/* Copyright 2015 The TensorFlow Authors. 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.
==============================================================================*/

#include "tensorflow/core/framework/common_shape_fns.h"
#include "tensorflow/core/framework/op.h"
#include "tensorflow/core/framework/shape_inference.h"

namespace tensorflow {

using shape_inference::DimensionHandle;
using shape_inference::InferenceContext;
using shape_inference::ShapeHandle;

namespace {

// Return in <out> the result of making the end of <s> a square matrix.
Status MakeBatchSquareMatrix(InferenceContext* c, ShapeHandle input,
                             ShapeHandle* out) {
  ShapeHandle s;
  TF_RETURN_IF_ERROR(c->WithRankAtLeast(input, 2, &s));

  DimensionHandle d;
  TF_RETURN_IF_ERROR(c->Merge(c->Dim(s, -2), c->Dim(s, -1), &d));

  ShapeHandle batch_shape;
  TF_RETURN_IF_ERROR(c->Subshape(s, 0, -2, &batch_shape));
  TF_RETURN_IF_ERROR(c->Concatenate(batch_shape, c->Matrix(d, d), out));
  return Status::OK();
}

Status BatchUnchangedSquareShapeFn(InferenceContext* c) {
  ShapeHandle out;
  TF_RETURN_IF_ERROR(MakeBatchSquareMatrix(c, c->input(0), &out));
  c->set_output(0, out);
  return Status::OK();
}

// The first input is [...,M,N] and second input is either [...,M,K] or [...,M].
// Output is [...,N,K] or [...,N]. If <square>, then input is [...,M,M].
Status MatrixSolveShapeFn(InferenceContext* c, bool square) {
  ShapeHandle lhs;
  ShapeHandle rhs;
  if (square) {
    TF_RETURN_IF_ERROR(MakeBatchSquareMatrix(c, c->input(0), &lhs));
  } else {
    TF_RETURN_IF_ERROR(c->WithRankAtLeast(c->input(0), 2, &lhs));
  }
  TF_RETURN_IF_ERROR(c->WithRankAtLeast(c->input(1), 2, &rhs));

  ShapeHandle lhs_batch_shape;
  ShapeHandle rhs_batch_shape;
  // Make the common batch subshape.
  TF_RETURN_IF_ERROR(c->Subshape(lhs, 0, -2, &lhs_batch_shape));
  TF_RETURN_IF_ERROR(c->Subshape(rhs, 0, -2, &rhs_batch_shape));
  // Make sure the batch dimensions match between lhs and rhs.
  TF_RETURN_IF_ERROR(
      c->Merge(lhs_batch_shape, rhs_batch_shape, &lhs_batch_shape));

  DimensionHandle m;
  // lhs and rhs have the same value for m to be compatible.
  TF_RETURN_IF_ERROR(c->Merge(c->Dim(lhs, -2), c->Dim(rhs, -2), &m));
  DimensionHandle n = c->Dim(lhs, -1);
  if (square) {
    TF_RETURN_IF_ERROR(c->Merge(m, n, &n));
  }

  ShapeHandle out;
  // Build final shape (batch_shape + n + k) in <out>.
  TF_RETURN_IF_ERROR(c->Concatenate(lhs_batch_shape, c->Vector(n), &out));
  TF_RETURN_IF_ERROR(c->Concatenate(out, c->Vector(c->Dim(rhs, -1)), &out));
  c->set_output(0, out);
  return Status::OK();
}

// Input is [...,N,N]. Outputs are:
//   [...,N];[0], if compute_v is false,
//   [...,N];[...,N,N], if compute_v is true.
Status SelfAdjointEigV2ShapeFn(InferenceContext* c) {
  ShapeHandle input;
  TF_RETURN_IF_ERROR(MakeBatchSquareMatrix(c, c->input(0), &input));
  DimensionHandle n;
  TF_RETURN_IF_ERROR(c->Merge(c->Dim(input, -2), c->Dim(input, -1), &n));
  ShapeHandle batch_shape;
  TF_RETURN_IF_ERROR(c->Subshape(input, 0, -2, &batch_shape));
  ShapeHandle e_shape;
  TF_RETURN_IF_ERROR(c->Concatenate(batch_shape, c->Vector(n), &e_shape));
  c->set_output(0, e_shape);
  bool compute_v;
  TF_RETURN_IF_ERROR(c->GetAttr("compute_v", &compute_v));
  if (compute_v) {
    ShapeHandle v_shape;
    TF_RETURN_IF_ERROR(c->Concatenate(batch_shape, c->Matrix(n, n), &v_shape));
    c->set_output(1, v_shape);
  } else {
    c->set_output(1, c->Vector(0ll));
  }
  return Status::OK();
}

