Source code for jax._src.scipy.linalg

# Copyright 2018 Google LLC
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.


from functools import partial

import numpy as np
import scipy.linalg
import textwrap
import warnings

import jax
from jax import jit, vmap, jvp
from jax import lax
from jax._src import dtypes
from jax._src.lax import linalg as lax_linalg
from jax._src.lax import qdwh
from jax._src.numpy.util import _wraps, _promote_dtypes_inexact, _promote_dtypes_complex
from jax._src.numpy import lax_numpy as jnp
from jax._src.numpy import linalg as np_linalg

_T = lambda x: jnp.swapaxes(x, -1, -2)
_no_chkfinite_doc = textwrap.dedent("""
Does not support the Scipy argument ``check_finite=True``,
because compiled JAX code cannot perform checks of array values at runtime.
""")
_no_overwrite_and_chkfinite_doc = _no_chkfinite_doc + "\nDoes not support the Scipy argument ``overwrite_*=True``."

@partial(jit, static_argnames=('lower',))
def _cholesky(a, lower):
  a, = _promote_dtypes_inexact(jnp.asarray(a))
  l = lax_linalg.cholesky(a if lower else jnp.conj(_T(a)), symmetrize_input=False)
  return l if lower else jnp.conj(_T(l))

[docs]@_wraps(scipy.linalg.cholesky, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'check_finite')) def cholesky(a, lower=False, overwrite_a=False, check_finite=True): del overwrite_a, check_finite return _cholesky(a, lower)
[docs]@_wraps(scipy.linalg.cho_factor, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'check_finite')) def cho_factor(a, lower=False, overwrite_a=False, check_finite=True): return (cholesky(a, lower=lower), lower)
@partial(jit, static_argnames=('lower',)) def _cho_solve(c, b, lower): c, b = _promote_dtypes_inexact(jnp.asarray(c), jnp.asarray(b)) lax_linalg._check_solve_shapes(c, b) b = lax_linalg.triangular_solve(c, b, left_side=True, lower=lower, transpose_a=not lower, conjugate_a=not lower) b = lax_linalg.triangular_solve(c, b, left_side=True, lower=lower, transpose_a=lower, conjugate_a=lower) return b
[docs]@_wraps(scipy.linalg.cho_solve, update_doc=False, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_b', 'check_finite')) def cho_solve(c_and_lower, b, overwrite_b=False, check_finite=True): del overwrite_b, check_finite c, lower = c_and_lower return _cho_solve(c, b, lower)
@partial(jit, static_argnames=('full_matrices', 'compute_uv')) def _svd(a, *, full_matrices, compute_uv): a, = _promote_dtypes_inexact(jnp.asarray(a)) return lax_linalg.svd(a, full_matrices=full_matrices, compute_uv=compute_uv)
[docs]@_wraps(scipy.linalg.svd, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'check_finite', 'lapack_driver')) def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False, check_finite=True, lapack_driver='gesdd'): del overwrite_a, check_finite, lapack_driver return _svd(a, full_matrices=full_matrices, compute_uv=compute_uv)
[docs]@_wraps(scipy.linalg.det, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'check_finite')) def det(a, overwrite_a=False, check_finite=True): del overwrite_a, check_finite return np_linalg.det(a)
@partial(jit, static_argnames=('lower', 'eigvals_only', 'eigvals', 'type')) def _eigh(a, b, lower, eigvals_only, eigvals, type): if b is not None: raise NotImplementedError("Only the b=None case of eigh is implemented") if type != 1: raise NotImplementedError("Only the type=1 case of eigh is implemented.") if eigvals is not None: raise NotImplementedError( "Only the eigvals=None case of eigh is implemented.") a, = _promote_dtypes_inexact(jnp.asarray(a)) v, w = lax_linalg.eigh(a, lower=lower) if eigvals_only: return w else: return w, v
