#include <ATen/ATen.h>
#include <ATen/ExpandUtils.h>
#include <ATen/Dispatch.h>
#include <ATen/NativeFunctions.h>
#include <ATen/native/CPUBlas.h>
#include <ATen/native/LinearAlgebraUtils.h>
#include <ATen/TensorUtils.h>
#include <ATen/Parallel.h>
#include <ATen/LegacyTHFunctionsCPU.h>
#include <ATen/core/grad_mode.h>
#include <functional>
#include <numeric>
#include <vector>
#include <limits>
#include <ATen/NamedTensorUtils.h>

namespace at {
namespace native {

// Helper function for det methods.
// For pivoted LU factorization A = P * L * U. Since we always have det(L) = 1,
// det(P) = \pm 1, this method returns a 3-tuple:
//   (det(P), diag(U), info),
// where info helps us identify singular matrices.
static inline std::tuple<Tensor, Tensor> _lu_det_P_diag_U(const Tensor& self) {
  Tensor pivs, lu, infos;
  std::tie(lu, pivs, infos) = at::_lu_with_info(self, /*pivot=*/true, /*check_errors=*/false);
  TORCH_CHECK(infos.ge(0).all().item<uint8_t>(), "Invalid argument passed to lu");
  auto n = self.size(-1);
  auto num_exchanges = (at::arange(1, n + 1, pivs.options()) != pivs).sum(-1, /*keepdim=*/false, /*dtype=*/self.scalar_type()).fmod_(2);
  // NB: the `.contiguous()` call is added due to the bug in `.prod()` as reported in
  // issue #https://github.com/pytorch/pytorch/issues/34061
  auto u_diagonal = lu.diagonal(/*offset=*/0, /*dim1=*/-2, /*dim2=*/-1).contiguous();
  return std::tuple<Tensor, Tensor>(num_exchanges.mul_(-2).add_(1), u_diagonal);
}

// torch.linalg.det, alias for torch.det
Tensor linalg_det(const Tensor& self) {
  return self.det();
}

Tensor det(const Tensor& self) {
  squareCheckInputs(self);
  TORCH_CHECK((at::isFloatingType(self.scalar_type()) || at::isComplexType(self.scalar_type())),
              "Expected a floating point tensor as input");

  Tensor det_P, diag_U;
  std::tie(det_P, diag_U) = _lu_det_P_diag_U(self);
  // complete_det is 0 when U is singular (U(i, i) = 0 for some i in [1, self.size(-1)]).
  // The product accumulation takes care of this case, and hence no special case handling is required.
  auto complete_det = diag_U.prod(-1).mul_(det_P);
  return complete_det;
}

Tensor logdet(const Tensor& self) {
  squareCheckInputs(self);
  TORCH_CHECK((at::isFloatingType(self.scalar_type()) || at::isComplexType(self.scalar_type())),
              "Expected a floating point tensor as input");

  Tensor det_P, diag_U;
  std::tie(det_P, diag_U) = _lu_det_P_diag_U(self);
  Tensor det_sign = diag_U.sign().prod(-1).mul_(det_P);

  // If det_sign > 0, diag_U.abs_().log_().sum(-1) gives logdet (this means U is not singular).
  // If det_sign <= 0, then we get proper nan (when det < 0, i.e., det_sign) or -inf (when det = 0, i.e., U is singular).
  // U is singular when U(i, i) = 0 for some i in [1, self.size(-1)].
  Tensor logdet_vals = diag_U.abs_().log_().sum(-1);
  if (self.dim() > 2) {
    logdet_vals.index_put_((det_sign < 0).nonzero_numpy(), at::full({}, NAN, self.options()));
  } else if (det_sign.item<double>() < 0) {
    logdet_vals.fill_(NAN);
  }
  return logdet_vals;
}

std::tuple<Tensor, Tensor> slogdet(const Tensor& self) {
  squareCheckInputs(self);
  TORCH_CHECK((at::isFloatingType(self.scalar_type()) || at::isComplexType(self.scalar_type())),
              "Expected a floating point tensor as input");

  Tensor det_P, diag_U;
  std::tie(det_P, diag_U) = _lu_det_P_diag_U(self);
  auto det_sign = diag_U.sign().prod(-1).mul_(det_P);
  // abslogdet_val is -inf if U is singular, in which case diag_U.abs_().log_().sum(-1) will return -inf.
  // U is singular when U(i, i) = 0 for some i in [1, self.size(-1)].
  // Since abslogdet_val cannot take nan, no special case handling is required.
  auto abslogdet_val = diag_U.abs_().log_().sum(-1);
  return std::make_tuple(det_sign, abslogdet_val);
}

Tensor pinverse(const Tensor& self, double rcond) {
  TORCH_CHECK((at::isFloatingType(self.scalar_type()) || at::isComplexType(self.scalar_type())) && self.dim() >= 2,
              "pinverse(", self.scalar_type(), "{", self.sizes(), "}): expected a tensor with 2 or more dimensions "
              "of floating types");
  if (self.numel() == 0) {
    // Match NumPy
    auto self_sizes = self.sizes().vec();
    std::swap(self_sizes[self.dim() - 1], self_sizes[self.dim() - 2]);
    return at::empty(self_sizes, self.options());
  }
  Tensor U, S, V;
  std::tie(U, S, V) = self.svd();
  Tensor max_val = at::narrow(S, /*dim=*/-1, /*start=*/0, /*length=*/1);
  Tensor S_pseudoinv = at::where(S > rcond * max_val, S.reciprocal(), at::zeros({}, self.options()));
  return at::matmul(V, at::matmul(S_pseudoinv.diag_embed(/*offset=*/0, /*dim1=*/-2, /*dim2=*/-1), U.transpose(-2, -1)));
}

static inline Tensor _matrix_rank_helper(const Tensor& self, bool symmetric) {
  Tensor S;
  if (!symmetric) {
    Tensor U, V;
    std::tie(U, S, V) = self.svd(/*some=*/true, /*compute_uv=*/false);
  } else {
    Tensor eigvecs;
    std::tie(S, eigvecs) = self.symeig(/*eigenvectors=*/false);
    S = S.abs();
  }
  return S;
}

Tensor matrix_rank(const Tensor& self, double tol, bool symmetric) {
  TORCH_CHECK((at::isFloatingType(self.scalar_type()) || at::isComplexType(self.scalar_type())) && self.dim() == 2,
              "matrix_rank(", self.scalar_type(), "{", self.sizes(), "}): expected a 2D tensor "
              "of floating types");

  Tensor S = _matrix_rank_helper(self, symmetric);
  return (S > tol).sum();
}

Tensor matrix_rank(const Tensor& self, bool symmetric) {
  TORCH_CHECK((at::isFloatingType(self.scalar_type()) || at::isComplexType(self.scalar_type())) && self.dim() == 2,
              "matrix_rank(", self.scalar_type(), "{", self.sizes(), "}): expected a 2D tensor "
              "of floating types");

  Tensor S = _matrix_rank_helper(self, symmetric);
  double tol = _get_epsilon(self.scalar_type()) * std::max(self.size(0), self.size(1));
  return (S > S.max().mul_(tol)).sum();
}

static void check_1d(const Tensor& t, const char* arg, const char* fn) {
 TORCH_CHECK(t.dim() == 1, fn, ": Expected 1-D argument ", arg, ", but got ", t.dim(), "-D");
}

Tensor addr(const Tensor& self, const Tensor& vec1, const Tensor& vec2, Scalar beta, Scalar alpha) {
  check_1d(vec1, "vec1", "addr");
  check_1d(vec2, "vec2", "addr");
  Tensor b_self;
  std::tie(b_self) = expand_size(self, {vec1.size(0), vec2.size(0)}, "addr");
  return at::_addr(b_self, vec1, vec2, beta, alpha);
}

