def gram_schmidt(
a: MatrixType, n: Optional[int] = None
) -> Tuple[MatrixType, List[int]]:
cnt_vectors = min(a.ncols, n) if n else a.ncols
q = a.copy()
perm = list(range(cnt_vectors))
for j in range(0, cnt_vectors):
pivot_col_idx, pivot_norm = max(
((c, norm(q[:, c])) for c in range(j, cnt_vectors)), key=lambda x: x[1]
)
# Swap columns
if j != pivot_col_idx:
temp = q[:, j].copy()
q[:, j] = q[:, pivot_col_idx]
q[:, pivot_col_idx] = temp
perm[pivot_col_idx], perm[j] = perm[j], perm[pivot_col_idx]
# Normalize
q[:, j] /= pivot_norm
u = q[:, j]
# Remove U component everywhere
q[:, j + 1 :] -= u @ (u.T @ q[:, j + 1 :])
return q, perm
def lu_decompose(a: MatrixType) -> LuDecomposition:
validate.is_true("matrix", "must be square", a.is_square())
n = a.nrows
# Init
lu = a.copy()
perm: List[int] = list(range(a.nrows))
cnt_row_exchanges = 0
for i in range(n):
# All work will be done on sub-matrix lu[i:, i:]
pivot_row_idx, pivot_value = max(
enumerate(lu[i:, i], start=i), key=lambda x: abs(x[1])
)
if abs(pivot_value) < epsilon:
raise ValueError(
"Matrix has not independent rows (or columns) "
"and hence cannot be LU decomposed"
)
if pivot_row_idx != i:
# Execute row exchange on perm
temp_idx = perm[pivot_row_idx]
perm[pivot_row_idx] = perm[i]
perm[i] = temp_idx
# Execute Row exchange on LU
temp_row = lu[pivot_row_idx].copy()
lu[pivot_row_idx] = lu[i]
lu[i] = temp_row
cnt_row_exchanges += 1
# Perform elimination
factors_column = lu[i + 1 :, i] / pivot_value
pivot_row = lu[i, i + 1 :]
lu[i + 1 :, i] = factors_column
lu[i + 1 :, i + 1 :] -= factors_column * pivot_row
return LuDecomposition(lu, perm, cnt_row_exchanges)
def echelon(a: MatrixType) -> EchelonForm:
m, n = a.shape
# Init
lu = a.copy()
perm: List[int] = list(range(m))
cnt_row_exchanges = 0
j = 0
i = 0
pivot_cols = []
while i < m and j < n:
# All work will be done on sub-matrix lu[i:, j:]
pivot_row_idx, pivot_value = max(
enumerate(lu[i:, j], start=i), key=lambda x: abs(x[1])
)
if abs(pivot_value) < epsilon:
j += 1
continue
if pivot_row_idx != i:
# Execute row exchange on perm
temp_idx = perm[pivot_row_idx]
perm[pivot_row_idx] = perm[i]
perm[i] = temp_idx
# Execute Row exchange on LU
temp_row = lu[pivot_row_idx].copy()
lu[pivot_row_idx] = lu[i]
lu[i] = temp_row
cnt_row_exchanges += 1
# Perform elimination
pivot_cols.append(j)
factors_column = lu[i + 1 :, j] / pivot_value
pivot_row = lu[i, j + 1 :]
lu[i + 1 :, j] = factors_column
lu[i + 1 :, j + 1 :] -= factors_column * pivot_row
i += 1
j += 1
return EchelonForm(a, lu, perm, pivot_cols)
def shaped(data: Collection[N], shape: Shape):
return MatrixType.from_flat_collection(data, shape)
def zeros(nrows: int = None, ncols: int = None, shape: Shape = None):
return MatrixType.zeros(_shape(nrows, ncols, shape))
def eye(nrows: int = None, ncols: int = None, shape: Shape = None):
return MatrixType.eye(_shape(nrows, ncols, shape))
def singleton(value: Number) -> MatrixType:
return MatrixType.singleton(value)
def det(m: MatrixType) -> float:
if not m.is_square():
raise ValueError("Determinants only make sense for square matrices")
lu = elimination.lu_decompose(m)
return lu.determinant()