from __future__ import absolute_import, division, print_function

import operator
from functools import wraps
from numbers import Number

import numpy as np
import toolz

from ..base import tokenize
from ..compatibility import apply
from .. import sharedict
from .core import top, dotmany, Array, concatenate
from .creation import eye
from .random import RandomState


def _cumsum_blocks(it):
    total = 0
    for x in it:
        total_previous = total
        total += x
        yield (total_previous, total)


def _cumsum_part(last, new):
    return (last[1], last[1] + new)


def _nanmin(m, n):
    k_0 = min([m, n])
    k_1 = m if np.isnan(n) else n
    return k_1 if np.isnan(k_0) else k_0


def tsqr(data, compute_svd=False, _max_vchunk_size=None):
    """ Direct Tall-and-Skinny QR algorithm

    As presented in:

        A. Benson, D. Gleich, and J. Demmel.
        Direct QR factorizations for tall-and-skinny matrices in
        MapReduce architectures.
        IEEE International Conference on Big Data, 2013.
        http://arxiv.org/abs/1301.1071

    This algorithm is used to compute both the QR decomposition and the
    Singular Value Decomposition.  It requires that the input array have a
    single column of blocks, each of which fit in memory.

    Parameters
    ----------
    data: Array
    compute_svd: bool
        Whether to compute the SVD rather than the QR decomposition
    _max_vchunk_size: Integer
        Used internally in recursion to set the maximum row dimension
        of chunks in subsequent recursive calls.

    Notes
    -----
    With ``k`` blocks of size ``(m, n)``, this algorithm has memory use that
    scales as ``m * n * k``.

    The implementation here is the recursive variant due to the ultimate
    need for one "single core" QR decomposition. Given m blocks, the
    "single core" QR decomposition will have to work with a ``(k * n, n)``
    matrix, and ``k * n`` may be substantially larger than ``m``.

    As such, (if ``m / n >= 2``) recursion will be applied as necessary
    to ensure that single core computations do not have to work on blocks
    larger than ``(m, n)``.

    Where blocks are irregular, the above logic is applied with the "height" of
    the "tallest" block used in place of ``m``.

    Consider use of the ``rechunk`` method to control this behavior. Blocks
    that are as tall as possible are recommended.

    See Also
    --------
    dask.array.linalg.qr - Powered by this algorithm
    dask.array.linalg.svd - Powered by this algorithm
    dask.array.linalg.sfqr - Variant for short-and-fat arrays
    """
    nr, nc = len(data.chunks[0]), len(data.chunks[1])
    cr_max, cc = max(data.chunks[0]), data.chunks[1][0]

    if not (data.ndim == 2 and  # Is a matrix
            nc == 1):           # Only one column block
        raise ValueError(
            "Input must have the following properties:\n"
            "  1. Have two dimensions\n"
            "  2. Have only one column of blocks\n\n"
            "Note: This function (tsqr) supports QR decomposition in the case of\n"
            "tall-and-skinny matrices (single column chunk/block; see qr)"
        )

    token = '-' + tokenize(data, compute_svd)

    m, n = data.shape
    numblocks = (nr, 1)

    qq, rr = np.linalg.qr(np.ones(shape=(1, 1), dtype=data.dtype))

    dsk = sharedict.ShareDict()
    dsk.update(data.dask)

    # Block qr
    name_qr_st1 = 'qr' + token
    dsk_qr_st1 = top(np.linalg.qr, name_qr_st1, 'ij', data.name, 'ij',
                     numblocks={data.name: numblocks})
    dsk.update_with_key(dsk_qr_st1, key=name_qr_st1)

    # Block qr[0]
    name_q_st1 = 'getitem' + token + '-q1'
    dsk_q_st1 = dict(((name_q_st1, i, 0),
                      (operator.getitem, (name_qr_st1, i, 0), 0))
                     for i in range(numblocks[0]))
    dsk.update_with_key(dsk_q_st1, key=name_q_st1)

    # Block qr[1]
    name_r_st1 = 'getitem' + token + '-r1'
    dsk_r_st1 = dict(((name_r_st1, i, 0),
                      (operator.getitem, (name_qr_st1, i, 0), 1))
                     for i in range(numblocks[0]))
    dsk.update_with_key(dsk_r_st1, key=name_r_st1)

