lapack.py

# Copyright 2018 The JAX Authors.
#
# 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.

# Shims that allow the XLA CPU backend to call scipy-provided LAPACK kernels
# via CustomCallWithLayout.

import jaxlib.mlir.ir as ir
import jaxlib.mlir.dialects.stablehlo as hlo

import numpy as np
from jaxlib import xla_client

from .hlo_helpers import custom_call
from .cpu import _lapack

for _name, _value in _lapack.registrations().items():
  xla_client.register_custom_call_target(_name, _value, platform="cpu")

# Function that lazily initializes the LAPACK kernels in the runtime on first
# use.
_initialize = _lapack.initialize


def _hlo_u8(x):
  return hlo.ConstantOp(
      ir.DenseElementsAttr.get(
          np.array(x, dtype=np.uint8),
          type=ir.IntegerType.get_unsigned(8))).result

def _hlo_s32(x):
  return hlo.ConstantOp(
      ir.DenseElementsAttr.get(
          np.array(x, dtype=np.int32),
          type=ir.IntegerType.get_signless(32))).result

# TODO(phawkins): it would be nice to avoid duplicating code for each type.

# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def trsm_mhlo(dtype, alpha, a, b, left_side=False, lower=False, trans_a=False,
              conj_a=False, diag=False):
  return trsm_hlo(dtype, alpha, a, b, left_side=left_side, lower=lower,
                  trans_a=trans_a, conj_a=conj_a, diag=diag)

# ?trsm(left_side, lower, trans_a, diag, m, n, alpha, a, b):
# triangular solve
def trsm_hlo(dtype, alpha, a, b, left_side=False, lower=False, trans_a=False,
             conj_a=False, diag=False):
  _initialize()
  a_type = ir.RankedTensorType(a.type)
  b_type = ir.RankedTensorType(b.type)

  dims = b_type.shape
  m, n = dims[-2:]
  k = m if left_side else n

  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  num_b = 1
  for d in batch_dims:
    num_b *= d

  if (batch_dims + (k, k) != tuple(a_type.shape) or
      a_type.element_type != b_type.element_type):
    raise ValueError("Argument mismatch for trsm, got {} and {}".format(
      a_type, b_type))

  if dtype == np.float32:
    fn = "blas_strsm"
  elif dtype == np.float64:
    fn = "blas_dtrsm"
  elif dtype == np.complex64:
    fn = "blas_ctrsm"
  elif dtype == np.complex128:
    fn = "blas_ztrsm"
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  if conj_a and not trans_a:
    raise NotImplementedError("Conjugation without transposition not supported")
  scalar_layout = []
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))
  return custom_call(
      fn,
      [b.type],
      [_hlo_s32(int(left_side)), _hlo_s32(int(lower)),
       _hlo_s32((2 if conj_a else 1) if trans_a else 0), _hlo_s32(int(diag)),
       _hlo_s32(m), _hlo_s32(n), _hlo_s32(num_b),
       alpha, a, b],
      operand_layouts=[scalar_layout] * 8 + [layout] * 2,
      result_layouts=[layout],
      operand_output_aliases={9: 0},
  )


# # ?getrf: LU decomposition

# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def getrf_mhlo(dtype, a):
  return getrf_hlo(dtype, a)

def getrf_hlo(dtype, a):
  _initialize()
  dims = ir.RankedTensorType(a.type).shape
  assert len(dims) >= 2
  m, n = dims[-2:]
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d

  if dtype == np.float32:
    fn = b"lapack_sgetrf"
  elif dtype == np.float64:
    fn = b"lapack_dgetrf"
  elif dtype == np.complex64:
    fn = b"lapack_cgetrf"
  elif dtype == np.complex128:
    fn = b"lapack_zgetrf"
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  scalar_layout = []
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))
  i32_type = ir.IntegerType.get_signless(32)
  return custom_call(
      fn,
      [
        a.type,
        ir.RankedTensorType.get(batch_dims + (min(m, n),), i32_type),
        ir.RankedTensorType.get(batch_dims, i32_type),
      ],
      [_hlo_s32(int(b)), _hlo_s32(m), _hlo_s32(n), a],
      operand_layouts=[scalar_layout] * 3 + [layout],
      result_layouts=[
        layout,
        tuple(range(num_bd, -1, -1)),
        tuple(range(num_bd - 1, -1, -1)),
      ],
      operand_output_aliases={3: 0},
  )


