1.MATH347 Homework 0

1.1.Background

This homework is meant to familiarize yourself with basic operations within TeXmacs, a public-domain scientific editing platform. The TeXmacs website provides several tutorials. The key features of TeXmacs that motivate adoption of the platform for this course are:

• Simple, efficient editing of mathematical content. The editor has a default text mode, and also a mathematics mode triggered by inserting an equation from the menu using Insert->Mathematics->(formula type), or the keyboard through key-strokes $, or Alt-Shift-$. Here is an example: the solution of the linear system $𝑨𝒙=𝒃$ with a symmetric matrix, ${𝑨}^T=𝑨$, can be found by gradient descent

  ${\displaystyle \varphi \left(𝒙\right)=\frac{1}{2}{𝒙}^T𝑨𝒙-{𝒃}^T𝒙,{𝒙}^{(k+1)}={𝒙}^{\left(k\right)}-\lambda \nabla \varphi \left({𝒙}^{\left(k\right)}\right).}$ ${\displaystyle \varphi \left(𝒙\right)=\frac{1}{2}{𝒙}^T𝑨𝒙-{𝒃}^T𝒙}$

• Sessions from other mathematical packages can be inserted directly into a document. Julia is used extensively in this course, and the menu item Insert->Session->Julia leads to creation of space within the document to execute Julia instructions. Define a matrix $𝑨$.

  ∴	A=[1 2 3; -1 0 1; 2 1 -2]

  ${\displaystyle \left[\begin{array}{ccc}1& 2& 3\\ -1& 0& 1\\ 2& 1& -2\end{array}\right]}$

  (1)

  Extend the Julia environment by adding a package to compute row echelon forms. This need be done only once.

  ∴	import Pkg; Pkg.add("RowEchelon");

  Resolving package versions…

  No Changes to ‘~/.julia/environments/v1.9/Project.toml‘

  No Changes to ‘~/.julia/environments/v1.9/Manifest.toml‘

  Load the RowEchelon package into the current session and invoke the rref function.

  ∴	using RowEchelon

  ∴

  rref(A)

  ${\displaystyle \left[\begin{array}{ccc}1.0& 0.0& 0.0\\ 0.0& 1.0& 0.0\\ 0.0& 0.0& 1.0\end{array}\right]}$

  (2)

  Compute the inverse of the matrix $𝑨$.

  ∴

  inv(A)

  ${\displaystyle \left[\begin{array}{ccc}0.25000000000000006& -1.7500000000000002& -0.5000000000000002\\ -1.1102230246251565e-16& 2.0000000000000004& 1.0000000000000002\\ 0.25000000000000006& -0.7500000000000002& -0.5000000000000001\end{array}\right]}$

  (3)

  ∴

• Documents can readily be converted to other formats: PDF, LaTeX, HTML. All course documents, including the website are produced with TeXmacs.

1.2.Theoretical questions

1.2.1.Text editing in TeXmacs

Problem

Write an itemized list of ingredients in your favorite dessert recipe. (Menu->Insert->Itemize)

Answer

Tort Diplomat ingredients:

• 4 eggs

• 1 cup flour

• 1 cup sugar

• 125 ml vegetable oil

• 1 teaspoon baking powder

• ¹⁄₂ lemon, juice of

• ¹⁄₂ orange, zest of

• 1 ¹⁄₂ teaspoons vanilla extract

Figure 1. Tort Diplomat

1.2.2.Inline mathematics

Problem

The fundamental theorem of calculus states ${\int }_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$ for $F^{\text{'}}\left(x\right)=f\left(x\right)$. Apply this result for $a=0$, $b=\pi$, $f\left(x\right)=\sin x$, $F\left(x\right)=-\cos x$. Write your answer inline.