    # Next step is to obtain a QR decomposition for the stacked R factors, so either:
    # - gather R factors into a single core and do a QR decomposition
    # - recurse with tsqr (if single core computation too large and a-priori "meaningful
    #   reduction" possible, meaning that chunks have to be well defined)

    single_core_compute_m = nr * cc
    chunks_well_defined = not any(np.isnan(c) for cs in data.chunks for c in cs)
    prospective_blocks = np.ceil(single_core_compute_m / cr_max)
    meaningful_reduction_possible = (cr_max if _max_vchunk_size is None else _max_vchunk_size) >= 2 * cc
    can_distribute = chunks_well_defined and int(prospective_blocks) > 1

    if chunks_well_defined and meaningful_reduction_possible and can_distribute:
        # stack chunks into blocks and recurse using tsqr

        # Prepare to stack chunks into blocks (from block qr[1])
        all_blocks = []
        curr_block = []
        curr_block_sz = 0
        for idx, a_m in enumerate(data.chunks[0]):
            m_q = a_m
            n_q = min(m_q, cc)
            m_r = n_q
            # n_r = cc
            if curr_block_sz + m_r > cr_max:
                all_blocks.append(curr_block)
                curr_block = []
                curr_block_sz = 0
            curr_block.append((idx, m_r))
            curr_block_sz += m_r
        if len(curr_block) > 0:
            all_blocks.append(curr_block)

        # R_stacked
        name_r_stacked = 'stack' + token + '-r1'
        dsk_r_stacked = dict(((name_r_stacked, i, 0),
                              (np.vstack, (tuple,
                                           [(name_r_st1, idx, 0)
                                            for idx, _ in sub_block_info])))
                             for i, sub_block_info in enumerate(all_blocks))
        dsk.update_with_key(dsk_r_stacked, key=name_r_stacked)

        # retrieve R_stacked for recursion with tsqr
        vchunks_rstacked = tuple([sum(map(lambda x: x[1], sub_block_info)) for sub_block_info in all_blocks])
        r_stacked = Array(dsk, name_r_stacked,
                          shape=(sum(vchunks_rstacked), n), chunks=(vchunks_rstacked, (n)), dtype=rr.dtype)

        # recurse
        q_inner, r_inner = tsqr(r_stacked, _max_vchunk_size=cr_max)
        dsk.update(q_inner.dask)
        dsk.update(r_inner.dask)

        # Q_inner: "unstack"
        name_q_st2 = 'getitem-' + token + '-q2'
        dsk_q_st2 = dict(((name_q_st2, j, 0),
                          (operator.getitem,
                           (q_inner.name, i, 0),
                           ((slice(e[0], e[1])), (slice(0, n)))))
                         for i, sub_block_info in enumerate(all_blocks)
                         for j, e in zip([x[0] for x in sub_block_info],
                                         _cumsum_blocks([x[1] for x in sub_block_info])))
        dsk.update_with_key(dsk_q_st2, key=name_q_st2)

        # R: R_inner
        name_r_st2 = 'r-inner-' + token
        dsk_r_st2 = {(name_r_st2, 0, 0): (r_inner.name, 0, 0)}
        dsk.update_with_key(dsk_r_st2, key=name_r_st2)

        # Q: Block qr[0] (*) Q_inner
        name_q_st3 = 'dot-' + token + '-q3'
        dsk_q_st3 = top(np.dot, name_q_st3, 'ij', name_q_st1, 'ij',
                        name_q_st2, 'ij', numblocks={name_q_st1: numblocks,
                                                     name_q_st2: numblocks})
        dsk.update_with_key(dsk_q_st3, key=name_q_st3)
    else:
        # Do single core computation

        # Stacking for in-core QR computation
        to_stack = [(name_r_st1, i, 0) for i in range(numblocks[0])]
        name_r_st1_stacked = 'stack' + token + '-r1'
        dsk_r_st1_stacked = {(name_r_st1_stacked, 0, 0): (np.vstack,
                                                          (tuple, to_stack))}
        dsk.update_with_key(dsk_r_st1_stacked, key=name_r_st1_stacked)