[docs]@_wraps(scipy.linalg.eigh, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'overwrite_b', 'check_finite')) def eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False, overwrite_b=False, turbo=True, eigvals=None, type=1, check_finite=True): del overwrite_a, overwrite_b, turbo, check_finite return _eigh(a, b, lower, eigvals_only, eigvals, type)
@partial(jit, static_argnames=('output',)) def _schur(a, output): if output == "complex": a = a.astype(dtypes._to_complex_dtype(a.dtype)) return lax_linalg.schur(a)
[docs]@_wraps(scipy.linalg.schur) def schur(a, output='real'): if output not in ('real', 'complex'): raise ValueError( f"Expected 'output' to be either 'real' or 'complex', got output={output}.") return _schur(a, output)
[docs]@_wraps(scipy.linalg.inv, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'check_finite')) def inv(a, overwrite_a=False, check_finite=True): del overwrite_a, check_finite return np_linalg.inv(a)
[docs]@_wraps(scipy.linalg.lu_factor, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'check_finite')) @partial(jit, static_argnames=('overwrite_a', 'check_finite')) def lu_factor(a, overwrite_a=False, check_finite=True): del overwrite_a, check_finite a, = _promote_dtypes_inexact(jnp.asarray(a)) lu, pivots, _ = lax_linalg.lu(a) return lu, pivots
[docs]@_wraps(scipy.linalg.lu_solve, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_b', 'check_finite')) @partial(jit, static_argnames=('trans', 'overwrite_b', 'check_finite')) def lu_solve(lu_and_piv, b, trans=0, overwrite_b=False, check_finite=True): del overwrite_b, check_finite lu, pivots = lu_and_piv m, n = lu.shape[-2:] perm = lax_linalg.lu_pivots_to_permutation(pivots, m) return lax_linalg.lu_solve(lu, perm, b, trans)
@partial(jit, static_argnums=(1,)) def _lu(a, permute_l): a, = _promote_dtypes_inexact(jnp.asarray(a)) lu, _, permutation = lax_linalg.lu(a) dtype = lax.dtype(a) m, n = jnp.shape(a) p = jnp.real(jnp.array(permutation[None, :] == jnp.arange(m, dtype=permutation.dtype)[:, None], dtype=dtype)) k = min(m, n) l = jnp.tril(lu, -1)[:, :k] + jnp.eye(m, k, dtype=dtype) u = jnp.triu(lu)[:k, :] if permute_l: return jnp.matmul(p, l), u else: return p, l, u
[docs]@_wraps(scipy.linalg.lu, update_doc=False, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'check_finite')) @partial(jit, static_argnames=('permute_l', 'overwrite_a', 'check_finite')) def lu(a, permute_l=False, overwrite_a=False, check_finite=True): del overwrite_a, check_finite return _lu(a, permute_l)
@partial(jit, static_argnames=('mode', 'pivoting')) def _qr(a, mode, pivoting): if pivoting: raise NotImplementedError( "The pivoting=True case of qr is not implemented.") if mode in ("full", "r"): full_matrices = True elif mode == "economic": full_matrices = False else: raise ValueError(f"Unsupported QR decomposition mode '{mode}'") a, = _promote_dtypes_inexact(jnp.asarray(a)) q, r = lax_linalg.qr(a, full_matrices=full_matrices) if mode == "r": return (r,) return q, r
[docs]@_wraps(scipy.linalg.qr, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'check_finite')) def qr(a, overwrite_a=False, lwork=None, mode="full", pivoting=False, check_finite=True): del overwrite_a, lwork, check_finite return _qr(a, mode, pivoting)