Tensor& addr_(Tensor& self, const Tensor& vec1, const Tensor& vec2, Scalar beta, Scalar alpha) {
  check_1d(vec1, "vec1", "addr");
  check_1d(vec2, "vec2", "addr");
  return at::_addr_(self, vec1, vec2, beta, alpha);
}

Tensor& addr_out(Tensor &result, const Tensor& self, const Tensor& vec1, const Tensor& vec2, Scalar beta, Scalar alpha) {
  check_1d(vec1, "vec1", "addr");
  check_1d(vec2, "vec2", "addr");
  Tensor b_self;
  std::tie(b_self) = expand_size(self, {vec1.size(0), vec2.size(0)}, "addr_out");
  return at::_addr_out(result, b_self, vec1, vec2, beta, alpha);
}

Tensor& ger_out(Tensor &result, const Tensor& self, const Tensor& vec2) {
  check_1d(self, "self", "ger");
  check_1d(vec2, "vec2", "ger");
  if (result.dim() != 2 || result.size(0) != self.size(0) || result.size(1) != vec2.size(0)) {
    result.resize_({ self.size(0), vec2.size(0) });
  }
  // resize_ does the "broadcasting", don't need to broadcast again.
  return at::_addr_out(result, result, self, vec2, Scalar(0), Scalar(1));
}

Tensor ger(const Tensor& self, const Tensor& vec2) {
  Tensor result = at::empty({0}, self.options());
  at::ger_out(result, self, vec2);
  return result;
}

// torch.outer, alias for torch.ger
Tensor& outer_out(Tensor &result, const Tensor& self, const Tensor& vec2) {
  return at::ger_out(result, self, vec2);
}

Tensor outer(const Tensor& self, const Tensor& vec2) {
  return self.ger(vec2);
}

static void addmm_impl_cpu_(
    Tensor &result, const Tensor &self, Tensor m1, Tensor m2, Scalar beta, Scalar alpha) {
  TORCH_INTERNAL_ASSERT(self.dim() == 2 && m1.dim() == 2 && m2.dim() == 2);

  // Array access is faster than .size(n) and .stride(n)
  const auto self_sizes = self.sizes();
  auto m1_strides = m1.strides();
  auto m1_sizes = m1.sizes();
  auto m2_strides = m2.strides();
  auto m2_sizes = m2.sizes();

  TORCH_CHECK(
      m1_sizes[1] == m2_sizes[0], "mat1 and mat2 shapes cannot be multiplied (",
      m1_sizes[0], "x", m1_sizes[1], " and ", m2_sizes[0], "x", m2_sizes[1], ")");

  TORCH_CHECK(
      self_sizes[0] == m1_sizes[0] && self_sizes[1] == m2_sizes[1],
      "input shape is incompatible with matrix multiplication (",
      m1_sizes[0], "x", m1_sizes[1], " @ ", m2_sizes[0], "x", m2_sizes[1], " != ",
      self_sizes[0], "x", self_sizes[1], ")");

  native::resize_(result, self_sizes);
  const auto result_strides = result.strides();
  const auto result_sizes = result.sizes();

  if (result.numel() == 0) {
    return;
  }

  if (beta.to<double>() != 0.0 && !self.is_same(result)) {
    result.copy_(self);
  }

  bool transpose_c = false;
  Tensor c;

  // Cast result as matrix a
  if (result_strides[0] == 1 &&
      (result_sizes[1] == 1 || result_strides[1] >= std::max(int64_t{1}, result_sizes[0]))) {
    transpose_c = false;
    c = result;
  } else if (result_strides[1] == 1 &&
             (result_sizes[0] == 1 || result_strides[0] >= std::max(int64_t{1}, result_sizes[1]))) {
    std::swap(m1, m2);
    std::swap(m1_sizes, m2_sizes);
    std::swap(m1_strides, m2_strides);
    transpose_c = true;
    c = result;
  } else {
    transpose_c = false;
    // make c FORTRAN contiguous
    c = result.transpose(0, 1).contiguous().transpose_(0, 1);
  }

  const int64_t m = result_sizes[transpose_c ? 1 : 0];
  const int64_t n = result_sizes[transpose_c ? 0 : 1];
  const int64_t k = m1_sizes[transpose_c ? 0 : 1];

  // Cast m1 as matrix a
  bool transpose_a = false;
  Tensor a;
  /* Need lda >= max(1, (transpose_a ? k : m)) */
  if (m1_strides[transpose_c ? 1 : 0] == 1 &&
      m1_strides[transpose_c ? 0 : 1] >= std::max(int64_t{1}, m)) {
    transpose_a = false;
    a = m1;
  } else if (m1_strides[transpose_c ? 0 : 1] == 1 &&
             m1_strides[transpose_c ? 1 : 0] >= std::max(int64_t{1}, k)) {
    transpose_a = true;
    a = m1;
  } else {
    transpose_a = !transpose_c;
    a = m1.clone(at::MemoryFormat::Contiguous);
  }

  // Cast m2 as matrix b
  bool transpose_b = false;
  Tensor b;
  /* Need ldm2_ >= max(1, (transpose_m2 == 'n' ? k : n)) */
  if (m2_strides[transpose_c ? 1 : 0] == 1 &&
      m2_strides[transpose_c ? 0 : 1] >= std::max(int64_t{1}, k)) {
    transpose_b = false;
    b = m2;
  } else if (m2_strides[transpose_c ? 0 : 1] == 1 &&
             m2_strides[transpose_c ? 1 : 0] >= std::max(int64_t{1}, n)) {
    transpose_b = true;
    b = m2;
  } else {
    transpose_b = !transpose_c;
    b = m2.clone(at::MemoryFormat::Contiguous);
  }

  const int64_t lda = a.strides()[(transpose_a == transpose_c) ? 1 : 0];
  const int64_t ldb = b.strides()[(transpose_b == transpose_c) ? 1 : 0];
  const int64_t ldc = c.strides()[transpose_c ? 0 : 1];

  // Apply BLAS routine
  AT_DISPATCH_ALL_TYPES_AND_COMPLEX_AND2(kHalf, kBFloat16,
      result.scalar_type(), "addmm_impl_cpu_",
      [&]{
        at::native::cpublas::gemm(
            transpose_a ? cpublas::Transpose : cpublas::NoTranspose,
            transpose_b ? cpublas::Transpose : cpublas::NoTranspose,
            m, n, k,
            alpha.to<scalar_t>(),
            a.data_ptr<scalar_t>(), lda,
            b.data_ptr<scalar_t>(), ldb,
            beta.to<scalar_t>(),
            c.data_ptr<scalar_t>(), ldc);
      });

  if (!c.is_same(result)) {
    result.copy_(c);
  }
}

static void addbmm_impl_cpu_(
    Tensor &result, const Tensor &self, const Tensor &batch1, const Tensor &batch2, Scalar beta, Scalar alpha) {
  TORCH_CHECK(batch1.dim() == 3, "batch1 must be a 3D tensor");
  TORCH_CHECK(batch2.dim() == 3, "batch2 must be a 3D tensor");
  TORCH_CHECK(batch1.size(0) == batch2.size(0),
      "batch1 and batch2 must have same number of batches, got ",
      batch1.size(0), " and ", batch2.size(0));
  TORCH_CHECK(batch1.size(2) == batch2.size(1),
      "Incompatible matrix sizes for bmm (",
      batch1.size(1), "x", batch1.size(2), " and ",
      batch2.size(1), "x", batch2.size(2), ")");

  const int64_t dim1 = batch1.size(1);
  const int64_t dim2 = batch2.size(2);
  TORCH_CHECK(self.size(0) == dim1 && self.size(1) == dim2,
      "self tensor does not match matmul output shape");

  result.resize_as_(self);

  if (beta.to<double>() != 0.0 && !self.is_same(result)) {
    result.copy_(self);
  }

  const int64_t num_batches = batch1.size(0);

  for (int64_t batch = 0; batch < num_batches; ++batch) {
    addmm_impl_cpu_(result, result, batch1[batch], batch2[batch], beta, alpha);
    beta = 1; // accumulate output once
  }
}