        # In-core QR computation
        name_qr_st2 = 'qr' + token + '-qr2'
        dsk_qr_st2 = top(np.linalg.qr, name_qr_st2, 'ij', name_r_st1_stacked, 'ij',
                         numblocks={name_r_st1_stacked: (1, 1)})
        dsk.update_with_key(dsk_qr_st2, key=name_qr_st2)

        # In-core qr[0]
        name_q_st2_aux = 'getitem' + token + '-q2-aux'
        dsk_q_st2_aux = {(name_q_st2_aux, 0, 0): (operator.getitem,
                                                  (name_qr_st2, 0, 0), 0)}
        dsk.update_with_key(dsk_q_st2_aux, key=name_q_st2_aux)

        if not any(np.isnan(c) for cs in data.chunks for c in cs):
            # when chunks are all known...
            # obtain slices on q from in-core compute (e.g.: (slice(10, 20), slice(0, 5)))
            q2_block_sizes = [min(e, n) for e in data.chunks[0]]
            block_slices = [(slice(e[0], e[1]), slice(0, n))
                            for e in _cumsum_blocks(q2_block_sizes)]
            dsk_q_blockslices = {}
        else:
            # when chunks are not already known...

            # request shape information: vertical chunk sizes & column dimension (n)
            name_q2bs = 'shape' + token + '-q2'
            dsk_q2_shapes = {(name_q2bs, i): (min, (getattr, (data.name, i, 0), 'shape'))
                             for i in range(numblocks[0])}
            name_n = 'getitem' + token + '-n'
            dsk_n = {name_n: (operator.getitem,
                              (getattr, (data.name, 0, 0), 'shape'), 1)}

            # cumulative sums (start, end)
            name_q2cs = 'cumsum' + token + '-q2'
            dsk_q2_cumsum = {(name_q2cs, 0): [0, (name_q2bs, 0)]}
            dsk_q2_cumsum.update({(name_q2cs, i): (_cumsum_part,
                                                   (name_q2cs, i - 1),
                                                   (name_q2bs, i))
                                  for i in range(1, numblocks[0])})

            # obtain slices on q from in-core compute (e.g.: (slice(10, 20), slice(0, 5)))
            name_blockslice = 'slice' + token + '-q'
            dsk_block_slices = {(name_blockslice, i): (tuple, [
                (apply, slice, (name_q2cs, i)), (slice, 0, name_n)])
                for i in range(numblocks[0])}

            dsk_q_blockslices = toolz.merge(dsk_n,
                                            dsk_q2_shapes,
                                            dsk_q2_cumsum,
                                            dsk_block_slices)

            block_slices = [(name_blockslice, i) for i in range(numblocks[0])]

        dsk.update_with_key(dsk_q_blockslices, key='q-blocksizes' + token)

        # In-core qr[0] unstacking
        name_q_st2 = 'getitem' + token + '-q2'
        dsk_q_st2 = dict(((name_q_st2, i, 0),
                          (operator.getitem, (name_q_st2_aux, 0, 0), b))
                         for i, b in enumerate(block_slices))
        dsk.update_with_key(dsk_q_st2, key=name_q_st2)

        # Q: Block qr[0] (*) In-core qr[0]
        name_q_st3 = 'dot' + token + '-q3'
        dsk_q_st3 = top(np.dot, name_q_st3, 'ij', name_q_st1, 'ij',
                        name_q_st2, 'ij', numblocks={name_q_st1: numblocks,
                                                     name_q_st2: numblocks})
        dsk.update_with_key(dsk_q_st3, key=name_q_st3)

        # R: In-core qr[1]
        name_r_st2 = 'getitem' + token + '-r2'
        dsk_r_st2 = {(name_r_st2, 0, 0): (operator.getitem, (name_qr_st2, 0, 0), 1)}
        dsk.update_with_key(dsk_r_st2, key=name_r_st2)

    if not compute_svd:
        is_unknown_m = np.isnan(data.shape[0]) or any(np.isnan(c) for c in data.chunks[0])
        is_unknown_n = np.isnan(data.shape[1]) or any(np.isnan(c) for c in data.chunks[1])

