linalg_test.py

# 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.

"""Tests for the LAPAX linear algebra module."""

from functools import partial
import unittest

import numpy as np
import scipy
import scipy as osp

from absl.testing import absltest
from absl.testing import parameterized

import jax
from jax import jit, grad, jvp, vmap
from jax import lax
from jax import numpy as jnp
from jax import scipy as jsp
from jax._src import test_util as jtu

from jax.config import config
config.parse_flags_with_absl()
FLAGS = config.FLAGS

T = lambda x: np.swapaxes(x, -1, -2)


float_types = jtu.dtypes.floating
complex_types = jtu.dtypes.complex


class NumpyLinalgTest(jtu.JaxTestCase):

  def testNotImplemented(self):
    for name in jnp.linalg._NOT_IMPLEMENTED:
      func = getattr(jnp.linalg, name)
      with self.assertRaises(NotImplementedError):
        func()

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (4, 4), (2, 5, 5), (200, 200), (1000, 0, 0)]
      for dtype in float_types + complex_types))
  def testCholesky(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    def args_maker():
      factor_shape = shape[:-1] + (2 * shape[-1],)
      a = rng(factor_shape, dtype)
      return [np.matmul(a, jnp.conj(T(a)))]

    self._CheckAgainstNumpy(np.linalg.cholesky, jnp.linalg.cholesky, args_maker,
                            tol=1e-3)
    self._CompileAndCheck(jnp.linalg.cholesky, args_maker)

    if jnp.finfo(dtype).bits == 64:
      jtu.check_grads(jnp.linalg.cholesky, args_maker(), order=2)

  def testCholeskyGradPrecision(self):
    rng = jtu.rand_default(self.rng())
    a = rng((3, 3), np.float32)
    a = np.dot(a, a.T)
    jtu.assert_dot_precision(
        lax.Precision.HIGHEST, partial(jvp, jnp.linalg.cholesky), (a,), (a,))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_n={}".format(jtu.format_shape_dtype_string((n,n), dtype)),
       "n": n, "dtype": dtype}
      for n in [0, 2, 3, 4, 5, 25]  # TODO(mattjj): complex64 unstable on large sizes?
      for dtype in float_types + complex_types))
  def testDet(self, n, dtype):
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng((n, n), dtype)]

    self._CheckAgainstNumpy(np.linalg.det, jnp.linalg.det, args_maker, tol=1e-3)
    self._CompileAndCheck(jnp.linalg.det, args_maker,
                          rtol={np.float64: 1e-13, np.complex128: 1e-13})

  def testDetOfSingularMatrix(self):
    x = jnp.array([[-1., 3./2], [2./3, -1.]], dtype=np.float32)
    self.assertAllClose(np.float32(0), jsp.linalg.det(x))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (3, 3), (2, 4, 4)]
      for dtype in float_types))
  @jtu.skip_on_devices("tpu")
  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def testDetGrad(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    a = rng(shape, dtype)
    jtu.check_grads(jnp.linalg.det, (a,), 2, atol=1e-1, rtol=1e-1)
    # make sure there are no NaNs when a matrix is zero
    if len(shape) == 2:
      pass
      jtu.check_grads(
        jnp.linalg.det, (jnp.zeros_like(a),), 1, atol=1e-1, rtol=1e-1)
    else:
      a[0] = 0
      jtu.check_grads(jnp.linalg.det, (a,), 1, atol=1e-1, rtol=1e-1)

  def testDetGradIssue6121(self):
    f = lambda x: jnp.linalg.det(x).sum()
    x = jnp.ones((16, 1, 1))
    jax.grad(f)(x)
    jtu.check_grads(f, (x,), 2, atol=1e-1, rtol=1e-1)

  def testDetGradOfSingularMatrixCorank1(self):
    # Rank 2 matrix with nonzero gradient
    a = jnp.array([[ 50, -30,  45],
                  [-30,  90, -81],
                  [ 45, -81,  81]], dtype=jnp.float32)
    jtu.check_grads(jnp.linalg.det, (a,), 1, atol=1e-1, rtol=1e-1)

  def testDetGradOfSingularMatrixCorank2(self):
    # Rank 1 matrix with zero gradient
    b = jnp.array([[ 36, -42,  18],
                  [-42,  49, -21],
                  [ 18, -21,   9]], dtype=jnp.float32)
    jtu.check_grads(jnp.linalg.det, (b,), 1, atol=1e-1, rtol=1e-1, eps=1e-1)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_m={}_n={}_q={}".format(
            jtu.format_shape_dtype_string((m,), dtype),
            jtu.format_shape_dtype_string((nq[0],), dtype),
            jtu.format_shape_dtype_string(nq[1], dtype)),
       "m": m, "nq": nq, "dtype": dtype}
      for m in [1, 5, 7, 23]
      for nq in zip([2, 4, 6, 36], [(1, 2), (2, 2), (1, 2, 3), (3, 3, 1, 4)])
      for dtype in float_types))
  def testTensorsolve(self, m, nq, dtype):
    rng = jtu.rand_default(self.rng())

    # According to numpy docs the shapes are as follows:
    # Coefficient tensor (a), of shape b.shape + Q.
    # And prod(Q) == prod(b.shape)
    # Therefore, n = prod(q)
    n, q = nq
    b_shape = (n, m)
    # To accomplish prod(Q) == prod(b.shape) we append the m extra dim
    # to Q shape
    Q = q + (m,)
    args_maker = lambda: [
        rng(b_shape + Q, dtype), # = a
        rng(b_shape, dtype)]     # = b
    a, b = args_maker()
    result = jnp.linalg.tensorsolve(*args_maker())
    self.assertEqual(result.shape, Q)

    self._CheckAgainstNumpy(np.linalg.tensorsolve,
                            jnp.linalg.tensorsolve, args_maker,
                            tol={np.float32: 1e-2, np.float64: 1e-3})
    self._CompileAndCheck(jnp.linalg.tensorsolve,
                          args_maker,
                          rtol={np.float64: 1e-13})

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(0, 0), (1, 1), (3, 3), (4, 4), (10, 10), (200, 200),
                    (2, 2, 2), (2, 3, 3), (3, 2, 2)]
      for dtype in float_types + complex_types))
  def testSlogdet(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng(shape, dtype)]

    self._CheckAgainstNumpy(np.linalg.slogdet, jnp.linalg.slogdet, args_maker,
                            tol=1e-3)
    self._CompileAndCheck(jnp.linalg.slogdet, args_maker)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (4, 4), (5, 5), (2, 7, 7)]
      for dtype in float_types + complex_types))
  @jtu.skip_on_devices("tpu")
  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def testSlogdetGrad(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    a = rng(shape, dtype)
    jtu.check_grads(jnp.linalg.slogdet, (a,), 2, atol=1e-1, rtol=2e-1)

  def testIssue1213(self):
    for n in range(5):
      mat = jnp.array([np.diag(np.ones([5], dtype=np.float32))*(-.01)] * 2)
      args_maker = lambda: [mat]
      self._CheckAgainstNumpy(np.linalg.slogdet, jnp.linalg.slogdet, args_maker,
                              tol=1e-3)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shape={}_leftvectors={}_rightvectors={}".format(
           jtu.format_shape_dtype_string(shape, dtype),
           compute_left_eigenvectors, compute_right_eigenvectors),
       "shape": shape, "dtype": dtype,
       "compute_left_eigenvectors": compute_left_eigenvectors,
       "compute_right_eigenvectors": compute_right_eigenvectors}
      for shape in [(0, 0), (4, 4), (5, 5), (50, 50), (2, 6, 6)]
      for dtype in float_types + complex_types
      for compute_left_eigenvectors, compute_right_eigenvectors in [
          (False, False),
          (True, False),
          (False, True),
          (True, True)
      ]))
  # TODO(phawkins): enable when there is an eigendecomposition implementation
  # for GPU/TPU.
  @jtu.skip_on_devices("gpu", "tpu")
  def testEig(self, shape, dtype, compute_left_eigenvectors,
              compute_right_eigenvectors):
    rng = jtu.rand_default(self.rng())
    n = shape[-1]
    args_maker = lambda: [rng(shape, dtype)]