// Input is [...,M,N].
// First and second outputs are:
//   [...,M,M]; [...,M,N], if full_matrices is true,
//   [...,M,P]; [...,P,N], if full_matrices is false,
// where P = min(M,N).
Status QrShapeFn(InferenceContext* c) {
  ShapeHandle input;
  TF_RETURN_IF_ERROR(c->WithRankAtLeast(c->input(0), 2, &input));
  DimensionHandle m = c->Dim(input, -2);
  DimensionHandle n = c->Dim(input, -1);
  DimensionHandle p;
  TF_RETURN_IF_ERROR(c->Min(m, n, &p));
  ShapeHandle batch_shape;
  TF_RETURN_IF_ERROR(c->Subshape(input, 0, -2, &batch_shape));
  ShapeHandle q_shape;
  ShapeHandle r_shape;
  bool full_matrices;
  TF_RETURN_IF_ERROR(c->GetAttr("full_matrices", &full_matrices));
  if (full_matrices) {
    TF_RETURN_IF_ERROR(c->Concatenate(batch_shape, c->Matrix(m, m), &q_shape));
    TF_RETURN_IF_ERROR(c->Concatenate(batch_shape, c->Matrix(m, n), &r_shape));
  } else {
    TF_RETURN_IF_ERROR(c->Concatenate(batch_shape, c->Matrix(m, p), &q_shape));
    TF_RETURN_IF_ERROR(c->Concatenate(batch_shape, c->Matrix(p, n), &r_shape));
  }
  c->set_output(0, q_shape);
  c->set_output(1, r_shape);
  return Status::OK();
}

// Input is [...,M,N].  First output is [...,min(M,N)].
// Second and third outputs are:
//   [0]; [0], if compute_uv is false.
//   [...,M,M]; [...,N,N], if compute_uv is true and full_matrices is true,
//   [...,M,P]; [...,N,P], if compute_uv is true and full_matrices is false,
// where P = min(M,N).
Status SvdShapeFn(InferenceContext* c) {
  ShapeHandle input;
  TF_RETURN_IF_ERROR(c->WithRankAtLeast(c->input(0), 2, &input));
  DimensionHandle m = c->Dim(input, -2);
  DimensionHandle n = c->Dim(input, -1);
  DimensionHandle p;
  TF_RETURN_IF_ERROR(c->Min(m, n, &p));
  ShapeHandle batch_shape;
  TF_RETURN_IF_ERROR(c->Subshape(input, 0, -2, &batch_shape));
  ShapeHandle e_shape;
  TF_RETURN_IF_ERROR(c->Concatenate(batch_shape, c->Vector(p), &e_shape));
  c->set_output(0, e_shape);
  bool compute_uv;
  TF_RETURN_IF_ERROR(c->GetAttr("compute_uv", &compute_uv));
  if (compute_uv) {
    ShapeHandle u_shape;
    ShapeHandle v_shape;
    bool full_matrices;
    TF_RETURN_IF_ERROR(c->GetAttr("full_matrices", &full_matrices));
    if (full_matrices) {
      TF_RETURN_IF_ERROR(
          c->Concatenate(batch_shape, c->Matrix(m, m), &u_shape));
      TF_RETURN_IF_ERROR(
          c->Concatenate(batch_shape, c->Matrix(n, n), &v_shape));
    } else {
      TF_RETURN_IF_ERROR(
          c->Concatenate(batch_shape, c->Matrix(m, p), &u_shape));
      TF_RETURN_IF_ERROR(
          c->Concatenate(batch_shape, c->Matrix(n, p), &v_shape));
    }
    c->set_output(1, u_shape);
    c->set_output(2, v_shape);
  } else {
    c->set_output(1, c->Vector(0ll));
    c->set_output(2, c->Vector(0ll));
  }
  return Status::OK();
}

