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# Python
| Name | Description |
| ---------------------------------------- | ------------------------------------------------------------------ |
| [tram-stat-problem](./tram-stat-problem) | Quick script to visualize a dumb stat problem I was thinking about |
| Name | Description |
| ---------------------------------------- | ------------------------------------------------------------------------ |
| [denoising](./denoising) | Uni homework implementation of a terrible image denoising implementation |
| [tram-stat-problem](./tram-stat-problem) | Quick script to visualize a dumb stat problem I was thinking about |

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.direnv
.ipynb_checkpoints
.envrc
*.pyc
*.zip

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under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License. Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".
You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
<https://www.gnu.org/licenses/>.
The GNU General Public License does not permit incorporating your program
into proprietary programs. If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library. If this is what you want to do, use the GNU Lesser General
Public License instead of this License. But first, please read
<https://www.gnu.org/licenses/why-not-lgpl.html>.

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{
"nodes": {
"flake-utils": {
"locked": {
"lastModified": 1676283394,
"narHash": "sha256-XX2f9c3iySLCw54rJ/CZs+ZK6IQy7GXNY4nSOyu2QG4=",
"owner": "numtide",
"repo": "flake-utils",
"rev": "3db36a8b464d0c4532ba1c7dda728f4576d6d073",
"type": "github"
},
"original": {
"owner": "numtide",
"repo": "flake-utils",
"type": "github"
}
},
"nixpkgs": {
"locked": {
"lastModified": 1676428076,
"narHash": "sha256-8caqXsVSUvZqp4/pOLjbWWabzLEV/ZfDUiEv32WumKw=",
"owner": "nixos",
"repo": "nixpkgs",
"rev": "e0a054002198445ef987b55c34cde892cecc0de7",
"type": "github"
},
"original": {
"owner": "nixos",
"ref": "release-22.11",
"repo": "nixpkgs",
"type": "github"
}
},
"root": {
"inputs": {
"flake-utils": "flake-utils",
"nixpkgs": "nixpkgs"
}
}
},
"root": "root",
"version": 7
}

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{
inputs = {
nixpkgs.url = "github:nixos/nixpkgs/release-22.11";
flake-utils.url = "github:numtide/flake-utils";
};
outputs = { self, nixpkgs, flake-utils }:
flake-utils.lib.eachDefaultSystem
(system:
let
pkgs = nixpkgs.legacyPackages.${system};
pyDeps = p: with p; [ numpy scipy matplotlib pillow ];
in
rec {
devShell = pkgs.mkShell {
buildInputs = with pkgs; [
(python3.withPackages pyDeps)
jupyter
pandoc
];
};
});
}

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import numpy as np
def decompose(M):
n = len(M)
U = M.copy()
L = np.identity(n)
for i in range(0, n - 1):
for j in range(i + 1, n):
L[j, i] = U[j, i] / U[i, i]
U[j, i:n] -= L[j, i] * U[i, i:n]
assert np.allclose(M, L@U)
return L, U
def solve_lower(m, v):
x = np.zeros(len(v))
for i in range(0, len(v)):
x[i] = (v[i] - m[i, :i] @ x[:i]) / m[i, i]
assert np.allclose(m@x, v)
return x
def solve_upper(m, v):
x = np.zeros(len(v))
for i in reversed(range(0, len(m))):
x[i] = (v[i] - m[i, i+1:] @ x[i+1:]) / m[i, i]
assert np.allclose(m@x, v)
return x
def solve(m, v):
L, U = decompose(m)
x = solve_upper(U, solve_lower(L, v))
assert np.allclose(m@x, v)
return x

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"""Project 2 tools for generating a noisy image"""
import matplotlib.pyplot as plt
import numpy as np
from PIL import Image # PIL is an image processing module
def create_noisy_image(path="/content/drive/My Drive/", file="sonic.jpg"):
"""Transform color image to a noisy grayscale image.
Plot and save the transformed images.
The file sonic.jpg must be located in your google drive and the drive must have been
"mounted" with:
from google.colab import drive
drive.mount("/content/drive")
"""
image = Image.open(path + file)
# Transform RGB image to grayscale
image = image.convert("L")
# get the filename without the extension
# (note: the pathlib module offers better ways to do this!)
name = file.removesuffix(".jpg")
image.save(path + name + "_gray.jpg")
# set images to be plotted in gray scale
plt.gray()
plt.imshow(image)
plt.title("Original image")
plt.show()
# add noise to grayscale image
sigma = 2.0
img_arr = np.array(image)
img_arr_noisy = np.clip(
img_arr + sigma * np.random.normal(0, 100, img_arr.shape), 0, 255
).astype(np.uint8)
plt.title("Noisy image")
plt.imshow(img_arr_noisy)
# create Image object from numpy array and save to disk
image = Image.fromarray(img_arr_noisy, mode="L")
image.save(path + name + "_gray_noisy.jpg")
plt.show()