    # Norm, adjusted for dimension and type.
    def norm(x):
      norm = np.linalg.norm(x, axis=(-2, -1))
      return norm / ((n + 1) * jnp.finfo(dtype).eps)

    def check_right_eigenvectors(a, w, vr):
      self.assertTrue(
        np.all(norm(np.matmul(a, vr) - w[..., None, :] * vr) < 100))

    def check_left_eigenvectors(a, w, vl):
      rank = len(a.shape)
      aH = jnp.conj(a.transpose(list(range(rank - 2)) + [rank - 1, rank - 2]))
      wC = jnp.conj(w)
      check_right_eigenvectors(aH, wC, vl)

    a, = args_maker()
    results = lax.linalg.eig(a, compute_left_eigenvectors,
                             compute_right_eigenvectors)
    w = results[0]

    if compute_left_eigenvectors:
      check_left_eigenvectors(a, w, results[1])
    if compute_right_eigenvectors:
      check_right_eigenvectors(a, w, results[1 + compute_left_eigenvectors])

    self._CompileAndCheck(partial(jnp.linalg.eig), args_maker,
                          rtol=1e-3)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shape={}".format(
           jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(4, 4), (5, 5), (8, 8), (7, 6, 6)]
      for dtype in float_types + complex_types))
  # TODO(phawkins): enable when there is an eigendecomposition implementation
  # for GPU/TPU.
  @jtu.skip_on_devices("gpu", "tpu")
  def testEigvalsGrad(self, shape, dtype):
    # This test sometimes fails for large matrices. I (@j-towns) suspect, but
    # haven't checked, that might be because of perturbations causing the
    # ordering of eigenvalues to change, which will trip up check_grads. So we
    # just test on small-ish matrices.
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng(shape, dtype)]
    a, = args_maker()
    tol = 1e-4 if dtype in (np.float64, np.complex128) else 1e-1
    jtu.check_grads(lambda x: jnp.linalg.eigvals(x), (a,), order=1,
                    modes=['fwd', 'rev'], rtol=tol, atol=tol)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shape={}".format(
           jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(4, 4), (5, 5), (50, 50)]
      for dtype in float_types + complex_types))
  # TODO: enable when there is an eigendecomposition implementation
  # for GPU/TPU.
  @jtu.skip_on_devices("gpu", "tpu")
  def testEigvals(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng(shape, dtype)]
    a, = args_maker()
    w1, _ = jnp.linalg.eig(a)
    w2 = jnp.linalg.eigvals(a)
    self.assertAllClose(w1, w2, rtol={np.complex128: 1e-14})

  @jtu.skip_on_devices("gpu", "tpu")
  def testEigvalsInf(self):
    # https://github.com/google/jax/issues/2661
    x = jnp.array([[jnp.inf]])
    self.assertTrue(jnp.all(jnp.isnan(jnp.linalg.eigvals(x))))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (4, 4), (5, 5)]
      for dtype in float_types + complex_types))
  @jtu.skip_on_devices("gpu", "tpu")
  def testEigBatching(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    shape = (10,) + shape
    args = rng(shape, dtype)
    ws, vs = vmap(jnp.linalg.eig)(args)
    self.assertTrue(np.all(np.linalg.norm(
        np.matmul(args, vs) - ws[..., None, :] * vs) < 1e-3))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_n={}_lower={}".format(
           jtu.format_shape_dtype_string((n,n), dtype), lower),
       "n": n, "dtype": dtype, "lower": lower}
      for n in [0, 4, 5, 50]
      for dtype in float_types + complex_types
      for lower in [True, False]))
  def testEigh(self, n, dtype, lower):
    rng = jtu.rand_default(self.rng())
    tol = 1e-3
    args_maker = lambda: [rng((n, n), dtype)]

    uplo = "L" if lower else "U"

    a, = args_maker()
    a = (a + np.conj(a.T)) / 2
    w, v = jnp.linalg.eigh(np.tril(a) if lower else np.triu(a),
                           UPLO=uplo, symmetrize_input=False)
    self.assertLessEqual(
        np.linalg.norm(np.eye(n) - np.matmul(np.conj(T(v)), v)), 1e-3)
    self.assertLessEqual(np.linalg.norm(np.matmul(a, v) - w * v),
                         tol * np.linalg.norm(a))

    self._CompileAndCheck(partial(jnp.linalg.eigh, UPLO=uplo), args_maker,
                          rtol=1e-3)

  def testEighZeroDiagonal(self):
    a = np.array([[0., -1., -1.,  1.],
                  [-1.,  0.,  1., -1.],
                  [-1.,  1.,  0., -1.],
                  [1., -1., -1.,  0.]], dtype=np.float32)
    w, v = jnp.linalg.eigh(a)
    self.assertLessEqual(np.linalg.norm(np.matmul(a, v) - w * v),
                         1e-3 * np.linalg.norm(a))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shape={}".format(
           jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(4, 4), (5, 5), (50, 50)]
      for dtype in float_types + complex_types))
  def testEigvalsh(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    n = shape[-1]
    def args_maker():
      a = rng((n, n), dtype)
      a = (a + np.conj(a.T)) / 2
      return [a]
    self._CheckAgainstNumpy(np.linalg.eigvalsh, jnp.linalg.eigvalsh, args_maker,
                            tol=1e-3)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}_lower={}".format(jtu.format_shape_dtype_string(shape, dtype),
                                   lower),
       "shape": shape, "dtype": dtype, "lower":lower}
      for shape in [(1, 1), (4, 4), (5, 5), (50, 50), (2, 10, 10)]
      for dtype in float_types + complex_types
      for lower in [True, False]))
  def testEighGrad(self, shape, dtype, lower):
    rng = jtu.rand_default(self.rng())
    self.skipTest("Test fails with numeric errors.")
    uplo = "L" if lower else "U"
    a = rng(shape, dtype)
    a = (a + np.conj(T(a))) / 2
    ones = np.ones((a.shape[-1], a.shape[-1]), dtype=dtype)
    a *= np.tril(ones) if lower else np.triu(ones)
    # Gradient checks will fail without symmetrization as the eigh jvp rule
    # is only correct for tangents in the symmetric subspace, whereas the
    # checker checks against unconstrained (co)tangents.
    if dtype not in complex_types:
      f = partial(jnp.linalg.eigh, UPLO=uplo, symmetrize_input=True)
    else:  # only check eigenvalue grads for complex matrices
      f = lambda a: partial(jnp.linalg.eigh, UPLO=uplo, symmetrize_input=True)(a)[0]
    jtu.check_grads(f, (a,), 2, rtol=1e-1)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}_lower={}".format(jtu.format_shape_dtype_string(shape, dtype),
                                   lower),
       "shape": shape, "dtype": dtype, "lower":lower, "eps":eps}
      for shape in [(1, 1), (4, 4), (5, 5), (50, 50)]
      for dtype in complex_types
      for lower in [True, False]
      for eps in [1e-4]))
  def testEighGradVectorComplex(self, shape, dtype, lower, eps):
    rng = jtu.rand_default(self.rng())
    # Special case to test for complex eigenvector grad correctness.
    # Exact eigenvector coordinate gradients are hard to test numerically for complex
    # eigensystem solvers given the extra degrees of per-eigenvector phase freedom.
    # Instead, we numerically verify the eigensystem properties on the perturbed
    # eigenvectors.  You only ever want to optimize eigenvector directions, not coordinates!
    uplo = "L" if lower else "U"
    a = rng(shape, dtype)
    a = (a + np.conj(a.T)) / 2
    a = np.tril(a) if lower else np.triu(a)
    a_dot = eps * rng(shape, dtype)
    a_dot = (a_dot + np.conj(a_dot.T)) / 2
    a_dot = np.tril(a_dot) if lower else np.triu(a_dot)
    # evaluate eigenvector gradient and groundtruth eigensystem for perturbed input matrix
    f = partial(jnp.linalg.eigh, UPLO=uplo)
    (w, v), (dw, dv) = jvp(f, primals=(a,), tangents=(a_dot,))
    self.assertTrue(jnp.issubdtype(w.dtype, jnp.floating))
    self.assertTrue(jnp.issubdtype(dw.dtype, jnp.floating))
    new_a = a + a_dot
    new_w, new_v = f(new_a)
    new_a = (new_a + np.conj(new_a.T)) / 2
    # Assert rtol eigenvalue delta between perturbed eigenvectors vs new true eigenvalues.
    RTOL = 1e-2
    assert np.max(
      np.abs((np.diag(np.dot(np.conj((v+dv).T), np.dot(new_a,(v+dv)))) - new_w) / new_w)) < RTOL
    # Redundant to above, but also assert rtol for eigenvector property with new true eigenvalues.
    assert np.max(
      np.linalg.norm(np.abs(new_w*(v+dv) - np.dot(new_a, (v+dv))), axis=0) /
      np.linalg.norm(np.abs(new_w*(v+dv)), axis=0)
    ) < RTOL