}  // namespace

REGISTER_OP("MatrixDeterminant")
    .Input("input: T")
    .Output("output: T")
    .Attr("T: {float, double, complex64, complex128}")
    .SetShapeFn([](InferenceContext* c) {
      ShapeHandle input;
      TF_RETURN_IF_ERROR(c->WithRankAtLeast(c->input(0), 2, &input));

      DimensionHandle unused;
      TF_RETURN_IF_ERROR(
          c->Merge(c->Dim(input, -1), c->Dim(input, -2), &unused));

      ShapeHandle out;
      TF_RETURN_IF_ERROR(c->Subshape(input, 0, -2, &out));
      c->set_output(0, out);
      return Status::OK();
    })
    .Doc(R"doc(
Computes the determinant of one or more square matrices.

The input is a tensor of shape `[..., M, M]` whose inner-most 2 dimensions
form square matrices. The output is a tensor containing the determinants
for all input submatrices `[..., :, :]`.

input: Shape is `[..., M, M]`.
output: Shape is `[...]`.
)doc");

REGISTER_OP("LogMatrixDeterminant")
    .Input("input: T")
    .Output("sign: T")
    .Output("log_abs_determinant: T")
    .Attr("T: {float, double, complex64, complex128}")
    .SetShapeFn([](InferenceContext* c) {
      ShapeHandle input;
      TF_RETURN_IF_ERROR(c->WithRankAtLeast(c->input(0), 2, &input));

      DimensionHandle unused;
      TF_RETURN_IF_ERROR(
          c->Merge(c->Dim(input, -1), c->Dim(input, -2), &unused));

      ShapeHandle s;
      TF_RETURN_IF_ERROR(c->Subshape(input, 0, -2, &s));
      c->set_output(0, s);

      ShapeHandle out;
      TF_RETURN_IF_ERROR(c->Subshape(input, 0, -2, &out));
      c->set_output(1, out);
      return Status::OK();
    })
    .Doc(R"doc(
Computes the sign and the log of the absolute value of the determinant of
one or more square matrices.

The input is a tensor of shape `[N, M, M]` whose inner-most 2 dimensions
form square matrices. The outputs are two tensors containing the signs and
absolute values of the log determinants for all N input submatrices
`[..., :, :]` such that the determinant = sign*exp(log_abs_determinant).
The log_abs_determinant is computed as det(P)*sum(log(diag(LU))) where LU
is the LU decomposition of the input and P is the corresponding
permutation matrix.

input: Shape is `[N, M, M]`.
sign: The signs of the log determinants of the inputs. Shape is `[N]`.
log_abs_determinant: The logs of the absolute values of the determinants
of the N input matrices.  Shape is `[N]`.
)doc");

REGISTER_OP("MatrixInverse")
    .Input("input: T")
    .Output("output: T")
    .Attr("adjoint: bool = False")
    .Attr("T: {double, float, complex64, complex128}")
    .SetShapeFn(BatchUnchangedSquareShapeFn)
    .Doc(R"doc(
Computes the inverse of one or more square invertible matrices or their
adjoints (conjugate transposes).

The input is a tensor of shape `[..., M, M]` whose inner-most 2 dimensions
form square matrices. The output is a tensor of the same shape as the input
containing the inverse for all input submatrices `[..., :, :]`.

The op uses LU decomposition with partial pivoting to compute the inverses.

If a matrix is not invertible there is no guarantee what the op does. It
may detect the condition and raise an exception or it may simply return a
garbage result.

input: Shape is `[..., M, M]`.
output: Shape is `[..., M, M]`.

@compatibility(numpy)
Equivalent to np.linalg.inv
@end_compatibility
)doc");

REGISTER_OP("MatrixExponential")
    .Input("input: T")
    .Output("output: T")
    .Attr("T: {double, float, complex64, complex128}")
    .SetShapeFn(BatchUnchangedSquareShapeFn)
    .Doc(R"doc(
Computes the matrix exponential of one or more square matrices:

exp(A) = \sum_{n=0}^\infty A^n/n!

The exponential is computed using a combination of the scaling and squaring
method and the Pade approximation. Details can be founds in:
Nicholas J. Higham, "The scaling and squaring method for the matrix exponential
revisited," SIAM J. Matrix Anal. Applic., 26:1179-1193, 2005.

