -1 & -1 & 5 \\
-1 & 7 & 1 & -1 \\
-1 & 1 & 3 & -3 \\
5 & -1 & -3 & 6
\end{array}\right) \mathbf{u}=\left(\begin{array}{r}
1 \\
2 \\
0 \\
-1
\end{array}\right), \quad(e)\left(\begin{array}{llll}
5 & 1 & 1 & 1 \\
1 & 5 & 1 & 1 \\
1 & 1 & 5 & 1 \\
1 & 1 & 1 & 5
\end{array}\right) \mathbf{u}=\left(\begin{array}{l}
4 \\
0 \\
0 \\
0
\end{array}\right) .
\end{gathered}
$$

Nick Johnson

Numerade Educator

Problem 7

Prove Lemma 9.54.

Check back soon!

00:41

Problem 8

Investigate the solutions of the linear iterative equation $u^{(k+1)}=\lambda u^{(k)}$ when $\lambda$ is a complex number with $|\lambda|=1$, and look for patterns.

Ankit Gupta

Numerade Educator

Problem 8

Discuss the asymptotic behavior of solutions to an iterative system that has two eigenvalues of largest modulus, e.g., $\lambda_1=-\lambda_2$, or $\lambda_1=\bar{\lambda}_2$ are complex conjugate eigenvalues. How would you detect this? How can you determine the eigenvalues and eigenvectors?

Check back soon!

02:16

Problem 8

A traveling salesman visits the three cities of Atlanta, Boston, and Chicago. The matrix $\left(\begin{array}{rrr}0 & .5 & .5 \\ 1 & 0 & .5 \\ 0 & .5 & 0\end{array}\right)$ describes the transition probabilities of his trips. Describe his travels in words, and calculate how often he visits each city on average.

Lucas Finney

Numerade Educator

05:10

Problem 8

Prove that $\mathbf{0} \neq \mathbf{u} \in$ ker $A$ if and only if $\mathbf{u}$ is an eigenvector of the Jacobi iteration matrix. with eigenvalue 1 . What does this imply about convergence?

Gennady Notowidigdo

Numerade Educator

Problem 8

Apply the Shifted Inverse Power Method of Exercise 9.5.7 to the find the eigenvalue closest to $\mu=.5$ of the matrices in Exercise 9.5.1.

Check back soon!

Problem 8

Use the Conjugate Gradient Method to solve the system in Exercise 9.4.33. How many iterations do you need to obtain the solution that is accurate to 2 decimal places? How does this compare to the Jacobi and SOR Methods?

Check back soon!

Problem 8

Answer Exercises 9.7.1 and 9.7.2 using the Daubechies wavelets instead of the Haar wavelets. Do you see any improvement in your approximations? Discuss the advantages and disadvantages of both in light of these examples.

Check back soon!

02:06

Problem 9

Let $\lambda, c \in \mathbb{R}$. Solve the affine or inhomogeneous linear iterative equation
$$
u^{(k+1)}=\lambda u^{(k)}+c, \quad u^{(0)}=a .
$$
Discuss the possible behaviors of the solutions, Hint: Write the solution in the form $u^{(k)}=u^*+v^{(k)}$, where $u^*$ is the equilibrium solution.

Ronald Prasad

Numerade Educator

01:18

Problem 9

Suppose $T$ has spectral radius $\rho(T)$. Can you predict the spectral radius of $a T+b \mathrm{I}$, where $a, b$ are scalars? If not, what additional information do you need?

Mitchel Vereen

Numerade Educator

03:04

Problem 9

Explain why the irregular Markov process with transition matrix $T=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$ does not reach a steady state. Use a population model, as in Exercise 9.3.4, to interpret what is going on.

Amany Waheeb

Numerade Educator

Problem 9

Prove that if $A$ is a nonsingular coefficient matrix, then one can always arrange that all its diagonal entries are nonzero by suitably permuting its rows.

Check back soon!

02:42

Problem 9

Suppose that $A \mathbf{u}^{(k)}=\mathbf{0}$ in the iterative procedure (9.82). What does this indicate?

Faizanullah Kazmi

Numerade Educator

Problem 9

According to Example 3.39, the $n \times n$ Hilbert matrix $H_n$ is positive definite, and hence we can apply the Conjugate Gradient Method to solve the linear system $H_n \mathbf{u}=\mathbf{f}$. For the values $n=5,10,30$, let $\mathbf{u}^* \in \mathbb{R}^n$ be the vector with all entries equal to 1 .
(a) Compute $\mathbf{f}=H_n \mathbf{u}^*$. (b) Use Gaussian Elimination to solve $H_n \mathbf{u}=\mathbf{f}$. How close is your solution to $\mathbf{u}^*$ ? ${ }^n$ (c) Does pivoting improve the solution in part (b)?
(d) Does the conjugate gradient algorithm do any better?

Check back soon!

View

Problem 9

Answer Exercise 9.7.3 using the Daubechies wavelets to compress the data. Compare your results.

Victor Salazar

Numerade Educator

01:33

Problem 10

A bank offers $5 \%$ interest compounded yearly. Suppose you deposit $\$ 120$ in the account each year. Set up an affine iterative equation $(9.5)$ to represent your bank balance. How much money do you have after 10 years? After you retire in 50 years? After 200 years?

Lourence Gonhovi

Numerade Educator

View

Problem 10

Prove that if $A$ is any square matrix, then there exists $c \neq 0$ such that the scalar multiple $c A$ is a convergent matrix. Find a formula for the largest possible such $c$.

Victor Salazar

Numerade Educator

01:10

Problem 10

A bug crawls along the edges of the pictured triangular lattice with six vertices. Upon arriving at a vertex, there is an equal probability of its choosing any edge to leave the vertex. Set up the Markov chain described by the bug's motion, and determine how often, on average, it visits each vertex.

Nick Johnson

Numerade Educator

07:13

Problem 10

Consider the iterative system $(9.42)$ with spectral radius $\rho(T)<1$. Explain why it takes roughly $-1 / \log _{10} \rho(T)$ iterations to produce one further decimal digit of accuracy in the solution.

Amy Jiang

Numerade Educator

Problem 10

(i) Explain how to use the Deflation Method of Exercise 8.2.51 to find the subdominant eigenvalue of a nonsingular matrix $A$. (ii) Apply your method to the matrices listed in Exercise 9.5.1.

Check back soon!

01:00

Problem 10

Try applying the Conjugate Gradient algorithm to the system $-x+2 y+z=-2$, $y+2 z=1,3 x+y-z=1$. Do you obtain the solution? Why or why not?

Hast Aggarwal

Numerade Educator

01:20

Problem 10

Verify formulas (9.139) and (9.141).

Darshan Maheshwari

Numerade Educator

01:29

Problem 11

Redo Exercise 9.1.10 in the case that the interest is compounded monthly and you deposit $\$ 10$ each month.

Nick Johnson

Numerade Educator

01:53

Problem 11

Let $M_n$ be the $n \times n$ tridiagonal matrix with all 1 's on the sub- and super-diagonals, and zeros on the main diagonal. (a) What is the spectral radius of $M_n$ ? Hint: Use Exercise 8.2.47. (b) Is $M_n$ convergent? (c) Find the general solution to the iterative system $\mathbf{u}^{(k+1)}=M_n \mathbf{u}^{(k)}$.

Michelle Z.

Numerade Educator

02:40

Problem 11

Answer Exercise 9.3.10 for the larger triangular lattice.

Aman Gupta

Numerade Educator

View

Problem 11

True or false: If a system $A \mathbf{u}=\mathbf{b}$ has a strictly diagonally dominant coefficient matrix $A$, then the equivalent system obtained by applying an elementary row operation to $A$ also has a strictly diagonally dominant coefficient matrix.

Donna Densmore

Numerade Educator

Problem 11

Apply the $Q R$ algorithm to the following symmetric matrices to find their eigenvalues and eigenvectors to 2 decimal places:
(a) $\left(\begin{array}{ll}1 & 2 \\ 2 & 6\end{array}\right)$,
(b) $\left(\begin{array}{rr}3 & -1 \\ -1 & 5\end{array}\right)$,
(c) $\left(\begin{array}{lll}2 & 1 & 0 \\ 1 & 2 & 3 \\ 0 & 3 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}2 & 5 & 0 \\ 5 & 0 & -3 \\ 0 & -3 & 3\end{array}\right)$
(e) $\left(\begin{array}{rrrr}3 & -1 & 0 & 0 \\ -1 & 3 & -1 & 0 \\ 0 & -1 & 3 & -1 \\ 0 & 0 & -1 & 3\end{array}\right)$,
(f) $\left(\begin{array}{rrrr}6 & 1 & -1 & 0 \\ 1 & 8 & 1 & -1 \\ -1 & 1 & 4 & 1 \\ 0 & -1 & 1 & 3\end{array}\right)$.

Check back soon!

00:51

Problem 11

True or folse: If the residual vector $\mathbf{r}=\mathbf{b}-A \mathbf{x}$ satisfies $\|\mathbf{r}\|<.01$, then $\mathbf{x}$ approximates the true solution to within two decimal places.

