void ComputeAsync(OpKernelContext* context, DoneCallback done) final {
    const Tensor& input = context->input(0);
    const Tensor& rhs = context->input(1);
    const int ndims = input.dims();
    const int64 n = input.dim_size(ndims - 1);
    const int64 nrhs = rhs.dim_size(ndims - 1);
    // Validate inputs.
    OP_REQUIRES_ASYNC(
        context, ndims >= 2,
        errors::InvalidArgument("Input must have rank >= 2, got ", ndims),
        done);
    OP_REQUIRES_ASYNC(context, rhs.dims() == ndims,
                      errors::InvalidArgument(
                          "Input and right-hand side must have same rank, got ",
                          ndims, " != ", rhs.dims()),
                      done);
    OP_REQUIRES_ASYNC(
        context, input.dim_size(ndims - 2) == n,
        errors::InvalidArgument("Input matrices must be squares, got ",
                                input.dim_size(ndims - 2), " != ", n),
        done);
    OP_REQUIRES_ASYNC(context, rhs.dim_size(ndims - 2) == n,
                      errors::InvalidArgument(
                          "Input matrix and right-hand side must have the "
                          "same number of rows, got ",
                          n, " != ", rhs.dim_size(ndims - 2)),
                      done);
    for (int dim = 0; dim < ndims - 2; dim++) {
      OP_REQUIRES_ASYNC(
          context, input.dim_size(dim) == rhs.dim_size(dim),
          errors::InvalidArgument(
              "All input tensors must have the same outer dimensions."),
          done);
    }

    // Allocate output.
    Tensor* output;
    OP_REQUIRES_OK_ASYNC(
        context,
        context->forward_input_or_allocate_output({1}, 0, rhs.shape(), &output),
        done);

    // To be consistent with the MatrixInverse op, we define the solution for
    // an empty set of equations as the empty matrix.
    if (input.NumElements() == 0 || rhs.NumElements() == 0) {
      done();
      return;
    }

    // TODO(rmlarsen): Convert to std::make_unique when available.
    std::unique_ptr<CudaSolver> solver(new CudaSolver(context));

    // Make a copy of the input for the factorization step, or, if adjoint_ is
    // false, try to reuse the input buffer if this op owns it exclusively.
    Tensor input_copy;
    const GPUDevice& device = context->eigen_device<GPUDevice>();
    if (adjoint_) {
      // For the adjoint case, it is simpler to always make a transposed copy up
      // front.
      OP_REQUIRES_OK_ASYNC(
          context,
          solver->allocate_scoped_tensor(DataTypeToEnum<Scalar>::value,
                                         input.shape(), &input_copy),
          done);
      OP_REQUIRES_OK_ASYNC(context,
                           DoMatrixTranspose(device, input, &input_copy), done);
    } else {
      OP_REQUIRES_OK_ASYNC(
          context,
          solver->forward_input_or_allocate_scoped_tensor(
              {0}, DataTypeToEnum<Scalar>::value, input.shape(), &input_copy),
          done);
      if (!input.SharesBufferWith(input_copy)) {
        device.memcpy(input_copy.flat<Scalar>().data(),
                      input.flat<Scalar>().data(),
                      input.NumElements() * sizeof(Scalar));
      }
    }
    auto input_copy_reshaped = input_copy.template flat_inner_dims<Scalar, 3>();
    const int64 batch_size = input_copy_reshaped.dimension(0);

    // Allocate pivots on the device.
    Tensor pivots;
    OP_REQUIRES_OK_ASYNC(
        context,
        solver->allocate_scoped_tensor(DataTypeToEnum<int>::value,
                                       TensorShape{batch_size, n}, &pivots),
        done);
    auto pivots_mat = pivots.template matrix<int>();

    // 1. Compute the partially pivoted LU factorization(s) of the
    // matrix/matrices.
    std::vector<DeviceLapackInfo> dev_info;
    auto input_copy_ptrs = solver->GetScratchSpace<uint8>(
        sizeof(Scalar*) * batch_size, "input_copt_ptrs",
        /* on_host */ true);
    const int kMaxMatrixSizeToBatchSizeRatio = 128;
    const bool use_batched_solver =
        n <= kMaxMatrixSizeToBatchSizeRatio * batch_size;
    if (use_batched_solver) {
      // For small matrices or large batch sizes, we use the batched interface
      // from cuBlas.
      const Scalar** input_copy_ptrs_base =
          reinterpret_cast<const Scalar**>(input_copy_ptrs.mutable_data());
      for (int batch = 0; batch < batch_size; ++batch) {
        input_copy_ptrs_base[batch] = &input_copy_reshaped(batch, 0, 0);
      }
      dev_info.push_back(
          solver->GetDeviceLapackInfo(batch_size, "getrfBatched"));
      OP_REQUIRES_OK_ASYNC(
          context,
          solver->GetrfBatched(n, input_copy_ptrs_base, n, pivots_mat.data(),
                               &dev_info.back(), batch_size),
          done);
    } else {
      // For small batch sizes or large matrices, we use the non-batched
      // interface from cuSolver, which is much faster for large matrices.
      dev_info.push_back(solver->GetDeviceLapackInfo(batch_size, "getrf"));
      for (int batch = 0; batch < batch_size; ++batch) {
        OP_REQUIRES_OK_ASYNC(
            context,
            solver->Getrf(n, n, &input_copy_reshaped(batch, 0, 0), n,
                          &pivots_mat(batch, 0), &dev_info.back()(batch)),
            done);
      }
    }

