From 8c2986026bc42d81a6d9386c9fe621fea8ff2d15 Mon Sep 17 00:00:00 2001
From: Dhruv Manilawala <dhruvmanila@gmail.com>
Date: Mon, 5 Apr 2021 19:07:38 +0530
Subject: [PATCH] fix(mypy): type annotations for linear algebra algorithms
 (#4317)

* fix(mypy): type annotations for linear algebra algorithms

* refactor: remove linear algebra directory from mypy exclude
---
 .github/workflows/build.yml               |  2 +-
 linear_algebra/src/conjugate_gradient.py  | 17 +++--
 linear_algebra/src/lib.py                 | 83 ++++++++++++++---------
 linear_algebra/src/polynom_for_points.py  | 13 ++--
 linear_algebra/src/power_iteration.py     |  7 +-
 linear_algebra/src/rayleigh_quotient.py   | 10 ++-
 linear_algebra/src/test_linear_algebra.py | 40 +++++------
 linear_algebra/src/transformations_2d.py  |  2 -
 8 files changed, 100 insertions(+), 74 deletions(-)

diff --git a/.github/workflows/build.yml b/.github/workflows/build.yml
index eb54d85ea..ca3e80922 100644
--- a/.github/workflows/build.yml
+++ b/.github/workflows/build.yml
@@ -23,7 +23,7 @@ jobs:
           python -m pip install mypy pytest-cov -r requirements.txt
       # FIXME: #4052 fix mypy errors in the exclude directories and remove them below
       - run: mypy --ignore-missing-imports
-          --exclude '(data_structures|digital_image_processing|dynamic_programming|graphs|linear_algebra|maths|matrix|other|project_euler|scripts|searches|strings*)/$' .
+          --exclude '(data_structures|digital_image_processing|dynamic_programming|graphs|maths|matrix|other|project_euler|scripts|searches|strings*)/$' .
       - name: Run tests
         run: pytest --doctest-modules --ignore=project_euler/ --ignore=scripts/ --cov-report=term-missing:skip-covered --cov=. .
       - if: ${{ success() }}
diff --git a/linear_algebra/src/conjugate_gradient.py b/linear_algebra/src/conjugate_gradient.py
index 1a65b8ccf..418ae88a5 100644
--- a/linear_algebra/src/conjugate_gradient.py
+++ b/linear_algebra/src/conjugate_gradient.py
@@ -3,10 +3,12 @@ Resources:
 - https://en.wikipedia.org/wiki/Conjugate_gradient_method
 - https://en.wikipedia.org/wiki/Definite_symmetric_matrix
 """
+from typing import Any
+
 import numpy as np
 
 
-def _is_matrix_spd(matrix: np.array) -> bool:
+def _is_matrix_spd(matrix: np.ndarray) -> bool:
     """
     Returns True if input matrix is symmetric positive definite.
     Returns False otherwise.
@@ -38,10 +40,11 @@ def _is_matrix_spd(matrix: np.array) -> bool:
     eigen_values, _ = np.linalg.eigh(matrix)
 
     # Check sign of all eigenvalues.
-    return np.all(eigen_values > 0)
+    # np.all returns a value of type np.bool_
+    return bool(np.all(eigen_values > 0))
 
 
-def _create_spd_matrix(dimension: np.int64) -> np.array:
+def _create_spd_matrix(dimension: int) -> Any:
     """
     Returns a symmetric positive definite matrix given a dimension.
 
@@ -64,11 +67,11 @@ def _create_spd_matrix(dimension: np.int64) -> np.array:
 
 
 def conjugate_gradient(
-    spd_matrix: np.array,
-    load_vector: np.array,
+    spd_matrix: np.ndarray,
+    load_vector: np.ndarray,
     max_iterations: int = 1000,
     tol: float = 1e-8,
-) -> np.array:
+) -> Any:
     """
     Returns solution to the linear system np.dot(spd_matrix, x) = b.
 
