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Homework 1: Extending polynomial the other way

Homework (2 points)

Extend the original polynomial function to the case where x is a square matrix. Create a function called circlemat, that returns nxn matrix $A(n)$ with the following elements

\[\left[A(n)\right]_{ij} = \begin{cases} 1 &\text{if } (i = j-1 \land j > 1) \lor (i = n \land j=1) \\ 1 &\text{if } (i = j+1 \land j < n) \lor (i = 1 \land j=n) \\ 0 & \text{ otherwise} \end{cases}\]

and evaluate the polynomial

\[f(A) = I + A + A^2 + A^3.\]

, at point $A = A(10)$.

HINTS for matrix definition: You can try one of these options:

  • create matrix with all zeros with zeros(n,n), use two nested for loops going in ranges 1:n and if condition with logical or ||, and &&
  • employ array comprehension with nested loops [expression for i in 1:n, j in 1:n] and ternary operator condition ? true branch : false

HINTS for polynomial extension:

  • extend the original example (one with for-loop) to initialize the accumulator variable with matrix of proper size (use size function to get the dimension), using argument typing for x is preferred to distinguish individual implementations <: AbstractMatrix

or

  • test later defined polynomial methods, that may work out of the box
Exercise (voluntary)

Install GraphRecipes and Plots packages into the environment defined during the lecture and figure out, how to plot the graph defined by adjacency matrix A from the homework.

HINTS:

  • There is help command inside the the pkg mod of the REPL. Type ? add to find out how to install a package. Note that both pkgs are registered.
  • Follow a guide in the Plots pkg's documentation, which is accessible through docs icon on top of the README in the GitHub repository. Direct link.
Solution:

Activate the environment in pkg mode, if it is not currently active.

pkg> activate .

Installing pkgs is achieved using the add command. Running ] ? add returns a short piece of documentation for this command:

pkg> ? add
[...]
  Examples

  pkg> add Example                                          # most commonly used for registered pkgs (installs usually the latest release)
  pkg> add Example@0.5                                      # install with some specific version (realized through git tags)
  pkg> add Example#master                                   # install from master branch directly
  pkg> add Example#c37b675                                  # install from specific git commit
  pkg> add https://github.com/JuliaLang/Example.jl#master   # install from specific remote repository (when pkg is not registered)
  pkg> add git@github.com:JuliaLang/Example.jl.git          # same as above but using the ssh protocol
  pkg> add Example=7876af07-990d-54b4-ab0e-23690620f79a     # when there are multiple pkgs with the same name

As the both Plots and GraphRecipes are registered and we don't have any version requirements, we will use the first option.

pkg> add Plots
pkg> add GraphRecipes

This process downloads the pkgs and triggers some build steps, if for example some binary dependencies are needed. The process duration depends on the "freshness" of Julia installation and the size of each pkg. With Plots being quite dependency heavy, expect few minutes. After the installation is complete we can check the updated environment with the status command.

pkg> status

The plotting itself as easy as calling the graphplot function on our adjacency matrix.

julia> using GraphRecipes, Plots
julia> graphplot(A)Plot{Plots.GRBackend() n=21}
Captured extra kwargs:
  Series{1}:
    num_edges_nodes: (10, 10)
  Series{2}:
    num_edges_nodes: (10, 10)
  Series{3}:
    num_edges_nodes: (10, 10)
  Series{4}:
    num_edges_nodes: (10, 10)
  Series{5}:
    num_edges_nodes: (10, 10)
  Series{6}:
    num_edges_nodes: (10, 10)
  Series{7}:
    num_edges_nodes: (10, 10)
  Series{8}:
    num_edges_nodes: (10, 10)
  Series{9}:
    num_edges_nodes: (10, 10)
  Series{10}:
    num_edges_nodes: (10, 10)

How to submit?

Isolate the code of the compulsory task into a script named hw.jl alongside with the Project.toml and Manifest.toml of the environment. Create a zipfile of the folder and send it to the lab instructor, who has assigned the task, via email (contact emails are located on the homepage of the course).