Smendowski / matrix-algorithms

Implementation of selected matrix algorithms i.e. Strassen, Binet, block matrix inversion and block LU factorization with recursive approach.

1. Matrix Multiplication

Implementation of matrix muliplication algorithms according to the following description: For matrices smaller than or equal to 2^l × 2^l use traditional algorithm. For matrices larger than 2^l × 2^l use Strassen recursive algorithm.

k = log2(matrix_order)

if k <= l:
    return traditional_algorithm(A, B)
elif k > l:
    return strassen_recursive_algorithm(A, B)

2. Matrix Inversion

Implementation of matrix inversion algorithm with block and recursive approach.

3. Matrix LU Factorization

Implementation of matrix LU factorization algorithm with block and recursive approach.

4. Usage

1. Command Line Interface

$ python main.py --help

(...)

options:
  -h, --help
  -m, --multiplication
  -i, --inversion
  -f, --factorization

2. Configuration file

FLOATING_POINT_OPERATIONS: int = 0
MEASURE_MULTIPLICATION_TIME = False
MEASURE_INVERSION_TIME = False
MEASURE_FACTORIZATION_TIME = False

3. Define matrix/matrices and threshold l

Recursive inversion and LU factorization use matrix multiplication. Specification on threshold l allow to choose the matrix multiplication method, according to the logic in the Matrix Multiplication section.
Functions run_multiplication_logic, run_inversion_logic and run_factorization_logic allow to specify aforementioned threshold and declare matrix/matrices to operate on. Example:

def run_multiplication_logic():
    matrix_mul_thr = 3
    A = np.array([
        [1, 2, 3, 4], [1, 2, 3, 4],
        [1, 2, 3, 4], [1, 2, 3, 4]])
    B = np.array([
        [1, 2, 3, 4], [1, 2, 3, 4],
        [1, 2, 3, 4], [1, 2, 3, 4]])

(...)

4. Sample Output

• Multiplication

Input matrix A:
 [1 2 3 4]
 [1 2 3 4]
 [1 2 3 4]
 [1 2 3 4]
Input matrix B:
 [1 2 3 4]
 [1 2 3 4]
 [1 2 3 4]
 [1 2 3 4]
A*B Multiplication Result: 
 [10 20 30 40]
 [10 20 30 40]
 [10 20 30 40]
 [10 20 30 40]
Result Shape: (4, 4))
FlOps: 80

• Inversion

Input matrix A:
 [6 8 1 1]
 [2 6 8 4]
 [1 1 5 9]
 [8 3 5 4]
Inversion Result:
 [ 0.   -0.07 -0.03  0.14]
 [ 0.13  0.04  0.02 -0.11]
 [-0.13  0.15 -0.08  0.07]
 [ 0.06 -0.08  0.16 -0.04]
Result Shape: (4, 4))
FlOps: 156

• LU Factorization

Input matrix A:
 [5 2 3 9]
 [2 4 6 8]
 [7 4 8 4]
 [3 4 5 5]
L Result:
 [ 1.    0.    0.    0.  ]
 [ 0.4   1.    0.    0.  ]
 [ 1.4   0.38  1.    0.  ]
 [ 0.6   0.88 -0.5   1.  ]
L Shape: (4, 4)
U Result:
 [  5.     2.     3.     9.  ]
 [  0.     3.2    4.8    4.4 ]
 [  0.     0.     2.   -10.25]
 [  0.     0.     0.    -9.38]
U Shape: (4, 4)
FlOps: 96
Determinant: -300.00

Each output is associated with the number of floating point operations (FlOps). Furthermore, to enrich results with execution time, provide necessary changes in config.py. Example

MEASURE_MULTIPLICATION_TIME = True

5. Vital conclusion

In the case of the Strassen algorithm, the theoretical computational complexity is significantly less than that of the traditional method. Due to the non-zero time of data transfer between the memory and the processor cache, the profit from the use of the algorithm in practice is visible only for large matrices. The analyzed algorithms are topping examples of confronting theoretical assumptions with their actual implementation.