@@ -141,6 +144,8 @@ def conjugate_gradient(
 
         # Update number of iterations.
         iterations += 1
+        if iterations > max_iterations:
+            break
 
     return x
 
diff --git a/linear_algebra/src/lib.py b/linear_algebra/src/lib.py
index 353c83340..5e2f82018 100644
--- a/linear_algebra/src/lib.py
+++ b/linear_algebra/src/lib.py
@@ -22,6 +22,7 @@ Overview:
 
 import math
 import random
+from typing import Collection, Optional, Union, overload
 
 
 class Vector:
@@ -45,7 +46,7 @@ class Vector:
     TODO: compare-operator
     """
 
-    def __init__(self, components=None):
+    def __init__(self, components: Optional[Collection[float]] = None) -> None:
         """
         input: components or nothing
         simple constructor for init the vector
@@ -54,7 +55,7 @@ class Vector:
             components = []
         self.__components = list(components)
 
-    def set(self, components):
+    def set(self, components: Collection[float]) -> None:
         """
         input: new components
         changes the components of the vector.
@@ -65,13 +66,13 @@ class Vector:
         else:
             raise Exception("please give any vector")
 
-    def __str__(self):
+    def __str__(self) -> str:
         """
         returns a string representation of the vector
         """
         return "(" + ",".join(map(str, self.__components)) + ")"
 
-    def component(self, i):
+    def component(self, i: int) -> float:
         """
         input: index (start at 0)
         output: the i-th component of the vector.
@@ -81,22 +82,22 @@ class Vector:
         else:
             raise Exception("index out of range")
 
-    def __len__(self):
+    def __len__(self) -> int:
         """
         returns the size of the vector
         """
         return len(self.__components)
 
-    def euclidLength(self):
+    def euclidLength(self) -> float:
         """
         returns the euclidean length of the vector
         """
-        summe = 0
+        summe: float = 0
         for c in self.__components:
             summe += c ** 2
         return math.sqrt(summe)
 
-    def __add__(self, other):
+    def __add__(self, other: "Vector") -> "Vector":
         """
         input: other vector
         assumes: other vector has the same size
@@ -109,7 +110,7 @@ class Vector:
         else:
             raise Exception("must have the same size")
 
-    def __sub__(self, other):
+    def __sub__(self, other: "Vector") -> "Vector":
         """
         input: other vector
         assumes: other vector has the same size
@@ -122,7 +123,15 @@ class Vector:
         else:  # error case
             raise Exception("must have the same size")
 
-    def __mul__(self, other):
+    @overload
+    def __mul__(self, other: float) -> "Vector":
+        ...
+
+    @overload
+    def __mul__(self, other: "Vector") -> float:
+        ...
+
+    def __mul__(self, other: Union[float, "Vector"]) -> Union[float, "Vector"]:
         """
         mul implements the scalar multiplication
         and the dot-product
@@ -132,20 +141,20 @@ class Vector:
             return Vector(ans)
         elif isinstance(other, Vector) and (len(self) == len(other)):
             size = len(self)
-            summe = 0
+            summe: float = 0
             for i in range(size):
                 summe += self.__components[i] * other.component(i)
             return summe
         else:  # error case
             raise Exception("invalid operand!")
 
-    def copy(self):
+    def copy(self) -> "Vector":
         """
         copies this vector and returns it.
         """
         return Vector(self.__components)
 
-    def changeComponent(self, pos, value):
+    def changeComponent(self, pos: int, value: float) -> None:
         """
         input: an index (pos) and a value
         changes the specified component (pos) with the
@@ -156,7 +165,7 @@ class Vector:
         self.__components[pos] = value
 
 
-def zeroVector(dimension):
+def zeroVector(dimension: int) -> Vector:
     """
     returns a zero-vector of size 'dimension'
     """
@@ -165,7 +174,7 @@ def zeroVector(dimension):
     return Vector([0] * dimension)
 