Monica Miller

Numerade Educator

01:44

Problem 11

Prove that the most general solution to the functional equation $\varphi(x)=2 \varphi(2 x)$ is $\varphi(x)=f\left(\log _2 x\right) / x$ where $f(z+1)=f(z)$ is any 1 periodic function.

Varsha Aggarwal

Numerade Educator

10:32

Problem 12

Each spring, the deer in Minnesota produce offspring at a rate of roughly 1.2 times the total population, while approximately $5 \%$ of the population dies as a result of predators and natural causes. In the fall, hunters are allowed to shoot 3,600 deer. This winter the Department of Natural Resources (DNR) estimates that there are 20,000 deer. Set up an affine iterative equation (9.5) to represent the deer population each subsequent year. Solve the system and find the population in the next 5 years. How many deer in the long term will there be? Using this information, formulate a reasonable policy of how many deer hunting licenses the DNR should allow each fall, assuming one kill per license.

Brittany Knowlton

Numerade Educator

Problem 12

Let $\alpha, \beta$ be scalars. Let $T_{\alpha, \beta}$ be the $n \times n$ tridiagonal matrix that has all $\alpha$ 's on the sub-and super-diagonals, and $\beta$ 's on the main diagonal. (a) Solve the iterative system $\mathbf{u}^{(k+1)}=T_{\alpha, \beta} \mathbf{u}^{(k)}$. (b) For which values of $\alpha, \beta$ is the system asymptotically stable? Hint: Combine Exercises 9.2.11 and 9.1.32.

Check back soon!

00:44

Problem 12

Suppose the bug of Exercise 9.3.10 crawls along the edges of the pictured square lattice. What can you say about its behavior?

Eugene Schneider

University of Minnesota - Twin Cities

Problem 12

Consider the linear system $A \mathbf{x}=\mathbf{b}$, where $A=\left(\begin{array}{rrr}4 & 1 & -2 \\ -1 & 4 & -1 \\ 1 & -1 & 4\end{array}\right), \mathbf{b}=\left(\begin{array}{r}-2 \\ -1 \\ 7\end{array}\right)$. (a) First, solve the equation directly by Gaussian Elimination. (b) Write the Jacobi iteration in the form $\mathbf{x}^{(k+1)}=T \mathbf{x}^{(k)}+\mathbf{c}$. Find the $3 \times 3$ matrix $T$ and the vector $\mathbf{c}$ explicitly. (c) Using the initial approximation $\mathbf{x}^{(0)}=\mathbf{0}$, carry out three iterations of the Jacobi algorithm to compute $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}$ and $\mathbf{x}^{(3)}$. How close are you to the exact solution? (d) Write the Gauss-Seidel iteration in the form $\mathbf{x}^{(k+1)}=\bar{T} \mathbf{x}^{(k)}+\overline{\mathbf{c}}$. Find the $3 \times 3$ matrix $\bar{T}$ and the vector $\overline{\mathbf{c}}$ explicitly. (e) Using the initial approximation $\mathbf{x}^{(0)}=\mathbf{0}$, carry out three iterations of the Gauss - Seidel algorithm. Which is a better approximation to the solution - Jacobi or Gauss-Seidel? (f) Determine the spectral radius of the Jacobi matrix $T$, and use this to prove that the Jacobi Method will converge to the solution of $A \mathbf{x}=\mathbf{b}$ for any choice of the initial approximation $\mathbf{x}^{(0)}$ (g) Determine the spectral radius of the Gauss-Seidel matrix $\bar{T}$. Which method converges faster? $(h)$ For the faster method, how many iterations would you expect to need to obtain 5 decimal place accuracy?
(i) Test your prediction by computing the solution to the desired accuracy.

Check back soon!

13:43

Problem 12

Show that applying the $Q R$ algorithm to the matrix $A=\left(\begin{array}{rrr}4 & -1 & 1 \\ -1 & 7 & 2 \\ 1 & 2 & 7\end{array}\right)$ results in a diagonal matrix with the eigenvalues on the diagonal, but not in decreasing order. Explain.

Cyrielle Lorio

Numerade Educator

View

Problem 12

How many arithmetic operations are needed to implement one iteration of the Conjugate Gradient Method? How many iterations can you perform before the method becomes more work than direct Gaussian Elimination?
Remark. If the matrix is sparse, the number of operations can decrease dramatically.

Victor Salazar

Numerade Educator

00:39

Problem 12

Consider the dilation equation (9.138) with $c_0=0, c_1=c_2=1$, so $\varphi(x)=$ $\varphi(2 x-1)+\varphi(2 x-2)$. Prove that $\psi(x)=\varphi(x+1)$ satisfies the Haar dilation equation (9.139). Generalize this result to prove that we can always, without loss of generality, assume that $c_0 \neq 0$ in the general dilation equation (9.138).

Kashif Qureshi

Numerade Educator

Problem 13

Find the explicit formula for the solution to the following linear iterative systems:
(a) $u^{(k+1)}=u^{(k)}-2 v^{(k)}, v^{(k+1)}=-2 u^{(k)}+v^{(k)}, u^{(0)}=1, v^{(0)}=0$.
(b) $u^{(k+1)}=u^{(k)}-\frac{2}{3} v^{(k)}, v^{(k+1)}=\frac{1}{2} u^{(k)}-\frac{1}{6} v^{(k)}, u^{(0)}=-2, v^{(0)}=3$.
(c) $u^{(k+1)}=u^{(k)}-v^{(k)}, v^{(k+1)}=-u^{(k)}+5 v^{(k)}, u^{(0)}=1, v^{(0)}=0$.
(d) $u^{(k+1)}=\frac{1}{2} u^{(k)}+v^{(k)}, v^{(k+1)}=v^{(k)}-2 w^{(k)}, w^{(k+1)}=\frac{1}{3} w^{(k)}$,
$$
u^{(0)}=1, v^{(0)}=-1, w^{(0)}=1 .
$$
$(e) u^{(k+1)}=-u^{(k)}+2 v^{(k)}-w^{(k)}, v^{(k+1)}=-6 u^{(k)}+7 v^{(k)}-4 w^{(k)}+$ $w^{(k+1)}=-6 u^{(k)}+6 v^{(k)}-4 w^{(k)}, \quad u^{(0)}=0, \quad v^{(0)}=1, \quad w^{(0)}=3$.

Check back soon!

View

Problem 13

(a) Prove that if $|\operatorname{det} T|>1$, then the iterative system $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}$ is unstable.
(b) If $\mid$ det $T \mid<1$, is the system asymptotically stable? Prove or give a counterexample.

Victor Salazar

Numerade Educator

02:02

Problem 13

Let $T$ be a regular transition matrix with probability eigenvector $\mathbf{v}$.
(a) Prove that $\lim _{k \rightarrow \infty} T^k=P=(\mathbf{v} \mathbf{v} \ldots \mathbf{v})$ is a matrix with every column equal to $\mathbf{v}$.
(b) Explain why $(\mathbf{v} \mathbf{v} \ldots \mathbf{v}) \mathbf{v}=\mathbf{v} . \quad$ (c) Prove directly that $P$ is idempotent: $P^2=P$.

Nicholas Barvinok

Numerade Educator

04:39

Problem 13

For the strictly diagonally dominant systems in Exercise 9.4.3, starting with the initial guess $x=y=z=0$, compute the solution to 3 decimal places using the Gauss-Seidel Method. Check your answer by solving the system directly by Gaussian Elimination.

Carson Merrill

Numerade Educator

Problem 13

Apply the $Q R$ algorithm to the following non-symmetric matrices to find their eigenvalues to 3 decimal places:
(a) $\left(\begin{array}{rr}-1 & -2 \\ 3 & 4\end{array}\right)$,
(b) $\left(\begin{array}{ll}2 & 3 \\ 1 & 5\end{array}\right)$,
(c) $\left(\begin{array}{rrr}2 & 1 & 0 \\ 2 & 0 & -3 \\ 0 & -2 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}2 & 5 & 1 \\ 2 & -1 & 3 \\ 4 & 5 & 3\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}6 & 1 & 7 & 9 \\ 6 & 8 & 14 & 9 \\ 3 & 1 & 4 & 6 \\ 3 & 2 & 5 & 3\end{array}\right)$

Check back soon!

Problem 13

Fill in the details in a direct derivation of the Conjugate Gradient algorithm following the ideas outlined in the text: starting with the initial guess $\mathrm{x}_0$ and corresponding residual vector $\mathbf{w}_1=\mathbf{r}_0=\mathbf{b}$, at the $k^{\text {th }}$ step in the algorithm, given the approximation $\mathbf{x}_k$ and residual $\mathbf{r}_k=\mathbf{b}-A \mathbf{x}_k$, the $k^{\text {th }}$ conjugate direction is chosen so that $\mathbf{w}_{k+1}=\mathbf{r}_k+s_k \mathbf{w}_k$ satisfies the conjugacy conditions (9.113). The next approximation $\mathbf{x}_{k+1}=\mathbf{x}_k+t_{k+1} \mathbf{w}_{k+1}$ is chosen so that its residual $\mathbf{r}_{k+1}=\mathbf{b}-A \mathbf{x}_{k+1}$ is as small as possible.

Check back soon!