# # ?geqrf: QR decomposition

# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def geqrf_mhlo(dtype, a):
  return geqrf_hlo(dtype, a)

def geqrf_hlo(dtype, a):
  _initialize()
  a_type = ir.RankedTensorType(a.type)
  dims = a_type.shape
  assert len(dims) >= 2
  m, n = dims[-2:]
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d

  if dtype == np.float32:
    fn = b"lapack_sgeqrf"
    lwork = _lapack.lapack_sgeqrf_workspace(m, n)
  elif dtype == np.float64:
    fn = b"lapack_dgeqrf"
    lwork = _lapack.lapack_dgeqrf_workspace(m, n)
  elif dtype == np.complex64:
    fn = b"lapack_cgeqrf"
    lwork = _lapack.lapack_cgeqrf_workspace(m, n)
  elif dtype == np.complex128:
    fn = b"lapack_zgeqrf"
    lwork = _lapack.lapack_zgeqrf_workspace(m, n)
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  scalar_layout = []
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))
  i32_type = ir.IntegerType.get_signless(32)
  out = custom_call(
      fn,
      [
        a.type,
        ir.RankedTensorType.get(batch_dims + (min(m, n),), a_type.element_type),
        ir.RankedTensorType.get(batch_dims, i32_type),
        ir.RankedTensorType.get([lwork], a_type.element_type),
      ],
      [_hlo_s32(int(b)), _hlo_s32(m), _hlo_s32(n), _hlo_s32(lwork), a],
      operand_layouts=[scalar_layout] * 4 + [layout],
      result_layouts=[
        layout,
        tuple(range(num_bd, -1, -1)),
        tuple(range(num_bd - 1, -1, -1)),
        [0],
      ],
      operand_output_aliases={4: 0},
  )
  return out[:3]


# # ?orgqr: product of elementary Householder reflectors:

# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def orgqr_mhlo(dtype, a, tau):
  return orgqr_hlo(dtype, a, tau)

def orgqr_hlo(dtype, a, tau):
  _initialize()
  a_type = ir.RankedTensorType(a.type)
  dims = a_type.shape
  assert len(dims) >= 2
  m, n = dims[-2:]
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d

  tau_dims = ir.RankedTensorType(tau.type).shape
  assert tau_dims[:-1] == dims[:-2], (tau.type, a.type)
  k = tau_dims[-1]

  if dtype == np.float32:
    fn = b"lapack_sorgqr"
    lwork = _lapack.lapack_sorgqr_workspace(m, n, k)
  elif dtype == np.float64:
    fn = b"lapack_dorgqr"
    lwork = _lapack.lapack_dorgqr_workspace(m, n, k)
  elif dtype == np.complex64:
    fn = b"lapack_cungqr"
    lwork = _lapack.lapack_cungqr_workspace(m, n, k)
  elif dtype == np.complex128:
    fn = b"lapack_zungqr"
    lwork = _lapack.lapack_zungqr_workspace(m, n, k)
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  scalar_layout = []
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))
  i32_type = ir.IntegerType.get_signless(32)
  out = custom_call(
      fn,
      [
        a.type,
        ir.RankedTensorType.get(batch_dims, i32_type),
        ir.RankedTensorType.get([lwork], a_type.element_type),
      ],
      [_hlo_s32(int(b)), _hlo_s32(m), _hlo_s32(n), _hlo_s32(k),
       _hlo_s32(lwork), a, tau],
      operand_layouts=[scalar_layout] * 5 + [
        layout,
        tuple(range(num_bd, -1, -1)),
      ],
      result_layouts=[
        layout,
        tuple(range(num_bd - 1, -1, -1)),
        [0],
      ],
      operand_output_aliases={5: 0},
  )
  return out[:2]