 
-def unitBasisVector(dimension, pos):
+def unitBasisVector(dimension: int, pos: int) -> Vector:
     """
     returns a unit basis vector with a One
     at index 'pos' (indexing at 0)
@@ -177,7 +186,7 @@ def unitBasisVector(dimension, pos):
     return Vector(ans)
 
 
-def axpy(scalar, x, y):
+def axpy(scalar: float, x: Vector, y: Vector) -> Vector:
     """
     input: a 'scalar' and two vectors 'x' and 'y'
     output: a vector
@@ -192,7 +201,7 @@ def axpy(scalar, x, y):
     return x * scalar + y
 
 
-def randomVector(N, a, b):
+def randomVector(N: int, a: int, b: int) -> Vector:
     """
     input: size (N) of the vector.
            random range (a,b)
@@ -200,7 +209,7 @@ def randomVector(N, a, b):
             random integer components between 'a' and 'b'.
     """
     random.seed(None)
-    ans = [random.randint(a, b) for i in range(N)]
+    ans = [random.randint(a, b) for _ in range(N)]
     return Vector(ans)
 
 
@@ -222,7 +231,7 @@ class Matrix:
            operator - _ implements the matrix-subtraction
     """
 
-    def __init__(self, matrix, w, h):
+    def __init__(self, matrix: list[list[float]], w: int, h: int) -> None:
         """
         simple constructor for initializing
         the matrix with components.
@@ -231,7 +240,7 @@ class Matrix:
         self.__width = w
         self.__height = h
 
-    def __str__(self):
+    def __str__(self) -> str:
         """
         returns a string representation of this
         matrix.
@@ -246,7 +255,7 @@ class Matrix:
                     ans += str(self.__matrix[i][j]) + "|\n"
         return ans
 
-    def changeComponent(self, x, y, value):
+    def changeComponent(self, x: int, y: int, value: float) -> None:
         """
         changes the x-y component of this matrix
         """
@@ -255,7 +264,7 @@ class Matrix:
         else:
             raise Exception("changeComponent: indices out of bounds")
 
-    def component(self, x, y):
+    def component(self, x: int, y: int) -> float:
         """
         returns the specified (x,y) component
         """
@@ -264,13 +273,13 @@ class Matrix:
         else:
             raise Exception("changeComponent: indices out of bounds")
 
-    def width(self):
+    def width(self) -> int:
         """
         getter for the width
         """
         return self.__width
 
-    def height(self):
+    def height(self) -> int:
         """
         getter for the height
         """
@@ -303,7 +312,15 @@ class Matrix:
         else:
             raise Exception("matrix is not square")
 
-    def __mul__(self, other):
+    @overload
+    def __mul__(self, other: float) -> "Matrix":
+        ...
+
+    @overload
+    def __mul__(self, other: Vector) -> Vector:
+        ...
+
+    def __mul__(self, other: Union[float, Vector]) -> Union[Vector, "Matrix"]:
         """
         implements the matrix-vector multiplication.
         implements the matrix-scalar multiplication
@@ -312,7 +329,7 @@ class Matrix:
             if len(other) == self.__width:
                 ans = zeroVector(self.__height)
                 for i in range(self.__height):
-                    summe = 0
+                    summe: float = 0
                     for j in range(self.__width):
                         summe += other.component(j) * self.__matrix[i][j]
                     ans.changeComponent(i, summe)
@@ -330,7 +347,7 @@ class Matrix:
             ]
             return Matrix(matrix, self.__width, self.__height)
 
-    def __add__(self, other):
+    def __add__(self, other: "Matrix") -> "Matrix":
         """
         implements the matrix-addition.
         """
@@ -345,7 +362,7 @@ class Matrix:
         else:
             raise Exception("matrix must have the same dimension!")
 
-    def __sub__(self, other):
+    def __sub__(self, other: "Matrix") -> "Matrix":
         """
         implements the matrix-subtraction.
         """
@@ -361,19 +378,21 @@ class Matrix:
             raise Exception("matrix must have the same dimension!")
 