  def testEighGradPrecision(self):
    rng = jtu.rand_default(self.rng())
    a = rng((3, 3), np.float32)
    jtu.assert_dot_precision(
        lax.Precision.HIGHEST, partial(jvp, jnp.linalg.eigh), (a,), (a,))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (4, 4), (5, 5)]
      for dtype in float_types + complex_types))
  def testEighBatching(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    shape = (10,) + shape
    args = rng(shape, dtype)
    args = (args + np.conj(T(args))) / 2
    ws, vs = vmap(jsp.linalg.eigh)(args)
    self.assertTrue(np.all(np.linalg.norm(
        np.matmul(args, vs) - ws[..., None, :] * vs) < 1e-3))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1,), (4,), (5,)]
      for dtype in (np.int32,)))
  def testLuPivotsToPermutation(self, shape, dtype):
    pivots_size = shape[-1]
    permutation_size = 2 * pivots_size

    pivots = jnp.arange(permutation_size - 1, pivots_size - 1, -1, dtype=dtype)
    pivots = jnp.broadcast_to(pivots, shape)
    actual = lax.linalg.lu_pivots_to_permutation(pivots, permutation_size)
    expected = jnp.arange(permutation_size - 1, -1, -1, dtype=dtype)
    expected = jnp.broadcast_to(expected, actual.shape)
    self.assertArraysEqual(actual, expected)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1,), (4,), (5,)]
      for dtype in (np.int32,)))
  def testLuPivotsToPermutationBatching(self, shape, dtype):
    shape = (10,) + shape
    pivots_size = shape[-1]
    permutation_size = 2 * pivots_size

    pivots = jnp.arange(permutation_size - 1, pivots_size - 1, -1, dtype=dtype)
    pivots = jnp.broadcast_to(pivots, shape)
    batched_fn = vmap(
        lambda x: lax.linalg.lu_pivots_to_permutation(x, permutation_size))
    actual = batched_fn(pivots)
    expected = jnp.arange(permutation_size - 1, -1, -1, dtype=dtype)
    expected = jnp.broadcast_to(expected, actual.shape)
    self.assertArraysEqual(actual, expected)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shape={}_ord={}_axis={}_keepdims={}".format(
         jtu.format_shape_dtype_string(shape, dtype), ord, axis, keepdims),
       "shape": shape, "dtype": dtype, "axis": axis, "keepdims": keepdims,
       "ord": ord}
      for axis, shape in [
        (None, (1,)), (None, (7,)), (None, (5, 8)),
        (0, (9,)), (0, (4, 5)), ((1,), (10, 7, 3)), ((-2,), (4, 8)),
        (-1, (6, 3)), ((0, 2), (3, 4, 5)), ((2, 0), (7, 8, 9)),
        (None, (7, 8, 11))]
      for keepdims in [False, True]
      for ord in (
          [None] if axis is None and len(shape) > 2
          else [None, 0, 1, 2, 3, -1, -2, -3, jnp.inf, -jnp.inf]
          if (axis is None and len(shape) == 1) or
             isinstance(axis, int) or
             (isinstance(axis, tuple) and len(axis) == 1)
          else [None, 'fro', 1, 2, -1, -2, jnp.inf, -jnp.inf, 'nuc'])
      for dtype in float_types + complex_types))  # type: ignore
  def testNorm(self, shape, dtype, ord, axis, keepdims):
    rng = jtu.rand_default(self.rng())
    if (ord in ('nuc', 2, -2) and (
        jtu.device_under_test() != "cpu" or
        (isinstance(axis, tuple) and len(axis) == 2))):
      raise unittest.SkipTest("No adequate SVD implementation available")

    args_maker = lambda: [rng(shape, dtype)]
    np_fn = partial(np.linalg.norm, ord=ord, axis=axis, keepdims=keepdims)
    jnp_fn = partial(jnp.linalg.norm, ord=ord, axis=axis, keepdims=keepdims)
    self._CheckAgainstNumpy(np_fn, jnp_fn, args_maker, check_dtypes=False,
                            tol=1e-3)
    self._CompileAndCheck(jnp_fn, args_maker)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_n={}_full_matrices={}_compute_uv={}_hermitian={}".format(
          jtu.format_shape_dtype_string(b + (m, n), dtype), full_matrices,
          compute_uv, hermitian),
       "b": b, "m": m, "n": n, "dtype": dtype, "full_matrices": full_matrices,
       "compute_uv": compute_uv, "hermitian": hermitian}
      for b in [(), (3,), (2, 3)]
      for m in [0, 2, 7, 29, 53]
      for n in [0, 2, 7, 29, 53]
      for dtype in float_types + complex_types
      for full_matrices in [False, True]
      for compute_uv in [False, True]
      for hermitian in ([False, True] if m == n else [False])))
  def testSVD(self, b, m, n, dtype, full_matrices, compute_uv, hermitian):
    if (jnp.issubdtype(dtype, np.complexfloating) and
        jtu.device_under_test() == "tpu"):
      raise unittest.SkipTest("No complex SVD implementation")
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng(b + (m, n), dtype)]

    # Norm, adjusted for dimension and type.
    def norm(x):
      norm = np.linalg.norm(x, axis=(-2, -1))
      return norm / (max(1, m, n) * jnp.finfo(dtype).eps)

    a, = args_maker()
    if hermitian:
      a = a + np.conj(T(a))
    out = jnp.linalg.svd(a, full_matrices=full_matrices, compute_uv=compute_uv,
                         hermitian=hermitian)
    if compute_uv:
      # Check the reconstructed matrices
      if full_matrices:
        k = min(m, n)
        if m < n:
          self.assertTrue(np.all(
              norm(a - np.matmul(out[1][..., None, :] * out[0], out[2][..., :k, :])) < 50))
        else:
          self.assertTrue(np.all(
              norm(a - np.matmul(out[1][..., None, :] * out[0][..., :, :k], out[2])) < 350))
      else:
        self.assertTrue(np.all(
          norm(a - np.matmul(out[1][..., None, :] * out[0], out[2])) < 350))

      # Check the unitary properties of the singular vector matrices.
      self.assertTrue(np.all(norm(np.eye(out[0].shape[-1]) - np.matmul(np.conj(T(out[0])), out[0])) < 15))
      if m >= n:
        self.assertTrue(np.all(norm(np.eye(out[2].shape[-1]) - np.matmul(np.conj(T(out[2])), out[2])) < 10))
      else:
        self.assertTrue(np.all(norm(np.eye(out[2].shape[-2]) - np.matmul(out[2], np.conj(T(out[2])))) < 20))

    else:
      self.assertTrue(np.allclose(np.linalg.svd(a, compute_uv=False), np.asarray(out), atol=1e-4, rtol=1e-4))

    self._CompileAndCheck(partial(jnp.linalg.svd, full_matrices=full_matrices, compute_uv=compute_uv),
                          args_maker)
    if not compute_uv:
      svd = partial(jnp.linalg.svd, full_matrices=full_matrices,
                    compute_uv=compute_uv)
      # TODO(phawkins): these tolerances seem very loose.
      if dtype == np.complex128:
        jtu.check_jvp(svd, partial(jvp, svd), (a,), rtol=1e-4, atol=1e-4, eps=1e-8)
      else:
        jtu.check_jvp(svd, partial(jvp, svd), (a,), rtol=5e-2, atol=2e-1)

    if jtu.device_under_test() == "tpu":
      raise unittest.SkipTest("TPU matmul does not have enough precision")
    # TODO(frederikwilde): Find the appropriate precision to use for this test on TPUs.