        if is_unknown_m and is_unknown_n:
            # assumption: m >= n
            q_shape = data.shape
            q_chunks = (data.chunks[0], (np.nan,))
            r_shape = (np.nan, np.nan)
            r_chunks = ((np.nan,), (np.nan,))
        elif is_unknown_m and not is_unknown_n:
            # assumption: m >= n
            q_shape = data.shape
            q_chunks = (data.chunks[0], (n,))
            r_shape = (n, n)
            r_chunks = (n, n)
        elif not is_unknown_m and is_unknown_n:
            # assumption: m >= n
            q_shape = data.shape
            q_chunks = (data.chunks[0], (np.nan,))
            r_shape = (np.nan, np.nan)
            r_chunks = ((np.nan,), (np.nan,))
        else:
            q_shape = data.shape if data.shape[0] >= data.shape[1] else (data.shape[0], data.shape[0])
            q_chunks = data.chunks if data.shape[0] >= data.shape[1] else (data.chunks[0], data.chunks[0])
            r_shape = (n, n) if data.shape[0] >= data.shape[1] else data.shape
            r_chunks = r_shape

        q = Array(dsk, name_q_st3,
                  shape=q_shape, chunks=q_chunks, dtype=qq.dtype)
        r = Array(dsk, name_r_st2,
                  shape=r_shape, chunks=r_chunks, dtype=rr.dtype)
        return q, r
    else:
        # In-core SVD computation
        name_svd_st2 = 'svd' + token + '-2'
        dsk_svd_st2 = top(np.linalg.svd, name_svd_st2, 'ij', name_r_st2, 'ij',
                          numblocks={name_r_st2: (1, 1)})
        # svd[0]
        name_u_st2 = 'getitem' + token + '-u2'
        dsk_u_st2 = {(name_u_st2, 0, 0): (operator.getitem,
                                          (name_svd_st2, 0, 0), 0)}
        # svd[1]
        name_s_st2 = 'getitem' + token + '-s2'
        dsk_s_st2 = {(name_s_st2, 0): (operator.getitem,
                                       (name_svd_st2, 0, 0), 1)}
        # svd[2]
        name_v_st2 = 'getitem' + token + '-v2'
        dsk_v_st2 = {(name_v_st2, 0, 0): (operator.getitem,
                                          (name_svd_st2, 0, 0), 2)}
        # Q * U
        name_u_st4 = 'getitem' + token + '-u4'
        dsk_u_st4 = top(dotmany, name_u_st4, 'ij', name_q_st3, 'ik',
                        name_u_st2, 'kj', numblocks={name_q_st3: numblocks,
                                                     name_u_st2: (1, 1)})

        dsk.update_with_key(dsk_svd_st2, key=name_svd_st2)
        dsk.update_with_key(dsk_u_st2, key=name_u_st2)
        dsk.update_with_key(dsk_u_st4, key=name_u_st4)
        dsk.update_with_key(dsk_s_st2, key=name_s_st2)
        dsk.update_with_key(dsk_v_st2, key=name_v_st2)

        uu, ss, vvh = np.linalg.svd(np.ones(shape=(1, 1), dtype=data.dtype))

        k = _nanmin(m, n)  # avoid RuntimeWarning with np.nanmin([m, n])

        m_u = m
        n_u = int(k) if not np.isnan(k) else k
        n_s = n_u
        m_vh = n_u
        n_vh = n
        d_vh = max(m_vh, n_vh)  # full matrix returned: but basically n
        u = Array(dsk, name_u_st4, shape=(m_u, n_u), chunks=(data.chunks[0], (n_u,)),
                  dtype=uu.dtype)
        s = Array(dsk, name_s_st2, shape=(n_s,), chunks=((n_s,),), dtype=ss.dtype)
        vh = Array(dsk, name_v_st2, shape=(d_vh, d_vh), chunks=((n,), (n,)),
                   dtype=vvh.dtype)
        return u, s, vh


def sfqr(data, name=None):
    """ Direct Short-and-Fat QR

    Currently, this is a quick hack for non-tall-and-skinny matrices which
    are one chunk tall and (unless they are one chunk wide) have chunks
    that are wider than they are tall

    Q [R_1 R_2 ...] = [A_1 A_2 ...]

    it computes the factorization Q R_1 = A_1, then computes the other
    R_k's in parallel.