 
-def squareZeroMatrix(N):
+def squareZeroMatrix(N: int) -> Matrix:
     """
     returns a square zero-matrix of dimension NxN
     """
-    ans = [[0] * N for i in range(N)]
+    ans: list[list[float]] = [[0] * N for _ in range(N)]
     return Matrix(ans, N, N)
 
 
-def randomMatrix(W, H, a, b):
+def randomMatrix(W: int, H: int, a: int, b: int) -> Matrix:
     """
     returns a random matrix WxH with integer components
     between 'a' and 'b'
     """
     random.seed(None)
-    matrix = [[random.randint(a, b) for j in range(W)] for i in range(H)]
+    matrix: list[list[float]] = [
+        [random.randint(a, b) for _ in range(W)] for _ in range(H)
+    ]
     return Matrix(matrix, W, H)
diff --git a/linear_algebra/src/polynom_for_points.py b/linear_algebra/src/polynom_for_points.py
index 7a363723d..091849542 100644
--- a/linear_algebra/src/polynom_for_points.py
+++ b/linear_algebra/src/polynom_for_points.py
@@ -1,6 +1,3 @@
-from __future__ import annotations
-
-
 def points_to_polynomial(coordinates: list[list[int]]) -> str:
     """
     coordinates is a two dimensional matrix: [[x, y], [x, y], ...]
@@ -55,12 +52,12 @@ def points_to_polynomial(coordinates: list[list[int]]) -> str:
 
     if check == 1:
         count_of_line = 0
-        matrix = []
+        matrix: list[list[float]] = []
         # put the x and x to the power values in a matrix
         while count_of_line < x:
             count_in_line = 0
             a = coordinates[count_of_line][0]
-            count_line: list[int] = []
+            count_line: list[float] = []
             while count_in_line < x:
                 count_line.append(a ** (x - (count_in_line + 1)))
                 count_in_line += 1
@@ -69,7 +66,7 @@ def points_to_polynomial(coordinates: list[list[int]]) -> str:
 
         count_of_line = 0
         # put the y values into a vector
-        vector: list[int] = []
+        vector: list[float] = []
         while count_of_line < x:
             vector.append(coordinates[count_of_line][1])
             count_of_line += 1
@@ -96,14 +93,14 @@ def points_to_polynomial(coordinates: list[list[int]]) -> str:
         # make solutions
         solution: list[str] = []
         while count < x:
-            solution.append(vector[count] / matrix[count][count])
+            solution.append(str(vector[count] / matrix[count][count]))
             count += 1
 
         count = 0
         solved = "f(x)="
 
         while count < x:
-            remove_e: list[str] = str(solution[count]).split("E")
+            remove_e: list[str] = solution[count].split("E")
             if len(remove_e) > 1:
                 solution[count] = remove_e[0] + "*10^" + remove_e[1]
             solved += "x^" + str(x - (count + 1)) + "*" + str(solution[count])
diff --git a/linear_algebra/src/power_iteration.py b/linear_algebra/src/power_iteration.py
index 476361e0d..2cf22838e 100644
--- a/linear_algebra/src/power_iteration.py
+++ b/linear_algebra/src/power_iteration.py
@@ -2,8 +2,11 @@ import numpy as np
 
 
 def power_iteration(
-    input_matrix: np.array, vector: np.array, error_tol=1e-12, max_iterations=100
-) -> [float, np.array]:
+    input_matrix: np.ndarray,
+    vector: np.ndarray,
+    error_tol: float = 1e-12,
+    max_iterations: int = 100,
+) -> tuple[float, np.ndarray]:
     """
     Power Iteration.
     Find the largest eignevalue and corresponding eigenvector
diff --git a/linear_algebra/src/rayleigh_quotient.py b/linear_algebra/src/rayleigh_quotient.py
index 69bbbac11..78083aa75 100644
--- a/linear_algebra/src/rayleigh_quotient.py
+++ b/linear_algebra/src/rayleigh_quotient.py
@@ -1,10 +1,12 @@
 """
 https://en.wikipedia.org/wiki/Rayleigh_quotient
 """
+from typing import Any
+
 import numpy as np
 