    if compute_uv and (not full_matrices):
      b, = args_maker()
      def f(x):
        u, s, v = jnp.linalg.svd(
          a + x * b,
          full_matrices=full_matrices,
          compute_uv=compute_uv)
        vdiag = jnp.vectorize(jnp.diag, signature='(k)->(k,k)')
        return jnp.matmul(jnp.matmul(u, vdiag(s)), v).real
      _, t_out = jvp(f, (1.,), (1.,))
      if dtype == np.complex128:
        atol = 1e-13
      else:
        atol = 5e-4
      self.assertArraysAllClose(t_out, b.real, atol=atol)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shape={}_fullmatrices={}".format(
          jtu.format_shape_dtype_string(shape, dtype), full_matrices),
       "shape": shape, "dtype": dtype, "full_matrices": full_matrices}
      for shape in [(1, 1), (3, 3), (3, 4), (2, 10, 5), (2, 200, 100)]
      for dtype in float_types + complex_types
      for full_matrices in [False, True]))
  def testQr(self, shape, dtype, full_matrices):
    rng = jtu.rand_default(self.rng())
    m, n = shape[-2:]

    if full_matrices:
      mode, k = "complete", m
    else:
      mode, k = "reduced", min(m, n)

    a = rng(shape, dtype)
    lq, lr = jnp.linalg.qr(a, mode=mode)

    # np.linalg.qr doesn't support batch dimensions. But it seems like an
    # inevitable extension so we support it in our version.
    nq = np.zeros(shape[:-2] + (m, k), dtype)
    nr = np.zeros(shape[:-2] + (k, n), dtype)
    for index in np.ndindex(*shape[:-2]):
      nq[index], nr[index] = np.linalg.qr(a[index], mode=mode)

    max_rank = max(m, n)

    # Norm, adjusted for dimension and type.
    def norm(x):
      n = np.linalg.norm(x, axis=(-2, -1))
      return n / (max_rank * jnp.finfo(dtype).eps)

    def compare_orthogonal(q1, q2):
      # Q is unique up to sign, so normalize the sign first.
      sum_of_ratios = np.sum(np.divide(q1, q2), axis=-2, keepdims=True)
      phases = np.divide(sum_of_ratios, np.abs(sum_of_ratios))
      q1 *= phases
      self.assertTrue(np.all(norm(q1 - q2) < 30))

    # Check a ~= qr
    self.assertTrue(np.all(norm(a - np.matmul(lq, lr)) < 30))

    # Compare the first 'k' vectors of Q; the remainder form an arbitrary
    # orthonormal basis for the null space.
    compare_orthogonal(nq[..., :k], lq[..., :k])

    # Check that q is close to unitary.
    self.assertTrue(np.all(
        norm(np.eye(k) - np.matmul(np.conj(T(lq)), lq)) < 5))

    if not full_matrices and m >= n:
        jtu.check_jvp(jnp.linalg.qr, partial(jvp, jnp.linalg.qr), (a,), atol=3e-3)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shape={}".format(
          jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(10, 4, 5), (5, 3, 3), (7, 6, 4)]
      for dtype in float_types + complex_types))
  def testQrBatching(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    args = rng(shape, jnp.float32)
    qs, rs = vmap(jsp.linalg.qr)(args)
    self.assertTrue(np.all(np.linalg.norm(args - np.matmul(qs, rs)) < 1e-3))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}_pnorm={}".format(jtu.format_shape_dtype_string(shape, dtype), pnorm),
       "shape": shape, "pnorm": pnorm, "dtype": dtype}
      for shape in [(1, 1), (4, 4), (2, 3, 5), (5, 5, 5), (20, 20), (5, 10)]
      for pnorm in [jnp.inf, -jnp.inf, 1, -1, 2, -2, 'fro']
      for dtype in float_types + complex_types))
  @jtu.skip_on_devices("gpu")  # TODO(#2203): numerical errors
  def testCond(self, shape, pnorm, dtype):
    if (jnp.issubdtype(dtype, np.complexfloating) and
        jtu.device_under_test() == "tpu"):
      raise unittest.SkipTest("No complex SVD implementation")

    def gen_mat():
      # arr_gen = jtu.rand_some_nan(self.rng())
      arr_gen = jtu.rand_default(self.rng())
      res = arr_gen(shape, dtype)
      return res

    def args_gen(p):
      def _args_gen():
        return [gen_mat(), p]
      return _args_gen

    args_maker = args_gen(pnorm)
    if pnorm not in [2, -2] and len(set(shape[-2:])) != 1:
      with self.assertRaises(np.linalg.LinAlgError):
        jnp.linalg.cond(*args_maker())
    else:
      self._CheckAgainstNumpy(np.linalg.cond, jnp.linalg.cond, args_maker,
                              check_dtypes=False, tol=1e-3)
      partial_norm = partial(jnp.linalg.cond, p=pnorm)
      self._CompileAndCheck(partial_norm, lambda: [gen_mat()],
                            check_dtypes=False, rtol=1e-03, atol=1e-03)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (4, 4), (200, 200), (7, 7, 7, 7)]
      for dtype in float_types))
  def testTensorinv(self, shape, dtype):
    rng = jtu.rand_default(self.rng())

    def tensor_maker():
      invertible = False
      while not invertible:
        a = rng(shape, dtype)
        try:
          np.linalg.inv(a)
          invertible = True
        except np.linalg.LinAlgError:
          pass
      return a

    args_maker = lambda: [tensor_maker(), int(np.floor(len(shape) / 2))]
    self._CheckAgainstNumpy(np.linalg.tensorinv, jnp.linalg.tensorinv, args_maker,
                            check_dtypes=False, tol=1e-3)
    partial_inv = partial(jnp.linalg.tensorinv, ind=int(np.floor(len(shape) / 2)))
    self._CompileAndCheck(partial_inv, lambda: [tensor_maker()], check_dtypes=False, rtol=1e-03, atol=1e-03)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_lhs={}_rhs={}".format(
           jtu.format_shape_dtype_string(lhs_shape, dtype),
           jtu.format_shape_dtype_string(rhs_shape, dtype)),
       "lhs_shape": lhs_shape, "rhs_shape": rhs_shape, "dtype": dtype}
      for lhs_shape, rhs_shape in [
          ((1, 1), (1, 1)),
          ((4, 4), (4,)),
          ((8, 8), (8, 4)),
          ((1, 2, 2), (3, 2)),
          ((2, 1, 3, 3), (1, 4, 3, 4)),
          ((1, 0, 0), (1, 0, 2)),
      ]
      for dtype in float_types + complex_types))
  def testSolve(self, lhs_shape, rhs_shape, dtype):
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng(lhs_shape, dtype), rng(rhs_shape, dtype)]

    self._CheckAgainstNumpy(np.linalg.solve, jnp.linalg.solve, args_maker,
                            tol=1e-3)
    self._CompileAndCheck(jnp.linalg.solve, args_maker)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (4, 4), (2, 5, 5), (200, 200), (5, 5, 5), (0, 0)]
      for dtype in float_types))
  def testInv(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    if jtu.device_under_test() == "gpu" and shape == (200, 200):
      raise unittest.SkipTest("Test is flaky on GPU")

    def args_maker():
      invertible = False
      while not invertible:
        a = rng(shape, dtype)
        try:
          np.linalg.inv(a)
          invertible = True
        except np.linalg.LinAlgError:
          pass
      return [a]

    self._CheckAgainstNumpy(np.linalg.inv, jnp.linalg.inv, args_maker,
                            tol=1e-3)
    self._CompileAndCheck(jnp.linalg.inv, args_maker)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (4, 4), (2, 70, 7), (2000, 7), (7, 1000), (70, 7, 2),
                    (2, 0, 0), (3, 0, 2), (1, 0)]
      for dtype in float_types + complex_types))
  def testPinv(self, shape, dtype):
    if (jnp.issubdtype(dtype, np.complexfloating) and
        jtu.device_under_test() == "tpu"):
      raise unittest.SkipTest("No complex SVD implementation")
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng(shape, dtype)]

