<commit_before># Copyright 2019 DeepMind Technologies Limited. All Rights Reserved.
#
# 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
#
#     http://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.
# ==============================================================================
"""Linear algebra utilities used in optimisation."""

from optparse import Option
import jax
from jax import lax
import jax.numpy as jnp
import chex
import numpy as np
from typing import Optional

from optax._src import base
from optax._src import numerics


def global_norm(updates: base.Updates) -> base.Updates:
  """Compute the global norm across a nested structure of tensors."""
  return jnp.sqrt(sum(
      jnp.sum(numerics.abs_sq(x)) for x in jax.tree_util.tree_leaves(updates)))


def power_iteration(
    matrix: chex.Array,
    num_iters: Optional[int]=100,
    error_tolerance: Optional[float]=1e-6,
    precision: Optional[jnp.float32]=lax.Precision.HIGHEST):
  r"""Power iteration algorithm.
<commit_msg>removing unused optparse import<commit_after># Copyright 2019 DeepMind Technologies Limited. All Rights Reserved.
#
# 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
#
#     http://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.
# ==============================================================================
"""Linear algebra utilities used in optimisation."""

import jax
from jax import lax
import jax.numpy as jnp
import chex
import numpy as np
from typing import Optional

from optax._src import base
from optax._src import numerics


def global_norm(updates: base.Updates) -> base.Updates:
  """Compute the global norm across a nested structure of tensors."""
  return jnp.sqrt(sum(
      jnp.sum(numerics.abs_sq(x)) for x in jax.tree_util.tree_leaves(updates)))


def power_iteration(
    matrix: chex.Array,
    num_iters: Optional[int]=100,
    error_tolerance: Optional[float]=1e-6,
    precision: Optional[jnp.float32]=lax.Precision.HIGHEST):
  r"""Power iteration algorithm.

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