02:56

Problem 13

Prove that a cubic $B$ spline, as defined in Exercise 5.5.76, solves the dilation equation (9.138) for $c_0=c_4=\frac{1}{8}, c_1=c_3=\frac{1}{2}, c_2=\frac{3}{4}$.

Adarsh Kumar

Numerade Educator

03:13

Problem 14

Find the explicit formula for the general solution to the linear iterative systems with the following coefficient matrices:
(a) $\left(\begin{array}{rr}-1 & 2 \\ 1 & -1\end{array}\right)$,
(b) $\left(\begin{array}{ll}-2 & 7 \\ -1 & 3\end{array}\right)$
(c) $\left(\begin{array}{rrr}-3 & 2 & -2 \\ -6 & 4 & -3 \\ 12 & -6 & -5\end{array}\right)$,
(d) $\left(\begin{array}{rrr}-\frac{5}{6} & \frac{1}{3} & -\frac{1}{6} \\ 0 & -\frac{1}{2} & \frac{1}{3} \\ 1 & -1 & \frac{2}{3}\end{array}\right)$.

Vishnu P

Numerade Educator

00:47

Problem 14

True or false: (a) $\rho(c A)=c \rho(A)$, (b) $\rho\left(S^{-1} A S\right)=\rho(A)$, (c) $\rho\left(A^2\right)=\rho(A)^2$, (d) $\rho\left(A^{-1}\right)=1 / \rho(A)$, (e) $\rho(A+B)=\rho(A)+\rho(B)$, (f) $\rho(A B)=\rho(A) \rho(B)$.

Kian Manafi

Numerade Educator

01:15

Problem 14

Find $\lim _{k \rightarrow \infty} T^k$ when $T=\left(\begin{array}{lll}.8 & .1 & .1 \\ .1 & .8 & .1 \\ 1 & .1 & .8\end{array}\right)$.

Suzanne W.

Numerade Educator

Problem 14

Which of the systems in Exercise 9.4.3 lead to convergent Gauss-Seidel algorithms? In each case, which converges faster, Jacobi or Gauss-Seidel?

Check back soon!

View

Problem 14

The matrix $A=\left(\begin{array}{lll}-1 & 2 & 1 \\ -2 & 3 & 1 \\ -2 & 2 & 2\end{array}\right)$ has a double eigenvalue of 1 , and so our proof of convergence of the $Q R$ algorithm doesn't apply. Does the $Q R$ algorithm find its eigenvalues?

Victor Salazar

Numerade Educator

Problem 14

In $(9.120)$, find the value of $d_k$ that minimizes $p\left(\mathbf{x}_{k+1}\right)$.

Check back soon!

01:09

Problem 14

Explain why the scaling function $\varphi(x)$ and the mother wavelet $w(x)$ have the same support: $\operatorname{supp} \varphi=\operatorname{supp} w$.

Darshan Maheshwari

Numerade Educator

03:09

Problem 15

Prove that all the Fibonaci integers $u^{(k)}, k \geq 0$, can be found by just computing the first term in the Binet formula $(9.17)$ and then rounding off to the nearest integer.

Mohamed Mohamed

Numerade Educator

00:21

Problem 15

True or false: (a) If $T$ is convergent, then $T^2$ is convergent.
(b) If $A$ is convergent, then $T=A^T A$ is convergent.

Nick Johnson

Numerade Educator

03:15

Problem 15

Prove that, for all $0 \leq p, q \leq 1$ with $p+q>0$, the probability eigenvector of the transition matrix $T=\left(\begin{array}{cc}1-p & q \\ p & 1-q\end{array}\right)$ is $\mathbf{v}=\left(\frac{q}{p+q}, \frac{p}{p+q}\right)^T$.

Mahnoor Amin

Numerade Educator

Problem 15

(a) Solve the positive definite linear systems in Exercise 9.4.6 using the Gauss-Seidel Method to achieve 4 decimal place accuracy.
(b) Compare the convergence rate with that of the Jacobi Method.

Check back soon!

Problem 15

Explain why the $Q R$ algorithm fails to find the eigenvalues of the matrices
(a) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$,
(b) $\left(\begin{array}{rrr}-2 & 1 & 0 \\ 0 & -2 & 1 \\ 1 & 0 & -2\end{array}\right)$,
(c) $\left(\begin{array}{rrr}5 & -4 & 2 \\ -4 & 5 & 2 \\ 2 & 2 & -1\end{array}\right)$.

Check back soon!

Problem 15

Use the direct gradient descent algorithm (9.120) using the value of $d_k$ found in Exercise 9.6.14 to solve the linear systems in Exercise 9.6.7. Compare the speed of convergence with that of the Conjugate Gradient Method.

Check back soon!

Problem 15

Prove that (9.147) implies $\langle\varphi(x-l), \varphi(x-m)\rangle=0$ for all $l \neq m$.

Check back soon!

07:31

Problem 16

The $k^{\text {th }}$ Lucas number is defined as $L^{(k)}=\left(\frac{1+\sqrt{5}}{2}\right)^k+\left(\frac{1-\sqrt{5}}{2}\right)^k$.
(a) Explain why the Lucas numbers satisfy the Fibonacci iterative equation $L^{(k+2)}=L^{(k+1)}+L^{(k)}$. (b) Write down the first 7 Lucas numbers.
(c) Prove that every Lucas number is a positive integer.

Chris Trentman

Numerade Educator

02:02

Problem 16

Suppose $T^k \rightarrow P$ as $k \rightarrow \infty$. (a) Prove that $P$ is idempotent: $P^2=P$.
(b) Can you characterize all such matrices $P$ ?
(c) What are the conditions on the matrix $A$ for this to happen?

Nicholas Barvinok

Numerade Educator

Problem 16

Describe the final state of a Markov chain with symmetric transition matrix $T=T^T$.

Check back soon!

Problem 16

Let $A=\left(\begin{array}{llll}c & 1 & 0 & 0 \\ 1 & c & 1 & 0 \\ 0 & 1 & c & 1 \\ 0 & 0 & 1 & c\end{array}\right)$. (a) For what values of $c$ is $A$ strictly diagonally dominant?
(b) Use a computer to find the smallest positive value of $c>0$ for which Jacobi iteration converges. (c) Find the smallest positive value of $c>0$ for which Gauss-Seidel iteration converges. Is your answer the same? (d) When they both converge, which converges faster - Jacobi or Gauss-Seidel? How much faster? Does your answer depend upon the value of c?

Check back soon!

Problem 16

Prove that all of the matrices $A_k$ defined in (9.83) have the same eigenvalues.

Check back soon!

Problem 16

Use GMRES to solve the system in Exercise 9.4.33. Compare the rate of convergence with the CG algorithm in Exercise 9.6.8.

Check back soon!

Problem 16

Let $\varphi(x)$ be any scaling function, $w(x)$ the corresponding mother wavelet and $w_{j, k}(x)$ the wavelet descendants. Prove that (a) $\|\varphi\|=\|w\|$.
(b) $\left\|w_{j, k}\right\|=2^{-j}\|\varphi\|$.

Check back soon!

00:41

Problem 17

What happens to the Fibonacci integers $u^{(k)}$ if we go "backward in time", i.e., for $k<0$ ? How is $u^{(-k)}$ related to $u^{(k)}$ ?

James Kiss

Numerade Educator

01:28

Problem 17

Prove that a matrix $T$ with all integer entries is convergent if and only if it is nilpotent. i.e., $T^k=\mathrm{O}$ for some $k \geq 0$. Give a nonzero example of such a matrix.

Nick Johnson

Numerade Educator

01:46

Problem 17

True or false: If $T$ and $T^T$ are both transition matrices, then $T=T^T$.

Zachary Mitchell

Numerade Educator

Problem 17

Consider the linear system
$$
2.4 x-.8 y+.8 z=1, \quad-.6 x+3.6 y-.6 z=0, \quad 15 x+14.4 y-3.6 z=0 .
$$
Show, by direct computation, that Jacobi iteration converges to the solution, but GaussSeidel does not.

Check back soon!

01:13

Problem 17

(a) Prove that if $A$ is symmetric and tridiagonal, then all matrices $A_k$ appearing in the $Q R$ algorithm are also symmetric and tridiagonal. Hint: First prove symmetry.
(b) Is the result true if $A$ is not symmetric - only tridiagonal?

Urvashi Arora

Numerade Educator

01:06

Problem 17

Is GMRES able to solve the system in Exercise 9.6.10?

Anurag Kumar

Numerade Educator

Problem 17

(a) Prove that the scaling function $\varphi(x)$ and the mother wavelet $w(x)$ are orthogonal.
(b) Prove that the integer translates $w(x-m)$ of the mother wavelet are mutually orthogonal. (c) Prove orthogonality of all the wavelet offspring $w_{j, k}(x)$.

Check back soon!

01:05

Problem 18

Use formula $(9.20)$ to compute the $k^{\text {th }}$ power of the following matrices:
(a) $\left(\begin{array}{ll}5 & 2 \\ 2 & 2\end{array}\right)$,
(b) $\left(\begin{array}{rr}4 & 1 \\ -2 & 1\end{array}\right)$.
(c) $\left(\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right)$,
(d) $\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 1\end{array}\right)$,
(e) $\left(\begin{array}{rrr}0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 2\end{array}\right)$.

