quatica.data_gen
quatica.data_gen
create_sparse_quat_matrix(m, n, density=0.1)
Create a random sparse quaternion matrix of size m×n using CSR format.
Generates a sparse quaternion matrix with random entries distributed according to the specified density. Each quaternion component (real, i, j, k) is stored as a separate CSR sparse matrix for efficient computation.
Parameters:
m : int Number of rows in the matrix n : int Number of columns in the matrix density : float, optional Density of non-zero elements (between 0 and 1) (default: 0.1)
Returns:
SparseQuaternionMatrix A sparse quaternion matrix with random entries stored in CSR format
Notes:
The sparse format is particularly efficient for large matrices with low density. Each quaternion component is stored separately to optimize sparse matrix operations.
create_test_matrix(m, n, rank=None, cond_number=None)
Generate a random dense quaternion matrix with optional rank and conditioning.
Creates a quaternion matrix by multiplying two factor matrices A (m×rank) and B (rank×n). This ensures the resulting matrix has exactly the specified rank. Optional conditioning can be applied to control the singular value distribution.
Parameters:
m : int Number of rows in the output matrix n : int Number of columns in the output matrix rank : int, optional Desired rank of the matrix (default: min(m, n)) cond_number : float, optional Condition number for the matrix (default: None, no conditioning applied)
Returns:
np.ndarray An m×n quaternion matrix with the specified rank and conditioning
Notes:
The matrix is generated as A @ B where A is m×rank and B is rank×n. When cond_number is specified, singular values are distributed logarithmically from 1 to 1/cond_number to achieve the desired conditioning.
generate_random_unitary_matrix(n)
Generate a random unitary quaternion matrix of size n×n.
This function creates a random unitary matrix by: 1. Generating a random n×n quaternion matrix 2. Computing its QR decomposition 3. Returning the Q matrix, which is guaranteed to be unitary
Parameters:
n : int Size of the square unitary matrix to generate
Returns:
np.ndarray An n×n unitary quaternion matrix Q satisfying Q^H * Q = I
Notes:
The generated matrix is truly unitary (orthogonal in quaternion space) and can be used for various applications including: - Random rotations in 4D space - Unitary transformations - Orthogonal basis generation - Testing unitary matrix algorithms
small_test_Mat()
Create a validation quaternion matrix with known theoretical properties.
This function returns a specific 2×3 quaternion matrix that serves as a validation test for our pseudoinverse implementation. The matrix has known theoretical properties that allow us to verify the correctness of our numerical algorithms.
The matrix corresponds to Example 5.2 from the reference paper:
A = [1 i+2k 3] [i 6+j 7]
The theoretical pseudoinverse A^† (computed using Maple package) is: ⎛ -47/347 + 21/694 i + 11/694 j - 21/694 k 63/347 - 28/347 i + 21/694 j - 101/694 k ⎞ ⎜ -11/694 - 347 i - 11/694 k 61/694 + 21/694 i - 6/347 j + 21/347 k ⎟ ⎝ 57/347 + 49/694 i + 77/694 k 21/347 - 21/694 i - 33/694 k ⎠
Returns:
np.ndarray A 2×3 quaternion matrix for validation testing
Notes:
This matrix is designed to test the accuracy and convergence properties of quaternion pseudoinverse algorithms. It should converge quickly and achieve high numerical accuracy, with results closely matching the theoretical pseudoinverse from the paper.
References:
[1] Huang, L., Wang, Q.-W., & Zhang, Y. (2015). The Moore–Penrose inverses of matrices over quaternion polynomial rings. Linear Algebra and its Applications, 475, 45-61. https://doi.org/10.1016/j.laa.2015.02.004
Example 5.2 provides the exact theoretical pseudoinverse for comparison.
theoretical_pseudoinverse_example_5_2()
Compute the theoretical pseudoinverse from Example 5.2 of the reference paper.
This function returns the exact theoretical pseudoinverse A^† for the matrix A = [1 i+2k 3; i 6+j 7] as computed using the Maple package in the paper.
Returns:
np.ndarray The 3×2 theoretical pseudoinverse matrix
Notes:
This provides the exact analytical solution for comparison with our numerical implementation. The values are computed using the exact fractions from the paper.
References:
[1] Huang, L., Wang, Q.-W., & Zhang, Y. (2015). The Moore–Penrose inverses of matrices over quaternion polynomial rings. Linear Algebra and its Applications, 475, 45-61. https://doi.org/10.1016/j.laa.2015.02.004