Answer

The integral is ${\int }_0^{\pi }\sin \left(x\right)dx=-\cos \left(\pi \right)+\cos \left(0\right).$

1.2.3.Displayed mathematics

Problem

A matrix is a row of column vectors, $𝑨=\left[\begin{array}{llll}{𝒂}_1& {𝒂}_2& \dots & {𝒂}_n\end{array}\right]\in {ℝ}^{m\times n}$, which can be expressed in terms of vector components as

${\displaystyle 𝑨=\left[\begin{array}{llll}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{\mathrm{mn}}\end{array}\right].}$

Look up the definition of a Hilbert matrix $𝑯$ and write in the above forms, both as a row of column vectors, and as components.

Answer

A Hibert matrix is a square matrix $𝑯\in {ℝ}^{m\times m}$ defined by

${\displaystyle 𝑯=\left[h_{i,j}\right]=\left[\frac{1}{i+j-1}\right]=\left[\begin{array}{llll}{𝒉}_1& {𝒉}_2& \dots & {𝒉}_m\end{array}\right],1⩽i,j⩽m.}$

1.2.4.Julia session - working with numbers

Problem

Insert a Julia session and produce a table of the squares and cubes of the first ten natural numbers.

Answer

The squares and cubes of $1,2,\dots ,10$ are:

∴	i=1:10; [i.^1 i.^2 i.^3]

${\displaystyle \left[\begin{array}{ccc}1& 1& 1\\ 2& 4& 8\\ 3& 9& 27\\ 4& 16& 64\\ 5& 25& 125\\ 6& 36& 216\\ 7& 49& 343\\ 8& 64& 512\\ 9& 81& 729\\ 10& 100& 1000\end{array}\right]}$

(4)

∴

1.2.5.Julia session - working with column vectors

Problem

Insert a Julia session and define the vectors

${\displaystyle 𝒖=\left[\begin{array}{l}1\\ 2\\ 3\end{array}\right],𝒗=\left[\begin{array}{l}-1\\ 0\\ 1\end{array}\right],𝒘=\left[\begin{array}{l}0\\ -1\\ 0\end{array}\right].}$

Answer

Define vectors

∴	u=[1 2 3]';

∴	v=[-1 0 1]';

∴	w=[0 -1 0]';

∴

[u v w]

${\displaystyle \left[\begin{array}{ccc}1& -1& 0\\ 2& 0& -1\\ 3& 1& 0\end{array}\right]}$

(5)

∴

1.2.6.Julia session - working with row vectors

Problem

Insert a Julia session and define the vectors

${\displaystyle 𝒂=\left[\begin{array}{lll}1& 2& 3\end{array}\right],𝒃=\left[\begin{array}{lll}-1& 0& 1\end{array}\right],𝒄=\left[\begin{array}{lll}0& -1& 0\end{array}\right].}$

Answer

Define vectors

∴	a=[1 2 3];

∴	b=[-1 0 1];

∴	c=[0 -1 0];

∴

[a; b; c]

${\displaystyle \left[\begin{array}{ccc}1& 2& 3\\ -1& 0& 1\\ 0& -1& 0\end{array}\right]}$

(6)

∴

1.2.7.Julia session - assembling column vectors into a matrix

Problem

Insert a Julia session and define the matrix $𝑿=\left[\begin{array}{lll}𝒖& 𝒗& 𝒘\end{array}\right].$

Answer

Define vectors

∴	u=[1 2 3]';

∴	v=[-1 0 1]';

∴	w=[0 -1 0]';

∴

X=[u v w]

${\displaystyle \left[\begin{array}{ccc}1& -1& 0\\ 2& 0& -1\\ 3& 1& 0\end{array}\right]}$

(7)

∴

1.2.8.Julia session - assembling row vectors into a matrix

Problem

Insert a Julia session and define the matrix

${\displaystyle 𝒀=\left[\begin{array}{l}𝒂\\ 𝒃\\ 3𝒄\end{array}\right]}$

Answer

Define vectors

∴	a=[1 2 3];

∴	b=[-1 0 1];

∴	c=[0 -1 0];

∴	Y=[a; b; 3*c]

${\displaystyle \left[\begin{array}{ccc}1& 2& 3\\ -1& 0& 1\\ 0& -3& 0\end{array}\right]}$

(8)

∴

1.2.9.Julia session - componentwise definition of a matrix

Problem

Insert a Julia session and display the Hilbert matrix $𝑯\in {ℝ}^{4\times 4}$.