@partial(jit, static_argnames=('sym_pos', 'lower')) def _solve(a, b, sym_pos, lower): if not sym_pos: return np_linalg.solve(a, b) a, b = _promote_dtypes_inexact(jnp.asarray(a), jnp.asarray(b)) lax_linalg._check_solve_shapes(a, b) # With custom_linear_solve, we can reuse the same factorization when # computing sensitivities. This is considerably faster. factors = cho_factor(lax.stop_gradient(a), lower=lower) custom_solve = partial( lax.custom_linear_solve, lambda x: lax_linalg._matvec_multiply(a, x), solve=lambda _, x: cho_solve(factors, x), symmetric=True) if a.ndim == b.ndim + 1: # b.shape == [..., m] return custom_solve(b) else: # b.shape == [..., m, k] return vmap(custom_solve, b.ndim - 1, max(a.ndim, b.ndim) - 1)(b)
[docs]@_wraps(scipy.linalg.solve, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_a', 'overwrite_b', 'debug', 'check_finite')) def solve(a, b, sym_pos=False, lower=False, overwrite_a=False, overwrite_b=False, debug=False, check_finite=True): del overwrite_a, overwrite_b, debug, check_finite return _solve(a, b, sym_pos, lower)
@partial(jit, static_argnames=('trans', 'lower', 'unit_diagonal')) def _solve_triangular(a, b, trans, lower, unit_diagonal): if trans == 0 or trans == "N": transpose_a, conjugate_a = False, False elif trans == 1 or trans == "T": transpose_a, conjugate_a = True, False elif trans == 2 or trans == "C": transpose_a, conjugate_a = True, True else: raise ValueError(f"Invalid 'trans' value {trans}") a, b = _promote_dtypes_inexact(jnp.asarray(a), jnp.asarray(b)) # lax_linalg.triangular_solve only supports matrix 'b's at the moment. b_is_vector = jnp.ndim(a) == jnp.ndim(b) + 1 if b_is_vector: b = b[..., None] out = lax_linalg.triangular_solve(a, b, left_side=True, lower=lower, transpose_a=transpose_a, conjugate_a=conjugate_a, unit_diagonal=unit_diagonal) if b_is_vector: return out[..., 0] else: return out
[docs]@_wraps(scipy.linalg.solve_triangular, lax_description=_no_overwrite_and_chkfinite_doc, skip_params=('overwrite_b', 'debug', 'check_finite')) def solve_triangular(a, b, trans=0, lower=False, unit_diagonal=False, overwrite_b=False, debug=None, check_finite=True): del overwrite_b, debug, check_finite return _solve_triangular(a, b, trans, lower, unit_diagonal)
[docs]@_wraps(scipy.linalg.tril) def tril(m, k=0): return jnp.tril(m, k)
[docs]@_wraps(scipy.linalg.triu) def triu(m, k=0): return jnp.triu(m, k)
_expm_description = textwrap.dedent(""" In addition to the original NumPy argument(s) listed below, also supports the optional boolean argument ``upper_triangular`` to specify whether the ``A`` matrix is upper triangular, and the optional argument ``max_squarings`` to specify the max number of squarings allowed in the scaling-and-squaring approximation method. Return nan if the actual number of squarings required is more than ``max_squarings``. The number of required squarings = max(0, ceil(log2(norm(A)) - c) where norm() denotes the L1 norm, and - c=2.42 for float64 or complex128, - c=1.97 for float32 or complex64 """)
[docs]@_wraps(scipy.linalg.expm, lax_description=_expm_description) @partial(jit, static_argnames=('upper_triangular', 'max_squarings')) def expm(A, *, upper_triangular=False, max_squarings=16): P, Q, n_squarings = _calc_P_Q(A) def _nan(args): A, *_ = args return jnp.full_like(A, jnp.nan) def _compute(args): A, P, Q = args R = _solve_P_Q(P, Q, upper_triangular) R = _squaring(R, n_squarings, max_squarings) return R R = lax.cond(n_squarings > max_squarings, _nan, _compute, (A, P, Q)) return R