Hast Aggarwal

Numerade Educator

Problem 18

Prove the inequality (9.28) when $T$ is incomplete. Use it to complete the proof of Theorem 9.14 in the incomplete case. Hint: Use Exercises 9.1.40, 9.2.22.

Check back soon!

View

Problem 18

True or false: If $T$ is a transition matrix, so is $T^{-1}$.

Nick Johnson

Numerade Educator

02:02

Problem 18

Discuss convergence of Gauss - Seidel iteration for the system
$$
\begin{aligned}
5 x+7 y+6 z+5 w & =23, & 6 x+8 y+10 z+9 w & =33, \\
7 x+10 y+8 z+7 w & =32, & 5 x+7 y+9 z+10 w & =31 .
\end{aligned}
$$

Ernest Castorena

Numerade Educator

03:44

Problem 18

Use Householder matrices to convert the following matrices into tridiagonal form:
(a) $\left(\begin{array}{rrr}8 & -7 & 2 \\ -7 & 17 & -7 \\ 2 & -7 & 8\end{array}\right)$,
(b) $\left(\begin{array}{rrrr}5 & 1 & -2 & 1 \\ 1 & 5 & 1 & -2 \\ -2 & 1 & 5 & 1 \\ 1 & -2 & 1 & 5\end{array}\right)$
(c) $\left(\begin{array}{rrrr}4 & 0 & -1 & 1 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 2 & 0 \\ 1 & -1 & 0 & 3\end{array}\right)$.

Kushal Mohnot

Numerade Educator

03:31

Problem 18

Explain in what sense the GMRES approximation $\mathbf{x}_{k+1}$ of order $k+1$ is a better approximation to the true solution than that of order $k$, namely $\mathbf{x}_{k^*}$

Joseph Lentino

Numerade Educator

02:03

Problem 18

Find the values of the Daubechies scaling function $\varphi(x)$ and mother wavelet $w(x)$ at $x=$ (a) $\frac{1}{2}$, (b) $\frac{1}{4}$, (c) $\frac{5}{16}$.

Uma Kumari

Numerade Educator

Problem 19

Use your answer from Exercise 9.1.18 to solve the following iterative systems:
(a) $u^{(k+1)}=5 u^{(k)}+2 v^{(k)}, v^{(k+1)}=2 u^{(k)}+2 v^{(k)}, u^{(0)}=-1, v^{(0)}=0$,
(b) $u^{(k+1)}=4 u^{(k)}+v^{(k)}, v^{(k+1)}=-2 u^{(k)}+v^{(k)}, u^{(0)}=1, v^{(0)}=-3$,
(c) $u^{(k+1)}=u^{(k)}+v^{(k)}, v^{(k+1)}=-u^{(k)}+v^{(k)}, u^{(0)}=0, v^{(0)}=2$,
(d) $u^{(k+1)}=u^{(k)}+v^{(k)}+2 w^{(k)}, v^{(k+1)}=u^{(k)}+2 v^{(k)}+w^{(k)}$,
$$
\begin{array}{r}
w^{\prime(k+1)}=2 u^{(k)}+v^{(k)}+w^{(k)}, u^{(0)}=1, \quad v^{(0)}=0, w^{(0)}=1, \\
(c) u^{(k+1)}=v^{(k)}, v^{(k+1)}=w^{(k)}, w^{(k+1)}=-u^{(k)}+2 w^{(k)}, u^{(0)}=1, v^{(0)}=0, w^{(0)}=0 .
\end{array}
$$

Check back soon!

06:35

Problem 19

Suppose that $M$ is a nonsingular matrix. (a) Prove that the implicit iterative system $M \mathbf{u}^{(n+1)}=\mathbf{u}^{(n)}$ has globally asymptotically stable zero solution if and only if all the eigenvalues of $M$ are strictly greater than one in magnitude: $\left|\mu_i\right|>1$. (b) Let $K$ be another matrix. Prove that more general implicit iterative system of the form $M \mathbf{u}^{(n+1)}=K \mathbf{u}^{(n)}$ has globally asymptotically stable zero solution if and only if all the generalized eigenvalues of the matrix pair $K, M$, as in Exercise 8.5 .8 , are strictly less than 1 in magnitude: $\left|\lambda_i\right|<1$.

Uma Kumari

Numerade Educator

03:59

Problem 19

A transition matrix is called doubly stochastic if both its row and column sums are equal to 1 . What is the limiting probability state of a Markov chain with doubly stochastic transition matrix?

Madi Sousa

Numerade Educator

01:13

Problem 19

Let $A=\left(\begin{array}{rrr}2 & 4 & -4 \\ 3 & 3 & 3 \\ 2 & 2 & 1\end{array}\right)$. Find the spectral radius of the Jacobi and Gauss-Seidel iteration matrices, and discuss their convergence.

Victor Salazar

Numerade Educator

01:31

Problem 19

Find the eigenvalues, to 2 decimal places, of the matrices in Exercise 9.5 .18 by applying the $Q R$ algorithm to the tridiagonal form.

James Kiss

Numerade Educator

02:13

Problem 19

(a) Explain what happens to the GMRES algorithm if the right-hand side $\mathbf{b}$ of the linear system $A \mathbf{x}=\mathbf{b}$ is an eigenvector of $A$. (b) More generally, prove that if the Krylov subspaces generated by $\mathbf{b}$ stabilize at order $m$, then the solution ot the linear system lies in $V^{(m)}$ and so the GMRES algorithm converges to the solution at order $m$.

Mahnoor Amin

Numerade Educator

Problem 19

Prove the formulas in Proposition 9.60 for the norms of the mother and daughter wavelets.

Check back soon!

Problem 20

(a) Given initial data $\mathbf{u}^{(0)}=(1,1,1)^T$, explain why the resulting solution $\mathbf{u}^{(k)}$ to the system in Example 9.7 has all integer entries. (b) Find the coefficients $c_1, c_2, c_3$ in the explicit solution formula (9.18). (c) Check the first few iterates to convince yourself that the solution formula does, in spite of appearances, always give an integer value.

Check back soon!

01:52

Problem 20

The stable subspace $S \subset \mathbb{R}^n$ for a linear iterative system $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}$ is defined as the set of all points a such that the solution with initial condition $\mathbf{u}^{(0)}=\mathbf{a}$ satisfies $u^{(k)} \rightarrow \mathbf{0}$ as $k \rightarrow \infty$. (a) Prove that $S$ is an invariant subspace for the matrix $T$.
(b) Determine necessary and sufficient conditions for $\mathbf{a} \in S$.
(c) Find the stable subspace for the linear systems in Exercise 9.1.14

E R

Numerade Educator

01:26

Problem 20

True or false: The set of all probability vectors forms a subspace of $\mathbb{R}^n$.

Victor Salazar

Numerade Educator

Problem 20

Consider the linear system $H_5 \mathbf{u}=\mathbf{e}_1$, where $H_5$ is the $5 \times 5$ Hilbert matrix. Does the Jacobi Method converge to the solution? If so, how fast? What about Gauss-Seidel?

Check back soon!

View

Problem 20

Use the tridiagonal $Q R$ Method to find the singular values of $A=\left(\begin{array}{rrrr}2 & 2 & 1 & -1 \\ 1 & -2 & 0 & 1 \\ 0 & -1 & 2 & 2\end{array}\right)$.

Michelle Z.

Numerade Educator

Problem 20

Write a computer program to zoom in on the Daubechies scaling function and discuss what you see.

Check back soon!

00:50

Problem 21

(a) Show how to convert the higher order linear iterative equation
$$
u^{(k+j)}=c_1 u^{(k+j-1)}+c_2 u^{(k+j-2)}+\cdots+c_j u^{(k)}
$$
into a first order system $\mathbf{u}^{(k)}=T \mathbf{u}^{(k)}$. Hint: See Example 9.6.
(b) Write down initial conditions that guarantee a unique solution $u^{(k)}$ for all $k \geq 0$.

Linh Vu

Numerade Educator

Problem 21

Consider a second order iterative system $\mathbf{u}^{(k+2)}=A \mathbf{u}^{(k+1)}+B \mathbf{u}^{(k)}$, where $A, B$ are $n \times n$ matrices. Define a quadratic eigenvalue to be a complex number that satisfies $\operatorname{det}\left(\lambda^2 \mathrm{I}-\lambda A-B\right)=0$. Prove that the zero solution is globally asymptotically stable if and only if all its quadratic eigenvalues satisfy $|\lambda|<1$.

Check back soon!

Problem 21

Multiple choice: Every probability vector in $\mathrm{R}^n$ lies on the unit sphere for the
(a) 1 norm, (b) 2 norm, (c) $\infty$ norm, (d) all of the above, (e) none of the above.

Check back soon!

Problem 21

How many arithmetic operations are needed to perform $k$ steps of the Jacobi iteration? What about Gauss-Seidel? Under what conditions is Jacobi or Gauss-Seidel more efficient than Gaussian Elimination?

Check back soon!