Tensor& addbmm_cpu_out(Tensor& result, const Tensor& self, const Tensor& batch1, const Tensor& batch2, Scalar beta, Scalar alpha) {
  Tensor b_self = std::get<0>(expand_size(self, {batch1.size(1), batch2.size(2)}, "addbmm_out"));
  {
    at::NoNamesGuard guard;
    addbmm_impl_cpu_(result, b_self, batch1, batch2, beta, alpha);
  }
  at::namedinference::propagate_names_for_addmm(result, batch1, batch2, self);
  return result;
}

Tensor &addbmm_cpu_(Tensor& self, const Tensor& batch1, const Tensor& batch2, Scalar beta, Scalar alpha) {
  return addbmm_cpu_out(self, self, batch1, batch2, beta, alpha);
}

Tensor addbmm_cpu(const Tensor& self, const Tensor& batch1, const Tensor& batch2, Scalar beta, Scalar alpha) {
  Tensor result = at::empty({0}, self.options());
  return addbmm_cpu_out(result, self, batch1, batch2, beta, alpha);
}

Tensor& addmm_cpu_out(Tensor &result, const Tensor& self, const Tensor& mat1, const Tensor& mat2, Scalar beta, Scalar alpha) {
  TORCH_CHECK(mat1.dim() == 2, "mat1 must be a matrix, got ", mat1.dim(), "-D tensor");
  TORCH_CHECK(mat2.dim() == 2, "mat2 must be a matrix, got ", mat2.dim(), "-D tensor");
  Tensor b_self = std::get<0>(expand_size(self, {mat1.sizes()[0], mat2.sizes()[1]}, "addmm_out"));
  {
    at::NoNamesGuard guard;
    addmm_impl_cpu_(result, b_self, mat1, mat2, beta, alpha);
  }
  at::namedinference::propagate_names_for_addmm(result, mat1, mat2, self);
  return result;
}

Tensor addmm_cpu(const Tensor& self, const Tensor& mat1, const Tensor& mat2, Scalar beta, Scalar alpha) {
  Tensor result = at::empty({0}, self.options());
  return addmm_cpu_out(result, self, mat1, mat2, beta, alpha);
}

Tensor &addmm_cpu_(Tensor& self, const Tensor& mat1, const Tensor& mat2, Scalar beta, Scalar alpha) {
  return addmm_cpu_out(self, self, mat1, mat2, beta, alpha);
}

Tensor& mm_cpu_out(Tensor & result, const Tensor & self, const Tensor & mat2) {
  TORCH_CHECK(self.dim() == 2, "self must be a matrix");
  TORCH_CHECK(mat2.dim() == 2, "mat2 must be a matrix");
  native::resize_(result, {self.sizes()[0], mat2.sizes()[1]});
  return addmm_cpu_out(result, result, self, mat2, 0, 1);
}

Tensor mm_cpu(const Tensor & self, const Tensor & mat2) {
  TORCH_CHECK(self.dim() == 2, "self must be a matrix");
  TORCH_CHECK(mat2.dim() == 2, "mat2 must be a matrix");
  Tensor result = at::empty({self.sizes()[0], mat2.sizes()[1]}, self.options());
  return addmm_cpu_out(result, result, self, mat2, 0, 1);
}

template <typename scalar_t, bool is_bmm>
inline void baddbmm_cpu_kernel(const Tensor& result, const Tensor& self, const Tensor& mat2, Scalar beta_, Scalar alpha_) {
  int64_t bs = result.size(0);
  int64_t is = result.size(1);
  int64_t js = result.size(2);
  int64_t ks = self.size(2);

  scalar_t alpha = alpha_.to<scalar_t>();
  scalar_t beta = beta_.to<scalar_t>();

  auto r0 = result.accessor<scalar_t, 3>();
  auto s0 = self.accessor<scalar_t, 3>();
  auto m0 = mat2.accessor<scalar_t, 3>();

  int64_t grain_size = std::min(internal::GRAIN_SIZE / (is * js * ks), (int64_t)1);
  parallel_for(0, bs, grain_size, [&](int64_t b_begin, int64_t b_end) {
      for (int64_t b = b_begin; b < b_end; b++) {
        auto r1 = r0[b];
        auto s1 = s0[b];
        auto m1 = m0[b];
        for (int64_t i = 0; i < is; i++) {
          auto r2 = r1[i];
          auto s2 = s1[i];
          for (int64_t j = 0; j < js; j++) {
            scalar_t &r = r2[j];
            if (is_bmm) {
              r = 0;
              for (int64_t k = 0; k < ks; k++) {
                r += s2[k] * m1[k][j];
              }
            } else {
              r *= beta;
              for (int64_t k = 0; k < ks; k++) {
                r += alpha * s2[k] * m1[k][j];
              }
            }
          }
        }
      }
    });
}

// This tries to apply some optimizations to bmm/baddbmm:
// - When the operand size is small, computation are parallelized over the batch
//   dimension using OMP and naive matrix multiplication is applied.
// - When the operand size is larger than the threshold, if compiled with MKL, MKL's batch gemm is used.
// - Otherwise, we use a series of matrix multiplications.
// The threshold of 400 for the first has not been thoroughly benchmarked yet and may have room for further
// optimization, it likely depends on the characteristics of the CPU, MKL will be different from non-MKL etc.,
// but this seems to be a first starting point.

static inline Tensor& bmm_out_or_baddbmm_(Tensor& self_or_result, const Tensor& batch1, const Tensor& batch2, Scalar beta, Scalar alpha, bool is_bmm_out) {
  // is_bmm_out: true for bmm_out, false for baddbmm_
  // self_or_result is "self" for baddbmm_ and "result" for bmm_out
  CheckedFrom c = (is_bmm_out ? "bmm" : "baddbmm");
  TensorArg self_arg(self_or_result, is_bmm_out ? "self" : "result", 0);
  TensorArg b1_arg(batch1, "batch1", 1);
  TensorArg b2_arg(batch2, "batch2", 2);
  checkBackend(c, {self_or_result, batch1, batch2}, Backend::CPU);
  checkDim(c, b1_arg, 3);
  checkDim(c, b2_arg, 3);

  int64_t bs = batch1.size(0);
  checkSize(c, b2_arg, 0, bs);
  int64_t contraction_size = batch1.size(2);
  int64_t res_rows = batch1.size(1);
  int64_t res_cols = batch2.size(2);
  checkSize(c, b2_arg, 1, contraction_size);

  if (is_bmm_out) {
    self_or_result.resize_({bs, res_rows, res_cols});
  } else {
    checkSize(c, self_arg, 0, bs);
    checkSize(c, self_arg, 1, res_rows);
    checkSize(c, self_arg, 2, res_cols);
  }

  // handle pathological cases that blas may not like
  if (self_or_result.numel() == 0) {
    return self_or_result;
  } else if (contraction_size == 0) {
    if (is_bmm_out) {
      return self_or_result.zero_();
    } else {
      return self_or_result.mul_(beta);
    }
  }

  auto batch_items_contiguous_or_transposed = [&](const Tensor& t) {
    return (t.stride(2) == 1 && t.stride(1) >= t.size(2))
            || (t.stride(1) == 1 && t.stride(2) >= t.size(1));
  };

  if (contraction_size * res_rows * res_cols < 400) {
    if (is_bmm_out) {
      AT_DISPATCH_ALL_TYPES_AND_COMPLEX(batch1.scalar_type(), "bmm", [&] {
          baddbmm_cpu_kernel<scalar_t, true>(self_or_result, batch1, batch2, beta, alpha);
        });
    } else {
      AT_DISPATCH_ALL_TYPES_AND_COMPLEX(batch1.scalar_type(), "baddbmm", [&] {
          baddbmm_cpu_kernel<scalar_t, false>(self_or_result, batch1, batch2, beta, alpha);
        });
    }
  } else if (at::hasMKL() && (at::native::is_floating_point(self_or_result) ||
            at::native::is_complex(self_or_result))
            && batch_items_contiguous_or_transposed(batch1)
            && batch_items_contiguous_or_transposed(batch2)
            && self_or_result.is_contiguous()) {
    at::native::_baddbmm_mkl_(self_or_result, batch1, batch2, beta, alpha);
  } else { // split along batch dimension
    if (is_bmm_out) {
      for (int64_t b = 0; b < bs; b++) {
        auto r = self_or_result.select(0, b);
        native::mm_cpu_out(r, batch1.select(0, b), batch2.select(0, b));
      }
    } else {
      for (int64_t b = 0; b < bs; b++) {
        self_or_result.select(0, b).addmm_(batch1.select(0, b), batch2.select(0, b), beta, alpha);
      }
    }
  }
  return self_or_result;
}