The input is a tensor of shape `[..., M, M]` whose inner-most 2 dimensions
form square matrices. The output is a tensor of the same shape as the input
containing the exponential for all input submatrices `[..., :, :]`.

input: Shape is `[..., M, M]`.
output: Shape is `[..., M, M]`.

@compatibility(scipy)
Equivalent to scipy.linalg.expm
@end_compatibility
)doc");

REGISTER_OP("Cholesky")
    .Input("input: T")
    .Output("output: T")
    .Attr("T: {double, float, complex64, complex128}")
    .SetShapeFn(BatchUnchangedSquareShapeFn)
    .Doc(R"doc(
Computes the Cholesky decomposition of one or more square matrices.

The input is a tensor of shape `[..., M, M]` whose inner-most 2 dimensions
form square matrices.

The input has to be symmetric and positive definite. Only the lower-triangular
part of the input will be used for this operation. The upper-triangular part
will not be read.

The output is a tensor of the same shape as the input
containing the Cholesky decompositions for all input submatrices `[..., :, :]`.

**Note**: The gradient computation on GPU is faster for large matrices but
not for large batch dimensions when the submatrices are small. In this
case it might be faster to use the CPU.

input: Shape is `[..., M, M]`.
output: Shape is `[..., M, M]`.
)doc");

REGISTER_OP("CholeskyGrad")
    .Input("l: T")
    .Input("grad: T")
    .Output("output: T")
    .Attr("T: {float, double}")
    .SetShapeFn(BatchUnchangedSquareShapeFn)
    .Doc(R"doc(
Computes the reverse mode backpropagated gradient of the Cholesky algorithm.

For an explanation see "Differentiation of the Cholesky algorithm" by
Iain Murray http://arxiv.org/abs/1602.07527.

l: Output of batch Cholesky algorithm l = cholesky(A). Shape is `[..., M, M]`.
  Algorithm depends only on lower triangular part of the innermost matrices of
  this tensor.
grad: df/dl where f is some scalar function. Shape is `[..., M, M]`.
  Algorithm depends only on lower triangular part of the innermost matrices of
  this tensor.
output: Symmetrized version of df/dA . Shape is `[..., M, M]`
)doc");

REGISTER_OP("SelfAdjointEig")
    .Input("input: T")
    .Output("output: T")
    .Attr("T: {double, float}")
    .Deprecated(11, "Use SelfAdjointEigV2 instead.")
    .SetShapeFn([](InferenceContext* c) {
      ShapeHandle input;
      TF_RETURN_IF_ERROR(MakeBatchSquareMatrix(c, c->input(0), &input));

      DimensionHandle d = c->Dim(input, -1);
      DimensionHandle d_plus_1;
      TF_RETURN_IF_ERROR(c->Add(d, 1, &d_plus_1));

      ShapeHandle s;
      TF_RETURN_IF_ERROR(c->Subshape(input, 0, -2, &s));
      TF_RETURN_IF_ERROR(c->Concatenate(s, c->Matrix(d_plus_1, d), &s));
      c->set_output(0, s);
      return Status::OK();
    })
    .Doc(R"doc(
Computes the Eigen Decomposition of a batch of square self-adjoint matrices.

The input is a tensor of shape `[..., M, M]` whose inner-most 2 dimensions
form square matrices, with the same constraints as the single matrix
SelfAdjointEig.

The result is a [..., M+1, M] matrix with [..., 0,:] containing the
eigenvalues, and subsequent [...,1:, :] containing the eigenvectors.

input: Shape is `[..., M, M]`.
output: Shape is `[..., M+1, M]`.
)doc");

REGISTER_OP("SelfAdjointEigV2")
    .Input("input: T")
    .Output("e: T")
    .Output("v: T")
    .Attr("compute_v: bool = True")
    .Attr("T: {double, float, complex64, complex128}")
    .SetShapeFn(SelfAdjointEigV2ShapeFn)
    .Doc(R"doc(
Computes the eigen decomposition of one or more square self-adjoint matrices.