@jit def _calc_P_Q(A): A = jnp.asarray(A) if A.ndim != 2 or A.shape[0] != A.shape[1]: raise ValueError('expected A to be a square matrix') A_L1 = np_linalg.norm(A,1) n_squarings = 0 if A.dtype == 'float64' or A.dtype == 'complex128': maxnorm = 5.371920351148152 n_squarings = jnp.maximum(0, jnp.floor(jnp.log2(A_L1 / maxnorm))) A = A / 2 ** n_squarings.astype(A.dtype) conds = jnp.array([1.495585217958292e-002, 2.539398330063230e-001, 9.504178996162932e-001, 2.097847961257068e+000], dtype=A_L1.dtype) idx = jnp.digitize(A_L1, conds) U, V = lax.switch(idx, [_pade3, _pade5, _pade7, _pade9, _pade13], A) elif A.dtype == 'float32' or A.dtype == 'complex64': maxnorm = 3.925724783138660 n_squarings = jnp.maximum(0, jnp.floor(jnp.log2(A_L1 / maxnorm))) A = A / 2 ** n_squarings.astype(A.dtype) conds = jnp.array([4.258730016922831e-001, 1.880152677804762e+000], dtype=A_L1.dtype) idx = jnp.digitize(A_L1, conds) U, V = lax.switch(idx, [_pade3, _pade5, _pade7], A) else: raise TypeError(f"A.dtype={A.dtype} is not supported.") P = U + V # p_m(A) : numerator Q = -U + V # q_m(A) : denominator return P, Q, n_squarings def _solve_P_Q(P, Q, upper_triangular=False): if upper_triangular: return solve_triangular(Q, P) else: return np_linalg.solve(Q, P) def _precise_dot(A, B): return jnp.dot(A, B, precision=lax.Precision.HIGHEST) @partial(jit, static_argnums=2) def _squaring(R, n_squarings, max_squarings): # squaring step to undo scaling def _squaring_precise(x): return _precise_dot(x, x) def _identity(x): return x def _scan_f(c, i): return lax.cond(i < n_squarings, _squaring_precise, _identity, c), None res, _ = lax.scan(_scan_f, R, jnp.arange(max_squarings, dtype=n_squarings.dtype)) return res def _pade3(A): b = (120., 60., 12., 1.) ident = jnp.eye(*A.shape, dtype=A.dtype) A2 = _precise_dot(A, A) U = _precise_dot(A, (b[3]*A2 + b[1]*ident)) V = b[2]*A2 + b[0]*ident return U, V def _pade5(A): b = (30240., 15120., 3360., 420., 30., 1.) ident = jnp.eye(*A.shape, dtype=A.dtype) A2 = _precise_dot(A, A) A4 = _precise_dot(A2, A2) U = _precise_dot(A, b[5]*A4 + b[3]*A2 + b[1]*ident) V = b[4]*A4 + b[2]*A2 + b[0]*ident return U, V def _pade7(A): b = (17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.) ident = jnp.eye(*A.shape, dtype=A.dtype) A2 = _precise_dot(A, A) A4 = _precise_dot(A2, A2) A6 = _precise_dot(A4, A2) U = _precise_dot(A, b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident) V = b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*ident return U,V def _pade9(A): b = (17643225600., 8821612800., 2075673600., 302702400., 30270240., 2162160., 110880., 3960., 90., 1.) ident = jnp.eye(*A.shape, dtype=A.dtype) A2 = _precise_dot(A, A) A4 = _precise_dot(A2, A2) A6 = _precise_dot(A4, A2) A8 = _precise_dot(A6, A2) U = _precise_dot(A, b[9]*A8 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident) V = b[8]*A8 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*ident return U,V def _pade13(A): b = (64764752532480000., 32382376266240000., 7771770303897600., 1187353796428800., 129060195264000., 10559470521600., 670442572800., 33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.) ident = jnp.eye(*A.shape, dtype=A.dtype) A2 = _precise_dot(A, A) A4 = _precise_dot(A2, A2) A6 = _precise_dot(A4, A2) U = _precise_dot(A, _precise_dot(A6, b[13]*A6 + b[11]*A4 + b[9]*A2) + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident) V = _precise_dot(A6, b[12]*A6 + b[10]*A4 + b[8]*A2) + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*ident return U,V _expm_frechet_description = textwrap.dedent(""" Does not currently support the Scipy argument ``jax.numpy.asarray_chkfinite``, because `jax.numpy.asarray_chkfinite` does not exist at the moment. Does not support the ``method='blockEnlarge'`` argument. """)