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import numpy as np
import scipy as sp
def assert_valid_tridiagonal(a, c, e):
"""
Validates a tridiagonal matrix.
"""
assert len(a) == len(c) + 1 == len(e) + 1
def create(a, c, e):
"""
Validates a tridiagonal matrix before creating it.
"""
assert_valid_tridiagonal(a, c, e)
return (a, c, e)
def decompose(a, c, e):
"""
Computes the LU decomposition of a tridiagonal matrix.
"""
assert_valid_tridiagonal(a, c, e)
α = np.zeros(len(c))
β = a.copy()
for i in range(len(c)):
α[i] = e[i] / β[i]
β[i + 1] -= c[i] * α[i]
# Sanity check
if len(a) <= 10:
assert np.allclose(
to_array(a, c, e),
to_array(*from_lower(α)) @ to_array(*from_upper(β, c))
)
return (α, β)
def to_csr(a, c, e):
"""
Converts a tridiagonal matrix into a scipy csr sparse matrix.
"""
assert_valid_tridiagonal(a, c, e)
return sp.sparse.diags((a, c, e), offsets=(0, 1, -1), format="csr")
def to_array(a, c, e):
"""
Converts a tridiagonal matrix into a numpy matrix.
"""
assert_valid_tridiagonal(a, c, e)
return to_csr(a, c, e).toarray()
def from_lower(α):
"""
Turns the lower vector of a decomposition into a tridiagonal matrix.
Example ussage:
```py
α, β = decompose(m)
print(from_lower(α))
```
"""
return create(np.ones(len(α) + 1), np.zeros(len(α)), α)
def from_upper(β, c):
"""
Turns the upper vectors of a decomposition into a tridiagonal matrix.
Example ussage:
```py
α, β = decompose((a, c, e))
print(from_upper(β, c))
```
"""
return create(β, c, np.zeros(len(c)))
def solve_lower(α, rhs):
"""
Solve a linear system of equations Mx = v
where M is a lower triangular matrix constructed
by LU decomposing a tridiagonal matrix.
"""
x = np.zeros_like(rhs)
x[0] = rhs[0]
for i in range(1, len(rhs)):
x[i] = rhs[i] - α[i - 1] * x[i - 1]
if len(α) <= 10:
assert np.allclose(to_array(*from_lower(α)) @ x, rhs)
return x
def solve_upper(β, c, rhs):
"""
Solve a linear system of equations Mx = v
where M is an upper triangular matrix constructed
by LU decomposing a tridiagonal matrix.
"""
x = np.zeros_like(rhs)
x[-1] = rhs[-1] / β[-1]
for i in reversed(range(len(rhs) - 1)):
x[i] = (rhs[i] - c[i] * x[i+1]) / β[i]
if len(β) <= 10:
assert np.allclose(to_array(*from_upper(β, c)) @ x, rhs)
return x
def solve(a, c, e, rhs):
"""
Solves a system of linear equations defined by a tridiagonal matrix.
"""
assert_valid_tridiagonal(a, c, e)
α, β = decompose(a, c, e)
x = solve_upper(β, c, solve_lower(α, rhs))
if len(α) <= 10:
assert np.allclose(to_array(a, c, e)@x, rhs)
return x
def multiply_vector(a, c, e, x):
"""
Performs a matrix-vector multiplication where the matrix is tridiagonal.
"""
assert_valid_tridiagonal(a, c, e)
assert len(x) == len(a)
result = np.zeros_like(x)
for i in range(len(x)):
result[i] = a[i] * x[i]
if i > 0:
result[i] += e[i - 1] * x[i - 1]
if i < len(x) - 1:
result[i] += c[i] * x[i + 1]
# Sanity check
if len(a) <= 10:
assert np.allclose(to_array(a, c, e) @ x, result)
return result
def largest_eigenvalue(a, c, e, initial_x, kmax):
"""
Computes the largest eigenvalue of a positive definite tridiagonal matrix.
"""
assert_valid_tridiagonal(a, c, e)
x = initial_x
for _ in range(kmax):
q = multiply_vector(a, c, e, x)
assert not np.allclose(q, 0) # Sanity check
x = q/np.linalg.norm(q)
# Computes the eigenvalue from the eigenvector
eigenvalue = (x @ multiply_vector(a, c, e, x)) / (x @ x)
# Sanity check
if len(initial_x) <= 10:
actual_eigenvalues, _ = np.linalg.eig(to_array(a, c, e))
assert np.allclose(
0,
np.min(
np.abs(
actual_eigenvalues - eigenvalue
)
)
)
return eigenvalue
def add(a, b):
"""
Adds two tridiagonal matrices.
"""
assert_valid_tridiagonal(*a)
assert_valid_tridiagonal(*b)
assert len(a[0]) == len(b[0])
result = create(a[0] + b[0], a[1] + b[1], a[2] + b[2])
# Sanity check
if len(a[0]) <= 10:
assert np.allclose(to_array(*result), to_array(*a) + to_array(*b))
return result
def identity(n):
"""
Returns the tridiagonal identity n*n matrix.
"""
assert n > 0 # Sanity check
return create(np.ones(n), np.zeros(n - 1), np.zeros(n - 1))
def scale(s, m):
"""
Multiplies a tridiagonal matrix by a scalar
"""
assert_valid_tridiagonal(*m)
result = create(s * m[0], s * m[1], s * m[2])
# Sanity check
if len(m[0]) <= 10:
assert np.allclose(result, s * to_array(*m))
return result
def transpose(a, c, e):
"""
Computes the transpose of a tridiagonal matrix.
"""
return create(a, e, c)
# main()