Tensor baddbmm_cpu(const Tensor& self, const Tensor& batch1, const Tensor& batch2, Scalar beta, Scalar alpha) {
  Tensor result = at::empty({0}, self.options());
  return at::native::baddbmm_out_cpu(result, self, batch1, batch2, beta, alpha);
}

Tensor& baddbmm_out_cpu(Tensor &result, const Tensor& self_, const Tensor& batch1, const Tensor& batch2, Scalar beta, Scalar alpha) {
  Tensor self;
  std::tie(self) = expand_size(self_, {batch1.size(0), batch1.size(1), batch2.size(2)}, "baddbmm");
  result.resize_(self.sizes());
  result.copy_(self);
  return at::native::baddbmm__cpu(result, batch1, batch2, beta, alpha);
}

Tensor& baddbmm__cpu(Tensor& self, const Tensor& batch1, const Tensor& batch2, Scalar beta, Scalar alpha) {
  return bmm_out_or_baddbmm_(self, batch1, batch2, beta, alpha, false);
}

Tensor bmm_cpu(const Tensor& self, const Tensor& mat2) {
  Tensor result = at::empty({0}, self.options());
  return at::native::bmm_out_cpu(result, self, mat2);
}

Tensor& bmm_out_cpu(Tensor &result, const Tensor& batch1, const Tensor& batch2) {
  Scalar beta(0.0);
  Scalar alpha(1.0);
  {
  NoNamesGuard guard;
  bmm_out_or_baddbmm_(result, batch1, batch2, beta, alpha, true);
  }
  namedinference::propagate_names_if_nonempty(
      result,
      namedinference::compute_bmm_outnames(result, batch1, batch2));
  return result;
}

Tensor& dot_out(Tensor& result, const Tensor& self, const Tensor& tensor) {
  result.resize_({});
  TORCH_CHECK(result.scalar_type() == self.scalar_type(),
           "result dtype ", result.scalar_type(), " does not match self dtype ", self.scalar_type());
  return result.fill_(self.dot(tensor));
}

/*
Matrix product of two Tensors.
The behavior depends on the dimensionality of the Tensors as follows:
- If both Tensors are 1-dimensional, the dot product (scalar) is returned.
- If both arguments are 2-dimensional, the matrix-matrix product is returned.
- If the first argument is 1-dimensional and the second argument is 2-dimensional,
  a 1 is prepended to its dimension for the purpose of the matrix multiply.
  After the matrix multiply, the prepended dimension is removed.
- If the first argument is 2-dimensional and the second argument is 1-dimensional,
  the matrix-vector product is returned.
- If both arguments are at least 1-dimensional and at least one argument is
  N-dimensional (where N > 2), then a batched matrix multiply is returned.  If the first
  argument is 1-dimensional, a 1 is prepended to its dimension for the purpose of the
  batched matrix multiply and removed after.  If the second argument is 1-dimensional, a
  1 is appended to its dimension for the purpose of the batched matrix multiple and removed after.
  The non-matrix (i.e. batch) dimensions are broadcasted (and thus
  must be broadcastable).  For example, if tensor1 is a (j x 1 x n x m) Tensor
  and tensor2 is a (k x m x p) Tensor, the returned tensor will be an (j x k x n x p) Tensor.
*/
Tensor matmul(
    c10::optional<Tensor> out_opt,
    const Tensor& tensor1,
    const Tensor& tensor2) {
  NoNamesGuard guard;
  auto dim_tensor1 = tensor1.dim();
  auto dim_tensor2 = tensor2.dim();
  auto has_out = out_opt.has_value();
  Tensor out = out_opt.value_or(Tensor());

  if (dim_tensor1 == 1 && dim_tensor2 == 1) {
    return has_out ? at::native::dot_out(out, tensor1, tensor2) : tensor1.dot(tensor2);
  } else if (dim_tensor1 == 2 && dim_tensor2 == 1) {
    return has_out ? at::mv_out(out, tensor1, tensor2) : tensor1.mv(tensor2);
  } else if (dim_tensor1 == 1 && dim_tensor2 == 2) {
    return has_out ? at::mm_out(out, tensor1.unsqueeze(0), tensor2).squeeze_(0)
                   : tensor1.unsqueeze(0).mm(tensor2).squeeze_(0);
  } else if (dim_tensor1 == 2 && dim_tensor2 == 2) {
    return has_out ? at::mm_out(out, tensor1, tensor2) : tensor1.mm(tensor2);
  } else if (dim_tensor1 >= 3 && (dim_tensor2 == 1 || dim_tensor2 == 2)) {
    // optimization: use mm instead of bmm by folding tensor1's batch into
    // its leading matrix dimension.

    Tensor t2 = dim_tensor2 == 1 ? tensor2.unsqueeze(-1) : tensor2;
    auto size1 = tensor1.sizes();
    auto size2 = t2.sizes();
    std::vector<int64_t> output_size;
    output_size.insert(output_size.end(), size1.begin(), size1.end() - 1);
    if (dim_tensor2 > 1) {
      output_size.push_back(size2[dim_tensor2 - 1]);
    }

    // fold the batch into the first dimension
    Tensor t1 = tensor1.contiguous().view({-1, size1[size1.size() - 1]});
    Tensor output = has_out ? at::_unsafe_view(at::mm_out(out, t1, t2), output_size)
                            : at::_unsafe_view(t1.mm(t2), output_size);
    return has_out ? out.set_(output) : output;
  } else if ((dim_tensor1 == 1 || dim_tensor1 == 2) && dim_tensor2 >= 3) {
    // optimization: transpose the inner dimensions of the arguments, call
    // matmul on the swapped arguments, then transpose the inner dimensions
    // of the result.
    const int64_t n = dim_tensor1 == 2 ? tensor1.size(-2) : 1;
    const int64_t m = tensor1.size(-1);
    const int64_t p = tensor2.size(-1);

    const Tensor t2_T = tensor2.transpose(-1, -2);
    const Tensor t1_T = dim_tensor1 == 2 ? tensor1.t() : tensor1.reshape({n, m}).t();
    const Tensor res_T = matmul(out_opt, t2_T, t1_T);

    if (dim_tensor1 == 2) {
      Tensor res = res_T.transpose(-1, -2).contiguous();
      return has_out ? out.set_(res) : res;
    }
    else {
      std::vector<int64_t> shape = tensor2.sizes().slice(0, dim_tensor2 - 2).vec();
      shape.push_back(p);