Computes the eigenvalues and (optionally) eigenvectors of each inner matrix in
`input` such that `input[..., :, :] = v[..., :, :] * diag(e[..., :])`.

```python
# a is a tensor.
# e is a tensor of eigenvalues.
# v is a tensor of eigenvectors.
e, v = self_adjoint_eig(a)
e = self_adjoint_eig(a, compute_v=False)
```

input: `Tensor` input of shape `[N, N]`.
compute_v: If `True` then eigenvectors will be computed and returned in `v`.
  Otherwise, only the eigenvalues will be computed.
e: Eigenvalues. Shape is `[N]`.
v: Eigenvectors. Shape is `[N, N]`.
)doc");

REGISTER_OP("MatrixSolve")
    .Input("matrix: T")
    .Input("rhs: T")
    .Output("output: T")
    .Attr("adjoint: bool = False")
    .Attr("T: {double, float, complex64, complex128}")
    .SetShapeFn([](InferenceContext* c) {
      return MatrixSolveShapeFn(c, true /* square (*/);
    })
    .Doc(R"doc(
Solves systems of linear equations.

`Matrix` is a tensor of shape `[..., M, M]` whose inner-most 2 dimensions
form square matrices. `Rhs` is a tensor of shape `[..., M, K]`. The `output` is
a tensor shape `[..., M, K]`.  If `adjoint` is `False` then each output matrix
satisfies `matrix[..., :, :] * output[..., :, :] = rhs[..., :, :]`.
If `adjoint` is `True` then each output matrix satisfies
`adjoint(matrix[..., :, :]) * output[..., :, :] = rhs[..., :, :]`.

matrix: Shape is `[..., M, M]`.
rhs: Shape is `[..., M, K]`.
output: Shape is `[..., M, K]`.
adjoint: Boolean indicating whether to solve with `matrix` or its (block-wise)
         adjoint.
)doc");

REGISTER_OP("MatrixTriangularSolve")
    .Input("matrix: T")
    .Input("rhs: T")
    .Output("output: T")
    .Attr("lower: bool = True")
    .Attr("adjoint: bool = False")
    .Attr("T: {double, float, complex64, complex128}")
    .SetShapeFn([](InferenceContext* c) {
      return MatrixSolveShapeFn(c, true /* square (*/);
    })
    .Doc(R"doc(
Solves systems of linear equations with upper or lower triangular matrices by
backsubstitution.

`matrix` is a tensor of shape `[..., M, M]` whose inner-most 2 dimensions form
square matrices. If `lower` is `True` then the strictly upper triangular part
of each inner-most matrix is assumed to be zero and not accessed.
If `lower` is False then the strictly lower triangular part of each inner-most
matrix is assumed to be zero and not accessed.
`rhs` is a tensor of shape `[..., M, K]`.

The output is a tensor of shape `[..., M, K]`. If `adjoint` is
`True` then the innermost matrices in `output` satisfy matrix equations
`matrix[..., :, :] * output[..., :, :] = rhs[..., :, :]`.
If `adjoint` is `False` then the strictly then the  innermost matrices in
`output` satisfy matrix equations
`adjoint(matrix[..., i, k]) * output[..., k, j] = rhs[..., i, j]`.

matrix: Shape is `[..., M, M]`.
rhs: Shape is `[..., M, K]`.
output: Shape is `[..., M, K]`.
lower: Boolean indicating whether the innermost matrices in `matrix` are
       lower or upper triangular.
adjoint: Boolean indicating whether to solve with `matrix` or its (block-wise)
         adjoint.

@compatibility(numpy)
Equivalent to np.linalg.triangular_solve
@end_compatibility
)doc");

REGISTER_OP("MatrixSolveLs")
    .Input("matrix: T")
    .Input("rhs: T")
    .Input("l2_regularizer: double")
    .Output("output: T")
    .Attr("T: {double, float, complex64, complex128}")
    .Attr("fast: bool = True")
    .SetShapeFn([](InferenceContext* c) {
      ShapeHandle l2_regularizer;
      TF_RETURN_IF_ERROR(c->WithRank(c->input(2), 0, &l2_regularizer));
      return MatrixSolveShapeFn(c, false /* square */);
    })
    .Doc(R"doc(
Solves one or more linear least-squares problems.

`matrix` is a tensor of shape `[..., M, N]` whose inner-most 2 dimensions
form real or complex matrices of size `[M, N]`. `Rhs` is a tensor of the same
type as `matrix` and shape `[..., M, K]`.
The output is a tensor shape `[..., N, K]` where each output matrix solves
each of the equations
`matrix[..., :, :]` * `output[..., :, :]` = `rhs[..., :, :]`
in the least squares sense.