Answer

Define a Julia function

∴	function hilb(m) H=ones(m,m) for i=1:m for j=1:m H[i,j]=1.0/(i+j-1) end end return H end

$hilb$

∴

hilb(4)

${\displaystyle \left[\begin{array}{cccc}1.0& 0.5& 0.3333333333333333& 0.25\\ 0.5& 0.3333333333333333& 0.25& 0.2\\ 0.3333333333333333& 0.25& 0.2& 0.16666666666666666\\ 0.25& 0.2& 0.16666666666666666& 0.14285714285714285\end{array}\right]}$

(9)

∴

1.2.10.Julia session - constructing plots

Problem

Insert a Julia session to plot the function $f\left(x\right)=\sin \left(\cos \left(x\right)\right)+\cos \left(\sin \left(x\right)\right)$.

Answer

Define $f$ and plot it in Fig. 2

Figure 2. Plot of $f\left(x\right)=\sin \left(\cos \left(x\right)\right)+\cos \left(\sin \left(x\right)\right)$.

∴	f(x) = sin(cos(x))+cos(sin(x));

∴	x=0:0.01:2*pi; y=f.(x);

∴

plot(x,y)

PyCall.PyObject[PyObject <matplotlib.lines.Line2D object at 0x33308acb0>]

∴	xlabel("x"); ylabel("f(x)"); title("Plot of function f"); grid("on");

∴	cd(homedir()*"/courses/MATH347DS/homework/hw00");

∴	savefig("H00Fig01.eps");

∴

1.3.Data Science Application

Carry out linear regression, i.e., fitting a line to data.

1.3.1.Generate synthetic data

Problem

The following generates data by random perturbation of points on a line $y=c_0+c_{1}x$.

∴	m=20; x=(0:m-1)/m; c0=-1; c1=1; yex=c0 .+ c1*x;

∴	y=yex .+ 0.5*(rand(m,1) .- 0.5);

∴	clf(); plot(x,yex,"k",x,y,"r.");

∴	cd(homedir()*"/courses/MATH347DS/homework/hw00");

∴	savefig("H00Fig01.eps");

∴

Repeat for different values of $m,c_0,c_1$.

Figure 3. Perturbation of points on a line.

Answer

1.3.2.Form the normal system

Problem

Define matrices $𝑿=\left[\begin{array}{ll}𝟏& 𝒙\end{array}\right]$, $𝑵={𝑿}^T𝑿$, and vector $𝒃={𝑿}^T𝒚$

Answer

Define $𝑿$

∴	X=[]; m=size(x)[1]; o=ones(m,1); X=[o x]; N=X'*X;

∴

b=X'*y;

∴

1.3.3.Solve the least square problem

Problem

Solve the system $𝑵𝒄=𝒃$ by use of the Octave backslash operator c=N\b. Display the coefficient vector $𝒄$, and compare to the values you chose in Question 3.1. Also compute $\stackrel{\sim }{𝒚}=𝑿𝒄$, using ytilde as a notation.

Answer

The coefficients are

∴

c = N \ b

${\displaystyle \left[\begin{array}{c}-1.0390506761500047\\ 1.0876860806750384\end{array}\right]}$

(10)

∴

1.3.4.Plot the result

Problem

Plot the original line, perturbed points and linear regression of the perturbed points.

Answer

Costruct the plots and present them in Fig. 4.

∴	yfit = c[1] .+ c[2]*x;

∴	clf(); plot(x,yex,"k",x,y,"r.",x,yfit,"g");

∴	xlabel("x"); ylabel("y");

∴	savefig("H00Fig02.eps");

∴

Figure 4. Comparison of least squares fit to original and perturbed data.

Submission instructions. Save your work, and also export to PDF (menu File->Export->Pdf). In Canvas submit the files:

• hw00.tm

• hw00.pdf