    self._CheckAgainstNumpy(np.linalg.pinv, jnp.linalg.pinv, args_maker,
                            tol=1e-2)
    self._CompileAndCheck(jnp.linalg.pinv, args_maker)
    if jtu.device_under_test() != "tpu":
      # TODO(phawkins): 1e-1 seems like a very loose tolerance.
      jtu.check_grads(jnp.linalg.pinv, args_maker(), 2, rtol=1e-1, atol=2e-1)

  def testPinvGradIssue2792(self):
    def f(p):
      a = jnp.array([[0., 0.],[-p, 1.]], jnp.float32) * 1 / (1 + p**2)
      return jnp.linalg.pinv(a)
    j = jax.jacobian(f)(jnp.float32(2.))
    self.assertAllClose(jnp.array([[0., -1.], [ 0., 0.]], jnp.float32), j)

    expected = jnp.array([[[[-1., 0.], [ 0., 0.]], [[0., -1.], [0.,  0.]]],
                         [[[0.,  0.], [-1., 0.]], [[0.,  0.], [0., -1.]]]],
                         dtype=jnp.float32)
    self.assertAllClose(
      expected, jax.jacobian(jnp.linalg.pinv)(jnp.eye(2, dtype=jnp.float32)))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shape={}_n={}".format(
          jtu.format_shape_dtype_string(shape, dtype), n),
       "shape": shape, "dtype": dtype, "n": n}
      for shape in [(1, 1), (2, 2), (4, 4), (5, 5),
                    (1, 2, 2), (2, 3, 3), (2, 5, 5)]
      for dtype in float_types + complex_types
      for n in [-5, -2, -1, 0, 1, 2, 3, 4, 5, 10]))
  @jtu.skip_on_devices("tpu")  # TODO(b/149870255): Bug in XLA:TPU?.
  def testMatrixPower(self, shape, dtype, n):
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng(shape, dtype)]
    tol = 1e-1 if jtu.device_under_test() == "tpu" else 1e-3
    self._CheckAgainstNumpy(partial(np.linalg.matrix_power, n=n),
                            partial(jnp.linalg.matrix_power, n=n),
                            args_maker, tol=tol)
    self._CompileAndCheck(partial(jnp.linalg.matrix_power, n=n), args_maker,
                          rtol=1e-3)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shape={}".format(
           jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(3, ), (1, 2), (8, 5), (4, 4), (5, 5), (50, 50)]
      for dtype in float_types + complex_types))
  def testMatrixRank(self, shape, dtype):
    if (jnp.issubdtype(dtype, np.complexfloating) and
        jtu.device_under_test() == "tpu"):
      raise unittest.SkipTest("No complex SVD implementation")
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng(shape, dtype)]
    a, = args_maker()
    self._CheckAgainstNumpy(np.linalg.matrix_rank, jnp.linalg.matrix_rank,
                            args_maker, check_dtypes=False, tol=1e-3)
    self._CompileAndCheck(jnp.linalg.matrix_rank, args_maker,
                          check_dtypes=False, rtol=1e-3)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_shapes={}".format(
           ','.join(jtu.format_shape_dtype_string(s, dtype) for s in shapes)),
       "shapes": shapes, "dtype": dtype}
      for shapes in [
        [(3, ), (3, 1)],  # quick-out codepath
        [(1, 3), (3, 5), (5, 2)],  # multi_dot_three codepath
        [(1, 3), (3, 5), (5, 2), (2, 7), (7, )]  # dynamic programming codepath
      ]
      for dtype in float_types + complex_types))
  def testMultiDot(self, shapes, dtype):
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [[rng(shape, dtype) for shape in shapes]]

    np_fun = np.linalg.multi_dot
    jnp_fun = partial(jnp.linalg.multi_dot, precision=lax.Precision.HIGHEST)
    tol = {np.float32: 1e-4, np.float64: 1e-10,
           np.complex64: 1e-4, np.complex128: 1e-10}

    self._CheckAgainstNumpy(np_fun, jnp_fun, args_maker, tol=tol)
    self._CompileAndCheck(jnp_fun, args_maker,
                          atol=tol, rtol=tol)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_lhs={}_rhs={}__rcond={}".format(
           jtu.format_shape_dtype_string(lhs_shape, dtype),
           jtu.format_shape_dtype_string(rhs_shape, dtype),
           rcond),
       "lhs_shape": lhs_shape, "rhs_shape": rhs_shape, "dtype": dtype, "rcond": rcond}
      for lhs_shape, rhs_shape in [
          ((1, 1), (1, 1)),
          ((4, 6), (4,)),
          ((6, 6), (6, 1)),
          ((8, 6), (8, 4)),
      ]
      for rcond in [-1, None, 0.5]
      for dtype in float_types + complex_types))
  @jtu.skip_on_devices("tpu")  # SVD not implemented on TPU.
  def testLstsq(self, lhs_shape, rhs_shape, dtype, rcond):
    rng = jtu.rand_default(self.rng())
    np_fun = partial(np.linalg.lstsq, rcond=rcond)
    jnp_fun = partial(jnp.linalg.lstsq, rcond=rcond)
    jnp_fun_numpy_resid = partial(jnp.linalg.lstsq, rcond=rcond, numpy_resid=True)
    tol = {np.float32: 1e-5, np.float64: 1e-12,
           np.complex64: 1e-5, np.complex128: 1e-12}
    args_maker = lambda: [rng(lhs_shape, dtype), rng(rhs_shape, dtype)]

    self._CheckAgainstNumpy(np_fun, jnp_fun_numpy_resid, args_maker, check_dtypes=False, tol=tol)
    self._CompileAndCheck(jnp_fun, args_maker, atol=tol, rtol=tol)

    # Disabled because grad is flaky for low-rank inputs.
    # TODO:
    # jtu.check_grads(lambda *args: jnp_fun(*args)[0], args_maker(), order=2, atol=1e-2, rtol=1e-2)

  # Regression test for incorrect type for eigenvalues of a complex matrix.
  def testIssue669(self):
    def test(x):
      val, vec = jnp.linalg.eigh(x)
      return jnp.real(jnp.sum(val))

    grad_test_jc = jit(grad(jit(test)))
    xc = np.eye(3, dtype=np.complex64)
    self.assertAllClose(xc, grad_test_jc(xc))

  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def testIssue1151(self):
    rng = self.rng()
    A = jnp.array(rng.randn(100, 3, 3), dtype=jnp.float32)
    b = jnp.array(rng.randn(100, 3), dtype=jnp.float32)
    x = jnp.linalg.solve(A, b)
    self.assertAllClose(vmap(jnp.dot)(A, x), b, atol=2e-3, rtol=1e-2)

    _ = jax.jacobian(jnp.linalg.solve, argnums=0)(A, b)
    _ = jax.jacobian(jnp.linalg.solve, argnums=1)(A, b)

    _ = jax.jacobian(jnp.linalg.solve, argnums=0)(A[0], b[0])
    _ = jax.jacobian(jnp.linalg.solve, argnums=1)(A[0], b[0])

  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def testIssue1383(self):
    seed = jax.random.PRNGKey(0)
    tmp = jax.random.uniform(seed, (2,2))
    a = jnp.dot(tmp, tmp.T)

    def f(inp):
      val, vec = jnp.linalg.eigh(inp)
      return jnp.dot(jnp.dot(vec, inp), vec.T)

    grad_func = jax.jacfwd(f)
    hess_func = jax.jacfwd(grad_func)
    cube_func = jax.jacfwd(hess_func)
    self.assertFalse(np.any(np.isnan(cube_func(a))))


class ScipyLinalgTest(jtu.JaxTestCase):