04:35

Problem 21

Use Householder matrices to convert the following matrices into upper Hessenberg form:
(a) $\left(\begin{array}{rrr}3 & -1 & 2 \\ 1 & 3 & -4 \\ 2 & -1 & -1\end{array}\right)$,
(b) $\left(\begin{array}{rrrr}3 & 2 & -1 & 1 \\ 2 & 4 & 0 & 1 \\ 0 & 1 & 2 & -6 \\ 1 & 0 & -5 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rrrr}1 & 0 & -1 & 1 \\ 2 & 1 & 1 & -1 \\ -1 & 0 & 1 & 3 \\ 3 & -1 & 1 & 4\end{array}\right)$.

Liuxi Sun

Numerade Educator

03:27

Problem 21

True or false: The iterative system $(9.156)$ is a Markov process.

Kamalesh Bagrecha

Numerade Educator

02:11

Problem 22

Apply the method of Exercise 9.1.21 to solve the following iterative equations:
(a) $u^{(k+2)}=-u^{(k+1)}+2 u^{(k)}, \quad u^{(0)}=1, \quad u^{(1)}=2$.
(b) $12 u^{(k+2)}=u^{(k+1)}+u^{(k)}, \quad u^{(0)}=-1, \quad u^{(1)}=2$.
(c) $u^{(k+2)}=4 u^{(k+1)}+u^{(k)}, \quad u^{(0)}=1, \quad u^{(1)}=-1$.
(d) $u^{(k+2)}=2 u^{(k+1)}-2 u^{(k)}, \quad u^{(0)}=1, \quad u^{(1)}=3$.
(e) $u^{(k+3)}=2 u^{(k+2)}+u^{(k+1)}-2 u^{(k)}, \quad u^{(0)}=0, \quad u^{(1)}=2, \quad u^{(2)}=3$.
(f) $u^{(k+3)}=u^{(k+2)}+2 u^{(k+1)}-2 u^{(k)}, \quad u^{(0)}=0, \quad u^{(1)}=1, \quad u^{(2)}=1$.

Manik Pulyani

Numerade Educator

02:33

Problem 22

Let $p(t)$ be a polynomial. Assume $0<\lambda<\mu$. Prove that there is a positive constant $C$ such that $p(n) \lambda^n<C \mu^n$ for all $n>0$.

Chris Trentman

Numerade Educator

00:17

Problem 22

True or false: Every probability eigenvector of a regular transition matrix has eigenvalue equal to 1 .

Jake Zanazzi

Numerade Educator

Problem 22

Consider the linear system $A \mathbf{x}=\mathbf{e}_1$ based on the $10 \times 10$ pentadiagonal matrix
$$
A=\left(\begin{array}{rrrrrrr}
z & -1 & 1 & 0 & & & \\
-1 & z & -1 & 1 & 0 & & \\
1 & -1 & z & -1 & 1 & 0 & \\
0 & 1 & -1 & z & -1 & 1 & \ddots \\
& 0 & 1 & -1 & z & -1 & \ddots \\
& & 0 & 1 & -1 & z & \ddots
\end{array}\right)
$$
(a) For what values of $z$ are the Jacobi and Gauss-Seidel Methods guaranteed to converge? (b) Set $z=4$. How many iterations are required to approximate the solution to 3 decimal places? (c) How small can $|z|$ be before the methods diverge?

Check back soon!

00:59

Problem 22

Find the eigenvalues, to 2 decimal places, of the matrices in Exercise 9.5 .21 by applying the $Q R$ algorithm to the upper Hessenberg form.

Raj Bala

Numerade Educator

Problem 22

Let $\varphi(x)$ satisfy the Daubechies scaling equation (9.153). Prove that if $\varphi(i) \neq 0$ for any $i \leq 0$ or $i \geq p$, then supp $\varphi$ is unbounded.

Check back soon!

01:29

Problem 23

Suppose you have $n$ dollars and can buy coffee for $\$ 1$, milk for $\$ 2$, and orange juice for \$2. Let $C^{(n)}$ count the number of different ways of spending all your money. (a) Explain why $C^{(n)}=C^{(n-1)}+2 C^{(n-2)}, C^{(0)}=C^{(1)}=1$. (b) Find an explicit formula for $C^{(n)}$.

Heather Zimmers

Numerade Educator

Problem 23

Find all fixed points for the iterative systems with the following coefficient matrices:
(a) $\left(\begin{array}{ll}7 & .3 \\ .2 & .8\end{array}\right)$,
(b) $\left(\begin{array}{rr}.6 & 1.0 \\ .3 & -.7\end{array}\right)$,
(c) $\left(\begin{array}{rrr}-1 & -1 & -4 \\ -2 & 0 & -4 \\ 1 & -1 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}2 & 1 & -1 \\ 2 & 3 & -2 \\ -1 & -1 & 2\end{array}\right)$.

Check back soon!

View

Problem 23

Write down an example of (a) an irregular transition matrix; (b) a regular transition matrix that has one or more zero entries.

Nick Johnson

Numerade Educator

06:29

Problem 23

The naïve iterative method for solving $A \mathbf{u}=\mathrm{b}$ is to rewrite it in fixed point form $\mathbf{u}=T \mathbf{u}+\mathbf{c}$, where $T=\mathrm{I}-A$ and $\mathbf{c}=\mathbf{b}$. (a) What conditions on the eigenvalues of $A$ ensure convergence of the naive method? (b) Use the Gershgorin Theorem 8.16 to prove that the naive method converges to the solution to (c) Check part (b) by implementing the method.
$$
\left(\begin{array}{rrr}
.8 & -.1 & -.1 \\
.2 & 1.5 & -.1 \\
.2 & -.1 & 1.0
\end{array}\right)\left(\begin{array}{l}
x \\
y \\
z
\end{array}\right)=\left(\begin{array}{r}
1 \\
-1 \\
2
\end{array}\right)
$$

Nicholas Barvinok

Numerade Educator

01:54

Problem 23

Prove that the effect of the first Householder reflection is as given in (9.90).

Ajay Singhal

Numerade Educator

10:51

Problem 23

(a) Use (9.142) to construct the "mother wavelet" corresponding to the hat function (9.140). (b) Is the hat function orthogonal to the mother wavelet? (c) Is the hat function orthogonal to its integer translates?

Donald Albin

Numerade Educator

Problem 24

Find the general solution to the iterative system $u_i^{(k+1)}=u_{i-1}^{(k)}+u_{i+1}^{(k)}, i=1, \ldots, n$, where we set $u_0^{(k)}=u_{n+1}^{(k)}=0$ for all $k$. Hint: Use Exercise 8.2.47.

Check back soon!

01:51

Problem 24

Discuss the stability of each fixed point and the asymptotic behavior(s) of the solutions to the systems in Exercise 9.2.23. Which fixed point, if any, does the solution with initial condition $\mathbf{u}^{(0)}=\mathbf{e}_1$ converge to?

Anand Jangid

Numerade Educator

View

Problem 24

Let $T$ be a transition matrix. Prove that if $\mathbf{u}$ is a probability vector, then so is $\mathbf{v}=T \mathbf{u}$.

Victor Salazar

Numerade Educator

Problem 24

Consider the linear system $A \mathbf{u}=\mathbf{b}$, where $A=\left(\begin{array}{ll}2 & 1 \\ 1 & 3\end{array}\right), \mathbf{b}=\left(\begin{array}{l}3 \\ 2\end{array}\right)$.
(a) What is the solution? (b) Discuss the convergence of the Jacobi iteration method. (c) Discuss the convergence of the Gauss-Seidel iteration method. (d) Write down the explicit formulas for the SOR Method. (e) What is the optimal value of the relaxation parameter $\omega$ for this system? How much faster is the convergence as compared to the Jacobi and Gauss-Seidel Methods? $(f)$ Suppose your initial guess is $\mathbf{u}^{(0)}=0$. Give an estimate as to how many steps each iterative method (Jacobi, Gauss-Seidel, SOR) would require in order to approximate the solution to the system to within 5 decimal places.
$(g)$ Verify your answer by direct computation.

Check back soon!

Problem 24

What is the effect of tridiagonalization on the eigenvectors of the matrix?

Check back soon!

01:23

Problem 24

Prove that a real number $x$ is dyadic if and only if its binary (base 2) expansion terminates, i.e., is eventually all zeros.

James Chok

Numerade Educator

04:21

Problem 25

Starting with $u^{(0)}=0, u^{(1)}=0, u^{(2)}=1$, define the sequence of tribonacci numbers $u^{(k)}$ by adding the previous three to get the next one. For instance, $u^{(3)}=u^{(0)}+u^{(1)}+u^{(2)}=1$. (a) Write out the next four tribonacci numbers. (b) Find a third order iterative equation for the $k^{\text {th }}$ tribonacci number. (c) Explain why the tribonacci numbers are all integers. (d) Find an explicit formula for the solution, using a computer to approximate the eigenvalues. (c) Do they grow faster than the usual Fibonacci numbers? What is their overall rate of growth?

Jacob Denson

Numerade Educator

03:31

Problem 25

Suppose $T$ is a symmetric matrix that satisfies the hypotheses of Proposition 9.17 with a simple eigenvalue $\lambda_1=1$. Prove that the solution $\mathbf{u}^{(k)}$ to the linear iterative system $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}$ has limiting value $\lim _{k \rightarrow \infty} \mathbf{u}^{(k)}=\frac{\mathbf{u}^{(0)} \cdot \mathbf{v}_1}{\left\|\mathbf{v}_1\right\|^2} \mathbf{v}_1$.