[docs]@_wraps(scipy.linalg.expm_frechet, lax_description=_expm_frechet_description) @partial(jit, static_argnames=('method', 'compute_expm')) def expm_frechet(A, E, *, method=None, compute_expm=True): A = jnp.asarray(A) E = jnp.asarray(E) if A.ndim != 2 or A.shape[0] != A.shape[1]: raise ValueError('expected A to be a square matrix') if E.ndim != 2 or E.shape[0] != E.shape[1]: raise ValueError('expected E to be a square matrix') if A.shape != E.shape: raise ValueError('expected A and E to be the same shape') if method is None: method = 'SPS' if method == 'SPS': bound_fun = partial(expm, upper_triangular=False, max_squarings=16) expm_A, expm_frechet_AE = jvp(bound_fun, (A,), (E,)) else: raise ValueError('only method=\'SPS\' is supported') if compute_expm: return expm_A, expm_frechet_AE else: return expm_frechet_AE
[docs]@_wraps(scipy.linalg.block_diag) @jit def block_diag(*arrs): if len(arrs) == 0: arrs = [jnp.zeros((1, 0))] arrs = jnp._promote_dtypes(*arrs) bad_shapes = [i for i, a in enumerate(arrs) if jnp.ndim(a) > 2] if bad_shapes: raise ValueError("Arguments to jax.scipy.linalg.block_diag must have at " "most 2 dimensions, got {} at argument {}." .format(arrs[bad_shapes[0]], bad_shapes[0])) arrs = [jnp.atleast_2d(a) for a in arrs] acc = arrs[0] dtype = lax.dtype(acc) for a in arrs[1:]: _, c = a.shape a = lax.pad(a, dtype.type(0), ((0, 0, 0), (acc.shape[-1], 0, 0))) acc = lax.pad(acc, dtype.type(0), ((0, 0, 0), (0, c, 0))) acc = lax.concatenate([acc, a], dimension=0) return acc
[docs]@_wraps(scipy.linalg.eigh_tridiagonal) @partial(jit, static_argnames=("eigvals_only", "select", "select_range")) def eigh_tridiagonal(d, e, *, eigvals_only=False, select='a', select_range=None, tol=None): if not eigvals_only: raise NotImplementedError("Calculation of eigenvectors is not implemented") def _sturm(alpha, beta_sq, pivmin, alpha0_perturbation, x): """Implements the Sturm sequence recurrence.""" n = alpha.shape[0] zeros = jnp.zeros(x.shape, dtype=jnp.int32) ones = jnp.ones(x.shape, dtype=jnp.int32) # The first step in the Sturm sequence recurrence # requires special care if x is equal to alpha[0]. def sturm_step0(): q = alpha[0] - x count = jnp.where(q < 0, ones, zeros) q = jnp.where(alpha[0] == x, alpha0_perturbation, q) return q, count # Subsequent steps all take this form: def sturm_step(i, q, count): q = alpha[i] - beta_sq[i - 1] / q - x count = jnp.where(q <= pivmin, count + 1, count) q = jnp.where(q <= pivmin, jnp.minimum(q, -pivmin), q) return q, count # The first step initializes q and count. q, count = sturm_step0() # Peel off ((n-1) % blocksize) steps from the main loop, so we can run # the bulk of the iterations unrolled by a factor of blocksize. blocksize = 16 i = 1 peel = (n - 1) % blocksize unroll_cnt = peel def unrolled_steps(args): start, q, count = args for j in range(unroll_cnt): q, count = sturm_step(start + j, q, count) return start + unroll_cnt, q, count i, q, count = unrolled_steps((i, q, count)) # Run the remaining steps of the Sturm sequence using a partially # unrolled while loop. unroll_cnt = blocksize def cond(iqc): i, q, count = iqc return jnp.less(i, n) _, _, count = lax.while_loop(cond, unrolled_steps, (i, q, count)) return count alpha = jnp.asarray(d) beta = jnp.asarray(e) supported_dtypes = (jnp.float32, jnp.float64, jnp.complex64, jnp.complex128) if alpha.dtype != beta.dtype: raise TypeError("diagonal and