      Tensor res = res_T.reshape(shape).contiguous();
      return has_out ? out.set_(res) : res;
    }
  } else if ((dim_tensor1 >= 1 && dim_tensor2 >= 1) && (dim_tensor1 >= 3 || dim_tensor2 >= 3)) {
    // We are multiplying b1 x n x m1 by x2 x m2 x p (where b1 can be a list);
    // we track m1 vs m2 separately even though they must match for nicer error messages
    int64_t n = dim_tensor1 > 1 ? tensor1.size(-2) : 1;
    int64_t m1 = tensor1.size(-1);
    IntArrayRef batch_tensor1(tensor1.sizes().data(), std::max<int64_t>(dim_tensor1 - 2, 0));
    int64_t m2 = dim_tensor2 > 1 ? tensor2.size(-2) : 1;
    int64_t p = tensor2.size(-1);
    IntArrayRef batch_tensor2(tensor2.sizes().data(), std::max<int64_t>(dim_tensor2 - 2, 0));

    // expand the batch portion (i.e. cut off matrix dimensions and expand rest)
    std::vector<int64_t> expand_batch_portion = infer_size(batch_tensor1, batch_tensor2);

    std::vector<int64_t> tensor1_expand_size(expand_batch_portion);
    tensor1_expand_size.insert(tensor1_expand_size.end(), {n, m1});

    std::vector<int64_t> tensor2_expand_size(expand_batch_portion);
    tensor2_expand_size.insert(tensor2_expand_size.end(), {m2, p});

    int expand_batch_product = std::accumulate(expand_batch_portion.begin(), expand_batch_portion.end(),
                                               1, std::multiplies<int64_t>());

    std::vector<int64_t> tensor1_bmm_view({expand_batch_product});
    tensor1_bmm_view.insert(tensor1_bmm_view.end(), {n, m1});

    std::vector<int64_t> tensor2_bmm_view({expand_batch_product});
    tensor2_bmm_view.insert(tensor2_bmm_view.end(), {m2, p});

    // flatten expanded batches
    Tensor tensor1_expanded = tensor1.expand(tensor1_expand_size).contiguous().view(tensor1_bmm_view);
    Tensor tensor2_expanded = tensor2.expand(tensor2_expand_size).contiguous().view(tensor2_bmm_view);

    // reshape batches back into result
    std::vector<int64_t> output_shape(expand_batch_portion);
    if (dim_tensor1 > 1) {
      output_shape.push_back(n);
    }
    if (dim_tensor2 > 1) {
      output_shape.push_back(p);
    }

    Tensor output = has_out ? at::_unsafe_view(at::bmm_out(out, tensor1_expanded, tensor2_expanded), output_shape)
                            : at::_unsafe_view(tensor1_expanded.bmm(tensor2_expanded), output_shape);

    return has_out ? out.set_(output) : output;
  }

 AT_ERROR("both arguments to matmul need to be at least 1D, but they are ",
          dim_tensor1, "D and ", dim_tensor2, "D");
}

Tensor matmul(const Tensor & tensor1, const Tensor & tensor2) {
  auto maybe_outnames = namedinference::compute_matmul_outnames(tensor1, tensor2);
  auto result = at::native::matmul(c10::nullopt, tensor1, tensor2);
  namedinference::propagate_names_if_nonempty(result, maybe_outnames);
  return result;
}

Tensor& matmul_out(Tensor &result, const Tensor & tensor1, const Tensor & tensor2) {
  auto maybe_outnames = namedinference::compute_matmul_outnames(tensor1, tensor2);
  at::native::matmul(c10::optional<Tensor>(result), tensor1, tensor2);
  namedinference::propagate_names_if_nonempty(result, maybe_outnames);
  return result;
}

// helper methods for matrix_exp
namespace {

template <typename scalar_t, int ROW, int COL>
using array2d = std::array<std::array<scalar_t, COL>, ROW>;

// we consider 6 Taylor expansions of degree
// 1, 2, 4, 8, 12, 18
constexpr int total_n_degs = 6;

Tensor operator_1_norm(const Tensor& tensor) {
  return std::get<0>(tensor.abs().sum(-2).max(-1));
}

// Allocates a buffers of uninitialized or zero values
// of shape [n_copies, a.size()]
Tensor _allocate_buffer(const Tensor& a, int n_copies, bool is_zero = false) {
  auto res = at::empty(
    {n_copies, a.size(0), a.size(1), a.size(2)},
    a.options().memory_format(at::MemoryFormat::Contiguous)
  );
  
  if (is_zero) {
    res.zero_();
  }

  return res;
}

// Makes `buffer` to store `num_matrices` number of matrices needed for
// compute the matrix exponentials of different orders, i.e.
// first `num_matrices` matrices from the list l := {I, A, A^2, A^3, A^6}
// in a contiguous block of memory such that
// buffer[0, ...] = l[0], // I
// buffer[1, ...] = l[1], // A
// ...
// buffer[num_matrices - 1, ...] = l[num_matries - 1]
void _fill_matrix_powers(Tensor& buffer, const Tensor& a, int num_matrices) {
  auto a_sizes_minus_last = a.sizes().vec();
  a_sizes_minus_last.pop_back();
  // fill I
  buffer.select(0, 0).copy_(
    at::diag_embed(
      at::ones({1}, buffer.options())
        .expand(a_sizes_minus_last)
    )
  );

  // fill a
  buffer.select(0, 1).copy_(a);

  // fill a^2
  if (2 <= num_matrices - 1) {
    at::native::matmul(
      buffer.select(0, 2), // out for a^2
      buffer.select(0, 1),
      buffer.select(0, 1)
    );
  }

  // fill a^3
  if (3 <= num_matrices - 1) {
    at::native::matmul(
      buffer.select(0, 3), // out for a^3
      buffer.select(0, 1),
      buffer.select(0, 2)
    );
  }

  // fill a^6
  if (4 <= num_matrices - 1) {
    at::native::matmul(
      buffer.select(0, 4),
      buffer.select(0, 3),
      buffer.select(0, 3)
    );
  }
}

inline Tensor _move_memory_if_cuda_input(
  const Tensor& mem,
  const Tensor& in
) {
  return (in.device().type() == at::kCUDA)
    ? mem.to(at::device_of(in).value())
    : mem;
}

// convert a 1D blob to a 2D Tensor of size [1, blob.size()]
// such that blob.device() == in.device())
// designed to be used with _compute_linear_combination
template <typename scalar_t>
inline Tensor _blob_to_Tensor(
  std::initializer_list<scalar_t> blob,
  const Tensor& in
) {
  // we convert to void* expecitly because begin() returns
  // a pointer to a constant.
  // Blob is assumed to be a 1D array, that is why
  // we also insert a fake dimension so that the result could directly
  // be used in _compute_linear_combination
  auto tensor = at::from_blob((void*)blob.begin(), blob.size(), in.dtype())
    .unsqueeze(0);
  return _move_memory_if_cuda_input(tensor, in);
}

// I + A
Tensor compute_T1(const Tensor& A) {
  // 2 for {I, A}
  auto As = _allocate_buffer(A, 2);
  _fill_matrix_powers(As, A, 2);
  return As.sum(0);
}

// I + A + A^2 / 2
Tensor compute_T2(const Tensor& A) {
  auto As = _allocate_buffer(A, 3);
  // 3 for {I, A, A^2}
  _fill_matrix_powers(As, A, 3);
  As.select(0, 2).div_(2.0);
  return As.sum(0);
}

// I + A + A^2 * (I / 2 + A / 6 + A^2 / 24)
template <typename scalar_t>
Tensor compute_T4(const Tensor& A) {
  auto As = _allocate_buffer(A, 4);
  // 3 for {I, A, A^2}
  _fill_matrix_powers(As, A, 3);
  
  at::native::matmul(
    // output for A^2 * (I / 2 + A / 6 + A^2 / 24)
    As.select(0, 3),
    // contains A^2
    As.select(0, 2),
    // computes (I / 2 + A / 6 + A^2 / 24)
    at::native::_compute_linear_combination(
      As.narrow(0, 0, 3),
      _blob_to_Tensor<scalar_t>({1 / 2.0, 1 / 6.0, 1 / 24.0}, A)
    )
  );

