def lu(a):
 import scipy.linalg
 if (a.ndim != 2):
  raise ValueError('Dimension must be 2 to perform lu decomposition')
 (xdim, ydim) = a.shape
 if (xdim != ydim):
  raise ValueError('Input must be a square matrix to perform lu decomposition')
 if (not (len(set((a.chunks[0] + a.chunks[1]))) == 1)):
  msg = 'All chunks must be a square matrix to perform lu decomposition. Use .rechunk method to change the size of chunks.'
  raise ValueError(msg)
 vdim = len(a.chunks[0])
 hdim = len(a.chunks[1])
 token = tokenize(a)
 name_lu = ('lu-lu-' + token)
 name_p = ('lu-p-' + token)
 name_l = ('lu-l-' + token)
 name_u = ('lu-u-' + token)
 name_p_inv = ('lu-p-inv-' + token)
 name_l_permuted = ('lu-l-permute-' + token)
 name_u_transposed = ('lu-u-transpose-' + token)
 name_plu_dot = ('lu-plu-dot-' + token)
 name_lu_dot = ('lu-lu-dot-' + token)
 dsk = {}
 for i in range(min(vdim, hdim)):
  target = (a.name, i, i)
  if (i > 0):
   prevs = []
   for p in range(i):
    prev = (name_plu_dot, i, p, p, i)
    dsk[prev] = (np.dot, (name_l_permuted, i, p), (name_u, p, i))
    prevs.append(prev)
   target = (operator.sub, target, (sum, prevs))
  dsk[(name_lu, i, i)] = (scipy.linalg.lu, target)
  for j in range((i + 1), hdim):
   target = (np.dot, (name_p_inv, i, i), (a.name, i, j))
   if (i > 0):
    prevs = []
    for p in range(i):
     prev = (name_lu_dot, i, p, p, j)
     dsk[prev] = (np.dot, (name_l, i, p), (name_u, p, j))
     prevs.append(prev)
    target = (operator.sub, target, (sum, prevs))
   dsk[(name_lu, i, j)] = (_solve_triangular_lower, (name_l, i, i), target)
  for k in range((i + 1), vdim):
   target = (a.name, k, i)
   if (i > 0):
    prevs = []
    for p in range(i):
     prev = (name_plu_dot, k, p, p, i)
     dsk[prev] = (np.dot, (name_l_permuted, k, p), (name_u, p, i))
     prevs.append(prev)
    target = (operator.sub, target, (sum, prevs))
   dsk[(name_lu, k, i)] = (np.transpose, (_solve_triangular_lower, (name_u_transposed, i, i), (np.transpose, target)))
 for i in range(min(vdim, hdim)):
  for j in range(min(vdim, hdim)):
   if (i == j):
    dsk[(name_p, i, j)] = (operator.getitem, (name_lu, i, j), 0)
    dsk[(name_l, i, j)] = (operator.getitem, (name_lu, i, j), 1)
    dsk[(name_u, i, j)] = (operator.getitem, (name_lu, i, j), 2)
    dsk[(name_l_permuted, i, j)] = (np.dot, (name_p, i, j), (name_l, i, j))
    dsk[(name_u_transposed, i, j)] = (np.transpose, (name_u, i, j))
    dsk[(name_p_inv, i, j)] = (np.transpose, (name_p, i, j))
   elif (i > j):
    dsk[(name_p, i, j)] = (np.zeros, (a.chunks[0][i], a.chunks[1][j]))
    dsk[(name_l, i, j)] = (np.dot, (name_p_inv, i, i), (name_lu, i, j))
    dsk[(name_u, i, j)] = (np.zeros, (a.chunks[0][i], a.chunks[1][j]))
    dsk[(name_l_permuted, i, j)] = (name_lu, i, j)
   else:
    dsk[(name_p, i, j)] = (np.zeros, (a.chunks[0][i], a.chunks[1][j]))
    dsk[(name_l, i, j)] = (np.zeros, (a.chunks[0][i], a.chunks[1][j]))
    dsk[(name_u, i, j)] = (name_lu, i, j)
 dsk.update(a.dask)
 (pp, ll, uu) = scipy.linalg.lu(np.ones(shape=(1, 1), dtype=a.dtype))
 p = Array(dsk, name_p, shape=a.shape, chunks=a.chunks, dtype=pp.dtype)
 l = Array(dsk, name_l, shape=a.shape, chunks=a.chunks, dtype=ll.dtype)
 u = Array(dsk, name_u, shape=a.shape, chunks=a.chunks, dtype=uu.dtype)
 return (p, l, u)