 
-def is_hermitian(matrix: np.array) -> bool:
+def is_hermitian(matrix: np.ndarray) -> bool:
     """
     Checks if a matrix is Hermitian.
     >>> import numpy as np
@@ -24,7 +26,7 @@ def is_hermitian(matrix: np.array) -> bool:
     return np.array_equal(matrix, matrix.conjugate().T)
 
 
-def rayleigh_quotient(A: np.array, v: np.array) -> float:
+def rayleigh_quotient(A: np.ndarray, v: np.ndarray) -> Any:
     """
     Returns the Rayleigh quotient of a Hermitian matrix A and
     vector v.
@@ -43,7 +45,9 @@ def rayleigh_quotient(A: np.array, v: np.array) -> float:
     array([[3.]])
     """
     v_star = v.conjugate().T
-    return (v_star.dot(A).dot(v)) / (v_star.dot(v))
+    v_star_dot = v_star.dot(A)
+    assert isinstance(v_star_dot, np.ndarray)
+    return (v_star_dot.dot(v)) / (v_star.dot(v))
 
 
 def tests() -> None:
diff --git a/linear_algebra/src/test_linear_algebra.py b/linear_algebra/src/test_linear_algebra.py
index 6eba3a163..0954a2d93 100644
--- a/linear_algebra/src/test_linear_algebra.py
+++ b/linear_algebra/src/test_linear_algebra.py
@@ -12,7 +12,7 @@ from .lib import Matrix, Vector, axpy, squareZeroMatrix, unitBasisVector, zeroVe
 
 
 class Test(unittest.TestCase):
-    def test_component(self):
+    def test_component(self) -> None:
         """
         test for method component
         """
@@ -21,28 +21,28 @@ class Test(unittest.TestCase):
         self.assertEqual(x.component(2), 3)
         _ = Vector()
 
-    def test_str(self):
+    def test_str(self) -> None:
         """
         test for toString() method
         """
         x = Vector([0, 0, 0, 0, 0, 1])
         self.assertEqual(str(x), "(0,0,0,0,0,1)")
 
-    def test_size(self):
+    def test_size(self) -> None:
         """
         test for size()-method
         """
         x = Vector([1, 2, 3, 4])
         self.assertEqual(len(x), 4)
 
-    def test_euclidLength(self):
+    def test_euclidLength(self) -> None:
         """
         test for the eulidean length
         """
         x = Vector([1, 2])
         self.assertAlmostEqual(x.euclidLength(), 2.236, 3)
 
-    def test_add(self):
+    def test_add(self) -> None:
         """
         test for + operator
         """
@@ -52,7 +52,7 @@ class Test(unittest.TestCase):
         self.assertEqual((x + y).component(1), 3)
         self.assertEqual((x + y).component(2), 4)
 
-    def test_sub(self):
+    def test_sub(self) -> None:
         """
         test for - operator
         """
@@ -62,7 +62,7 @@ class Test(unittest.TestCase):
         self.assertEqual((x - y).component(1), 1)
         self.assertEqual((x - y).component(2), 2)
 
-    def test_mul(self):
+    def test_mul(self) -> None:
         """
         test for * operator
         """
@@ -72,19 +72,19 @@ class Test(unittest.TestCase):
         self.assertEqual(str(x * 3.0), "(3.0,6.0,9.0)")
         self.assertEqual((a * b), 0)
 
-    def test_zeroVector(self):
+    def test_zeroVector(self) -> None:
         """
         test for the global function zeroVector(...)
         """
         self.assertTrue(str(zeroVector(10)).count("0") == 10)
 
-    def test_unitBasisVector(self):
+    def test_unitBasisVector(self) -> None:
         """
         test for the global function unitBasisVector(...)
         """
         self.assertEqual(str(unitBasisVector(3, 1)), "(0,1,0)")
 