  // I + A + A^2 * (I / 2 + A / 6 + A^2 / 24)
  return at::native::_compute_linear_combination(
    As, _blob_to_Tensor<scalar_t>({1.0, 1.0, 0.0, 1.0}, A)
  );
}

template <typename scalar_t>
Tensor compute_T8(const Tensor& A) {
  constexpr scalar_t sqrt_177 = 0.1330413469565007072504e+2;
  constexpr scalar_t x3 = 2. / 3.;
  constexpr scalar_t x1 = x3 * ((1. + sqrt_177) / 88.);
  constexpr scalar_t x2 = x3 * ((1. + sqrt_177) / 352.);
  constexpr scalar_t x4 = (-271. + 29. * sqrt_177) / (315. * x3);
  constexpr scalar_t x5 = (-11. + 11. * sqrt_177) / (1260. * x3);
  constexpr scalar_t x6 = (-99. + 11. * sqrt_177) / (5040. * x3);
  constexpr scalar_t x7 = (89. - sqrt_177) / (5040. * x3);
  constexpr scalar_t y2 = (857. - 58. * sqrt_177) / 630.;

  auto As = _allocate_buffer(A, 5);
  // 3 for {I, A, A^2}
  _fill_matrix_powers(As, A, 3);

  // A4 =  A2 * (x1 * A + x2 * A2)
  at::native::matmul(
    // output for A4
    As.select(0, 3),
    // As.select(0, 2) = A^2
    As.select(0, 2),
    at::native::_compute_linear_combination(
      // extract {A, A^2} from As
      As.narrow(0, 1, 2),
      _blob_to_Tensor<scalar_t>({x1, x2}, A)
    )
  );

  // A8 = (x3 * A2 + A4) * (x4 * I + x5 * A + x6 * A2 + x7 * A4)
  at::native::matmul(
    // output for A8
    As.select(0, 4),
    // x3 * A2 + A4
    at::native::_compute_linear_combination(
      As.narrow(0, 2, 2),
      _blob_to_Tensor<scalar_t>({x3, 1.0}, A)
    ),
    at::native::_compute_linear_combination(
      As.narrow(0, 0, 4),
      _blob_to_Tensor<scalar_t>({x4, x5, x6, x7}, A)
    )
  );

  // return I + A + y2 * A2 + A8;
  return at::native::_compute_linear_combination(
    As,
    _blob_to_Tensor<scalar_t>({1.0, 1.0, y2, 0.0, 1.0}, A)
  );
}

template <typename scalar_t>
Tensor compute_T12(const Tensor& A) {
  constexpr int num_prods = 4;
  array2d<scalar_t, num_prods, num_prods> b = {{
    {
      9.0198e-16,
      0.46932117595418237389,
      -0.20099424927047284052,
      -0.04623946134063071740
    },
    {
      5.31597895759871264183,
      1.19926790417132231573,
      0.01179296240992997031,
      0.01108844528519167989
    },
    {
      0.18188869982170434744,
      0.05502798439925399070,
      0.09351590770535414968,
      0.00610700528898058230
    },
    {
      -2.0861320e-13,
      -0.13181061013830184015,
      -0.02027855540589259079,
      -0.00675951846863086359
    }
  }};

  // gather coefficients `b` from above into a tensor,
  // and move them to device `device_of(A)`
  auto bs = at::from_blob(
    reinterpret_cast<void*>(&b),
    {num_prods, num_prods},
    {num_prods, 1},
    A.dtype()
  );
  bs = _move_memory_if_cuda_input(bs, A);

  auto As = _allocate_buffer(A, num_prods);
  _fill_matrix_powers(As, A, num_prods);

  auto Bs = at::native::_compute_linear_combination(As, bs);

  // compute A6
  Bs.select(0, 2).add_(at::native::matmul(
    // tmp buffer for this matrix product
    As.select(0, 0),
    Bs.select(0, 3),
    Bs.select(0, 3)
  ));

  return Bs.select(0,0).add_(at::native::matmul(
    // tmp buffer for this matrix product
    As.select(0, 0),
    Bs.select(0, 1).add_(Bs.select(0, 2)),
    Bs.select(0, 2)
  ));
}

template <typename scalar_t>
Tensor compute_T18(const Tensor& A) {
  constexpr int num_prods = 5;
  array2d<scalar_t, num_prods, num_prods> b = {{
    {
      0.,
      -1.00365581030144618291e-01,
      -8.02924648241156932449e-03,
      -8.92138498045729985177e-04,
      0.
    },
    {
      0.,
      3.97849749499645077844e-01,
      1.36783778460411720168e+00,
      4.98289622525382669416e-01,
      -6.37898194594723280150e-04
    },
    {
      -1.09676396052962061844e+01,
      1.68015813878906206114e+00,
      5.71779846478865511061e-02,
      -6.98210122488052056106e-03,
      3.34975017086070470649e-05
    },
    {
      -9.04316832390810593223e-02,
      -6.76404519071381882256e-02,
      6.75961301770459654925e-02,
      2.95552570429315521194e-02,
      -1.39180257516060693404e-05
    },
    {
      0.,
      0.,
      -9.23364619367118555360e-02,
      -1.69364939002081722752e-02,
      -1.40086798182036094347e-05
    }
  }};

  // gather coefficients `b` from above into a tensor,
  // and move them to device `device_of(A)`
  auto bs = at::from_blob(
    reinterpret_cast<void*>(&b),
    {num_prods, num_prods},
    {num_prods, 1},
    A.dtype()
  );
  bs = _move_memory_if_cuda_input(bs, A);

  auto As = _allocate_buffer(A, num_prods);
  _fill_matrix_powers(As, A, num_prods);

  auto Bs = at::native::_compute_linear_combination(As, bs);

  // compute A9
  Bs.select(0, 3).add_(at::native::matmul(
    // tmp buffer for this matrix product
    As.select(0, 0),
    Bs.select(0, 0),
    Bs.select(0, 4))
  );

  return Bs.select(0, 1).add_(at::native::matmul(
    // tmp buffer for this matrix product
    As.select(0, 0),
    Bs.select(0, 2).add_(Bs.select(0, 3)),
    Bs.select(0, 3)
  ));
}

template <typename scalar_t>
void compute_T18_scale_square(
  Tensor& mexp_out,
  const Tensor& a,
  const Tensor& norm,
  scalar_t theta
) {
  // Scale
  const auto s = at::max(
    at::zeros_like(norm),
    at::ceil(at::log2(norm / theta))
  ).unsqueeze(-1).unsqueeze(-1).to(at::kLong);
  const auto pow2s = at::pow(2, s);
  const auto a_scaled = a / pow2s;

  // Square
  auto mexp_scaled = at::native::compute_T18<scalar_t>(a_scaled);
  auto s_cpu = (s.device().type() == at::kCPU)
    ? s : s.to(at::kCPU);
  for (int64_t i = 0; i < mexp_scaled.size(0); ++i) {
    auto s_val = s_cpu.select(0, i).template item<int64_t>();
    auto mexp = mexp_scaled.select(0, i);
    for (int64_t p = 0; p < s_val; ++p) {
      mexp = at::matmul(mexp, mexp);
    }
    mexp_out.select(0, i).copy_(mexp);
  }
}

template <typename scalar_t>
Tensor mexp_impl(
  const Tensor& a,
  std::array<scalar_t, total_n_degs> thetas,
  bool compute_highest_degree_approx = false
) {
  auto res = at::empty_like(a);
  const auto norm = operator_1_norm(a);
  // `norm_cpu` is used to decide which Tensors require which approximation
  // based on their norm. This decision takes place on CPU.
  // It requires moving data back and forth between devices when `a` is on CUDA,
  // but at the cost of only one sigle CPU-CUDA synchronization (instead of 6),
  // and better performance overall (benchmarked).
  const auto norm_cpu = (a.device().type() == at::kCUDA)
    ? norm.to(at::kCPU) : norm;

  if (!compute_highest_degree_approx) {
    constexpr std::array<
      Tensor(*)(const Tensor&),
      total_n_degs - 1> 
    compute_Ts = {
      compute_T1, compute_T2, compute_T4<scalar_t>,
      compute_T8<scalar_t>, compute_T12<scalar_t>
    };

    for (int i = 0; i < total_n_degs - 1; ++i) {
      auto norm_lower_bound = (i == 0) ? static_cast<scalar_t>(-1) : thetas[i - 1];
      auto norm_upper_bound = thetas[i];
      // nonzero returns a 2D tensor, hence squeeze(-1) to make it 1D
      auto idx_curr_norm_interval = (
        (norm_lower_bound < norm_cpu) * (norm_cpu <= norm_upper_bound)
      ).nonzero().squeeze(-1);

      if (idx_curr_norm_interval.numel()) {
        auto idx_to_device = _move_memory_if_cuda_input(
          idx_curr_norm_interval, a
        );
        auto sub_a = at::index_select(a, 0, idx_to_device);
        res.index_put_({idx_to_device}, compute_Ts[i](sub_a));
      }
    }