    // 2. Make a transposed copy of the right-hand sides. This is necessary
    // because cuBLAS assumes column-major storage while TensorFlow TF uses
    // row-major.
    TensorShape transposed_rhs_shape(rhs.shape());
    transposed_rhs_shape.RemoveLastDims(2);
    transposed_rhs_shape.AddDim(nrhs);
    transposed_rhs_shape.AddDim(n);
    Tensor transposed_rhs;
    OP_REQUIRES_OK_ASYNC(
        context,
        solver->allocate_scoped_tensor(DataTypeToEnum<Scalar>::value,
                                       transposed_rhs_shape, &transposed_rhs),
        done);
    if (nrhs > 1) {
      OP_REQUIRES_OK_ASYNC(
          context, DoMatrixTranspose(device, rhs, &transposed_rhs), done);
    } else {
      device.memcpy(transposed_rhs.flat<Scalar>().data(),
                    rhs.flat<Scalar>().data(),
                    rhs.NumElements() * sizeof(Scalar));
    }

    // 3. Solve op(A) X = B (in column major form).
    // We use a trick here: If adjoint_ is true, we converted A to column major
    // form above. If adjoint is false then I leave A in row-major form and use
    // trans_a = CUBLAS_OP_T to effectively transform it to column-major on the
    // fly. (This means that we actually use the LU-factorization of A^T in that
    // case, but that is equally good for solving AX=B). This way we save an
    // explicit transpose in the more common case of adjoint_ == false.
    auto input_copy_ptr_array = solver->GetScratchSpace<uint8>(
        sizeof(Scalar*) * batch_size, "input_copy_ptr_array",
        /* on_host */ true);
    auto transposed_rhs_ptr_array = solver->GetScratchSpace<uint8>(
        sizeof(Scalar*) * batch_size, "transposed_rhs_ptr_array",
        /* on_host */ true);
    auto transposed_rhs_reshaped =
        transposed_rhs.template flat_inner_dims<Scalar, 3>();
    if (use_batched_solver) {
      const Scalar** input_copy_ptrs_base =
          reinterpret_cast<const Scalar**>(input_copy_ptr_array.mutable_data());
      const Scalar** transposed_rhs_ptrs_base =
          reinterpret_cast<const Scalar**>(
              transposed_rhs_ptr_array.mutable_data());
      for (int batch = 0; batch < batch_size; ++batch) {
        input_copy_ptrs_base[batch] = &input_copy_reshaped(batch, 0, 0);
        transposed_rhs_ptrs_base[batch] = &transposed_rhs_reshaped(batch, 0, 0);
      }
      int host_info = 0;
      OP_REQUIRES_OK_ASYNC(
          context,
          solver->GetrsBatched(adjoint_ ? CUBLAS_OP_C : CUBLAS_OP_T, n, nrhs,
                               input_copy_ptrs_base, n, pivots_mat.data(),
                               transposed_rhs_ptrs_base, n, &host_info,
                               batch_size),
          done);
      OP_REQUIRES_ASYNC(
          context, host_info == 0,
          errors::InvalidArgument("The ", -host_info,
                                  "'th argument to cublas*getrsBatched had "
                                  "an illegal value."),
          done);
    } else {
      dev_info.push_back(solver->GetDeviceLapackInfo(batch_size, "getrs"));
      for (int batch = 0; batch < batch_size; ++batch) {
        OP_REQUIRES_OK_ASYNC(
            context,
            solver->Getrs(adjoint_ ? CUBLAS_OP_C : CUBLAS_OP_T, n, nrhs,
                          &input_copy_reshaped(batch, 0, 0), n,
                          &pivots_mat(batch, 0),
                          &transposed_rhs_reshaped(batch, 0, 0), n,
                          &dev_info.back()(batch)),
            done);
      }
    }

    // 4. Transpose X to get the final result in row-major form.
    if (nrhs > 1) {
      OP_REQUIRES_OK_ASYNC(
          context, DoMatrixTranspose(device, transposed_rhs, output), done);
    } else {
      device.memcpy(output->flat<Scalar>().data(),
                    transposed_rhs.flat<Scalar>().data(),
                    transposed_rhs.NumElements() * sizeof(Scalar));
    }

    // Callback for checking info after kernels finish. Also capture the
    // temporary Tensors/ScratchSpace so they don't get deallocated before the
    // kernels run. TODO(rmlarsen): Use move capture once C++14 becomes
    // available.
    auto info_checker = [context, done, dev_info](
                            const Status& status,
                            const std::vector<HostLapackInfo>& host_infos) {
      if (!status.ok() && errors::IsInvalidArgument(status) &&
          !host_infos.empty()) {
        for (int i = 0; i < host_infos[0].size(); ++i) {
          // Match the CPU error message for singular matrices. Otherwise
          // just print the original error message from the status below.
          OP_REQUIRES_ASYNC(context, host_infos[0].data()[i] <= 0,
                            errors::InvalidArgument(kErrMsg), done);
        }
      }
      OP_REQUIRES_OK_ASYNC(context, status, done);
      done();
    };
    CudaSolver::CheckLapackInfoAndDeleteSolverAsync(std::move(solver), dev_info,
                                                    std::move(info_checker));
  }