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_i={}".format(i), "args": args}
      for i, args in enumerate([
        (),
        (1,),
        (7, -2),
        (3, 4, 5),
        (np.ones((3, 4), dtype=jnp.float_), 5,
         np.random.randn(5, 2).astype(jnp.float_)),
      ])))
  def testBlockDiag(self, args):
    args_maker = lambda: args
    self._CheckAgainstNumpy(osp.linalg.block_diag, jsp.linalg.block_diag,
                            args_maker)
    self._CompileAndCheck(jsp.linalg.block_diag, args_maker)


  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (4, 5), (10, 5), (50, 50)]
      for dtype in float_types + complex_types))
  def testLu(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng(shape, dtype)]
    x, = args_maker()
    p, l, u = jsp.linalg.lu(x)
    self.assertAllClose(x, np.matmul(p, np.matmul(l, u)),
                        rtol={np.float32: 1e-3, np.float64: 1e-12,
                              np.complex64: 1e-3, np.complex128: 1e-12})
    self._CompileAndCheck(jsp.linalg.lu, args_maker)

  def testLuOfSingularMatrix(self):
    x = jnp.array([[-1., 3./2], [2./3, -1.]], dtype=np.float32)
    p, l, u = jsp.linalg.lu(x)
    self.assertAllClose(x, np.matmul(p, np.matmul(l, u)))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(1, 1), (4, 5), (10, 5), (10, 10), (6, 7, 7)]
      for dtype in float_types + complex_types))
  @jtu.skip_on_devices("tpu")  # TODO(phawkins): precision problems on TPU.
  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def testLuGrad(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    a = rng(shape, dtype)
    lu = vmap(jsp.linalg.lu) if len(shape) > 2 else jsp.linalg.lu
    jtu.check_grads(lu, (a,), 2, atol=5e-2, rtol=3e-1)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
       "shape": shape, "dtype": dtype}
      for shape in [(4, 5), (6, 5)]
      for dtype in [jnp.float32]))
  def testLuBatching(self, shape, dtype):
    rng = jtu.rand_default(self.rng())
    args = [rng(shape, jnp.float32) for _ in range(10)]
    expected = list(osp.linalg.lu(x) for x in args)
    ps = np.stack([out[0] for out in expected])
    ls = np.stack([out[1] for out in expected])
    us = np.stack([out[2] for out in expected])

    actual_ps, actual_ls, actual_us = vmap(jsp.linalg.lu)(jnp.stack(args))
    self.assertAllClose(ps, actual_ps)
    self.assertAllClose(ls, actual_ls, rtol=5e-6)
    self.assertAllClose(us, actual_us)

  @jtu.skip_on_devices("cpu", "tpu")
  def testLuCPUBackendOnGPU(self):
    # tests running `lu` on cpu when a gpu is present.
    jit(jsp.linalg.lu, backend="cpu")(np.ones((2, 2)))  # does not crash

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_n={}".format(jtu.format_shape_dtype_string((n,n), dtype)),
       "n": n, "dtype": dtype}
      for n in [1, 4, 5, 200]
      for dtype in float_types + complex_types))
  def testLuFactor(self, n, dtype):
    rng = jtu.rand_default(self.rng())
    args_maker = lambda: [rng((n, n), dtype)]

    x, = args_maker()
    lu, piv = jsp.linalg.lu_factor(x)
    l = np.tril(lu, -1) + np.eye(n, dtype=dtype)
    u = np.triu(lu)
    for i in range(n):
      x[[i, piv[i]],] = x[[piv[i], i],]
    self.assertAllClose(x, np.matmul(l, u), rtol=1e-3,
                        atol=1e-3)
    self._CompileAndCheck(jsp.linalg.lu_factor, args_maker)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_lhs={}_rhs={}_trans={}".format(
           jtu.format_shape_dtype_string(lhs_shape, dtype),
           jtu.format_shape_dtype_string(rhs_shape, dtype),
           trans),
       "lhs_shape": lhs_shape, "rhs_shape": rhs_shape, "dtype": dtype,
       "trans": trans}
      for lhs_shape, rhs_shape in [
          ((1, 1), (1, 1)),
          ((4, 4), (4,)),
          ((8, 8), (8, 4)),
      ]
      for trans in [0, 1, 2]
      for dtype in float_types + complex_types))
  @jtu.skip_on_devices("cpu")  # TODO(frostig): Test fails on CPU sometimes
  def testLuSolve(self, lhs_shape, rhs_shape, dtype, trans):
    rng = jtu.rand_default(self.rng())
    osp_fun = lambda lu, piv, rhs: osp.linalg.lu_solve((lu, piv), rhs, trans=trans)
    jsp_fun = lambda lu, piv, rhs: jsp.linalg.lu_solve((lu, piv), rhs, trans=trans)

    def args_maker():
      a = rng(lhs_shape, dtype)
      lu, piv = osp.linalg.lu_factor(a)
      return [lu, piv, rng(rhs_shape, dtype)]

    self._CheckAgainstNumpy(osp_fun, jsp_fun, args_maker, tol=1e-3)
    self._CompileAndCheck(jsp_fun, args_maker)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_lhs={}_rhs={}_sym_pos={}_lower={}".format(
           jtu.format_shape_dtype_string(lhs_shape, dtype),
           jtu.format_shape_dtype_string(rhs_shape, dtype),
           sym_pos, lower),
       "lhs_shape": lhs_shape, "rhs_shape": rhs_shape, "dtype": dtype,
       "sym_pos": sym_pos, "lower": lower}
      for lhs_shape, rhs_shape in [
          ((1, 1), (1, 1)),
          ((4, 4), (4,)),
          ((8, 8), (8, 4)),
      ]
      for sym_pos, lower in [
        (False, False),
        (True, False),
        (True, True),
      ]
      for dtype in float_types + complex_types))
  def testSolve(self, lhs_shape, rhs_shape, dtype, sym_pos, lower):
    rng = jtu.rand_default(self.rng())
    osp_fun = lambda lhs, rhs: osp.linalg.solve(lhs, rhs, sym_pos=sym_pos, lower=lower)
    jsp_fun = lambda lhs, rhs: jsp.linalg.solve(lhs, rhs, sym_pos=sym_pos, lower=lower)

    def args_maker():
      a = rng(lhs_shape, dtype)
      if sym_pos:
        a = np.matmul(a, np.conj(T(a)))
        a = np.tril(a) if lower else np.triu(a)
      return [a, rng(rhs_shape, dtype)]

    self._CheckAgainstNumpy(osp_fun, jsp_fun, args_maker, tol=1e-3)
    self._CompileAndCheck(jsp_fun, args_maker)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_lhs={}_rhs={}_lower={}_transposea={}_unit_diagonal={}".format(
           jtu.format_shape_dtype_string(lhs_shape, dtype),
           jtu.format_shape_dtype_string(rhs_shape, dtype),
           lower, transpose_a, unit_diagonal),
       "lower": lower, "transpose_a": transpose_a,
       "unit_diagonal": unit_diagonal, "lhs_shape": lhs_shape,
       "rhs_shape": rhs_shape, "dtype": dtype}
      for lower in [False, True]
      for transpose_a in [False, True]
      for unit_diagonal in [False, True]
      for lhs_shape, rhs_shape in [
          ((4, 4), (4,)),
          ((4, 4), (4, 3)),
          ((2, 8, 8), (2, 8, 10)),
      ]
      for dtype in float_types))
  def testSolveTriangular(self, lower, transpose_a, unit_diagonal, lhs_shape,
                          rhs_shape, dtype):
    rng = jtu.rand_default(self.rng())
    k = rng(lhs_shape, dtype)
    l = np.linalg.cholesky(np.matmul(k, T(k))
                            + lhs_shape[-1] * np.eye(lhs_shape[-1]))
    l = l.astype(k.dtype)
    b = rng(rhs_shape, dtype)

    if unit_diagonal:
      a = np.tril(l, -1) + np.eye(lhs_shape[-1], dtype=dtype)
    else:
      a = l
    a = a if lower else T(a)

    inv = np.linalg.inv(T(a) if transpose_a else a).astype(a.dtype)
    if len(lhs_shape) == len(rhs_shape):
      np_ans = np.matmul(inv, b)
    else:
      np_ans = np.einsum("...ij,...j->...i", inv, b)