Ryan Williams

Numerade Educator

View

Problem 25

(a) Prove that if $T$ and $S$ are transition matrices, then so is their product $T S$.
(b) Prove that if $T$ is a transition matrix, then so is $T^k$ for all $k \geq 0$.

Nick Johnson

Numerade Educator

03:47

Problem 25

In Exercise 9.4.18 you were asked to solve a system by Gauss-Seidel. How much faster can you design an SOR scheme to converge? Experiment with several values of the relaxation parameter $\omega$, and discuss what you find.

Nick Johnson

Numerade Educator

01:29

Problem 25

(a) How many arithmetic operations - multiplications/divisions and additions/ subtractions - are required to place a generic $n \times n$ symmetric matrix into tridiagonal form? (b) How many operations are needed to perform one iteration of the $Q R$ algorithm on an $n \times n$ tridiagonal matrix? (c) How much faster, on average, is the tridiagonal algorithm than the direct $Q R$ algorithm for finding the eigenvalues of a symmetric matrix?

Nick Johnson

Numerade Educator

01:02

Problem 25

Find dyadic approximations, with error at most $2^{-8}$, to
(a) $\frac{3}{4}$,
(b) $\frac{1}{3}$,
(c) $\sqrt{2}$,
(d) $e$,
(e) $\pi$.

Harsh Gadhiya

Numerade Educator

01:03

Problem 26

Suppose that Fibonacci's rabbits live for only eight years, [44]. (a) Write out an iterative equation to describe the rabbit population. (b) Write down the first few terms.
(c) Convert your equation into a first order iterative system, using the method of Exercise 9.1.21. (d) At what rate does the rabbit population grow?

Ankit Gupta

Numerade Educator

00:28

Problem 26

True or false: If $T$ has a stable nonzero fixed point, then it is a convergent matrix.

Monica Miller

Numerade Educator

Problem 26

Investigate the three basic iterative techniques - Jacobi, Gauss Seidel, SOR - for solving the linear system $K^{\star} \mathbf{u}^{\star}=\mathrm{f}^{\star}$ for the cubical circuit in Example 6.4.

Check back soon!

02:06

Problem 26

Write out a pseudocode program to tridiagonalize a matrix. The input should be an $n \times n$ matrix $A$, and the output should be the Householder unit vectors $\mathbf{u}_1, \ldots, \mathbf{u}_{n-1}$ and the tridiagonal matrix $R$. Does your program produce the upper Hessenberg form when the input matrix is not symmetric?

Adriano Chikande

Numerade Educator

05:19

Problem 27

A well-known method of generating a sequence of "pseudo-random" integers $u^{(0)}, u^{(1)}, u^{(2)}, \ldots$ satisfying $0 \leq u^{(i)}<n$ is based on the modular Fibonacci equation $u^{(k+2)}=u^{(k+1)}+u^{(k)} \bmod n$, with suitably chosen initial values $0 \leq u^{(0)}, u^{(1)}<n$.
(a) Generate the sequence of pseudo-random numbers that result from the choices $n=10$, $u^{(0)}=3, u^{(1)}=7$. Keep iterating until the sequence starts repeating.
(b) Experiment with other sequences of pseudo-random numbers generated by the method.

Prathan Jarupoonphol

Numerade Educator

00:23

Problem 27

True or false: If every point $\mathbf{u} \in \mathbb{R}^n$ is a fixed point, then they are all stable. Can you characterize such systems?

Cory Kuzinski

Numerade Educator

04:39

Problem 27

Consider the linear system
$$
4 x-y-z=1,-x+4 y-w=2,-x+4 z-w=0,-y-z+4 w=1 .
$$
(a) Find the solution by using Gaussian Elimination and Back Substitution. (b) Using $\mathbf{0}$ as your initial guess, how many iterations are required to approximate the solution to within five decimal places using (i) Jacobi iteration? (ii) Gauss Seidel iteration? Can you estimate the spectral radii of the relevant matrices in each case? (c) Try to find the solution by using the SOR Method with the parameter $\omega$ taking various values between .5 and 1.5. Which value of $\omega$ gives the fastest convergence? What is the spectral radius of the SOR matrix?

Carson Merrill

Numerade Educator

Problem 27

Prove that in the $H=L U$ factorization of a regular upper Hessenberg matrix, the lower triangular factor $L$ is bidiagonal, as in (1.67).

Check back soon!

01:05

Problem 28

Prove that the curves $E_k=\left\{T^k \mathbf{x} \mid\|\mathbf{x}\|=1\right\}, k=0,1,2, \ldots$, sketched in Figure 9.2 form a family of ellipses with the same principal axes. What are the individual semi-axes? Hint: Use Exercise 8.7.23.

Carson Merrill

Numerade Educator

05:28

Problem 28

Prove Theorem 9.18: (a) assuming $T$ is complete, (b) for general $T$.
Hint: Use Exercise 9.1.40.

Aqib Basheer

Numerade Educator

Problem 28

(a) Find the spectral radius of the Jacobi and Gauss-Seidel iteration matrices when $A=\left(\begin{array}{llll}2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2\end{array}\right)$
(b) Is A strictly diagonally dominant? (c) Use (9.76) to fix the optimal value of the SOR parameter. Verify that the spectral radius of the resulting iteration matrix agrees with the second formula in (9.76). (d) For each iterative method, predict how many iterations are needed to solve the linear system $A \mathbf{x}=\mathbf{e}_1$ to 4 decimal places, and then verify your predictions by direct computation.

Check back soon!

00:55

Problem 29

Plot the ellipses $E_k=\left\{T^k \mathbf{x} \mid\|\mathbf{x}\|=1\right\}$ for $k=1,2,3,4$ for the following matrices $T$. Then determine their principal axes, semi-axes, and areas. Hint: Use Exercise 8.7.23.
(a) $\left(\begin{array}{rr}\frac{2}{3} & -\frac{1}{3} \\ -\frac{1}{3} & \frac{2}{3}\end{array}\right)$,
(b) $\left(\begin{array}{cc}0 & -1.2 \\ .4 & 0\end{array}\right)$,
(c) $\left(\begin{array}{ll}\frac{3}{5} & \frac{1}{5} \\ 2 & 4 \\ 5 & 5\end{array}\right)$.

Kayleah Tsai

Numerade Educator

02:11

Problem 29

(a) Under what conditions does the linear iterative system $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}$ have a period 2 solution, meaning that the iterates repeat after every other iterate: $\mathbf{u}^{(k+2)}=$ $\mathbf{u}^{(k)} \neq \mathbf{u}^{(k+1)}$ ? Give an example of such a system. (b) Under what conditions is there a unique period 2 solution?
(c) What about a period $m$ solution for $2<m \in \mathbb{N}$ ?

Manik Pulyani

Numerade Educator

03:33

Problem 29

Change the matrix in Exercise 9.4 .28 to $A=\left(\begin{array}{rrrr}2 & -1 & 0 & 0 \\ 1 & 2 & -1 & 0 \\ 0 & 1 & 2 & -1 \\ 0 & 0 & 1 & 2\end{array}\right)$, and answer the same questions. Does the SOR Method with parameter given by (9.76) speed the iterations up? Why not? Can you find a value of the SOR parameter that does?

Chandra Jain

Numerade Educator

03:07

Problem 30

Let $T$ be a positive definite $2 \times 2$ matrix. Let $E_n=\left\{T^n \mathbf{x} \mid\|\mathbf{x}\|=1\right\}, n=0,1,2, \ldots$, be the image of the unit circle under the $n^{\text {th }}$ power of $T$. (a) Prove that $E_n$ is an ellipse. True or false: (b) The ellipses $E_n$ all have the same principal axes. (c) The semi-axes are given by $r_n=r_1^n, s_n=s_1^n$. (d) The areas are given by $A_n=\pi \alpha^n$ where $\alpha=A_1 / \pi$.

Dorcas Attuabea Addo

Numerade Educator

Problem 30

Compute the $\infty$ matrix norm of the following matrices. Which are guaranteed to be convergent?
(a) $\left(\begin{array}{ll}\frac{1}{2} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{6}\end{array}\right)$,
(b) $\left(\begin{array}{rr}\frac{5}{3} & \frac{4}{3} \\ -\frac{7}{6} & -\frac{5}{6}\end{array}\right)$,
(c) $\left(\begin{array}{rr}\frac{2}{7} & -\frac{2}{7} \\ -\frac{2}{7} & \frac{6}{7}\end{array}\right)$,
(d)
$\left(\begin{array}{rr}\frac{1}{4} & \frac{3}{2} \\ -\frac{1}{2} & \frac{5}{4}\end{array}\right)$
(e) $\left(\begin{array}{rrr}\frac{2}{7} & \frac{2}{7} & -\frac{4}{7} \\ 0 & \frac{2}{7} & \frac{6}{7} \\ 2 & \frac{4}{7} & \frac{2}{7}\end{array}\right)$,
(f) $\left(\begin{array}{rrr}0 & .1 & .8 \\ -.1 & 0 & .1 \\ -.8 & -.1 & 0\end{array}\right)$,
(g) $\left(\begin{array}{rrr}1 & -\frac{2}{3} & -\frac{2}{3} \\ 1 & -\frac{1}{3} & -1 \\ \frac{1}{3} & -\frac{2}{3} & 0\end{array}\right)$,
(h) $\left(\begin{array}{rrr}\frac{1}{3} & 0 & 0 \\ -\frac{1}{3} & 0 & \frac{1}{3} \\ 0 & 2 & \frac{1}{3}\end{array}\right)$

Check back soon!