    // nonzero returns a 2D tensor, hence squeeze(-1) to make it 1D
    auto idx_large_norm = (norm_cpu >= thetas[total_n_degs - 2])
      .nonzero().squeeze(-1);

    if (idx_large_norm.numel()) {
      auto idx_to_device = _move_memory_if_cuda_input(
        idx_large_norm, a
      );
      auto a_large_norm = at::index_select(a, 0, idx_to_device);
      auto large_norm_subset = at::index_select(norm, 0, idx_to_device);
      auto mexp_out = at::empty_like(a_large_norm);

      compute_T18_scale_square(
        mexp_out,
        a_large_norm,
        large_norm_subset,
        thetas[total_n_degs - 1]
      );
      res.index_put_({idx_large_norm}, mexp_out);
    }

    return res;
  }

  compute_T18_scale_square(
    res, a, norm,
    thetas[total_n_degs - 1]
  );

  return res;
}

// matrix exponential
Tensor mexp(const Tensor& a, bool compute_highest_degree_approx = false) {
  // squash batch dimensions to one dimension for simplicity
  const auto a_3d = a.view({-1, a.size(-2), a.size(-1)});

  if (a.scalar_type() == at::ScalarType::Float
      || a.scalar_type() == at::ScalarType::ComplexFloat) {
    constexpr std::array<float, total_n_degs> thetas_float = {
      1.192092800768788e-07, // deg 1
      5.978858893805233e-04, // deg 2
      5.116619363445086e-02, // deg 4
      5.800524627688768e-01, // deg 8
      1.461661507209034e+00, // deg 12
      3.010066362817634e+00  // deg 18
    };

    return mexp_impl<float>(a_3d, thetas_float, compute_highest_degree_approx)
      .view(a.sizes());
  }
  else { // if Double or ComplexDouble
    constexpr std::array<double, total_n_degs> thetas_double = {
      2.220446049250313e-16, // deg 1
      2.580956802971767e-08, // deg 2
      3.397168839976962e-04, // deg 4
      4.991228871115323e-02, // deg 8
      2.996158913811580e-01, // deg 12
      1.090863719290036e+00  // deg 18
    };

    return mexp_impl<double>(a_3d, thetas_double, compute_highest_degree_approx)
      .view(a.sizes());
  }
}

// Based on:
//
// Mathias, Roy. 
// A Chain Rule for Matrix Functions and Applications.
// SIAM J. Matrix Anal. Appl. 17 (1996): 610-620.
//
template <typename func_t>
Tensor backward_analytic_function_of_a_matrix(
    const Tensor& self, const Tensor& grad,
    const func_t& function_of_a_matrix
  ) {
  auto self_transposed = self.transpose(-2, -1);
  auto self_transposed_sizes = self_transposed.sizes().vec();
  self_transposed_sizes[self.dim() - 2] <<= 1;
  self_transposed_sizes[self.dim() - 1] <<= 1;

  auto n = self_transposed.size(-1);
  auto meta_grad = at::zeros(self_transposed_sizes, grad.options());
  meta_grad.narrow(-2, 0, n).narrow(-1, 0, n).copy_(self_transposed);
  meta_grad.narrow(-2, n, n).narrow(-1, n, n).copy_(self_transposed);
  meta_grad.narrow(-2, 0, n).narrow(-1, n, n).copy_(grad);

  auto grad_input = function_of_a_matrix(meta_grad)
    .narrow(-2, 0, n).narrow(-1, n, n);
  return grad_input;
}

};

// Computes the matrix exponential for a given batch of squared matrices.
// The implementaion is based on:
//
// Bader, P.; Blanes, S.; Casas, F.
// Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation.
// Mathematics 2019, 7, 1174.
//
Tensor matrix_exp(const Tensor& a) {
  TORCH_CHECK(a.dim() >= 2 
          && (at::isFloatingType(a.scalar_type()) 
           || at::isComplexType(a.scalar_type())),
              "matrix_exp(", a.scalar_type(), "{", a.sizes(), "}): expected a tensor "
              "of floating or complex types with dim at least 2");
  TORCH_CHECK(a.size(-1) == a.size(-2),
              "matrix_exp(", a.scalar_type(), "{", a.sizes(), "}): expected a tensor "
              "of squared matrices");

  if (a.size(-1) == 1) {
    return a.exp();
  }

  return mexp(a);
}

Tensor matrix_exp_backward(const Tensor& self, const Tensor& grad) {
  return backward_analytic_function_of_a_matrix(
    self, grad,
    [](const Tensor& a) {
      return a.matrix_exp();
    }
  );
}

Tensor matrix_power(const Tensor& a, int64_t n) {
  TORCH_CHECK(a.dim() >= 2 && (at::isFloatingType(a.scalar_type()) || at::isComplexType(a.scalar_type())),
              "matrix_power(", a.scalar_type(), "{", a.sizes(), "}): expected a tensor "
              "of floating types with dim at least 2");
  if (n == 0) {
    return a.clone(at::MemoryFormat::Contiguous).copy_(at::eye(a.size(-2), a.options()).expand_as(a));
  } else if (n < 0) {
    Tensor a_ = at::inverse(a);
    n *= -1;
    return at::native::matrix_power(a_, n);
  } else if (n == 1) {
    return a.clone(at::MemoryFormat::Contiguous);
  } else if (n == 2) {
    return at::native::matmul(a, a);
  } else if (n == 3) {
    return at::native::matmul(at::native::matmul(a, a), a);
  }

  // This is a binary decomposition of n.
  // Moving from the least significant bit to the most significant bit
  // This is done to reduce the number of matrix multiplications
  // by raising the input matrix in powers of 2
  // The total number of matrix multiplications are
  // number of bits + number of bits that equal 1 ~ O(log n)
  // instead of O(n)
  Tensor result, z;
  int64_t r;
  while (n > 0) {
    z = (!z.defined()) ? a.clone(at::MemoryFormat::Contiguous) : at::native::matmul(z, z);
    r = n % 2;
    n = n / 2;
    if (r == 1) {
      result = (!result.defined()) ? z.clone(at::MemoryFormat::Contiguous) : at::native::matmul(result, z);
    }
  }
  return result;
}

Tensor frobenius_norm(const Tensor& self) {
  TORCH_CHECK(!self.is_complex(), "frobenius norm not supported for complex tensors");
  return at::norm(self);
}

Tensor frobenius_norm(const Tensor& self, IntArrayRef dim, bool keepdim) {
  TORCH_CHECK(!self.is_complex(), "frobenius norm not supported for complex tensors");
  TORCH_CHECK(
      dim.size() <= 2,
      "Expected at most 2 dimensions, but got ",
      dim.size(),
      " dimensions instead.");
  if (dim.size() == 1 || dim.size() == 0) {
    return at::norm(self, 2, dim, keepdim);
  }
  if (self.is_complex()){
    return at::sqrt(at::sum(at::real(self.conj() * self), dim, keepdim));
  } else {
    return at::sqrt(at::sum((self * self), dim, keepdim));
  }
}

