Octave tasks

Create matrices

Solve tasks from this block in a folder named octave-matrices. Name files as written in the beginning of each task, files should have the .m extension.

Each task asks you to create a certain matrix. Please use only methods that were discussed at the lecture or in lecture notes available on this site. That is, for example, don’t use loops and functions such as repmat. And try to find a short solution, especially, don’t enter all the values manually.

  1. task_10x10

    The \(10\times10\) matrix:

    \( \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ % 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ % 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ % 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ % 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ % 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ % 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ % 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ % 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ % 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} \)

  2. task_border

Create a border-matrix \(10\times10\), it should have 0s everywhere except 1s at the border:

\( \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ % 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ % 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ % 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ % 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ % 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ % 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ % 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ % 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ % 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{pmatrix} \)

  1. task_2_5x5

    The \(5\times5\) matrix:

    \( \begin{pmatrix} 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 \\ % 2 & 2 & 2 & 2 & 2 \\ % 2 & 2 & 2 & 2 & 2 \\ % 2 & 2 & 2 & 2 & 2 \end{pmatrix} \)

  2. task_0123

    The \(10\times10\) matrix:

    \( \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ % 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ % 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ % 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ % 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 3 \\ 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 3 \\ % 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 3 \\ % 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 3 \\ % 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 3 \end{pmatrix} \)

  3. task_chess

    The \(10\times10\) matrix with the chess-board order of 0’s and 1’s:

    \( \begin{pmatrix} 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ % 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ % 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ % 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ % 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0
    \end{pmatrix} \)

  4. task_1_to_9_lines

    The \(9\times9\) matrix:

    \( \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 \\ 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 \\ 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 \\ 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\ 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 \\ 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 \\ 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 \end{pmatrix} \)

  5. task_1_to_100

    The \(10\times10\) matrix with a range of numbers from 1 to 100:

    \( \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & 99 & 100 \end{pmatrix} \)

  6. task_multiplication_table

    The \(9\times9\) matrix with a “multiplication table”. That is, the element a(i, j) = i * j:

    \( \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 \\ 3 & 6 & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & 56 & . \\ . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & 72 & 81 \end{pmatrix} \)

  7. task_3_diags

    You are given numbers n, a, b. Create a matrix of size $n\times n$, that has a on the main diagonal, b strictly below and above the main diagonal, and 0 in all other places. Here is an example for arguments (4, 1, 2):

    1 2 0 0
2 1 2 0
0 2 1 2
0 0 2 1

  8. task_double_chess

    This task is similar to the chess-board task, but the size of the matrix should be $20\times20$, and cells with ones or zero should have size $2\times2$:

     0 0 1 1 0 0 .....
 0 0 1 1 0 0 .....
 1 1 0 0 1 1 .....
 1 1 0 0 1 1 .....
 .................

  9. task_chukh.

    You are given a number n. Create the matrix of size $n\times n$ of the following structure: Its main diagonal has numbers 1 and 2 that go. Дальше от каждой клетки главной диагонали направо и вниз расставляется то же числа, что и в самой этой клетке. Пример для аргумента (5):

    1 1 1 1 1
1 2 2 2 2
1 2 1 1 1
1 2 1 2 2
1 2 1 2 1

    Try to avoid loops. Additional question. What is a determinant of this matrix? Can you explain why the determinant is as it is?

  10. task_domino. Based on http://www.math.cornell.edu/~levine/18.312/alg-comb-lecture-18.pdf

Additional task. You may use loops in this task.

You are given natural numbers $m$ and $n$, they are the size of a rectangles made of cells. There are $m\times n$ cells, they are numbered from $1$ to $m\cdot n$.

Create a matrix $A$ of size $m\cdot n\times m\cdot n$, in which the element at $i,j$ contains:

* 1, if cells $i$ and $j$ are next to each other horizontally,
* complex unit ($\texttt i$), if cells $i$ and $j$ are next to each other vertically,
* 0 otherwise

Evaluate the determinant of $A$ and take the square root of it.
You will get the number of domino tilings of an $m\times n$ rectangle.