Problem 30

Consider the linear system $A \mathbf{u}=\mathrm{e}_1$ in which $A$ is the $8 \times 8$ tridiagonal matrix with all 2's on the main diagonal and all -1 's on the sub-and super-diagonals. (a) Use Exercise 8.2.47 to find the spectral radius of the Jacobi iteration method to solve $A \mathbf{u}=\mathbf{b}$. Does the Jacobi Method converge? (b) What is the optimal value of the SOR parameter based on $(9.76)$ ? How many Jacobi iterations are needed to match the effect of a single SOR step?
(c) Test out your conclusions by using both Jacobi and SOR to approximate the solution to 3 decimal places.

Check back soon!

04:36

Problem 31

Answer Exercise 9.1.30 when $T$ is an arbitrary nonsingular $2 \times 2$ matrix. Hint: Use Exercise 8.7.23.

Sam Stansfield

Numerade Educator

Problem 31

Compute the Euclidean matrix norm of each matrix in Exercise 9.2.30. Have your convergence conclusions changed?

Check back soon!

Problem 31

How much can you speed up the convergence of the iterative solution to the pentadiagonal linear system in Exercise 9.4 .22 when $z=4$ using SOR? Discuss.

Check back soon!

Problem 32

Given the general solution (9.9) of the iterative system $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}$, write down the solution to $\mathbf{v}^{(k+1)}=\alpha T \mathbf{v}^{(k)}+\beta \mathbf{v}^{(k)}$, where $\alpha, \beta \in \mathbb{R}$.

Check back soon!

Problem 32

Compute the spectral radii of the matrices in Exercise 9.2.30. Which are convergent? Compare your conclusions with those of Exercises 9.2.30 and 9.2.31.

Check back soon!

02:06

Problem 32

For the matrix treated in Example 9.40, prove that (a) as $\omega$ increases from 1 to $8-4 \sqrt{3}$, the two eigenvalues move towards each other, with the larger one decreasing in magnitude; (b) if $\omega>8-4 \sqrt{3}$, the eigenvalues are complex conjugates, with larger modulus than the optimal value. (c) Can you conclude that $\omega_*=8-4 \sqrt{3}$ is the optimal value for the SOR parameter?
$$
\left(\begin{array}{rrrrrrrrr}
4 & -1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
-1 & 4 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & -1 & 4 & 0 & 0 & -1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 4 & -1 & 0 & -1 & 0 & 0 \\
0 & -1 & 0 & -1 & 4 & -1 & 0 & -1 & 0 \\
0 & 0 & -1 & 0 & -1 & 4 & 0 & 0 & -1 \\
0 & 0 & 0 & -1 & 0 & 0 & 4 & -1 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & -1 & 4 & -1 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 4
\end{array}\right) \text { arises in the finite }
$$
difference (and finite element) discretization of the Poisson equation on a nine point square grid. Solve the linear system $A \mathbf{u}=\mathbf{e}_5$ using (a) Gaussian Elimination; (b) Jacobi iteration; (c) Gauss-Seidel iteration; (d) SOR based on the Jacobi spectral radius.

Victor Salazar

Numerade Educator

01:19

Problem 32

Victor Salazar

Numerade Educator

01:10

Problem 33

Prove directly that if the coefficient matrix of a linear iterative system is real, both the real and imaginary parts of a complex solution are real solutions.

Brandon Cleary

Numerade Educator

01:29

Problem 33

Let $k$ be an integer and set $A_k=\left(\begin{array}{cc}k & -1 \\ k^2 & -k\end{array}\right)$. Compute (a) $\left\|A_k\right\|_{\infty}$, (b) $\left\|A_k\right\|_2$, (c) $\rho\left(A_k\right) . \quad$ (d) Explain why every $A_k$ is a convergent matrix, even though their matrix norms can be arbitrarily large. (e) Why does this not contradict Corollary 9.27 ?

Anurag Kumar

Numerade Educator

Problem 33

The matrix $A=\left(\begin{array}{rrrrrrrrr}4 & -1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 4 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 4 & 0 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 4 & -1 & 0 & -1 & 0 & 0 \\ 0 & -1 & 0 & -1 & 4 & -1 & 0 & -1 & 0 \\ 0 & 0 & -1 & 0 & -1 & 4 & 0 & 0 & -1 \\ 0 & 0 & 0 & -1 & 0 & 0 & 4 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & -1 & 4 & -1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 4\end{array}\right)$ arises in the finite difference (and finite element) discretization of the Poisson equation on a nine point square grid. Solve the linear system $A \mathbf{u}=\mathbf{e}_5$ using (a) Gaussian Elimination; (b) Jacobi iteration; (c) Gauss-Seidel iteration; (d) SOR based on the Jacobi spectral radius.

Check back soon!

02:03

Problem 34

Explain why the solution $\mathbf{u}^{(k)}, k \geq 0$, to the initial value problem $(9.6)$ exists and is uniquely defined. Does this hold if we allow negative $k<0$ ?

Michael Jacobsen

Numerade Educator

03:05

Problem 34

Show that if $|c|<1 /\|A\|$, then $c A$ is a convergent matrix.

Shahab Ullah

Numerade Educator

01:58

Problem 34

The generalization of Exercise 9.4 .33 to an $n \times n$ grid results in an $n^2 \times n^2$ matrix in block tridiagonal form $A=\left(\begin{array}{rrrrr}K & -I & & & \\ -1 & K & -I & \\ & -I & K & -I & \\ & & \ddots & \ddots & \ddots\end{array}\right)$, in which $K$ is the tridiagonal $n \times n$ matrix with 4's on the main diagonal and -1 's on the sub-and super-diagonals, while I denotes the $n \times n$ identity matrix. Use the known value of the Jacobi spectral radius $\rho_J=\cos \frac{\pi}{n+1},[86]$, to design an SOR Method to solve the linear system $A \mathbf{u}=\mathbf{f}$. Run your method on the cases $n=5$ and $\mathbf{f}=\mathbf{e}_{13}$ and $n=25$ and $\mathbf{f}=\mathbf{e}_{313}$ corresponding to a unit force at the center of the grid. How much faster is the convergence rate of SOR than Jacobi and Gauss-Seidel?

Victor Salazar

Numerade Educator

02:40

Problem 35

Prove that if $T$ is a symmetric matrix, then the coefficients in (9.9) are given by the formula $c_j=\mathbf{a}^T \mathbf{v}_j / \mathbf{v}_j^T \mathbf{v}_j$

Urvashi Arora

Numerade Educator

Problem 35

Prove that the spectral radius function does not satisfy the triangle inequality by finding matrices $A, B$ such that $\rho(A+B)>\rho(A)+\rho(B)$.

Check back soon!

02:25

Problem 35

If $\mathbf{u}^{(k)}$ is an approximation to the solution to $A \mathbf{u}=\mathbf{b}$, then the residual vector $\mathbf{r}^{(k)}=\mathbf{b}-A \mathbf{u}^{(k)}$ measures how accurately the approximation solves the system.
(a) Show that the Jacobi iteration can be written in the form $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+D^{-1} \mathbf{r}^{(k)}$.
(b) Show that the Gauss-Seidel iteration has the form $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+(L+D)^{-1} \mathbf{r}^{(k)}$.
(c) Show that the SOR iteration has the form $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+(\omega L+D)^{-1} \mathbf{r}^{(k)}$.
(d) If $\| \mathbf{r}^{(k)} \mid$ is small, does this mean that $\mathbf{u}^{(k)}$ is close to the solution? Explain your answer and illustrate with a couple of examples.

Stanley Enemuo

Numerade Educator

01:52

Problem 36

Explain why the $j^{\text {th }}$ column $c_j^{(k)}$ of the matrix power $T^k$ satisfies the linear iterative system $\mathbf{c}_j^{(k+1)}=T \mathbf{c}_j^{(k)}$ with initial data $\mathbf{c}_j^{(0)}=\mathbf{e}_j$, the $j^{\text {th }}$ standard basis vector.

Doruk Isik

Numerade Educator

00:36

Problem 36

Find a convergent matrix that has dominant singular value $\sigma_1>1$.

Victor Salazar

Numerade Educator

Problem 36

Let $K$ be a positive definite $n \times n$ matrix with eigenvalues $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n>0$. For what values of $\varepsilon$ does the iterative system $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+\varepsilon \mathbf{r}^{(k)}$, where $\mathbf{r}^{(k)}=\mathbf{f}-K \mathbf{u}^{(k)}$ is the current residual vector, converge to the solution to the linear system $K \mathbf{u}=\mathbf{f}$ ? What is the optimal value of $\varepsilon$, and what is the convergence rate?

Check back soon!