Tensor &frobenius_norm_out(
    Tensor& result,
    const Tensor& self,
    IntArrayRef dim,
    bool keepdim) {
  TORCH_CHECK(!self.is_complex(), "frobenius norm not supported for complex tensors");
  TORCH_CHECK(
      dim.size() <= 2,
      "Expected at most 2 dimensions, but got ",
      dim.size(),
      " dimensions instead.");
  if (dim.size() == 1 || dim.size() == 0) {
    return at::norm_out(result, self, 2, dim, keepdim, self.scalar_type());
  }
  if (self.is_complex()){
    return at::sqrt_out(result, at::sum(at::real(self.conj() * self), dim, keepdim));
  } else {
    return at::sqrt_out(result, at::sum((self * self), dim, keepdim));
  }
}

Tensor nuclear_norm(const Tensor& self, bool keepdim) {
  TORCH_CHECK(
      self.dim() == 2,
      "Expected a tensor with 2 dimensions, but got a tensor with ",
      self.dim(), " dimension", self.dim()==1 ? "" : "s", " instead.");
  // Since we error out on svd_backward when we don't compute U and V, the backward pass for nuclear_norm
  // would end up throwing an error as a result if U and V aren't computed.
  // Due to this, we have to compute U and V conditionally.
  Tensor result = at::sum(std::get<1>(at::svd(self, /*some=*/true,
                 /*compute_uv=*/at::GradMode::is_enabled() && self.requires_grad())), 0, keepdim);
  if (keepdim) {
    result.unsqueeze_(0);
  }
  return result;
}

Tensor &nuclear_norm_out(Tensor& result, const Tensor& self, bool keepdim) {
  TORCH_CHECK(
      self.dim() == 2,
      "Expected a tensor with 2 dimensions, but got a tensor with ",
      self.dim(), " dimension", self.dim()==1 ? "" : "s", " instead.");
  at::sum_out(result, std::get<1>(at::svd(self, /*some=*/true, /*compute_uv=*/false)), 0, keepdim);
  if (keepdim) {
    result.unsqueeze_(0);
  }
  return result;
}

Tensor nuclear_norm(const Tensor& self, IntArrayRef dim, bool keepdim) {
  TORCH_CHECK(dim.size() == 2, "nuclear norm requires a 'dim' argument of size 2");

  auto permutation = create_dim_backshift_permutation(dim[0], dim[1], self.dim());
  auto permutation_reverse = create_reverse_permutation(permutation);
  Tensor p = self.permute(permutation);
  // Since we error out on svd_backward when we don't compute U and V, the backward pass for nuclear_norm
  // would end up throwing an error as a result if U and V aren't computed.
  // Due to this, we have to compute U and V conditionally.
  Tensor result = at::sum(std::get<1>(at::svd(p, /*some=*/true,
                 /*compute_uv=*/at::GradMode::is_enabled() && self.requires_grad())), -1, keepdim);
  if (keepdim) {
    result.unsqueeze_(-1);
    result = result.permute(permutation_reverse);
  }
  return result;
}

Tensor& nuclear_norm_out(Tensor& result, const Tensor& self, IntArrayRef dim, bool keepdim) {
  TORCH_CHECK(dim.size() == 2, "nuclear norm requires a 'dim' argument of size 2");

  auto permutation = create_dim_backshift_permutation(dim[0], dim[1], self.dim());
  auto permutation_reverse = create_reverse_permutation(permutation);

  Tensor p = self.permute(permutation);
  at::sum_out(result, std::get<1>(at::svd(p, /*some=*/true, /*compute_uv=*/false)), -1, keepdim);
  if (keepdim) {
    result.unsqueeze_(-1);
    result = result.permute(permutation_reverse);
  }
  return result;
}

static inline Tensor _chain_matmul_general(TensorList matrices, std::vector<std::vector<int64_t>>& order, int64_t i, int64_t j) {
  if (i == j)
    return matrices[i];
  else
    return at::mm(_chain_matmul_general(matrices, order, i, order[i][j]), _chain_matmul_general(matrices, order, order[i][j] + 1, j));
}

// Why the separate implementation for 3 matrices?
// The logic for three matrices is much faster when done directly
// Requires 1 comparison to 4 comparisons and lesser arithmetic operations
static inline Tensor _chain_matmul_three_matrices(TensorList matrices) {
  int64_t a = matrices[0].size(0);  // This is the first dimension
  int64_t b = matrices[1].size(0);  // This is the common dimension between the first two matrices
  int64_t c = matrices[2].size(0);  // This is the common dimension between the last two matrices
  int64_t d = matrices[2].size(1);  // This is the last dimension

  // The matrices are of size (a x b), (b x c), (c x d)
  // cost_1 is the cost of parenthesizing (a x b) and (b x c) and then combining (c x d)
  // cost_2 is the cost of parenthesizing (b x c) and (c x d) and then combining (a x b)
  int64_t cost_1 = (a * c) * (b + d);
  int64_t cost_2 = (b * d) * (a + c);

  if (cost_1 > cost_2) {
    return at::mm(matrices[0], at::mm(matrices[1], matrices[2]));
  } else {
    return at::mm(at::mm(matrices[0], matrices[1]), matrices[2]);
  }
}

Tensor chain_matmul(TensorList matrices) {
  checkAllSameDim(matrices, 2);

  TORCH_CHECK(matrices.size() > 0, "chain_matmul: Expected one or more matrices");
  if (matrices.size() == 1) {
    return matrices[0];
  } else if (matrices.size() == 2) {
    return at::mm(matrices[0], matrices[1]);
  } else if (matrices.size() == 3) {
    return _chain_matmul_three_matrices(matrices);
  } else {

    // Following the algorithm in Chapter 15.2 : Introduction to Algorithms, Cormen et al.
    // Minor modifications have been made to accommodate zero-indexing
    auto n = matrices.size();

    // Dim vector - the length of which is n + 1. Note that for matrix multiplication, there
    // needs to a common dimension between the multiplicands, hence for n matrices, there are
    // n + 1 values. The values p_{i} and p_{i + 1} correspond to the dimensions of matrix i in
    // the chain (zero-indexed)
    std::vector<int64_t> p;
    p.push_back(matrices[0].size(0));
    for (size_t i = 0; i < n; i++) {
      p.push_back(matrices[i].size(1));
    }

    // Cost matrix - an element m[i, j] of this matrix corresponds to the minimum cost of
    // parenthesizing matrices A_{i} to A_{j}. By this definition m[i, i] = 0 for all i
    // m[i, j] is filled using the substructure property of the algorithm, meaning:
    // m[i, j] = min_{i <= k < j} m[i, k] + m[k, j] + p_{i-1}p_{k}p_{j}
    std::vector<std::vector<int64_t>> m(n, std::vector<int64_t>(n, 0));

    // Auxiliary table for constructing the order
    // s[i, j] stores the index k at which the optimal split is obtained
    std::vector<std::vector<int64_t>> s(n, std::vector<int64_t>(n));

    // j and q are used repetitively in the algorithm below
    int64_t j, q;

    for (int64_t l = 1; l < n; l++) {
      for (int64_t i = 0; i < n - l; i++) {
        j = i + l;
        m[i][j] = std::numeric_limits<int64_t>::max();
        for (int64_t k = i; k < j; k++) {
          q = m[i][k] + m[k + 1][j] + p[i] * p[k + 1] * p[j + 1];
          if (q < m[i][j]) {
            m[i][j] = q;
            s[i][j] = k;
          }
        }
      }
    }

    // We use the result from the algorithm to compute the matrix chain product via recursion
    return _chain_matmul_general(matrices, s, 0, n - 1);
  }
}

} // namespace native
} // namespace at