Indexing octave-indexing

  1. task_left_top You are given a square matrix of an even size. (you are not supposed to test that the matrix is actually even and square, just use it). Imagine we split a matrix vertically and horizontally through the center. Return the matrix at the left-top.
  2. task_bottom_half You are given a matrix with an even number of rows. Return a new matrix where the top half is exchanged with the bottom half of the initial matrix.
  3. task_mod_3 You are given a vector with integer numbers. Return a vector that contains only that numbers of the initial vector, that are divisible by 3. Use the mod function to test divisibilty.
  4. task_div_by_2 Given a vector of integer numbers, return a vector with the same numbers, but where even numbers are divided by 2. Will your solution work if the matrix is 2 dimensional?

  5. Create two functions filter_multiples and seive. The first one is filter_multiples(a, k), it removes all numbers from the vector a that both are divisible by k and are strictly greater than k. For example, filter_multiples([1 2 3 4 5 6 7 8 9 10], 3) yields [1 2 3 4 5 7 8 10].

    The second one is sieve(n), and it implements the Sieve of Eratosthenes. You will probably need to use the while loop.

  6. task_animals You are given a matrix with two columns. Each column describes an animal: its height in meters and weight in kg. Return the following set of answers:
    • sum of animals’ heights
    • mean height of animals
    • mean height of animals that are higher than 100 kg
    • one column matrix with the body mass index (BMI) of animals. BMI is equal to weight divided by the square of height.
    • two columns matrix with animals, that are higher than 10 meters, and heaver than 100 kg.

Vectorization octave-vectorization

  1. task_polyval You are given a row matrix, for example, a = [1, 2, -2, 4] and a number x, for example, 1. The row a specifies coefficients of a polinomial, starting from the lowest degree, that is, the row from the example specifies the polynomial \(1 + 2x - 2x^2 + 4x^3\). Compute the value of the polinomial at the point x. The answer in the example is 5.
  2. task_sub_ij You are given a column matrix \(x\). Create a square matrix of the same size, its value in the row number \(i\) and the column number \(j\) should be equal to \(a_{i,j} = x_i – x_j\).
  3. task_convert_line_equations You are given a three columns matrix. Imagine, that each row specifies a line of the form \(ax + by + c = 0\), whith the columns being, correspondingly, a, b, c. Return a matrix, that has two columns, correspondingly, k и b, that define the same lines, but in the form \(y = kx + b\). For example, the row [1 1 1] should be converted to the row [-1 -1].
  4. task_all_dets. You are given a matrix x of two columns. Return a new square matrix with the size equal to the number of rows in x. An element in the row number \(i\) and the column number \(j\) should be equal to \(a_{i,j} = \left|\begin{matrix}x_{i,1}&x_{i,2}\\x_{j,1} &x_{j,2}\end{matrix}\right|\). This is a determinant of a matrix composed from the i-th and the j-th rows of x. I remind that \(\left|\begin{matrix}a&b\\c&d\end{matrix}\right|=ad - bc\).

  5. task_all_lines_intersections. You are given a matrix а of three columns, it describes a set of lines (as previously). The short task statement is: intersect each line with each line. To be more specific: the intersection of the line from the i-th row and the line from j-th row has two coordinates \(x\) and \(y\). The function should return two matrices x and y. The first one contains the x-coordinate of this intersection in the row \(i\) and the column \(j\). The second one contains, correspondingly, the y-coordinate. Don’t consider the case of parallel lines.

    Here is the formula for intersection of lines \(a_1x+b_1y+c_1\) = 0 and \(a_2x+b_2y+c_2 = 0\):

     Δ = det([a1 b1; a2 b2])
 Δx = det([-c1 b1; -c2 b2])
 Δy = det([a1 -c1; a2 -c2])
 x = Δx / Δ
 y = Δy / Δ

    So, you will need a previously implemented all2dets function.

Plotting octave-plotting

Don’t use loops, unless the opposite is explicitly stated. Your functions will have the side effect: they draw some plot. They should not return anything.