We use the following notation for (complex) matrix and right-hand sides
in the batch:

`matrix`=\\(A \in \mathbb{C}^{m \times n}\\),
`rhs`=\\(B  \in \mathbb{C}^{m \times k}\\),
`output`=\\(X  \in \mathbb{C}^{n \times k}\\),
`l2_regularizer`=\\(\lambda \in \mathbb{R}\\).

If `fast` is `True`, then the solution is computed by solving the normal
equations using Cholesky decomposition. Specifically, if \\(m \ge n\\) then
\\(X = (A^H A + \lambda I)^{-1} A^H B\\), which solves the least-squares
problem \\(X = \mathrm{argmin}_{Z \in \Re^{n \times k} } ||A Z - B||_F^2 +
\lambda ||Z||_F^2\\). If \\(m \lt n\\) then `output` is computed as
\\(X = A^H (A A^H + \lambda I)^{-1} B\\), which (for \\(\lambda = 0\\)) is the
minimum-norm solution to the under-determined linear system, i.e.
\\(X = \mathrm{argmin}_{Z \in \mathbb{C}^{n \times k} } ||Z||_F^2 \\),
subject to \\(A Z = B\\). Notice that the fast path is only numerically stable
when \\(A\\) is numerically full rank and has a condition number
\\(\mathrm{cond}(A) \lt \frac{1}{\sqrt{\epsilon_{mach} } }\\) or\\(\lambda\\) is
sufficiently large.

If `fast` is `False` an algorithm based on the numerically robust complete
orthogonal decomposition is used. This computes the minimum-norm
least-squares solution, even when \\(A\\) is rank deficient. This path is
typically 6-7 times slower than the fast path. If `fast` is `False` then
`l2_regularizer` is ignored.

matrix: Shape is `[..., M, N]`.
rhs: Shape is `[..., M, K]`.
output: Shape is `[..., N, K]`.
l2_regularizer: Scalar tensor.

@compatibility(numpy)
Equivalent to np.linalg.lstsq
@end_compatibility
)doc");

REGISTER_OP("Qr")
    .Input("input: T")
    .Output("q: T")
    .Output("r: T")
    .Attr("full_matrices: bool = False")
    .Attr("T: {double, float, complex64, complex128}")
    .SetShapeFn(QrShapeFn)
    .Doc(R"doc(
Computes the QR decompositions of one or more matrices.

Computes the QR decomposition of each inner matrix in `tensor` such that
`tensor[..., :, :] = q[..., :, :] * r[..., :,:])`

```python
# a is a tensor.
# q is a tensor of orthonormal matrices.
# r is a tensor of upper triangular matrices.
q, r = qr(a)
q_full, r_full = qr(a, full_matrices=True)
```

input: A tensor of shape `[..., M, N]` whose inner-most 2 dimensions
  form matrices of size `[M, N]`. Let `P` be the minimum of `M` and `N`.
q: Orthonormal basis for range of `a`. If `full_matrices` is `False` then
  shape is `[..., M, P]`; if `full_matrices` is `True` then shape is
  `[..., M, M]`.
r: Triangular factor. If `full_matrices` is `False` then shape is
  `[..., P, N]`. If `full_matrices` is `True` then shape is `[..., M, N]`.
full_matrices: If true, compute full-sized `q` and `r`. If false
  (the default), compute only the leading `P` columns of `q`.
)doc");

REGISTER_OP("Svd")
    .Input("input: T")
    .Output("s: T")
    .Output("u: T")
    .Output("v: T")
    .Attr("compute_uv: bool = True")
    .Attr("full_matrices: bool = False")
    .Attr("T: {double, float, complex64, complex128}")
    .SetShapeFn(SvdShapeFn)
    .Doc(R"doc(
Computes the singular value decompositions of one or more matrices.