    # The standard scipy.linalg.solve_triangular doesn't support broadcasting.
    # But it seems like an inevitable extension so we support it.
    ans = jsp.linalg.solve_triangular(
        l if lower else T(l), b, trans=1 if transpose_a else 0, lower=lower,
        unit_diagonal=unit_diagonal)

    self.assertAllClose(np_ans, ans,
                        rtol={np.float32: 1e-4, np.float64: 1e-11})

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_A={}_B={}_lower={}_transposea={}_conja={}_unitdiag={}_leftside={}".format(
           jtu.format_shape_dtype_string(a_shape, dtype),
           jtu.format_shape_dtype_string(b_shape, dtype),
           lower, transpose_a, conjugate_a, unit_diagonal, left_side),
       "lower": lower, "transpose_a": transpose_a, "conjugate_a": conjugate_a,
       "unit_diagonal": unit_diagonal, "left_side": left_side,
       "a_shape": a_shape, "b_shape": b_shape, "dtype": dtype}
      for lower in [False, True]
      for unit_diagonal in [False, True]
      for dtype in float_types + complex_types
      for transpose_a in [False, True]
      for conjugate_a in (
          [False] if jnp.issubdtype(dtype, jnp.floating) else [False, True])
      for left_side, a_shape, b_shape in [
          (False, (4, 4), (4,)),
          (False, (4, 4), (1, 4,)),
          (False, (3, 3), (4, 3)),
          (True, (4, 4), (4,)),
          (True, (4, 4), (4, 1)),
          (True, (4, 4), (4, 3)),
          (True, (2, 8, 8), (2, 8, 10)),
      ]))
  def testTriangularSolveGrad(
      self, lower, transpose_a, conjugate_a, unit_diagonal, left_side, a_shape,
      b_shape, dtype):
    rng = jtu.rand_default(self.rng())
    # Test lax.linalg.triangular_solve instead of scipy.linalg.solve_triangular
    # because it exposes more options.
    A = jnp.tril(rng(a_shape, dtype) + 5 * np.eye(a_shape[-1], dtype=dtype))
    A = A if lower else T(A)
    B = rng(b_shape, dtype)
    f = partial(lax.linalg.triangular_solve, lower=lower, transpose_a=transpose_a,
                conjugate_a=conjugate_a, unit_diagonal=unit_diagonal,
                left_side=left_side)
    jtu.check_grads(f, (A, B), 2, rtol=4e-2, eps=1e-3)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_A={}_B={}_bdim={}_leftside={}".format(
           a_shape, b_shape, bdims, left_side),
       "left_side": left_side, "a_shape": a_shape, "b_shape": b_shape,
       "bdims": bdims}
      for left_side, a_shape, b_shape, bdims in [
          (False, (4, 4), (2, 3, 4,), (None, 0)),
          (False, (2, 4, 4), (2, 2, 3, 4,), (None, 0)),
          (False, (2, 4, 4), (3, 4,), (0, None)),
          (False, (2, 4, 4), (2, 3, 4,), (0, 0)),
          (True, (2, 4, 4), (2, 4, 3), (0, 0)),
          (True, (2, 4, 4), (2, 2, 4, 3), (None, 0)),
      ]))
  def testTriangularSolveBatching(self, left_side, a_shape, b_shape, bdims):
    rng = jtu.rand_default(self.rng())
    A = jnp.tril(rng(a_shape, np.float32)
                + 5 * np.eye(a_shape[-1], dtype=np.float32))
    B = rng(b_shape, np.float32)
    solve = partial(lax.linalg.triangular_solve, lower=True, transpose_a=False,
                    conjugate_a=False, unit_diagonal=False, left_side=left_side)
    X = vmap(solve, bdims)(A, B)
    matmul = partial(jnp.matmul, precision=lax.Precision.HIGHEST)
    Y = matmul(A, X) if left_side else matmul(X, A)
    np.testing.assert_allclose(Y - B, 0, atol=1e-4)

  def testTriangularSolveGradPrecision(self):
    rng = jtu.rand_default(self.rng())
    a = jnp.tril(rng((3, 3), np.float32))
    b = rng((1, 3), np.float32)
    jtu.assert_dot_precision(
        lax.Precision.HIGHEST,
        partial(jvp, lax.linalg.triangular_solve),
        (a, b),
        (a, b))

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
       "_n={}".format(jtu.format_shape_dtype_string((n,n), dtype)),
       "n": n, "dtype": dtype}
      for n in [1, 4, 5, 20, 50, 100]
      for dtype in float_types + complex_types))
  def testExpm(self, n, dtype):
    rng = jtu.rand_small(self.rng())
    args_maker = lambda: [rng((n, n), dtype)]

    osp_fun = lambda a: osp.linalg.expm(a)
    jsp_fun = lambda a: jsp.linalg.expm(a)
    self._CheckAgainstNumpy(osp_fun, jsp_fun, args_maker)
    self._CompileAndCheck(jsp_fun, args_maker)

    args_maker_triu = lambda: [np.triu(rng((n, n), dtype))]
    jsp_fun_triu = lambda a: jsp.linalg.expm(a, upper_triangular=True)
    self._CheckAgainstNumpy(osp_fun, jsp_fun_triu, args_maker_triu)
    self._CompileAndCheck(jsp_fun_triu, args_maker_triu)

  @parameterized.named_parameters(jtu.cases_from_list(
    {"testcase_name":
     "_n={}".format(jtu.format_shape_dtype_string((n,n), dtype)),
     "n": n, "dtype": dtype}
    for n in [1, 4, 5, 20, 50, 100]
    for dtype in float_types + complex_types
  ))
  def testIssue2131(self, n, dtype):
    args_maker_zeros = lambda: [np.zeros((n, n), dtype)]
    osp_fun = lambda a: osp.linalg.expm(a)
    jsp_fun = lambda a: jsp.linalg.expm(a)
    self._CheckAgainstNumpy(osp_fun, jsp_fun, args_maker_zeros)
    self._CompileAndCheck(jsp_fun, args_maker_zeros)

  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name": "_lhs={}_rhs={}_lower={}".format(
          jtu.format_shape_dtype_string(lhs_shape, dtype),
          jtu.format_shape_dtype_string(rhs_shape, dtype),
          lower),
       "lhs_shape": lhs_shape, "rhs_shape": rhs_shape, "dtype": dtype,
       "lower": lower}
      for lhs_shape, rhs_shape in [
          [(1, 1), (1,)],
          [(4, 4), (4,)],
          [(4, 4), (4, 4)],
      ]
      for dtype in float_types
      for lower in [True, False]))
  def testChoSolve(self, lhs_shape, rhs_shape, dtype, lower):
    rng = jtu.rand_default(self.rng())
    def args_maker():
      b = rng(rhs_shape, dtype)
      if lower:
        L = np.tril(rng(lhs_shape, dtype))
        return [(L, lower), b]
      else:
        U = np.triu(rng(lhs_shape, dtype))
        return [(U, lower), b]
    self._CheckAgainstNumpy(osp.linalg.cho_solve, jsp.linalg.cho_solve,
                            args_maker, tol=1e-3)


  @parameterized.named_parameters(jtu.cases_from_list(
      {"testcase_name":
        "_n={}".format(jtu.format_shape_dtype_string((n,n), dtype)),
       "n": n, "dtype": dtype}
      for n in [1, 4, 5, 20, 50, 100]
      for dtype in float_types + complex_types))
  def testExpmFrechet(self, n, dtype):
    rng = jtu.rand_small(self.rng())
    if dtype == np.float64 or dtype == np.complex128:
      target_norms = [1.0e-2, 2.0e-1, 9.0e-01, 2.0, 3.0]
      # TODO(zhangqiaorjc): Reduce tol to default 1e-15.
      tol = {
        np.dtype(np.float64): 1e-14,
        np.dtype(np.complex128): 1e-14,
      }
    elif dtype == np.float32 or dtype == np.complex64:
      target_norms = [4.0e-1, 1.0, 3.0]
      tol = None
    else:
      raise TypeError("dtype={} is not supported.".format(dtype))
    for norm in target_norms:
      def args_maker():
        a = rng((n, n), dtype)
        a = a / np.linalg.norm(a, 1) * norm
        e = rng((n, n), dtype)
        return [a, e, ]