    Parameters
    ----------
    data: Array

    See Also
    --------
    dask.array.linalg.qr - Main user API that uses this function
    dask.array.linalg.tsqr - Variant for tall-and-skinny case
    """
    nr, nc = len(data.chunks[0]), len(data.chunks[1])
    cr, cc = data.chunks[0][0], data.chunks[1][0]

    if not ((data.ndim == 2) and  # Is a matrix
            (nr == 1) and         # Has exactly one block row
            ((cr <= cc) or        # Chunking dimension on rows is at least that on cols or...
             (nc == 1))):         # ... only one block col
        raise ValueError(
            "Input must have the following properties:\n"
            "  1. Have two dimensions\n"
            "  2. Have only one row of blocks\n"
            "  3. Either one column of blocks or (first) chunk size on cols\n"
            "     is at most that on rows (e.g.: for a 5x20 matrix,\n"
            "     chunks=((5), (8,4,8)) is fine, but chunks=((5), (4,8,8)) is not;\n"
            "     still, prefer something simple like chunks=(5,10) or chunks=5)\n\n"
            "Note: This function (sfqr) supports QR decomposition in the case\n"
            "of short-and-fat matrices (single row chunk/block; see qr)"
        )

    prefix = name or 'sfqr-' + tokenize(data)
    prefix += '_'

    m, n = data.shape

    qq, rr = np.linalg.qr(np.ones(shape=(1, 1), dtype=data.dtype))

    dsk = sharedict.ShareDict()
    dsk.update(data.dask)

    # data = A = [A_1 A_rest]
    name_A_1 = prefix + 'A_1'
    name_A_rest = prefix + 'A_rest'
    dsk.update_with_key({
        (name_A_1, 0, 0): (data.name, 0, 0)
    }, key=name_A_1)
    dsk.update_with_key({
        (name_A_rest, 0, idx): (data.name, 0, 1 + idx)
        for idx in range(nc - 1)
    }, key=name_A_rest)

    # Q R_1 = A_1
    name_Q_R1 = prefix + 'Q_R_1'
    name_Q = prefix + 'Q'
    name_R_1 = prefix + 'R_1'
    dsk.update_with_key({
        (name_Q_R1, 0, 0): (np.linalg.qr, (name_A_1, 0, 0))
    }, key=name_Q_R1)
    dsk.update_with_key({
        (name_Q, 0, 0): (operator.getitem, (name_Q_R1, 0, 0), 0)
    }, key=name_Q)
    dsk.update_with_key({
        (name_R_1, 0, 0): (operator.getitem, (name_Q_R1, 0, 0), 1),
    }, key=name_R_1)

    Q = Array(dsk, name_Q,
              shape=(m, min(m, n)), chunks=(m, min(m, n)), dtype=qq.dtype)
    R_1 = Array(dsk, name_R_1,
                shape=(min(m, n), cc), chunks=(cr, cc), dtype=rr.dtype)

    # R = [R_1 Q'A_rest]
    Rs = [R_1]

    if nc > 1:
        A_rest = Array(dsk, name_A_rest,
                       shape=(min(m, n), n - cc), chunks=((cr), data.chunks[1][1:]), dtype=rr.dtype)
        Rs.append(Q.T.dot(A_rest))

    R = concatenate(Rs, axis=1)

    return Q, R


def compression_level(n, q, oversampling=10, min_subspace_size=20):
    """ Compression level to use in svd_compressed

    Given the size ``n`` of a space, compress that that to one of size
    ``q`` plus oversampling.

    The oversampling allows for greater flexibility in finding an
    appropriate subspace, a low value is often enough (10 is already a
    very conservative choice, it can be further reduced).
    ``q + oversampling`` should not be larger than ``n``.  In this
    specific implementation, ``q + oversampling`` is at least
    ``min_subspace_size``.

    >>> compression_level(100, 10)
    20
    """
    return min(max(min_subspace_size, q + oversampling), n)


def compression_matrix(data, q, n_power_iter=0, seed=None):
    """ Randomly sample matrix to find most active subspace

    This compression matrix returned by this algorithm can be used to
    compute both the QR decomposition and the Singular Value
    Decomposition.

