The eqs package implements numeric methods to solve systems of equations. This package also defines the Matrix and Vector classes to define the systems of equations.
Contents:
Matrices are created using the constructor, which takes as parameters the number of rows and columns:
from eqs.matrix import Matrix
matrix = Matrix(2, 3)A newly created Matrix is initialized full of zeroes:
⎡ 0 0 0 ⎤
⎣ 0 0 0 ⎦
To set values in a Matrix, you can set all of them at once (make sure the size of the list matches the total number of items in the matrix):
from eqs.matrix import Matrix
matrix = Matrix(2, 3).set_data([1, 2, 3, 4, 5, 6])which defines the matrix:
⎡ 1 2 3 ⎤
⎣ 4 5 6 ⎦
Values can also be set individually:
from eqs.matrix import Matrix
matrix = Matrix(2, 3).set_value(50, 1, 2)which defines the matrix:
⎡ 0 0 0 ⎤
⎣ 0 0 50 ⎦
To get a value from a matrix you can use the value_at method:
matrix.value_at(1, 2)All the methods that modify the matrix (set_value, set_identity_row, scale ...) return the matrix, so that several modifications can be chained like this:
from eqs.matrix import Matrix
matrix = Matrix(2, 3) \
.set_value(50, 1, 2) \
.set_value(25, 2, 2)Matrices can be added and subtracted. For example, to add the matrices:
⎡ 1 2 ⎤ ⎡ 3 4 ⎤ ⎡ 4 6 ⎤
⎜ 3 4 ⎟ + ⎜ 5 6 ⎟ = ⎜ 8 10 ⎟
⎣ 4 5 ⎦ ⎣ 7 8 ⎦ ⎣ 12 14 ⎦
we can do:
from eqs.matrix import Matrix
mat_a = Matrix(3, 2).set_data([1, 2, 3, 4, 5, 6])
mat_b = Matrix(3, 2).set_data([3, 4, 5, 6, 7, 8])
result = mat_a + mat_bwhich yields the matrix:
⎡ 4 6 ⎤
⎜ 8 10 ⎟
⎣ 12 14 ⎦
To subtract the matrices:
⎡ 1 2 ⎤ ⎡ 3 4 ⎤ ⎡ -2 -2 ⎤
⎜ 3 4 ⎟ - ⎜ 5 6 ⎟ = ⎜ -2 -2 ⎟
⎣ 4 5 ⎦ ⎣ 7 8 ⎦ ⎣ -2 -2 ⎦
we can, using the matrices defined above:
result = mat_a - mat_bwhich yields the matrix:
⎡ -2 -2 ⎤
⎜ -2 -2 ⎟
⎣ -2 -2 ⎦
Matrices can also be multiplied. For example:
⎡ 1 2 ⎤ ⎡ 15 18 21 ⎤
⎜ 3 4 ⎟ * ⎡ 1 2 3 ⎤ = ⎜ 33 40 47 ⎟
⎣ 4 5 ⎦ ⎣ 4 5 6 ⎦ ⎣ 51 62 73 ⎦
can be multiplied:
from eqs.matrix import Matrix
mat_a = Matrix(3, 2).set_data([1, 2, 3, 4, 5, 6])
mat_b = Matrix(2, 3).set_data([3, 4, 5, 6, 7, 8])
result = mat_a * mat_bwhich yields the matrix:
⎡ 15 18 21 ⎤
⎜ 33 40 47 ⎟
⎣ 51 62 73 ⎦
A matrix can also be multiplied with a vector. For example:
⎡1 2 3⎤ ⎧1⎫ ⎧14⎫
⎢4 5 6⎥ ⎨2⎬ = ⎨32⎬
⎣7 8 9⎦ ⎩3⎭ ⎩50⎭
can be done:
from eqs.matrix import Matrix, Vector
mat = Matrix(3, 3).set_data([1, 2, 3, 4, 5, 6, 7, 8, 9])
vec = Vector(3).set_data([1, 2, 3])
result = mat.times_vector(vec)which results in the vector:
⎧14⎫
⎨32⎬
⎩50⎭
Vectors are created passing the Vector class constructor a size:
from eqs.vector import Vector
vec = Vector(3)A newly created Vector is initialized with all zeroes:
⎧0⎫
⎨0⎬
⎩0⎭
We can set the values of a vector:
from eqs.vector import Vector
vec = Vector(3).set_data([1, 2, 3])which would result in:
⎧1⎫
⎨2⎬
⎩3⎭
We can add to a given value in the vector:
from eqs.vector import Vector
vec = Vector(3).set_data([1, 2, 3])
vec.add_to_value(10, 2)results in:
⎧1 ⎫
⎨2 ⎬
⎩13⎭
We can use the value_at method to get a value:
from eqs.vector import Vector
vec = Vector(3).set_data([1, 2, 3])
vec.value_at(2) # returns 3Vectors can be added, subtracted an multiplied:
from eqs.vector import Vector
vec_a = Vector(2).set_data([1, 2])
vec_b = Vector(2).set_data([4, 3])
addition = vec_a + vec_b
subtraction = vec_a - vec_b
multiplication = vec_a * vec_bThe Cholesky factorization is a direct numerical method that can be used to solve systems of linear equations whose matrix is positive-definite. We can use it to solve systems of equations like the following:
⎡4 -2 4⎤ ⎧x1⎫ ⎧ 0 ⎫
⎢-2 10 -2⎥ ⎨x2⎬ = ⎨-3 ⎬
⎣4 -2 8⎦ ⎩x3⎭ ⎩-15⎭
in our code:
from eqs.vector import Vector
from eqs.matrix import Matrix
from eqs.cholesky import cholesky_solve
mat = Matrix(3, 3).set_data((4, -2, 4, -2, 10, -2, 4, -2, 8))
vec = Vector(3).set_data((0, -3, -15))
solution = cholesky_solve(mat, vec)here, solution is the vector:
⎧ 3.583⎫
⎨-0.333⎬
⎩-3.75 ⎭
The Cholesky's lower matrix decomposition can also be obtained. The Cholesky numerical method computes this lower triangular matrix as part of its process:
from eqs.matrix import Matrix
from eqs.cholesky import lower_matrix_decomposition
mat = Matrix(3, 3).set_data((4, -2, 4, -2, 10, -2, 4, -2, 8))
lower_triangular = lower_matrix_decomposition(mat)which yields the matrix:
⎡ 2 0 0⎤
⎢-1 3 0⎥
⎣ 2 0 2⎦
The conjugate gradient method is a iterative numeric method to solve systems of linear equations. Let's use it to solve the system:
⎡4 -2 4⎤ ⎧x1⎫ ⎧ 0 ⎫
⎢-2 10 -2⎥ ⎨x2⎬ = ⎨-3 ⎬
⎣4 -2 8⎦ ⎩x3⎭ ⎩-15⎭
in our code:
from eqs.vector import Vector
from eqs.matrix import Matrix
from eqs.conjugate_gradient import conjugate_gradient_solve
mat = Matrix(3, 3).set_data((4, -2, 4, -2, 10, -2, 4, -2, 8))
vec = Vector(3).set_data((0, -3, -15))
solution = conjugate_gradient_solve(
sys_mat=mat,
sys_vec=vec,
max_iter=100,
max_error=1E-5
)here, solution is the vector:
⎧ 3.583⎫
⎨-0.333⎬
⎩-3.75 ⎭