# ?potrf: Cholesky decomposition

# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def potrf_mhlo(dtype, a, lower=False):
  return potrf_hlo(dtype, a, lower=lower)

def potrf_hlo(dtype, a, lower=False):
  _initialize()
  a_type = ir.RankedTensorType(a.type)
  dims = a_type.shape
  m, n = dims[-2:]
  if m != n:
    raise ValueError(f"potrf expects a square matrix, got {a_type}")
  if dtype == np.float32:
    fn = b"lapack_spotrf"
  elif dtype == np.float64:
    fn = b"lapack_dpotrf"
  elif dtype == np.complex64:
    fn = b"lapack_cpotrf"
  elif dtype == np.complex128:
    fn = b"lapack_zpotrf"
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d

  scalar_layout = []
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))
  info_layout = tuple(range(num_bd - 1, -1, -1))
  out = custom_call(
      fn,
      [a.type,
       ir.RankedTensorType.get(batch_dims, ir.IntegerType.get_signless(32))],
      [_hlo_s32(int(lower)), _hlo_s32(b), _hlo_s32(n), a],
      operand_layouts=[scalar_layout] * 3 + [layout],
      result_layouts=[layout, info_layout],
      operand_output_aliases={3: 0},
  )
  return out[:2]



# # ?gesdd: Singular value decomposition

# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def gesdd_mhlo(dtype, a, full_matrices=True, compute_uv=True):
  return gesdd_hlo(dtype, a, full_matrices=full_matrices, compute_uv=compute_uv)

def gesdd_hlo(dtype, a, full_matrices=True, compute_uv=True):
  _initialize()
  a_type = ir.RankedTensorType(a.type)
  dims = a_type.shape
  assert len(dims) >= 2
  m, n = dims[-2:]
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d

  i32_type = ir.IntegerType.get_signless(32)
  if dtype == np.float32:
    fn = b"lapack_sgesdd"
    singular_vals_type = ir.F32Type.get()
    lwork = _lapack.sgesdd_work_size(m, n, compute_uv, full_matrices)
    workspace = [
        ir.RankedTensorType.get([_lapack.gesdd_iwork_size(m, n)], i32_type),
        ir.RankedTensorType.get([lwork], a_type.element_type),
    ]
    workspace_layouts = [[0], [0]]
  elif dtype == np.float64:
    fn = b"lapack_dgesdd"
    singular_vals_type = ir.F64Type.get()
    lwork = _lapack.dgesdd_work_size(m, n, compute_uv, full_matrices)
    workspace = [
        ir.RankedTensorType.get([_lapack.gesdd_iwork_size(m, n)], i32_type),
        ir.RankedTensorType.get([lwork], a_type.element_type),
    ]
    workspace_layouts = [[0], [0]]
  elif dtype == np.complex64:
    fn = b"lapack_cgesdd"
    singular_vals_type = ir.F32Type.get()
    lwork = _lapack.cgesdd_work_size(m, n, compute_uv, full_matrices)
    workspace = [
        ir.RankedTensorType.get([_lapack.gesdd_iwork_size(m, n)], i32_type),
        ir.RankedTensorType.get(
            [_lapack.cgesdd_rwork_size(m, n, int(compute_uv))],
            ir.F32Type.get()),
        ir.RankedTensorType.get([lwork], a_type.element_type),
    ]
    workspace_layouts = [[0], [0], [0]]
  elif dtype == np.complex128:
    fn = b"lapack_zgesdd"
    singular_vals_type = ir.F64Type.get()
    lwork = _lapack.zgesdd_work_size(m, n, compute_uv, full_matrices)
    workspace = [
        ir.RankedTensorType.get([_lapack.gesdd_iwork_size(m, n)], i32_type),
        ir.RankedTensorType.get(
            [_lapack.cgesdd_rwork_size(m, n, int(compute_uv))],
            ir.F64Type.get()),
        ir.RankedTensorType.get([lwork], a_type.element_type),
    ]
    workspace_layouts = [[0], [0], [0]]
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  scalar_layout = []
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))
  out = custom_call(
      fn,
      [
          a.type,
          ir.RankedTensorType.get(batch_dims + (min(m, n),), singular_vals_type),
          ir.RankedTensorType.get(
            batch_dims + (m, m if full_matrices else min(m, n)),
            a_type.element_type),
          ir.RankedTensorType.get(
            batch_dims + (n if full_matrices else min(m, n), n),
            a_type.element_type),
          ir.RankedTensorType.get(batch_dims, i32_type),
      ] + workspace,
      [_hlo_s32(int(full_matrices)), _hlo_s32(int(compute_uv)), _hlo_s32(b),
       _hlo_s32(m), _hlo_s32(n), _hlo_s32(lwork), a],
      operand_layouts=[scalar_layout] * 6 + [layout],
      result_layouts=[
          layout,
          (num_bd,) + tuple(range(num_bd - 1, -1, -1)),
          layout,
          layout,
          tuple(range(num_bd - 1, -1, -1)),
      ] + workspace_layouts,
      operand_output_aliases={6: 0},
  )
  return out[1:5]