View

Problem 37

Let $z^{(k+1)}=\lambda z^{(k)}$ be a complex scalar iterative equation with $\lambda=\mu+i \nu$. Show that its real and imaginary parts $x^{(k)}=\operatorname{Re} z^{(k)}, y^{(k)}=\operatorname{Im} z^{(k)}$, satisfy a two-dimensional real linear iterative system. Use the eigenvalue method to solve the real $2 \times 2$ system, and verify that your solution coincides with the solution to the original complex equation.

Victor Salazar

Numerade Educator

03:26

Problem 37

Prove that if $A$ is a real symmetric matrix, then its Euclidean matrix norm is equal to its spectral radius.

Ekaveera Kumar

Numerade Educator

01:28

Problem 38

Suppose $V \subset \mathbb{R}^n$ is an invariant subspace for the $n \times n$ matrix $T$ governing the linear iterative system $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}$. Prove that if $\mathbf{u}^{(0)} \in V$, then so is the solution: $\mathbf{u}^{(k)} \in V$.

Harshita Goel

Numerade Educator

04:15

Problem 38

Let $A$ be a square matrix. Let $s=\max \left\{s_1, \ldots, s_n\right\}$ be the maximal absolute row sum of $A$ and let $t=\min \left\{\left|a_{i i}\right|-r_i\right\}$, with $r_i$ given by (8.27). Prove that $\max \{0, t\} \leq \rho(A) \leq$ s.

Jack Chen

Numerade Educator

01:48

Problem 39

Suppose $\mathbf{u}^{(k)}$ and $\overline{\mathbf{u}}^{(k)}$ are two solutions to the same iterative system $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}$.
(a) Suppose $\mathbf{u}^{\left(k_0\right)}=\overline{\mathbf{u}}^{\left(k_0\right)}$ for some $k_0 \geq 0$. Can you conclude that these are the same solution: $\mathbf{u}^{(k)}=\overline{\mathbf{u}}^{(k)}$ for all $k$ ? (b) What can you say if $\mathbf{u}^{\left(k_0\right)}=\overline{\mathbf{u}}^{\left(k_1\right)}$ for $k_0 \neq k_1$ ?

Ankit Gupta

Numerade Educator

04:29

Problem 39

Suppose the largest entry (in modulus) of $A$ is $\left|a_{i j}\right|=a_*$. Can you bound its radius of convergence?

Ahmed Ibrahim

Numerade Educator

08:38

Problem 40

Let $T$ be an incomplete matrix, and suppose $\mathbf{w}_1, \ldots, \mathbf{w}_j$ is a Jordan chain associated with an incomplete eigenvalue $\lambda$. (a) Prove that, for $i=1, \ldots, j$,
$$
T^k \mathbf{w}_i=\lambda^k \mathbf{w}_i+k \lambda^{k-1} \mathbf{w}_{i-1}+\left(\begin{array}{l}
k \\
2
\end{array}\right) \lambda^{k-2} \mathbf{w}_{i-2}+\cdots .
$$
(b) Explain how to use a Jordan basis of $T$ to construct the general solution to the linear iterative system $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}$.

Lucía Guerrero

Numerade Educator

Problem 40

(a) Suppose that every entry of the $n \times n$ matrix $A$ is bounded by $\left|a_{i j}\right|<1 / n$. Prove that $A$ is a convergent matrix. Hint: Use Exercise 9.2.38. (b) Produce a matrix of size $n \times n$ with one or more entries satisfying $\left|a_{i j}\right|=1 / n$ that is not convergent.

Check back soon!

Problem 41

Use the method Exercise 9.1.40 to find the general real solution to the following linear iterative systems:
(a) $u^{(k+1)}=2 u^{(k)}+3 v^{(k)}, v^{(k+1)}=2 v^{(k)}$
(b) $u^{(k+1)}=u^{(k)}+v^{(k)}, v^{(k+1)}=-4 u^{(k)}+5 v^{(k)}$,
(c) $u^{(k+1)}=-u^{(k)}+v^{(k)}+w^{(k)}+v^{(k+1)}=-v^{(k)}+w^{(k)}, w^{(k+1)}=-w^{(k)}$,
(d) $u^{(k+1)}=3 u^{(k)}-v^{(k)}, v^{(k+1)}=-u^{(k)}+3 v^{(k)}+w^{(k)}, w^{(k+1)}=-v^{(k)}+3 w^{(k)}$,
(e) $u^{(k+1)}=u^{(k)}-v^{(k)}-w^{(k)}, v^{(k+1)}=2 u^{(k)}+2 v^{(k)}+2 w^{(k)}, w^{(k+1)}=-u^{(k)}+v^{(k)}+w^{(k)}$,
(f) $u^{(k+1)}=v^{(k)}+z^{(k)}, v^{(k+1)}=-u^{(k)}+w^{(k)}, w^{(k+1)}=z^{(k)}, z^{(k+1)}=-w^{(k)}$.

Check back soon!

00:47

Problem 41

Write down an example of a strictly diagonally dominant matrix that is also convergent.

Nez Nikoo

Numerade Educator

06:10

Problem 42

Find a formula for the $k^{\text {th }}$ power of a Jordan block matrix. Hint: Use Exercise 9.1.40.

Supratim Roy

Numerade Educator

03:33

Problem 42

True or false: If $B=S^{-1} A S$ are similar matrices, then
(a) $\|B\|_{\infty}=\|A\|_{\infty}$,
(b) $\|B\|_2=\|A\|_2$,
(c) $\rho(B)=\rho(A)$.

Patrick Burns

Numerade Educator

05:02

Problem 43

An affine itemtive system has the form $\mathbf{u}^{(k+1)}=T \mathbf{u}^{(k)}+\mathbf{b}, \mathbf{u}^{(0)}=\mathbf{c}$.
(a) Under what conditions does the system have an equilibrium solution $\mathbf{u}^{(k)} \equiv \mathbf{u}^{\star}$ ?
(b) In such cases, find a formula for the general solution. Hint: Look at $\mathbf{v}^{(k)}=\mathbf{u}^{(k)}-\mathbf{u}^*$.
(c) Solve the following affine iterative systems:
(i) $\mathbf{u}^{(k+1)}=\left(\begin{array}{rr}6 & 3 \\ -3 & -4\end{array}\right) \mathbf{u}^{(k)}+\left(\begin{array}{l}1 \\ 2\end{array}\right), \quad \mathbf{u}^{(0)}=\left(\begin{array}{r}4 \\ -3\end{array}\right)$,
(ii) $\mathbf{u}^{(k+1)}=\left(\begin{array}{rr}-1 & 2 \\ 1 & -1\end{array}\right) \mathbf{u}^{(k)}+\left(\begin{array}{l}1 \\ 0\end{array}\right), \quad \mathbf{u}^{(0)}=\left(\begin{array}{l}0 \\ 1\end{array}\right)$.
(iki) $\mathbf{u}^{(k+1)}=\left(\begin{array}{rrr}-3 & 2 & -2 \\ -6 & 4 & -3 \\ 12 & -6 & -5\end{array}\right) \mathbf{u}^{(k)}+\left(\begin{array}{r}1 \\ -3 \\ 0\end{array}\right), \quad \mathbf{u}^{(0)}=\left(\begin{array}{r}1 \\ 0 \\ -1\end{array}\right)$,
(iv) $\mathbf{u}^{(k+1)}=\left(\begin{array}{rrr}-\frac{5}{6} & \frac{1}{3} & -\frac{1}{6} \\ 0 & -\frac{1}{2} & \frac{1}{3} \\ 1 & -1 & \frac{2}{3}\end{array}\right) \mathbf{u}^{(k)}+\left(\begin{array}{r}\frac{1}{6} \\ -\frac{1}{3} \\ -\frac{1}{2}\end{array}\right), \quad \mathbf{u}^{(0)}=\left(\begin{array}{r}\frac{1}{6} \\ -\frac{2}{3} \\ \frac{1}{3}\end{array}\right)$.
(d) Discuss what happens in cases in which there is no fixed point, assuming that $T$ is complete.

Vikash Ranjan

Numerade Educator

05:02

Problem 43

Vikash Ranjan

Numerade Educator

00:54

Problem 43

Prove that the curve parametrized in (9.33) is an ellipse. What are its semi-axes?

Joseph Liao

Numerade Educator

Problem 44

(a) Prove that the individual entries $a_{i j}$ of a matrix $A$ are bounded in absolute value by its $\infty$ matrix norm: $\left|a_{i j}\right| \leq\|A\|_{\infty}$.
(b) Prove that if the series $\sum_{n=0}^{\infty}\left\|A_n\right\|_{\infty}<\infty$ converges, then the matrix series $\sum_{n=0}^{\infty} A_n=A^*$ converges to some matrix $A^*$.
(c) Let $\|A\|$ denote any natural matrix norm. Prove that if the series $\sum_{n=0}^{\infty}\left\|A_n\right\|<\infty$ converges, then the matrix series $\sum_{n=0}^{\infty} A_n=A^*$ converges.

Check back soon!

01:38

Problem 45

(a) Use Exercise 9.2.44 to prove that the geometric matrix series $\sum_{n=0}^{\infty} A^n$ converges whenever $\rho(A)<1$. Hint: Apply Corollary 9.27.
(b) Prove that the sum equals $(I-A)^{-1}$. How do you know $\mathrm{I}-A$ is invertible?

Angelo Rendina

Numerade Educator