    Parameters
    ----------
    data: Array
    q: int
        Size of the desired subspace (the actual size will be bigger,
        because of oversampling, see ``da.linalg.compression_level``)
    n_power_iter: int
        number of power iterations, useful when the singular values of
        the input matrix decay very slowly.

    References
    ----------
    N. Halko, P. G. Martinsson, and J. A. Tropp.
    Finding structure with randomness: Probabilistic algorithms for
    constructing approximate matrix decompositions.
    SIAM Rev., Survey and Review section, Vol. 53, num. 2,
    pp. 217-288, June 2011
    http://arxiv.org/abs/0909.4061
    """
    n = data.shape[1]
    comp_level = compression_level(n, q)
    state = RandomState(seed)
    omega = state.standard_normal(size=(n, comp_level), chunks=(data.chunks[1],
                                                                (comp_level,)))
    mat_h = data.dot(omega)
    for j in range(n_power_iter):
        mat_h = data.dot(data.T.dot(mat_h))
    q, _ = tsqr(mat_h)
    return q.T


def svd_compressed(a, k, n_power_iter=0, seed=None):
    """ Randomly compressed rank-k thin Singular Value Decomposition.

    This computes the approximate singular value decomposition of a large
    array.  This algorithm is generally faster than the normal algorithm
    but does not provide exact results.  One can balance between
    performance and accuracy with input parameters (see below).

    Parameters
    ----------
    a: Array
        Input array
    k: int
        Rank of the desired thin SVD decomposition.
    n_power_iter: int
        Number of power iterations, useful when the singular values
        decay slowly. Error decreases exponentially as n_power_iter
        increases. In practice, set n_power_iter <= 4.

    Examples
    --------

    >>> u, s, vt = svd_compressed(x, 20)  # doctest: +SKIP

    Returns
    -------
    u:  Array, unitary / orthogonal
    s:  Array, singular values in decreasing order (largest first)
    v:  Array, unitary / orthogonal

    References
    ----------
    N. Halko, P. G. Martinsson, and J. A. Tropp.
    Finding structure with randomness: Probabilistic algorithms for
    constructing approximate matrix decompositions.
    SIAM Rev., Survey and Review section, Vol. 53, num. 2,
    pp. 217-288, June 2011
    http://arxiv.org/abs/0909.4061
    """
    comp = compression_matrix(a, k, n_power_iter=n_power_iter, seed=seed)
    a_compressed = comp.dot(a)
    v, s, u = tsqr(a_compressed.T, compute_svd=True)
    u = comp.T.dot(u)
    v = v.T
    u = u[:, :k]
    s = s[:k]
    v = v[:k, :]
    return u, s, v


def qr(a):
    """
    Compute the qr factorization of a matrix.

    Examples
    --------

    >>> q, r = da.linalg.qr(x)  # doctest: +SKIP

    Returns
    -------

    q:  Array, orthonormal
    r:  Array, upper-triangular

    See Also
    --------

    np.linalg.qr : Equivalent NumPy Operation
    dask.array.linalg.tsqr: Implementation for tall-and-skinny arrays
    dask.array.linalg.sfqr: Implementation for short-and-fat arrays
    """

    if len(a.chunks[1]) == 1 and len(a.chunks[0]) > 1:
        return tsqr(a)
    elif len(a.chunks[0]) == 1:
        return sfqr(a)
    else:
        raise NotImplementedError(
            "qr currently supports only tall-and-skinny (single column chunk/block; see tsqr)\n"
            "and short-and-fat (single row chunk/block; see sfqr) matrices\n\n"
            "Consider use of the rechunk method. For example,\n\n"
            "x.rechunk({0: -1, 1: 'auto'}) or x.rechunk({0: 'auto', 1: -1})\n\n"
            "which rechunk one shorter axis to a single chunk, while allowing\n"
            "the other axis to automatically grow/shrink appropriately."
        )


def svd(a):
    """
    Compute the singular value decomposition of a matrix.

