lu_factor

detkit.lu_factor(A, shape=None, overwrite=False)

LU decomposition of a sub-matrix.

Parameters:

Anumpy.ndarray

A generic rectangular matrix with either float64 or float32 data type stored with either row-major or column major ordering.

shapetuple, default=None

A tuple of size two, determining the shape of an upper-left sub-matrix of A to be referenced. Namely, if shape is given as the tuple (m, n), the sub-matrix A[:m, :n] is considered. If None, the full shape of A[:, :] is used.

overwritebool, default=False

If True, the input matrix A will be overwritten as the output, allowing to save memory. In this case, the output matrix lu will point to the same location in the memory as the input matrix A.

Note

When overwrite is set to True, the matrix A should have column-major (Fortran) ordering.

Returns:

lunumpy.ndarray

A 2D matrix of the same shape as the input matrix A (and not the shape of the sub-matrix). The upper-left sub-matrix of lu contains the LU factor of the sub-matrix of A. Namely, the upper and lower triangular parts of the sub-matrix of lu consist of both matrices L and U. The diagonals of L (which consists of 1) are not stored (see notes for details). If overwrite is set to True, the output matrix lu is becomes a view for the matrix A.

permnumpy.array

Permutation indices of row pivoting (zero-based indexing). This array is of the size A.shape[0], regardless of the given shape of the sub-matrix.

See also

detkit.ldl_factor

detkit.cho_factor

detkit.lu_solve

Notes

LU decomposition of sub-matrix:

Let \(\mathbf{A}_{[:p,:q]}\) denote the sub-matrix of the size \(p \times q\) to be the upper-left corner of matrix \(\mathbf{A}\). Its LU decomposition is given by:

\[\mathbf{A}_{[:p, :q]} = \mathbf{P} \mathbf{L} \mathbf{U},\]

where \(\mathbf{P}\) is a permutation matrix of the size \(p \times p\), \(\mathbf{L}\) is a lower-triangular matrix of the size \(p \times k\) with diagonals \(1\), where \(k = \min(p, q)\), and \(\mathbf{U}\) is an upper-triangular matrix of the size \(k \times q\).

Interpreting L and U factors from the output lu:

Note that the output matrix lu is given as the same shape as A, and only its upper-left sub-matrix should be used to extract the matrices \(\mathbf{L}\) and \(\mathbf{U}\) as follows.

Let p, q = shape be the given shape of the upper-left sub-matrix, which may be smaller than the shape of A. Also, let k = min(p, q). Then, L = lu[:p, :k] (excluding the diagonals) and U = lu[:k, :q].

The diagonals of L are not stored in the above matrix, and you should assume the diagonals of L are 1 (see example below to adjust the diagonals of L to one manually).

Interpreting permutation array:

The output perm is a 1D array of the size of the number of rows of the input matrix A (and not the number of rows of the sub-matrix). The array A[perm, :] is equivalent of \(\mathbf{P}^{\intercal} \mathbf{A}\). As such, the relation relation \(\mathbf{P}^{\intercal} \mathbf{A} = \mathbf{L} \mathbf{U}\) can be achieved by A[perm[:p], :] = L @ U.

Comparison with scipy.linalg.lu_factor:

To compute the LU factorization of a sub-matrix of the input matrix using scipy.linalg.lu_factor function, you should pass a slice of the matrix to the function. This approach is not memory-efficient since the sliced array allocates new memory.

In contrast, using detkit.lu_factor together with the shape argument, no memory slice is created during the inner computation, rather, the data from the original input matrix is accessed efficiently.

Implementation:

This function is a wrapper around LAPACK’s sgetrf (for 32-bit precision) and dgetrf (for 64-bit precision).

This function is internally used for detkit.memdet() for efficient computation of matrix determinant under memory constraint.

References

• LAPACK

Examples

>>> from detkit import lu_factor, Memory
>>> import numpy

>>> # Create a matrix with 32-bit precision and column-major ordering
>>> A = numpy.random.randn(1000, 900) + 100 * numpy.eye(1000, 900)
>>> A = numpy.asfortranarray(A)
>>> A = A.astype(numpy.float32)

>>> # Get a copy of A (for the purpose of comparison) since we will
>>> # overwrite A
>>> A_copy = numpy.copy(A)

>>> # Track memory allocation to check if LU decomposition is not
>>> # creating any new memory.
>>> mem = Memory()
>>> mem.set()

>>> # LU factorization of the upper-left sub-matrix of smaller shape
>>> p, q = 800, 700
>>> lu, perm = lu_factor(A, shape=(p, q), overwrite=True)

>>> # Check peak memory allocation (compared to memory of a sum-matrix)
>>> slice_nbytes = p * q * A.dtype.itemsize
>>> print(mem.peak() / slice_nbytes)
0.001

>>> # Extract L and U factors from lu matrix
>>> k = min(p, q)
>>> U = numpy.triu(lu[:k, :q])
>>> L = numpy.tril(lu[:p, :k], -1)
>>> for i in range(k):
...    L[i, i] = 1.0

>>> # Check if A_copy = PLU holds.
>>> atol = numpy.finfo(A.dtype).resolution
>>> print(numpy.allclose(A_copy[perm[:p], :q], L @ U, atol=10*atol))
True

>>> # When overwrite is set to True, check if lu is indeed a view of A
>>> print(lu.base == A.base)
True

In the above example, the object mem of class detkit.Memory tracks memory allocation. The peak of allocated memory during the matrix multiplication is three orders of magnitude smaller than the size of one of the matrices slices, confirming that no new array slice was created during the operation.