# # syevd: Symmetric eigendecomposition

# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def syevd_mhlo(dtype, a, lower=False):
  return syevd_hlo(dtype, a, lower=lower)

def syevd_hlo(dtype, a, lower=False):
  _initialize()
  a_type = ir.RankedTensorType(a.type)
  dims = a_type.shape
  assert len(dims) >= 2
  m, n = dims[-2:]
  assert m == n
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))

  i32_type = ir.IntegerType.get_signless(32)
  if dtype == np.float32:
    fn = b"lapack_ssyevd"
    eigvals_type = ir.F32Type.get()
    workspace = [
        ir.RankedTensorType.get([_lapack.syevd_work_size(n)],
                                a_type.element_type),
        ir.RankedTensorType.get([_lapack.syevd_iwork_size(n)], i32_type),
    ]
    workspace_layouts = [[0], [0]]
  elif dtype == np.float64:
    fn = b"lapack_dsyevd"
    eigvals_type = ir.F64Type.get()
    workspace = [
        ir.RankedTensorType.get([_lapack.syevd_work_size(n)],
                                a_type.element_type),
        ir.RankedTensorType.get([_lapack.syevd_iwork_size(n)], i32_type),
    ]
    workspace_layouts = [[0], [0]]
  elif dtype == np.complex64:
    fn = b"lapack_cheevd"
    eigvals_type = ir.F32Type.get()
    workspace = [
        ir.RankedTensorType.get([_lapack.heevd_work_size(n)],
                                a_type.element_type),
        ir.RankedTensorType.get([_lapack.heevd_rwork_size(n)], eigvals_type),
        ir.RankedTensorType.get([_lapack.syevd_iwork_size(n)], i32_type),
    ]
    workspace_layouts = [[0], [0], [0]]
  elif dtype == np.complex128:
    fn = b"lapack_zheevd"
    eigvals_type = ir.F64Type.get()
    workspace = [
        ir.RankedTensorType.get([_lapack.heevd_work_size(n)],
                                a_type.element_type),
        ir.RankedTensorType.get([_lapack.heevd_rwork_size(n)], eigvals_type),
        ir.RankedTensorType.get([_lapack.syevd_iwork_size(n)], i32_type),
    ]
    workspace_layouts = [[0], [0], [0]]
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  scalar_layout = []
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))
  out = custom_call(
      fn,
      [
          a.type,
          ir.RankedTensorType.get(batch_dims + (n,), eigvals_type),
          ir.RankedTensorType.get(batch_dims, i32_type),
      ] + workspace,
      [_hlo_s32(1 if lower else 0), _hlo_s32(b), _hlo_s32(n), a],
      operand_layouts=[scalar_layout] * 3 + [layout],
      result_layouts=[
          layout,
          tuple(range(num_bd, -1, -1)),
          tuple(range(num_bd - 1, -1, -1)),
      ] + workspace_layouts,
      operand_output_aliases={3: 0},
  )
  return out[:3]