    Examples
    --------

    >>> u, s, v = da.linalg.svd(x)  # doctest: +SKIP

    Returns
    -------

    u:  Array, unitary / orthogonal
    s:  Array, singular values in decreasing order (largest first)
    v:  Array, unitary / orthogonal

    See Also
    --------

    np.linalg.svd : Equivalent NumPy Operation
    dask.array.linalg.tsqr: Implementation for tall-and-skinny arrays
    """
    return tsqr(a, compute_svd=True)


def _solve_triangular_lower(a, b):
    import scipy.linalg
    return scipy.linalg.solve_triangular(a, b, lower=True)


def lu(a):
    """
    Compute the lu decomposition of a matrix.

    Examples
    --------

    >>> p, l, u = da.linalg.lu(x)  # doctest: +SKIP

    Returns
    -------

    p:  Array, permutation matrix
    l:  Array, lower triangular matrix with unit diagonal.
    u:  Array, upper triangular matrix
    """

    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

    # for internal calculation
    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))
        # diagonal block
        dsk[name_lu, i, i] = (scipy.linalg.lu, target)

        # sweep to horizontal
        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)

        # sweep to vertical
        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))
            # solving x.dot(u) = target is equal to u.T.dot(x.T) = target.T
            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)

                # permuted l is required to be propagated to i > j blocks
                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))
                # transposed permutation matrix is equal to its inverse
                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]))
                # calculations are performed using permuted l,
                # thus the result should be reverted by inverted (=transposed) p
                # to have the same row order as diagonal blocks
                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)
                # l_permuted is not referred in upper triangulars

    dsk = sharedict.merge(a.dask, ('lu-' + token, dsk))
    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 solve_triangular(a, b, lower=False):
    """
    Solve the equation `a x = b` for `x`, assuming a is a triangular matrix.

    Parameters
    ----------
    a : (M, M) array_like
        A triangular matrix
    b : (M,) or (M, N) array_like
        Right-hand side matrix in `a x = b`
    lower : bool, optional
        Use only data contained in the lower triangle of `a`.
        Default is to use upper triangle.

    Returns
    -------
    x : (M,) or (M, N) array
        Solution to the system `a x = b`. Shape of return matches `b`.
    """

    import scipy.linalg

    if a.ndim != 2:
        raise ValueError('a must be 2 dimensional')
    if b.ndim <= 2:
        if a.shape[1] != b.shape[0]:
            raise ValueError('a.shape[1] and b.shape[0] must be equal')
        if a.chunks[1] != b.chunks[0]:
            msg = ('a.chunks[1] and b.chunks[0] must be equal. '
                   'Use .rechunk method to change the size of chunks.')
            raise ValueError(msg)
    else:
        raise ValueError('b must be 1 or 2 dimensional')

    vchunks = len(a.chunks[1])
    hchunks = 1 if b.ndim == 1 else len(b.chunks[1])
    token = tokenize(a, b, lower)
    name = 'solve-triangular-' + token

    # for internal calculation
    # (name, i, j, k, l) corresponds to a_ij.dot(b_kl)
    name_mdot = 'solve-tri-dot-' + token

    def _b_init(i, j):
        if b.ndim == 1:
            return b.name, i
        else:
            return b.name, i, j

    def _key(i, j):
        if b.ndim == 1:
            return name, i
        else:
            return name, i, j

    dsk = {}
    if lower:
        for i in range(vchunks):
            for j in range(hchunks):
                target = _b_init(i, j)
                if i > 0:
                    prevs = []
                    for k in range(i):
                        prev = name_mdot, i, k, k, j
                        dsk[prev] = (np.dot, (a.name, i, k), _key(k, j))
                        prevs.append(prev)
                    target = (operator.sub, target, (sum, prevs))
                dsk[_key(i, j)] = (_solve_triangular_lower, (a.name, i, i), target)
    else:
        for i in range(vchunks):
            for j in range(hchunks):
                target = _b_init(i, j)
                if i < vchunks - 1:
                    prevs = []
                    for k in range(i + 1, vchunks):
                        prev = name_mdot, i, k, k, j
                        dsk[prev] = (np.dot, (a.name, i, k), _key(k, j))
                        prevs.append(prev)
                    target = (operator.sub, target, (sum, prevs))
                dsk[_key(i, j)] = (scipy.linalg.solve_triangular, (a.name, i, i), target)

    dsk = sharedict.merge(a.dask, b.dask, (name, dsk))
    res = _solve_triangular_lower(np.array([[1, 0], [1, 2]], dtype=a.dtype),
                                  np.array([0, 1], dtype=b.dtype))
    return Array(dsk, name, shape=b.shape, chunks=b.chunks, dtype=res.dtype)


def solve(a, b, sym_pos=False):
    """
    Solve the equation ``a x = b`` for ``x``. By default, use LU
    decomposition and forward / backward substitutions. When ``sym_pos`` is
    ``True``, use Cholesky decomposition.