def lu ( a ) : import scipy . linalg if ( a . ndim != 2 ) : raise ValueError ( 'Dimension must be 2 to perform lu decomposition' ) ( xdim , ydim ) = a . shape if ( xdim != ydim ) : raise ValueError ( 'Input must be a square matrix to perform lu decomposition' ) if ( not ( len ( set ( ( a . chunks [ 0 ] + a . chunks [ 1 ] ) ) ) == 1 ) ) : msg = 'All chunks must be a square matrix to perform lu decomposition. Use .rechunk method to change the size of chunks.' raise ValueError ( msg ) vdim = len ( a . chunks [ 0 ] ) hdim = len ( a . chunks [ 1 ] ) token = tokenize ( a ) name_lu = ( 'lu-lu-' + token ) name_p = ( 'lu-p-' + token ) name_l = ( 'lu-l-' + token ) name_u = ( 'lu-u-' + token ) name_p_inv = ( 'lu-p-inv-' + token ) name_l_permuted = ( 'lu-l-permute-' + token ) name_u_transposed = ( 'lu-u-transpose-' + token ) name_plu_dot = ( 'lu-plu-dot-' + token ) name_lu_dot = ( 'lu-lu-dot-' + token ) dsk = { } for i in range ( min ( vdim , hdim ) ) : target = ( a . name , i , i ) if ( i > 0 ) : prevs = [ ] for p in range ( i ) : prev = ( name_plu_dot , i , p , p , i ) dsk [ prev ] = ( np . dot , ( name_l_permuted , i , p ) , ( name_u , p , i ) ) prevs . append ( prev ) target = ( operator . sub , target , ( sum , prevs ) ) dsk [ ( name_lu , i , i ) ] = ( scipy . linalg . lu , target ) for j in range ( ( i + 1 ) , hdim ) : target = ( np . dot , ( name_p_inv , i , i ) , ( a . name , i , j ) ) if ( i > 0 ) : prevs = [ ] for p in range ( i ) : prev = ( name_lu_dot , i , p , p , j ) dsk [ prev ] = ( np . dot , ( name_l , i , p ) , ( name_u , p , j ) ) prevs . append ( prev ) target = ( operator . sub , target , ( sum , prevs ) ) dsk [ ( name_lu , i , j ) ] = ( _solve_triangular_lower , ( name_l , i , i ) , target ) for k in range ( ( i + 1 ) , vdim ) : target = ( a . name , k , i ) if ( i > 0 ) : prevs = [ ] for p in range ( i ) : prev = ( name_plu_dot , k , p , p , i ) dsk [ prev ] = ( np . dot , ( name_l_permuted , k , p ) , ( name_u , p , i ) ) prevs . append ( prev ) target = ( operator . sub , target , ( sum , prevs ) ) dsk [ ( name_lu , k , i ) ] = ( np . transpose , ( _solve_triangular_lower , ( name_u_transposed , i , i ) , ( np . transpose , target ) ) ) for i in range ( min ( vdim , hdim ) ) : for j in range ( min ( vdim , hdim ) ) : if ( i == j ) : dsk [ ( name_p , i , j ) ] = ( operator . getitem , ( name_lu , i , j ) , 0 ) dsk [ ( name_l , i , j ) ] = ( operator . getitem , ( name_lu , i , j ) , 1 ) dsk [ ( name_u , i , j ) ] = ( operator . getitem , ( name_lu , i , j ) , 2 ) dsk [ ( name_l_permuted , i , j ) ] = ( np . dot , ( name_p , i , j ) , ( name_l , i , j ) ) dsk [ ( name_u_transposed , i , j ) ] = ( np . transpose , ( name_u , i , j ) ) dsk [ ( name_p_inv , i , j ) ] = ( np . transpose , ( name_p , i , j ) ) elif ( i > j ) : dsk [ ( name_p , i , j ) ] = ( np . zeros , ( a . chunks [ 0 ] [ i ] , a . chunks [ 1 ] [ j ] ) ) dsk [ ( name_l , i , j ) ] = ( np . dot , ( name_p_inv , i , i ) , ( name_lu , i , j ) ) dsk [ ( name_u , i , j ) ] = ( np . zeros , ( a . chunks [ 0 ] [ i ] , a . chunks [ 1 ] [ j ] ) ) dsk [ ( name_l_permuted , i , j ) ] = ( name_lu , i , j ) else : dsk [ ( name_p , i , j ) ] = ( np . zeros , ( a . chunks [ 0 ] [ i ] , a . chunks [ 1 ] [ j ] ) ) dsk [ ( name_l , i , j ) ] = ( np . zeros , ( a . chunks [ 0 ] [ i ] , a . chunks [ 1 ] [ j ] ) ) dsk [ ( name_u , i , j ) ] = ( name_lu , i , j ) dsk . update ( a . dask ) ( pp , ll , uu ) = scipy . linalg . lu ( np . ones ( shape = ( 1 , 1 ) , dtype = a . dtype ) ) p = Array ( dsk , name_p , shape = a . shape , chunks = a . chunks , dtype = pp . dtype ) l = Array ( dsk , name_l , shape = a . shape , chunks = a . chunks , dtype = ll . dtype ) u = Array ( dsk , name_u , shape = a . shape , chunks = a . chunks , dtype = uu . dtype ) return ( p , l , u )
compute pivoted lu decomposition of a matrix .