# # geev: Nonsymmetric eigendecomposition

# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def geev_mhlo(dtype, a, jobvl=True, jobvr=True):
  return geev_hlo(dtype, a, jobvl=jobvl, jobvr=jobvr)

def geev_hlo(dtype, a, jobvl=True, jobvr=True):
  _initialize()
  dims = ir.RankedTensorType(a.type).shape
  assert len(dims) >= 2
  m, n = dims[-2:]
  assert m == n
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))

  jobvl_c = ord('V' if jobvl else 'N')
  jobvr_c = ord('V' if jobvr else 'N')

  if dtype == np.float32:
    fn = b"lapack_sgeev"
    real = True
    eigvecs_type = ir.ComplexType.get(ir.F32Type.get())
    workspaces = [ir.RankedTensorType.get([n, n], ir.F32Type.get()),
                  ir.RankedTensorType.get([n, n], ir.F32Type.get()),
                  ir.RankedTensorType.get([n, n], ir.F32Type.get())]
    workspace_layouts = [[0, 1]] * 3
    eigvals = [ir.RankedTensorType.get(batch_dims + (n,), ir.F32Type.get()),
               ir.RankedTensorType.get(batch_dims + (n,), ir.F32Type.get())]
    eigvals_layouts = [tuple(range(num_bd, -1, -1))] * 2
  elif dtype == np.float64:
    fn = b"lapack_dgeev"
    real = True
    eigvecs_type = ir.ComplexType.get(ir.F64Type.get())
    workspaces = [ir.RankedTensorType.get([n, n], ir.F64Type.get()),
                  ir.RankedTensorType.get([n, n], ir.F64Type.get()),
                  ir.RankedTensorType.get([n, n], ir.F64Type.get())]
    workspace_layouts = [[0, 1]] * 3
    eigvals = [ir.RankedTensorType.get(batch_dims + (n,), ir.F64Type.get()),
               ir.RankedTensorType.get(batch_dims + (n,), ir.F64Type.get())]
    eigvals_layouts = [tuple(range(num_bd, -1, -1))] * 2
  elif dtype == np.complex64:
    fn = b"lapack_cgeev"
    real = False
    eigvecs_type = ir.ComplexType.get(ir.F32Type.get())
    workspaces = [ir.RankedTensorType.get([n, n],
                                          ir.ComplexType.get(ir.F32Type.get())),
                  ir.RankedTensorType.get([2 * n], ir.F32Type.get())]
    workspace_layouts = [[0, 1], [0]]
    eigvals = [ir.RankedTensorType.get(batch_dims + (n,),
                                       ir.ComplexType.get(ir.F32Type.get()))]
    eigvals_layouts = [tuple(range(num_bd, -1, -1))]
  elif dtype == np.complex128:
    fn = b"lapack_zgeev"
    real = False
    eigvecs_type = ir.ComplexType.get(ir.F64Type.get())
    workspaces = [ir.RankedTensorType.get([n, n],
                                          ir.ComplexType.get(ir.F64Type.get())),
                  ir.RankedTensorType.get([2 * n], ir.F64Type.get())]
    workspace_layouts = [[0, 1], [0]]
    eigvals = [ir.RankedTensorType.get(batch_dims + (n,),
                                       ir.ComplexType.get(ir.F64Type.get()))]
    eigvals_layouts = [tuple(range(num_bd, -1, -1))]
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  i32_type = ir.IntegerType.get_signless(32)
  scalar_layout = []
  info_layout = tuple(range(num_bd - 1, -1, -1))
  out = custom_call(
      fn,
      workspaces + eigvals + [
        ir.RankedTensorType.get(dims, eigvecs_type),
        ir.RankedTensorType.get(dims, eigvecs_type),
        ir.RankedTensorType.get(batch_dims, i32_type),
      ],
      [_hlo_s32(b), _hlo_s32(n), _hlo_u8(jobvl_c), _hlo_u8(jobvr_c), a],
      operand_layouts=[scalar_layout] * 4 + [layout],
      result_layouts=(workspace_layouts + eigvals_layouts + [layout] * 2 +
                      [info_layout])
  )
  if real:
    return (hlo.ComplexOp(out[3], out[4]).result, out[5], out[6], out[7])
  else:
    return out[2:6]