off-diagonal values must have same dtype, " f"got {alpha.dtype} and {beta.dtype}") if alpha.dtype not in supported_dtypes or beta.dtype not in supported_dtypes: raise TypeError("Only float32 and float64 inputs are supported as inputs " "to jax.scipy.linalg.eigh_tridiagonal, got " f"{alpha.dtype} and {beta.dtype}") n = alpha.shape[0] if n <= 1: return jnp.real(alpha) if jnp.issubdtype(alpha.dtype, jnp.complexfloating): alpha = jnp.real(alpha) beta_sq = jnp.real(beta * jnp.conj(beta)) beta_abs = jnp.sqrt(beta_sq) else: beta_abs = jnp.abs(beta) beta_sq = jnp.square(beta) # Estimate the largest and smallest eigenvalues of T using the Gershgorin # circle theorem. off_diag_abs_row_sum = jnp.concatenate( [beta_abs[:1], beta_abs[:-1] + beta_abs[1:], beta_abs[-1:]], axis=0) lambda_est_max = jnp.amax(alpha + off_diag_abs_row_sum) lambda_est_min = jnp.amin(alpha - off_diag_abs_row_sum) # Upper bound on 2-norm of T. t_norm = jnp.maximum(jnp.abs(lambda_est_min), jnp.abs(lambda_est_max)) # Compute the smallest allowed pivot in the Sturm sequence to avoid # overflow. finfo = np.finfo(alpha.dtype) one = np.ones([], dtype=alpha.dtype) safemin = np.maximum(one / finfo.max, (one + finfo.eps) * finfo.tiny) pivmin = safemin * jnp.maximum(1, jnp.amax(beta_sq)) alpha0_perturbation = jnp.square(finfo.eps * beta_abs[0]) abs_tol = finfo.eps * t_norm if tol is not None: abs_tol = jnp.maximum(tol, abs_tol) # In the worst case, when the absolute tolerance is eps*lambda_est_max and # lambda_est_max = -lambda_est_min, we have to take as many bisection steps # as there are bits in the mantissa plus 1. # The proof is left as an exercise to the reader. max_it = finfo.nmant + 1 # Determine the indices of the desired eigenvalues, based on select and # select_range. if select == 'a': target_counts = jnp.arange(n, dtype=jnp.int32) elif select == 'i': if select_range[0] > select_range[1]: raise ValueError('Got empty index range in select_range.') target_counts = jnp.arange(select_range[0], select_range[1] + 1, dtype=jnp.int32) elif select == 'v': # TODO(phawkins): requires dynamic shape support. raise NotImplementedError("eigh_tridiagonal(..., select='v') is not " "implemented") else: raise ValueError("'select must have a value in {'a', 'i', 'v'}.") # Run binary search for all desired eigenvalues in parallel, starting from # the interval lightly wider than the estimated # [lambda_est_min, lambda_est_max]. fudge = 2.1 # We widen starting interval the Gershgorin interval a bit. norm_slack = jnp.array(n, alpha.dtype) * fudge * finfo.eps * t_norm lower = lambda_est_min - norm_slack - 2 * fudge * pivmin upper = lambda_est_max + norm_slack + fudge * pivmin # Pre-broadcast the scalars used in the Sturm sequence for improved # performance. target_shape = jnp.shape(target_counts) lower = jnp.broadcast_to(lower, shape=target_shape) upper = jnp.broadcast_to(upper, shape=target_shape) mid = 0.5 * (upper + lower) pivmin = jnp.broadcast_to(pivmin, target_shape) alpha0_perturbation = jnp.broadcast_to(alpha0_perturbation, target_shape) # Start parallel binary searches. def cond(args): i, lower, _, upper = args return jnp.logical_and( jnp.less(i, max_it), jnp.less(abs_tol, jnp.amax(upper - lower))) def body(args): i, lower, mid, upper = args counts = _sturm(alpha, beta_sq, pivmin, alpha0_perturbation, mid) lower = jnp.where(counts <= target_counts, mid, lower) upper = jnp.where(counts > target_counts, mid, upper) mid = 0.5 * (lower + upper) return i + 1, lower, mid, upper _, _, mid, _ = lax.while_loop(cond, body, (0, lower, mid, upper)) return mid