-    def test_axpy(self):
+    def test_axpy(self) -> None:
         """
         test for the global function axpy(...) (operation)
         """
@@ -92,7 +92,7 @@ class Test(unittest.TestCase):
         y = Vector([1, 0, 1])
         self.assertEqual(str(axpy(2, x, y)), "(3,4,7)")
 
-    def test_copy(self):
+    def test_copy(self) -> None:
         """
         test for the copy()-method
         """
@@ -100,7 +100,7 @@ class Test(unittest.TestCase):
         y = x.copy()
         self.assertEqual(str(x), str(y))
 
-    def test_changeComponent(self):
+    def test_changeComponent(self) -> None:
         """
         test for the changeComponent(...)-method
         """
@@ -109,43 +109,43 @@ class Test(unittest.TestCase):
         x.changeComponent(1, 1)
         self.assertEqual(str(x), "(0,1,0)")
 
-    def test_str_matrix(self):
+    def test_str_matrix(self) -> None:
         A = Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]], 3, 3)
         self.assertEqual("|1,2,3|\n|2,4,5|\n|6,7,8|\n", str(A))
 
-    def test_determinate(self):
+    def test_determinate(self) -> None:
         """
         test for determinate()
         """
         A = Matrix([[1, 1, 4, 5], [3, 3, 3, 2], [5, 1, 9, 0], [9, 7, 7, 9]], 4, 4)
         self.assertEqual(-376, A.determinate())
 
-    def test__mul__matrix(self):
+    def test__mul__matrix(self) -> None:
         A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]], 3, 3)
         x = Vector([1, 2, 3])
         self.assertEqual("(14,32,50)", str(A * x))
         self.assertEqual("|2,4,6|\n|8,10,12|\n|14,16,18|\n", str(A * 2))
 
-    def test_changeComponent_matrix(self):
+    def test_changeComponent_matrix(self) -> None:
         A = Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]], 3, 3)
         A.changeComponent(0, 2, 5)
         self.assertEqual("|1,2,5|\n|2,4,5|\n|6,7,8|\n", str(A))
 
-    def test_component_matrix(self):
+    def test_component_matrix(self) -> None:
         A = Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]], 3, 3)
         self.assertEqual(7, A.component(2, 1), 0.01)
 
-    def test__add__matrix(self):
+    def test__add__matrix(self) -> None:
         A = Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]], 3, 3)
         B = Matrix([[1, 2, 7], [2, 4, 5], [6, 7, 10]], 3, 3)
         self.assertEqual("|2,4,10|\n|4,8,10|\n|12,14,18|\n", str(A + B))
 
-    def test__sub__matrix(self):
+    def test__sub__matrix(self) -> None:
         A = Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]], 3, 3)
         B = Matrix([[1, 2, 7], [2, 4, 5], [6, 7, 10]], 3, 3)
         self.assertEqual("|0,0,-4|\n|0,0,0|\n|0,0,-2|\n", str(A - B))
 
-    def test_squareZeroMatrix(self):
+    def test_squareZeroMatrix(self) -> None:
         self.assertEqual(
             "|0,0,0,0,0|\n|0,0,0,0,0|\n|0,0,0,0,0|\n|0,0,0,0,0|" + "\n|0,0,0,0,0|\n",
             str(squareZeroMatrix(5)),
diff --git a/linear_algebra/src/transformations_2d.py b/linear_algebra/src/transformations_2d.py
index 6a15189c5..cdf42100d 100644
--- a/linear_algebra/src/transformations_2d.py
+++ b/linear_algebra/src/transformations_2d.py
@@ -11,8 +11,6 @@ projection(45) = [[0.27596319193541496, 0.446998331800279],
 reflection(45) = [[0.05064397763545947, 0.893996663600558],
                   [0.893996663600558, 0.7018070490682369]]
 """
-from __future__ import annotations
-
 from math import cos, sin