# # gees : Schur factorization

# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def gees_mhlo(dtype, a, jobvs=True, sort=False, select=None):
  return gees_hlo(dtype, a, jobvs=jobvs, sort=sort, select=select)

def gees_hlo(dtype, a, jobvs=True, sort=False, select=None):
  _initialize()
  a_type = ir.RankedTensorType(a.type)
  etype = a_type.element_type
  dims = a_type.shape
  assert len(dims) >= 2
  m, n = dims[-2:]
  assert m == n
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d
  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))

  if sort:
    raise NotImplementedError(
        "The sort feature of LAPACK's gees routine is not implemented.")

  jobvs = ord('V' if jobvs else 'N')
  sort = ord('S' if sort else 'N')

  if dtype == np.float32:
    fn = "lapack_sgees"
  elif dtype == np.float64:
    fn = "lapack_dgees"
  elif dtype == np.complex64:
    fn = "lapack_cgees"
  elif dtype == np.complex128:
    fn = "lapack_zgees"
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  if not np.issubdtype(dtype, np.complexfloating):
    workspaces = [ir.RankedTensorType.get(dims, etype)]
    workspace_layouts = [layout]
    eigvals = [ir.RankedTensorType.get(batch_dims + (n,), etype)] * 2
    eigvals_layouts = [tuple(range(num_bd, -1, -1))] * 2
  else:
    workspaces = [
        ir.RankedTensorType.get(dims, etype),
        ir.RankedTensorType.get([n], ir.ComplexType(etype).element_type),
    ]
    workspace_layouts = [layout, [0]]
    eigvals = [ir.RankedTensorType.get(batch_dims + (n,), etype)]
    eigvals_layouts = [tuple(range(num_bd, -1, -1))]

  i32_type = ir.IntegerType.get_signless(32)

  scalar_layout = []
  out = custom_call(
      fn,
      workspaces + eigvals + [
        ir.RankedTensorType.get(dims, etype),
        ir.RankedTensorType.get(batch_dims, i32_type),
        ir.RankedTensorType.get(batch_dims, i32_type),
      ],
      [
        _hlo_s32(b),
        _hlo_s32(n),
        _hlo_u8(np.uint8(jobvs)),
        _hlo_u8(np.uint8(sort)),
        # TODO: figure out how to put the callable select function here
        a
      ],
      operand_layouts=[scalar_layout] * 4 + [layout],
      result_layouts=workspace_layouts + eigvals_layouts + [
        layout,
        tuple(range(num_bd - 1, -1, -1)),
        tuple(range(num_bd - 1, -1, -1)),
      ],
      operand_output_aliases={4: 0},
  )
  if sort == ord('S'):
    return (out[0], out[3], out[4], out[5])
  else:
    return (out[0], out[3], out[5])


# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def gehrd_mhlo(dtype, a):
  return gehrd_hlo(dtype, a)

# gehrd: Reduction of a non-symmetric square matrix to upper Hessenberg form.
def gehrd_hlo(dtype, a):
  _initialize()
  a_type = ir.RankedTensorType(a.type)
  dims = a_type.shape
  assert len(dims) >= 2
  m, n = dims[-2:]
  assert m == n, (m, n)
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d