[docs]@partial(jit, static_argnames=('side', 'method')) @jax.default_matmul_precision("float32") def polar(a, side='right', *, method='qdwh', eps=None, max_iterations=None): r"""Computes the polar decomposition. Given the :math:`m \times n` matrix :math:`a`, returns the factors of the polar decomposition :math:`u` (also :math:`m \times n`) and :math:`p` such that :math:`a = up` (if side is ``"right"``; :math:`p` is :math:`n \times n`) or :math:`a = pu` (if side is ``"left"``; :math:`p` is :math:`m \times m`), where :math:`p` is positive semidefinite. If :math:`a` is nonsingular, :math:`p` is positive definite and the decomposition is unique. :math:`u` has orthonormal columns unless :math:`n > m`, in which case it has orthonormal rows. Writing the SVD of :math:`a` as :math:`a = u_\mathit{svd} \cdot s_\mathit{svd} \cdot v^h_\mathit{svd}`, we have :math:`u = u_\mathit{svd} \cdot v^h_\mathit{svd}`. Thus the unitary factor :math:`u` can be constructed as the application of the sign function to the singular values of :math:`a`; or, if :math:`a` is Hermitian, the eigenvalues. Several methods exist to compute the polar decomposition. Currently two are supported: * ``method="svd"``: Computes the SVD of :math:`a` and then forms :math:`u = u_\mathit{svd} \cdot v^h_\mathit{svd}`. * ``method="qdwh"``: Applies the `QDWH`_ (QR-based Dynamically Weighted Halley) algorithm. Args: a: The :math:`m \times n` input matrix. side: Determines whether a right or left polar decomposition is computed. If ``side`` is ``"right"`` then :math:`a = up`. If ``side`` is ``"left"`` then :math:`a = pu`. The default is ``"right"``. method: Determines the algorithm used, as described above. precision: :class:`~jax.lax.Precision` object specifying the matmul precision. eps: The final result will satisfy :math:`\left|x_k - x_{k-1}\right| < \left|x_k\right| (4\epsilon)^{\frac{1}{3}}`, where :math:`x_k` are the QDWH iterates. Ignored if ``method`` is not ``"qdwh"``. max_iterations: Iterations will terminate after this many steps even if the above is unsatisfied. Ignored if ``method`` is not ``"qdwh"``. Returns: A ``(unitary, posdef)`` tuple, where ``unitary`` is the unitary factor (:math:`m \times n`), and ``posdef`` is the positive-semidefinite factor. ``posdef`` is either :math:`n \times n` or :math:`m \times m` depending on whether ``side`` is ``"right"`` or ``"left"``, respectively. .. _QDWH: https://epubs.siam.org/doi/abs/10.1137/090774999 """ a = jnp.asarray(a) if a.ndim != 2: raise ValueError("The input `a` must be a 2-D array.") if side not in ["right", "left"]: raise ValueError("The argument `side` must be either 'right' or 'left'.") m, n = a.shape if method == "qdwh": # TODO(phawkins): return info also if the user opts in? if m >= n and side == "right": unitary, posdef, _, _ = qdwh.qdwh(a, is_hermitian=False, eps=eps) elif m < n and side == "left": a = a.T.conj() unitary, posdef, _, _ = qdwh.qdwh(a, is_hermitian=False, eps=eps) posdef = posdef.T.conj() unitary = unitary.T.conj() else: raise NotImplementedError("method='qdwh' only supports mxn matrices " "where m < n where side='right' and m >= n " f"side='left', got {a.shape} with side={side}") elif method == "svd": u_svd, s_svd, vh_svd = lax_linalg.svd(a, full_matrices=False) s_svd = s_svd.astype(u_svd.dtype) unitary = u_svd @ vh_svd if side == "right": # a = u * p posdef = (vh_svd.T.conj() * s_svd[None, :]) @ vh_svd else: # a = p * u posdef = (u_svd * s_svd[None, :]) @ (u_svd.T.conj()) else: raise ValueError(f"Unknown polar decomposition method {method}.") return unitary, posdef
