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# -*- coding: utf-8 -*-
"""
Created on Mon Feb 26 14:29:11 2018
@author: Christian Bender
@license: MIT-license
This module contains some useful classes and functions for dealing
with linear algebra in python.
Overview:
- class Vector
- function zeroVector(dimension)
- function unitBasisVector(dimension,pos)
- function axpy(scalar,vector1,vector2)
- function randomVector(N,a,b)
- class Matrix
- function squareZeroMatrix(N)
- function randomMatrix(W,H,a,b)
"""
import math
import random
class Vector(object):
"""
This class represents a vector of arbitray size.
You need to give the vector components.
Overview about the methods:
constructor(components : list) : init the vector
set(components : list) : changes the vector components.
__str__() : toString method
component(i : int): gets the i-th component (start by 0)
size() : gets the size of the vector (number of components)
euclidLength() : returns the eulidean length of the vector.
operator + : vector addition
operator - : vector subtraction
operator * : scalar multiplication and dot product
copy() : copies this vector and returns it.
changeComponent(pos,value) : changes the specified component.
TODO: compare-operator
"""
def __init__(self, components):
"""
input: components or nothing
simple constructor for init the vector
"""
self.__components = components
def set(self, components):
"""
input: new components
changes the components of the vector.
replace the components with newer one.
"""
if len(components) > 0:
self.__components = components
else:
raise Exception("please give any vector")
def __str__(self):
"""
returns a string representation of the vector
"""
ans = "("
length = len(self.__components)
for i in range(length):
if i != length - 1:
ans += str(self.__components[i]) + ","
else:
ans += str(self.__components[i]) + ")"
if len(ans) == 1:
ans += ")"
return ans
def component(self, i):
"""
input: index (start at 0)
output: the i-th component of the vector.
"""
if i < len(self.__components) and i >= 0:
return self.__components[i]
else:
raise Exception("index out of range")
def size(self):
"""
returns the size of the vector
"""
return len(self.__components)
def eulidLength(self):
"""
returns the eulidean length of the vector
"""
summe = 0
for c in self.__components:
summe += c ** 2
return math.sqrt(summe)
def __add__(self, other):
"""
input: other vector
assumes: other vector has the same size
returns a new vector that represents the sum.
"""
size = self.size()
result = []
if size == other.size():
for i in range(size):
result.append(self.__components[i] + other.component(i))
else:
raise Exception("must have the same size")
return Vector(result)
def __sub__(self, other):
"""
input: other vector
assumes: other vector has the same size
returns a new vector that represents the differenz.
"""
size = self.size()
result = []
if size == other.size():
for i in range(size):
result.append(self.__components[i] - other.component(i))
else: # error case
raise Exception("must have the same size")
return Vector(result)
def __mul__(self, other):
"""
mul implements the scalar multiplication
and the dot-product
"""
ans = []
if isinstance(other, float) or isinstance(other, int):
for c in self.__components:
ans.append(c * other)
elif isinstance(other, Vector) and (self.size() == other.size()):
size = self.size()
summe = 0
for i in range(size):
summe += self.__components[i] * other.component(i)
return summe
else: # error case
raise Exception("invalide operand!")
return Vector(ans)
def copy(self):
"""
copies this vector and returns it.
"""
components = [x for x in self.__components]
return Vector(components)
def changeComponent(self, pos, value):
"""
input: an index (pos) and a value
changes the specified component (pos) with the
'value'
"""
# precondition
assert pos >= 0 and pos < len(self.__components)
self.__components[pos] = value
def norm(self):
"""
normalizes this vector and returns it.
"""
eLength = self.eulidLength()
quotient = 1.0 / eLength
for i in range(len(self.__components)):
self.__components[i] = self.__components[i] * quotient
return self
def __eq__(self, other):
"""
returns true if the vectors are equal otherwise false.
"""
ans = True
SIZE = self.size()
if SIZE == other.size():
for i in range(SIZE):
if self.__components[i] != other.component(i):
ans = False
break
else:
ans = False
return ans
def zeroVector(dimension):
"""
returns a zero-vector of size 'dimension'
"""
# precondition
assert isinstance(dimension, int)
ans = []
for i in range(dimension):
ans.append(0)
return Vector(ans)
def unitBasisVector(dimension, pos):
"""
returns a unit basis vector with a One
at index 'pos' (indexing at 0)
"""
# precondition
assert isinstance(dimension, int) and (isinstance(pos, int))
ans = []
for i in range(dimension):
if i != pos:
ans.append(0)
else:
ans.append(1)
return Vector(ans)
def axpy(scalar, x, y):
"""
input: a 'scalar' and two vectors 'x' and 'y'
output: a vector
computes the axpy operation
"""
# precondition
assert (
isinstance(x, Vector)
and (isinstance(y, Vector))
and (isinstance(scalar, int) or isinstance(scalar, float))
)
return x * scalar + y
def randomVector(N, a, b):
"""
input: size (N) of the vector.
random range (a,b)
output: returns a random vector of size N, with
random integer components between 'a' and 'b'.