  if dtype == np.float32:
    fn = b"lapack_sgehrd"
    lwork = _lapack.lapack_sgehrd_workspace(n, n, 1, n)
  elif dtype == np.float64:
    fn = b"lapack_dgehrd"
    lwork = _lapack.lapack_dgehrd_workspace(n, n, 1, n)
  elif dtype == np.complex64:
    fn = b"lapack_cgehrd"
    lwork = _lapack.lapack_cgehrd_workspace(n, n, 1, n)
  elif dtype == np.complex128:
    fn = b"lapack_zgehrd"
    lwork = _lapack.lapack_zgehrd_workspace(n, n, 1, n)
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))
  i32_type = ir.IntegerType.get_signless(32)
  out = custom_call(
      fn,
      [
        a.type,
        ir.RankedTensorType.get(batch_dims + (n - 1,), a_type.element_type),
        ir.RankedTensorType.get(batch_dims, i32_type),
        ir.RankedTensorType.get([lwork], a_type.element_type),
      ],
      [_hlo_s32(n), _hlo_s32(1), _hlo_s32(n), _hlo_s32(n), _hlo_s32(b),
       _hlo_s32(lwork), a],
      operand_layouts=[[]] * 6 + [layout],
      result_layouts=[
        layout,
        (num_bd,) + tuple(range(num_bd - 1, -1, -1)),
        tuple(range(num_bd - 1, -1, -1)),
        [0],
      ],
      operand_output_aliases={6: 0},
  )
  return out[:3]


# TODO(burmako): Remove this compatibility shim when mlir_api_version >= 41.
def sytrd_mhlo(dtype, a, *, lower):
  return sytrd_hlo(dtype, a, lower=lower)

# sytrd: Reduction of a symmetric (Hermitian) matrix to tridiagonal form.
def sytrd_hlo(dtype, a, *, lower):
  _initialize()
  a_type = ir.RankedTensorType(a.type)
  dims = a_type.shape
  assert len(dims) >= 2
  m, n = dims[-2:]
  assert m == n, (m, n)
  batch_dims = tuple(dims[:-2])
  num_bd = len(batch_dims)
  b = 1
  for d in batch_dims:
    b *= d

  if dtype == np.float32:
    fn = b"lapack_ssytrd"
    lwork = _lapack.lapack_ssytrd_workspace(n, n)
    diag_type = a_type.element_type
  elif dtype == np.float64:
    fn = b"lapack_dsytrd"
    lwork = _lapack.lapack_dsytrd_workspace(n, n)
    diag_type = a_type.element_type
  elif dtype == np.complex64:
    fn = b"lapack_chetrd"
    lwork = _lapack.lapack_chetrd_workspace(n, n)
    diag_type = ir.F32Type.get()
  elif dtype == np.complex128:
    fn = b"lapack_zhetrd"
    lwork = _lapack.lapack_zhetrd_workspace(n, n)
    diag_type = ir.F64Type.get()
  else:
    raise NotImplementedError(f"Unsupported dtype {dtype}")

  layout = (num_bd, num_bd + 1) + tuple(range(num_bd - 1, -1, -1))
  i32_type = ir.IntegerType.get_signless(32)
  out = custom_call(
      fn,
      [
        a.type,
        ir.RankedTensorType.get(batch_dims + (n,), diag_type),
        ir.RankedTensorType.get(batch_dims + (n - 1,), diag_type),
        ir.RankedTensorType.get(batch_dims + (n - 1,), a_type.element_type),
        ir.RankedTensorType.get(batch_dims, i32_type),
        ir.RankedTensorType.get([lwork], a_type.element_type),
      ],
      [_hlo_s32(n), _hlo_s32(1 if lower else 0), _hlo_s32(max(1, n)),
       _hlo_s32(b), _hlo_s32(lwork), a],
      operand_layouts=[[]] * 5 + [layout],
      result_layouts=[
        layout,
        (num_bd,) + tuple(range(num_bd - 1, -1, -1)),
        (num_bd,) + tuple(range(num_bd - 1, -1, -1)),
        (num_bd,) + tuple(range(num_bd - 1, -1, -1)),
        tuple(range(num_bd - 1, -1, -1)),
        [0],
      ],
      operand_output_aliases={5: 0},
  )
  return out[:5]