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Solve the problems in hw01.pdf:

- Solve tasks 1, 2 and 3.a in hand.
- Implement the root finding function for a univariate polynomial via companion matrices.
- Solve tasks 3.b and 4 using the function implemented in 2. In task 4 find the roots of the polynomial specified for you in Brute.
**The given polynomials have integer roots and we want you to upload them as integers, including multiplicities.**

**Forbidden methods/classes**: numpy.roots, numpy.polynomial, numpy.poly1d

You are not allowed to use other libraries than the ones indicated in the list **Allowed libraries**.

Create a function `roots(coeffs)`

which takes the coefficients of a univariate polynomial and returns its (complex) roots.

Input/Output specifications for `roots(coeffs: numpy.ndarray) → numpy.ndarray`

:

`coeffs`

: 1-dimensional`numpy.ndarray`

that defines the coefficients of the polynomial starting from the term of the highest degree.**Return value**: 1-dimensional`numpy.ndarray`

containing the roots of the given polynomial.

Implement the solution in a single file `hw01.py`

. The file must contain the `roots`

function, such that it can be imported (by the automatic evaluation) as

import hw01 import numpy as np coeffs = np.array([1, 2, 3]) res = hw01.roots(coeffs)

Upload a zip archive `hw01.zip`

(via the course ware) containing:

`hw01.json`

- json file containing the answers to tasks 1, 2, 3.b and 4 (see below for the description of how to create it)`hw01.py`

- python script containing the implemented function`roots`

`hw01.pdf`

- pdf file containing the solution to tasks 1, 2, 3.a together with the companion matrices for tasks 3.b and 4 (LaTex, photos, scans, …).

All the files must be contained in the root of

`hw01.zip`

.
**Creating** `hw01.json`

:

Create an empty dictionary in Python:

solution = {}

The keys for this dictionary are `“task1”`

, `“task2”`

, `“task3”`

(for task 3.b) and `“task4”`

.

The value for the key `“task1”`

is a list of coefficients of the product polynomial starting from the term of the highest degree, i.e. for the product polynomial `2*x^4 + 5*x^3 + 3*x^2 + 1`

do

solution["task1"] = [2, 5, 3, 0, 1]

The value for the key `“task2”`

is a list of lists of coefficients of the quotient polynomial and the remainder, respectively, starting from the term of the highest degree, i.e. for the quotient polynomial `2*x^3 + 1`

and the remainder `x + 2`

do

solution["task2"] = [[2, 0, 0, 1], [1, 2]]

The value for the key `“task3”`

is a list of roots to the polynomial from task 3.b (**including multiplicities**), i.e. for the sequence of integer roots `1, 1, 2, 3`

do

solution["task3"] = [1, 1, 2, 3]

The value for the key `“task4”`

is created in the same way as for the key `“task3”`

.

Finally, save `solution`

to `hw01.json`

:

import json with open("hw01.json", "w") as outfile: json.dump(solution, outfile)

courses/pkr/labs/hw01.txt · Last modified: 2023/09/26 12:39 by korotvik