🧮 pybandee ⚡️

⚠⚠⚠ Important Note ⚠⚠⚠

This package is currently only an experimental alpha version that is missing

• Python interfaces with input validation
• documentation
• tests that cover all possible edge cases
• citation of the original papers

⚠⚠⚠ Important Note ⚠⚠⚠

A Python package specialised in factorising and solving banded matrices. Currently, the package supports the following functionalities:

• pentadiagonal factorisation and solving
• computation of the log-determinant and inverse elements of a pentadiagonal matrix

☁️➡️📦 Installation

While this package can be installed and run without the optional dependency numba as

pip install pybandee

It is recommended to install the package together with numba because the low-level style code will be significantly faster. This can be done with

pip install pybandee["fast"]

5️⃣ Pentadiagonal Matrices

⚠ Currently, only the Numba low-level implementation is available ⚠
Please install the package with pip install pybandee["fast"] to get the performance benefits

For a well-conditioned pentadiagonal matrix consisting of

• 2 sub-diagonals,
• 1 main diagonal, and
• 2 super-diagonals

a simple $\mathbf{L}\mathbf{U}$ factorisation can be computed using the PTRANS-I algorithm. This can be solved efficiently and allows for the computation of additional properties such as the log-determinant and inverse elements.

For the numba low-level implementations, the pentadiagonal matrix

$$ \mathbf{A} = \begin{bmatrix} c_{0} & d_{0} & e_{0} & 0 & 0 & 0 & 0 & 0 \ b_{1} & c_{1} & d_{1} & e_{1} & 0 & 0 & 0 & 0 \ a_{2} & b_{2} & c_{2} & d_{2} & e_{2} & 0 & 0 & 0 \ 0 & a_{3} & b_{3} & c_{3} & d_{3} & e_{3} & 0 & 0 \ 0 & 0 & a_{4} & b_{4} & c_{4} & d_{4} & e_{4} & 0 \ 0 & 0 & 0 & a_{5} & b_{5} & c_{5} & d_{5} & e_{5} \ 0 & 0 & 0 & 0 & a_{6} & b_{6} & c_{6} & d_{6} \ 0 & 0 & 0 & 0 & 0 & a_{7} & b_{7} & c_{7} \ \end{bmatrix} $$

needs to be stored in the row-major banded storage format as

a = np.array(
    [   #  sub2   sub1   main   sup1   sup2
        [     *,     *,    c0,    d0,    e0     ],
        [     *,    b1,    c1,    d1,    e1     ],
        [    a2,    b2,    c2,    d2,    e2     ],
        [    a3,    b3,    c3,    d3,    e3     ],
        [    a4,    b4,    c4,    d4,    e4     ],
        [    a5,    b5,    c5,    d5,    e5     ],
        [    a6,    b6,    c6,    d6,     *     ],
        [    a7,    b7,    c7,     *,     *     ],
    ],
    order="C",  # ← this is important
)

where the entries marked with * are not used and can be any value.

A linear system can be solved like

 === Imports ===

import numpy as np

from pybandee.penta import numba as jit_penta

# === Setup ===

np.random.seed(0)

# a symmetric positive definite pentadiagonal matrix is created in row-major banded
# storage format
lhs_matrix = np.zeros(shape=(50, 5), dtype=np.float64)
column = np.random.rand(lhs_matrix.shape[0] - 2)
lhs_matrix[2:, 0] = column.copy()
lhs_matrix[0:-2, 4] = column.copy()
column = np.random.rand(lhs_matrix.shape[0] - 1)
lhs_matrix[1:, 1] = column.copy()
lhs_matrix[0:-1, 3] = column.copy()
column = np.random.rand(lhs_matrix.shape[0])
lhs_matrix[:, 2] = column.copy() + 2.0  # to ensure positive definiteness
lhs_matrix_original = lhs_matrix.copy()

rhs_vector = np.random.rand(lhs_matrix.shape[0])  # ← will be overwritten by the solve
rhs_vector_original = rhs_vector.copy()

# === Factorisation and Solve ===

info = jit_penta.ptrans1_factorize(matrix=lhs_matrix)
assert info == 0  # ← factorization successful

jit_penta.ptrans1_solve_single_rhs(
    factorization=lhs_matrix,  # ← is now the factorization
    rhs=rhs_vector,  # ← will become the solution
)

# === Comparison against Numpy ===

lhs_matrix_dense = np.zeros(
    shape=(lhs_matrix_original.shape[0], lhs_matrix_original.shape[0]),
    dtype=np.float64,
)
lhs_matrix_dense += np.diag(lhs_matrix_original[2::, 0], k=-2)
lhs_matrix_dense += np.diag(lhs_matrix_original[1::, 1], k=-1)
lhs_matrix_dense += np.diag(lhs_matrix_original[:, 2])
lhs_matrix_dense += np.diag(lhs_matrix_original[:-1, 3], k=1)
lhs_matrix_dense += np.diag(lhs_matrix_original[:-2, 4], k=2)

assert np.allclose(
    np.linalg.solve(lhs_matrix_dense, rhs_vector_original),
    rhs_vector,
)

Further properties can be computed such as

• the log-determinant of the matrix

# === Log-Determinant ===

sign, logabsdet = jit_penta.ptrans1_slogdet(factorization=lhs_matrix)
numpy_sign, numpy_logabsdet = np.linalg.slogdet(lhs_matrix_dense)

assert np.isclose(sign, numpy_sign)
assert np.isclose(logabsdet, numpy_logabsdet)

• the central pentadiagonal band of the inverse in the same layout as the original matrix (symmetric matrices only)

# === Central pentadiagonal band of the inverse ===

inverse_band = jit_penta.ptrans1_symmetric_inverse_central_penta_bands(
    factorization=lhs_matrix
)

numpy_inverse = np.linalg.inv(lhs_matrix_dense)
numpy_inverse_band = np.zeros_like(inverse_band)

numpy_inverse_band[2::, 0] = np.diag(numpy_inverse, k=-2)
numpy_inverse_band[1::, 1] = np.diag(numpy_inverse, k=-1)
numpy_inverse_band[:, 2] = np.diag(numpy_inverse)
numpy_inverse_band[:-1, 3] = np.diag(numpy_inverse, k=1)
numpy_inverse_band[:-2, 4] = np.diag(numpy_inverse, k=2)

assert np.allclose(inverse_band, numpy_inverse_band)

All of the functions can be invoked in a numba.jit-decorated function to get the maximum performance benefits ⚡️