Computes the SVD of each inner matrix in `input` such that
`input[..., :, :] = u[..., :, :] * diag(s[..., :, :]) * transpose(v[..., :, :])`

```python
# a is a tensor containing a batch of matrices.
# s is a tensor of singular values for each matrix.
# u is the tensor containing of left singular vectors for each matrix.
# v is the tensor containing of right singular vectors for each matrix.
s, u, v = svd(a)
s, _, _ = svd(a, compute_uv=False)
```

input: A tensor of shape `[..., M, N]` whose inner-most 2 dimensions
  form matrices of size `[M, N]`. Let `P` be the minimum of `M` and `N`.
s: Singular values. Shape is `[..., P]`.
u: Left singular vectors. If `full_matrices` is `False` then shape is
  `[..., M, P]`; if `full_matrices` is `True` then shape is
  `[..., M, M]`. Undefined if `compute_uv` is `False`.
v: Left singular vectors. If `full_matrices` is `False` then shape is
  `[..., N, P]`. If `full_matrices` is `True` then shape is `[..., N, N]`.
  Undefined if `compute_uv` is false.
compute_uv: If true, left and right singular vectors will be
  computed and returned in `u` and `v`, respectively.
  If false, `u` and `v` are not set and should never referenced.
full_matrices: If true, compute full-sized `u` and `v`. If false
  (the default), compute only the leading `P` singular vectors.
  Ignored if `compute_uv` is `False`.
)doc");

// Deprecated op registrations:

// Can be deleted after 3feb2017.
REGISTER_OP("BatchSelfAdjointEig")
    .Input("input: T")
    .Output("output: T")
    .Attr("T: {double, float}")
    .Deprecated(11, "Use SelfAdjointEigV2 instead.")
    .SetShapeFn(shape_inference::UnknownShape);

// Can all be deleted after 9mar2017.
REGISTER_OP("BatchMatrixDeterminant")
    .Input("input: T")
    .Output("output: T")
    .Attr("T: {float, double, complex64, complex128}")
    .Deprecated(13, "Use MatrixDeterminant instead.")
    .SetShapeFn(shape_inference::UnknownShape);

REGISTER_OP("BatchMatrixInverse")
    .Input("input: T")
    .Output("output: T")
    .Attr("adjoint: bool = False")
    .Attr("T: {double, float}")
    .Deprecated(13, "Use MatrixInverse instead.")
    .SetShapeFn(shape_inference::UnknownShape);

REGISTER_OP("BatchCholesky")
    .Input("input: T")
    .Output("output: T")
    .Attr("T: {double, float}")
    .Deprecated(13, "Use Cholesky instead.")
    .SetShapeFn(shape_inference::UnknownShape);

REGISTER_OP("BatchCholeskyGrad")
    .Input("l: T")
    .Input("grad: T")
    .Output("output: T")
    .Attr("T: {float, double}")
    .Deprecated(13, "Use CholeskyGrad instead.")
    .SetShapeFn(shape_inference::UnknownShape);

REGISTER_OP("BatchSelfAdjointEigV2")
    .Input("input: T")
    .Output("e: T")
    .Output("v: T")
    .Attr("compute_v: bool = True")
    .Attr("T: {double, float}")
    .Deprecated(13, "Use SelfAdjointEigV2 instead.")
    .SetShapeFn(shape_inference::UnknownShape);

REGISTER_OP("BatchMatrixSolve")
    .Input("matrix: T")
    .Input("rhs: T")
    .Output("output: T")
    .Attr("adjoint: bool = False")
    .Attr("T: {double, float}")
    .Deprecated(13, "Use MatrixSolve instead.")
    .SetShapeFn(shape_inference::UnknownShape);

REGISTER_OP("BatchMatrixTriangularSolve")
    .Input("matrix: T")
    .Input("rhs: T")
    .Output("output: T")
    .Attr("lower: bool = True")
    .Attr("adjoint: bool = False")
    .Attr("T: {double, float}")
    .Deprecated(13, "Use MatrixTriangularSolve instead.")
    .SetShapeFn(shape_inference::UnknownShape);

REGISTER_OP("BatchMatrixSolveLs")
    .Input("matrix: T")
    .Input("rhs: T")
    .Input("l2_regularizer: double")
    .Output("output: T")
    .Attr("T: {double, float}")
    .Attr("fast: bool = True")
    .Deprecated(13, "Use MatrixSolveLs instead.")
    .SetShapeFn(shape_inference::UnknownShape);

REGISTER_OP("BatchSvd")
    .Input("input: T")
    .Output("s: T")
    .Output("u: T")
    .Output("v: T")
    .Attr("compute_uv: bool = True")
    .Attr("full_matrices: bool = False")
    .Attr("T: {double, float, complex64, complex128}")
    .Deprecated(13, "Use Svd instead.")
    .SetShapeFn(shape_inference::UnknownShape);

}  // namespace tensorflow