  1. task_house Draw a house. You should draw just a few lines, of the house body, its roof, may be a window or a door. Don’t draw a lot to not spend a lot of time for this task.
  2. task_plot_sin Plot the function \(\sin(x) + \sin(3x)\) from \(-2\pi\) to \(2\pi\).
  3. task_plot_10_sin Plot \(\sin(x) + \frac1{2}\sin(2x) + \frac1{3}\sin(3x) + \cdots + \frac1{10}\sin(10x)\). Think how to do it without loops. Remember, that sin accepts matrices and evaluates results for each element separately.
  4. task_plot_A You are given a matrix A of the size \(N\times2\), each row contains coordinates of one point on the plane.
    1. Draw set A
    2. Draw the center of mass for A.
    3. Create and draw the set B that is a translation of A in such a way that its center of mass is in \((0, 0)\). Remind: translated by the vector \((dx, dy)\), point \((x, y)\) goes to \((x + dx, y + dy)\).
    4. Create and draw the set C: a rotation of A around (0, 0) by \(5^\circ\). Remind: to rotate the point \((x, y)\) by \(\varphi\) around zero, one should left-multiply \(\begin{pmatrix}x\\y\end{pmatrix}\) by the matrix \(\begin{pmatrix}\cos(\varphi)&\sin(\varphi)\\-\sin(\varphi)&\cos(\varphi)\end{pmatrix}\).
    5. Create and draw the set D: A rotation of А around its center of mass by \(5^\circ\).

    Draw each set with a different color and a different marker.

  5. task_10circles Draw 10 concentric circles with radiuses: 1, 2, …, 10. Use the fact, that points with coordinates \((\cos(\varphi), \sin(\varphi))\) lay on a circle of radius 1, when \(\varphi\) changes from 0 to \(2\pi\).

  6. task_plot_line this is a hard task You are given a linear matrix [a, b, c]. You are also given matrices xrange=[xmin, xmax] and yrange=[ymin, ymax]. Plot a line \(ax + by + c = 0\), and draw only the part of the line, that is contained inside a rectangle \(x_{min} \leq x \leq x_{max}\), and \(y_{min} \leq y \leq y_{max}\). Consider all the cases, including vertical lines.

    One of the ways to solve the task is as follows: intersect the line with each of the rectangle sides. They are horizontal and vertical, so it is quite easy to find intersections. Usually, you are left with only two intersections, so you now can call plot() for the line between the two points.

  7. task_plot_intersections. You are given a matrix with lines as in the task 3 from octave-vectorization. Plot all intersections of these lines (do not draw lines), use the function all_lines_intersections() from octave-vectorization.
  8. task_plot_lines. You are given a matrix a with lines (the same as matrices from the block octave-vectorization, you are also given xrange, yrange with ranges to draw the lines by means of the plot_line function from the previous task. Draw all of the lines from a. Use a loop to call plotting for each line.
  9. task_plot_lines_and_intersections You are given the same as in the previous task. Draw all lines from the matrix a and draw all the intersections. Remember, that you have already implemented evaluating of intersections in the block octave-vectorization.

Some fun with random numbers octave-random

  1. task_random_walk_2d Random walk on a plane. You are given an integer steps. Generate steps times a pair of numbers, this pair may be either (1 0), (-1 0), (0 1), or (0 -1). Put all this pairs in one matrix with two columns, so you get a matrix of size $size \times 2$. This pairs correspond to movements of a point on a plane, a pair is a change for x and y coordinates correspondingly. That is a point moves either up, left, right or down. Finally, sum all x coordinates, then sum all y coordinates, and you will get two coordinates of where a point had come after its random walk.

    you could additionaly try to plot this walk.

  2. task_kinder_surprise There are n possible toys inside a kinder surprise. How many kinder surprises should one buy in average to get all the toys? Make 10000 experiments, where you repeat opening kinder surprises until you get all the toys. Then compute an average number of steps you needed to finish.