"""
ans = zeroVector(N)
random.seed(None)
for i in range(N):
ans.changeComponent(i, random.randint(a, b))
return ans
class Matrix(object):
"""
class: Matrix
This class represents a arbitrary matrix.
Overview about the methods:
__str__() : returns a string representation
operator * : implements the matrix vector multiplication
implements the matrix-scalar multiplication.
changeComponent(x,y,value) : changes the specified component.
component(x,y) : returns the specified component.
width() : returns the width of the matrix
height() : returns the height of the matrix
operator + : implements the matrix-addition.
operator - _ implements the matrix-subtraction
"""
def __init__(self, matrix, w, h):
"""
simple constructor for initialzes
the matrix with components.
"""
self.__matrix = matrix
self.__width = w
self.__height = h
def __str__(self):
"""
returns a string representation of this
matrix.
"""
ans = ""
for i in range(self.__height):
ans += "|"
for j in range(self.__width):
if j < self.__width - 1:
ans += str(self.__matrix[i][j]) + ","
else:
ans += str(self.__matrix[i][j]) + "|\n"
return ans
def changeComponent(self, x, y, value):
"""
changes the x-y component of this matrix
"""
if x >= 0 and x < self.__height and y >= 0 and y < self.__width:
self.__matrix[x][y] = value
else:
raise Exception("changeComponent: indices out of bounds")
def component(self, x, y):
"""
returns the specified (x,y) component
"""
if x >= 0 and x < self.__height and y >= 0 and y < self.__width:
return self.__matrix[x][y]
else:
raise Exception("changeComponent: indices out of bounds")
def width(self):
"""
getter for the width
"""
return self.__width
def height(self):
"""
getter for the height
"""
return self.__height
def __mul__(self, other):
"""
implements the matrix-vector multiplication.
implements the matrix-scalar multiplication
"""
if isinstance(other, Vector): # vector-matrix
if other.size() == self.__width:
ans = zeroVector(self.__height)
for i in range(self.__height):
summe = 0
for j in range(self.__width):
summe += other.component(j) * self.__matrix[i][j]
ans.changeComponent(i, summe)
summe = 0
return ans
else:
raise Exception(
"vector must have the same size as the "
+ "number of columns of the matrix!"
)
elif isinstance(other, int) or isinstance(other, float): # matrix-scalar
matrix = []
for i in range(self.__height):
row = []
for j in range(self.__width):
row.append(self.__matrix[i][j] * other)
matrix.append(row)
return Matrix(matrix, self.__width, self.__height)
def __add__(self, other):
"""
implements the matrix-addition.
"""
if self.__width == other.width() and self.__height == other.height():
matrix = []
for i in range(self.__height):
row = []
for j in range(self.__width):
row.append(self.__matrix[i][j] + other.component(i, j))
matrix.append(row)
return Matrix(matrix, self.__width, self.__height)
else:
raise Exception("matrix must have the same dimension!")
def __sub__(self, other):
"""
implements the matrix-subtraction.
"""
if self.__width == other.width() and self.__height == other.height():
matrix = []
for i in range(self.__height):
row = []
for j in range(self.__width):
row.append(self.__matrix[i][j] - other.component(i, j))
matrix.append(row)
return Matrix(matrix, self.__width, self.__height)
else:
raise Exception("matrix must have the same dimension!")
def __eq__(self, other):
"""
returns true if the matrices are equal otherwise false.
"""
ans = True
if self.__width == other.width() and self.__height == other.height():
for i in range(self.__height):
for j in range(self.__width):
if self.__matrix[i][j] != other.component(i, j):
ans = False
break
else:
ans = False
return ans
def squareZeroMatrix(N):
"""
returns a square zero-matrix of dimension NxN
"""
ans = []
for i in range(N):
row = []
for j in range(N):
row.append(0)
ans.append(row)
return Matrix(ans, N, N)
def randomMatrix(W, H, a, b):
"""
returns a random matrix WxH with integer components
between 'a' and 'b'
"""
matrix = []
random.seed(None)
for i in range(H):
row = []
for j in range(W):
row.append(random.randint(a, b))
matrix.append(row)
return Matrix(matrix, W, H)
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