def _sqrtm_triu(T):
"""
Implements Björck, Å., & Hammarling, S. (1983).
"A Schur method for the square root of a matrix". Linear algebra and
its applications", 52, 127-140.
"""
diag = jnp.sqrt(jnp.diag(T))
n = diag.size
U = jnp.diag(diag)
def i_loop(l, data):
j, U = data
i = j - 1 - l
s = lax.fori_loop(i + 1, j, lambda k, val: val + U[i, k] * U[k, j],
0.0)
value = jnp.where(T[i, j] == s, 0.0,
(T[i, j] - s) / (diag[i] + diag[j]))
return j, U.at[i, j].set(value)
def j_loop(j, U):
_, U = lax.fori_loop(0, j, i_loop, (j, U))
return U
U = lax.fori_loop(0, n, j_loop, U)
return U
def funm(A, func, disp=True):
A = jnp.asarray(A)
if A.ndim != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected square array_like input')
T, Z = schur(A)
T, Z = rsf2csf(T, Z)
F = jnp.diag(func(jnp.diag(T)))
F = F.astype(T.dtype.char)
F, minden = _algorithm_11_1_1(F, T)
F = Z @ F @ Z.conj().T
if disp:
return F
if F.dtype.char.lower() == 'e':
tol = jnp.finfo(jnp.float16).eps
if F.dtype.char.lower() == 'f':
tol = jnp.finfo(jnp.float32).eps
else:
tol = jnp.finfo(jnp.float64).eps
minden = jnp.where(minden == 0.0, tol, minden)
err = jnp.where(jnp.any(jnp.isinf(F)), jnp.inf, jnp.minimum(1, jnp.maximum(
tol, (tol / minden) * norm(jnp.triu(T, 1), 1))))
return F, err
def _roots_no_zeros(p):
# build companion matrix and find its eigenvalues (the roots)
if p.size < 2:
return array([], dtype=dtypes._to_complex_dtype(p.dtype))
A = diag(ones((p.size - 2, ), p.dtype), -1)
A = A.at[0, :].set(-p[1:] / p[0])
return linalg.eigvals(A)
def recursive_case(B, offset, b, agenda, blocks, eigenvectors):
# The recursive case of the algorithm, specialized to a static block size
# of B.
H = _slice(blocks, (offset, 0), (b, b), (B, B))
V = _slice(eigenvectors, (0, offset), (n, b), (N, B))
split_point = jnp.nanmedian(
_mask(jnp.diag(jnp.real(H)), (b, ),
jnp.nan)) # TODO: Improve this?
H_minus, V_minus, H_plus, V_plus, rank = split_spectrum(H,
b,
split_point,
V0=V)
blocks = _update_slice(blocks, H_minus, (offset, 0), (rank, rank))
blocks = _update_slice(blocks, H_plus, (offset + rank, 0),
(b - rank, b - rank))
eigenvectors = _update_slice(eigenvectors, V_minus, (0, offset),
(n, rank))
eigenvectors = _update_slice(eigenvectors, V_plus, (0, offset + rank),
(n, b - rank))
agenda = agenda.push(_Subproblem(offset + rank, (b - rank)))
agenda = agenda.push(_Subproblem(offset, rank))
return agenda, blocks, eigenvectors
def _roots_no_zeros(p):
# assume: p does not have leading zeros and has length > 1
p, = _promote_dtypes_inexact(p)
# build companion matrix and find its eigenvalues (the roots)
A = diag(ones((p.size - 2, ), p.dtype), -1)
A = A.at[0, :].set(-p[1:] / p[0])
roots = linalg.eigvals(A)
return roots