[docs]def polar_unitary(a, *, method="qdwh", eps=None, max_iterations=None): """ Computes the unitary factor u in the polar decomposition ``a = u p`` (or ``a = p u``). .. warning:: This function is deprecated. Use :func:`jax.scipy.linalg.polar` instead. """ # TODO(phawkins): delete this function after 2022/8/11. warnings.warn("jax.scipy.linalg.polar_unitary is deprecated. Call " "jax.scipy.linalg.polar instead.", DeprecationWarning) unitary, _ = polar(a, method, eps, max_iterations) return unitary
@jit 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 @jit def _sqrtm(A): T, Z = schur(A, output='complex') sqrt_T = _sqrtm_triu(T) return jnp.matmul(jnp.matmul(Z, sqrt_T, precision=lax.Precision.HIGHEST), jnp.conj(Z.T), precision=lax.Precision.HIGHEST)
[docs]@_wraps(scipy.linalg.sqrtm, lax_description=""" This differs from ``scipy.linalg.sqrtm`` in that the return type of ``jax.scipy.linalg.sqrtm`` is always ``complex64`` for 32-bit input, and ``complex128`` for 64-bit input. This function implements the complex Schur method described in [A]. It does not use recursive blocking to speed up computations as a Sylvester Equation solver is not available yet in JAX. [A] Björck, Å., & Hammarling, S. (1983). "A Schur method for the square root of a matrix". Linear algebra and its applications, 52, 127-140. """) def sqrtm(A, blocksize=1): if blocksize > 1: raise NotImplementedError("Blocked version is not implemented yet.") return _sqrtm(A)
[docs]@_wraps(scipy.linalg.rsf2csf, lax_description=_no_chkfinite_doc) @partial(jit, static_argnames=('check_finite',)) def rsf2csf(T, Z, check_finite=True): T = jnp.asarray(T) Z = jnp.asarray(Z) for ind, X in enumerate([Z, T]): if X.ndim != 2 or X.shape[0] != X.shape[1]: arg = 'ZT'[ind] raise ValueError(f"Input '{arg}' must be square.") if T.shape[0] != Z.shape[0]: raise ValueError(f"Input array shapes must match: Z: {Z.shape} vs. T: {T.shape}") T, Z = _promote_dtypes_complex(T, Z) eps = jnp.finfo(T.dtype).eps N = T.shape[0] if N == 1: return T, Z def _update_T_Z(m, T, Z): mu = np_linalg.eigvals(lax.dynamic_slice(T, (m-1, m-1), (2, 2))) - T[m, m] r = np_linalg.norm(jnp.array([mu[0], T[m, m-1]])).astype(T.dtype) c = mu[0] / r s = T[m, m-1] / r G = jnp.array([[c.conj(), s], [-s, c]], dtype=T.dtype) # T[m-1:m+1, m-1:] = G @ T[m-1:m+1, m-1:] T_rows = lax.dynamic_slice_in_dim(T, m-1, 2, axis=0) col_mask = jnp.arange(N) >= m-1 G_dot_T_zeroed_cols = G @ jnp.where(col_mask, T_rows, 0) T_rows_new = jnp.where(~col_mask, T_rows, G_dot_T_zeroed_cols) T = lax.dynamic_update_slice_in_dim(T, T_rows_new, m-1, axis=0) # T[:m+1, m-1:m+1] = T[:m+1, m-1:m+1] @ G.conj().T T_cols = lax.dynamic_slice_in_dim(T, m-1, 2, axis=1) row_mask = jnp.arange(N)[:, jnp.newaxis] < m+1 T_zeroed_rows_dot_GH = jnp.where(row_mask, T_cols, 0) @ G.conj().T T_cols_new = jnp.where(~row_mask, T_cols, T_zeroed_rows_dot_GH) T = lax.dynamic_update_slice_in_dim(T, T_cols_new, m-1, axis=1) # Z[:, m-1:m+1] = Z[:, m-1:m+1] @ G.conj().T Z_cols = lax.dynamic_slice_in_dim(Z, m-1, 2, axis=1) Z = lax.dynamic_update_slice_in_dim(Z, Z_cols @ G.conj().T, m-1, axis=1) return T, Z def _rsf2scf_iter(i, TZ): m = N-i T, Z = TZ T, Z = lax.cond( jnp.abs(T[m, m-1]) > eps*(jnp.abs(T[m-1, m-1]) + jnp.abs(T[m, m])), _update_T_Z, lambda m, T, Z: (T, Z), m, T, Z) T = T.at[m, m-1].set(0.0) return T, Z return lax.fori_loop(1, N, _rsf2scf_iter, (T, Z))