      #compute_expm is True
      osp_fun = lambda a,e: osp.linalg.expm_frechet(a,e,compute_expm=True)
      jsp_fun = lambda a,e: jsp.linalg.expm_frechet(a,e,compute_expm=True)
      self._CheckAgainstNumpy(osp_fun, jsp_fun, args_maker,
                              check_dtypes=False, tol=tol)
      self._CompileAndCheck(jsp_fun, args_maker, check_dtypes=False)
      #compute_expm is False
      osp_fun = lambda a,e: osp.linalg.expm_frechet(a,e,compute_expm=False)
      jsp_fun = lambda a,e: jsp.linalg.expm_frechet(a,e,compute_expm=False)
      self._CheckAgainstNumpy(osp_fun, jsp_fun, args_maker,
                              check_dtypes=False, tol=tol)
      self._CompileAndCheck(jsp_fun, args_maker, check_dtypes=False)

  @parameterized.named_parameters(jtu.cases_from_list(
     {"testcase_name":
       "_n={}".format(jtu.format_shape_dtype_string((n,n), dtype)),
      "n": n, "dtype": dtype}
     for n in [1, 4, 5, 20, 50]
     for dtype in float_types + complex_types))
  def testExpmGrad(self, n, dtype):
    rng = jtu.rand_small(self.rng())
    a = rng((n, n), dtype)
    if dtype == np.float64 or dtype == np.complex128:
      target_norms = [1.0e-2, 2.0e-1, 9.0e-01, 2.0, 3.0]
    elif dtype == np.float32 or dtype == np.complex64:
      target_norms = [4.0e-1, 1.0, 3.0]
    else:
      raise TypeError("dtype={} is not supported.".format(dtype))
    # TODO(zhangqiaorjc): Reduce tol to default 1e-5.
    # Lower tolerance is due to 2nd order derivative.
    tol = {
      # Note that due to inner_product, float and complex tol are coupled.
      np.dtype(np.float32): 0.02,
      np.dtype(np.complex64): 0.02,
      np.dtype(np.float64): 1e-4,
      np.dtype(np.complex128): 1e-4,
    }
    for norm in target_norms:
      a = a / np.linalg.norm(a, 1) * norm
      def expm(x):
        return jsp.linalg.expm(x, upper_triangular=False, max_squarings=16)
      jtu.check_grads(expm, (a,), modes=["fwd", "rev"], order=1, atol=tol,
                      rtol=tol)

class LaxLinalgTest(jtu.JaxTestCase):

  def run_test(self, alpha, beta):
    n = alpha.shape[-1]
    # scipy.linalg.eigh_tridiagonal doesn't support complex inputs, so for
    # this we call the slower numpy.linalg.eigh.
    if np.issubdtype(alpha.dtype, np.complexfloating):
      tridiagonal = np.diag(alpha) + np.diag(beta, 1) + np.diag(
          np.conj(beta), -1)
      eigvals_expected, _ = np.linalg.eigh(tridiagonal)
    else:
      eigvals_expected = scipy.linalg.eigh_tridiagonal(
          alpha, beta, eigvals_only=True)
    eigvals = jax.scipy.linalg.eigh_tridiagonal(
        alpha, beta, eigvals_only=True)
    finfo = np.finfo(alpha.dtype)
    atol = 4 * np.sqrt(n) * finfo.eps * np.amax(np.abs(eigvals_expected))
    self.assertAllClose(eigvals_expected, eigvals, atol=atol, rtol=1e-4)

  @parameterized.named_parameters(jtu.cases_from_list(
     {"testcase_name": f"_n={n}_dtype={dtype.__name__}",
      "n": n, "dtype": dtype}
     for n in [1, 2, 3, 7, 8, 100]
     for dtype in float_types + complex_types))
  def testToeplitz(self, n, dtype):
    for a, b in [[2, -1], [1, 0], [0, 1], [-1e10, 1e10], [-1e-10, 1e-10]]:
      alpha = a * np.ones([n], dtype=dtype)
      beta = b * np.ones([n - 1], dtype=dtype)
      self.run_test(alpha, beta)

  @parameterized.named_parameters(jtu.cases_from_list(
     {"testcase_name": f"_n={n}_dtype={dtype.__name__}",
      "n": n, "dtype": dtype}
     for n in [1, 2, 3, 7, 8, 100]
     for dtype in float_types + complex_types))
  def testRandomUniform(self, n, dtype):
    alpha = jtu.rand_uniform(self.rng())((n,), dtype)
    beta = jtu.rand_uniform(self.rng())((n - 1,), dtype)
    self.run_test(alpha, beta)

  @parameterized.named_parameters(jtu.cases_from_list(
     {"testcase_name": f"_dtype={dtype.__name__}",
      "dtype": dtype}
     for dtype in float_types + complex_types))
  def testSelect(self, dtype):
    n = 5
    alpha = jtu.rand_uniform(self.rng())((n,), dtype)
    beta = jtu.rand_uniform(self.rng())((n - 1,), dtype)
    eigvals_all = jax.scipy.linalg.eigh_tridiagonal(alpha, beta, select="a",
                                                    eigvals_only=True)
    eps = np.finfo(alpha.dtype).eps
    atol = 2 * n * eps
    for first in range(n - 1):
      for last in range(first + 1, n - 1):
        # Check that we get the expected eigenvalues by selecting by
        # index range.
        eigvals_index = jax.scipy.linalg.eigh_tridiagonal(
            alpha, beta, select="i", select_range=(first, last),
            eigvals_only=True)
        self.assertAllClose(
            eigvals_all[first:(last + 1)], eigvals_index, atol=atol)

  @parameterized.parameters(np.float32, np.float64)
  def test_tridiagonal_solve(self, dtype):
    dl = np.array([0.0, 2.0, 3.0], dtype=dtype)
    d = np.ones(3, dtype=dtype)
    du = np.array([1.0, 2.0, 0.0], dtype=dtype)
    m = 3
    B = np.ones([m, 1], dtype=dtype)
    X = lax.linalg.tridiagonal_solve(dl, d, du, B)
    A = np.eye(3, dtype=dtype)
    A[[1, 2], [0, 1]] = dl[1:]
    A[[0, 1], [1, 2]] = du[:-1]
    np.testing.assert_allclose(A @ X, B, rtol=1e-6, atol=1e-6)

  @parameterized.named_parameters(
        jtu.cases_from_list({
            "testcase_name":
            "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
            "shape": shape, "dtype": dtype
        } for shape in [(4, 4), (15, 15), (50, 50), (100, 100)]
                            for dtype in float_types + complex_types))
  @jtu.skip_on_devices("gpu", "tpu")
  def testSchur(self, shape, dtype):
      rng = jtu.rand_default(self.rng())
      args_maker = lambda: [rng(shape, dtype)]

      self._CheckAgainstNumpy(osp.linalg.schur, lax.linalg.schur, args_maker)
      self._CompileAndCheck(lax.linalg.schur, args_maker)

  @parameterized.named_parameters(
      jtu.cases_from_list({
          "testcase_name":
          "_shape={}".format(jtu.format_shape_dtype_string(shape, dtype)),
          "shape": shape, "dtype": dtype
      } for shape in [(2, 2), (4, 4), (15, 15), (50, 50), (100, 100)]
                          for dtype in float_types + complex_types))
  @jtu.skip_on_devices("gpu", "tpu")
  def testSchurBatching(self, shape, dtype):
      rng = jtu.rand_default(self.rng())
      batch_size = 10
      shape = (batch_size, ) + shape
      args = rng(shape, dtype)
      reconstruct = vmap(lambda S, T: S @ T @ jnp.conj(S.T))

      Ts, Ss = vmap(lax.linalg.schur)(args)
      self.assertAllClose(reconstruct(Ss, Ts), args, atol=1e-4)

if __name__ == "__main__":
  absltest.main(testLoader=jtu.JaxTestLoader())