    Parameters
    ----------
    a : (M, M) array_like
        A square matrix.
    b : (M,) or (M, N) array_like
        Right-hand side matrix in ``a x = b``.
    sym_pos : bool
        Assume a is symmetric and positive definite. If ``True``, use Cholesky
        decomposition.

    Returns
    -------
    x : (M,) or (M, N) Array
        Solution to the system ``a x = b``.  Shape of the return matches the
        shape of `b`.
    """
    if sym_pos:
        l, u = _cholesky(a)
    else:
        p, l, u = lu(a)
        b = p.T.dot(b)
    uy = solve_triangular(l, b, lower=True)
    return solve_triangular(u, uy)


def inv(a):
    """
    Compute the inverse of a matrix with LU decomposition and
    forward / backward substitutions.

    Parameters
    ----------
    a : array_like
        Square matrix to be inverted.

    Returns
    -------
    ainv : Array
        Inverse of the matrix `a`.
    """
    return solve(a, eye(a.shape[0], chunks=a.chunks[0][0]))


def _cholesky_lower(a):
    import scipy.linalg
    return scipy.linalg.cholesky(a, lower=True)


def cholesky(a, lower=False):
    """
    Returns the Cholesky decomposition, :math:`A = L L^*` or
    :math:`A = U^* U` of a Hermitian positive-definite matrix A.

    Parameters
    ----------
    a : (M, M) array_like
        Matrix to be decomposed
    lower : bool, optional
        Whether to compute the upper or lower triangular Cholesky
        factorization.  Default is upper-triangular.

    Returns
    -------
    c : (M, M) Array
        Upper- or lower-triangular Cholesky factor of `a`.
    """

    l, u = _cholesky(a)
    if lower:
        return l
    else:
        return u


def _cholesky(a):
    """
    Private function to perform Cholesky decomposition, which returns both
    lower and upper triangulars.
    """
    import scipy.linalg

    if a.ndim != 2:
        raise ValueError('Dimension must be 2 to perform cholesky decomposition')

    xdim, ydim = a.shape
    if xdim != ydim:
        raise ValueError('Input must be a square matrix to perform cholesky decomposition')
    if not len(set(a.chunks[0] + a.chunks[1])) == 1:
        msg = ('All chunks must be a square matrix to perform cholesky 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 = 'cholesky-' + token

    # (name_lt_dot, i, j, k, l) corresponds to l_ij.dot(l_kl.T)
    name_lt_dot = 'cholesky-lt-dot-' + token
    # because transposed results are needed for calculation,
    # we can build graph for upper triangular simultaneously
    name_upper = 'cholesky-upper-' + token

    # calculates lower triangulars because subscriptions get simpler
    dsk = {}
    for i in range(vdim):
        for j in range(hdim):
            if i < j:
                dsk[name, i, j] = (np.zeros, (a.chunks[0][i], a.chunks[1][j]))
                dsk[name_upper, j, i] = (name, i, j)
            elif i == j:
                target = (a.name, i, j)
                if i > 0:
                    prevs = []
                    for p in range(i):
                        prev = name_lt_dot, i, p, i, p
                        dsk[prev] = (np.dot, (name, i, p), (name_upper, p, i))
                        prevs.append(prev)
                    target = (operator.sub, target, (sum, prevs))
                dsk[name, i, i] = (_cholesky_lower, target)
                dsk[name_upper, i, i] = (np.transpose, (name, i, i))
            else:
                # solving x.dot(L11.T) = (A21 - L20.dot(L10.T)) is equal to
                # L11.dot(x.T) = A21.T - L10.dot(L20.T)
                # L11.dot(x.T) = A12 - L10.dot(L02)
                target = (a.name, j, i)
                if j > 0:
                    prevs = []
                    for p in range(j):
                        prev = name_lt_dot, j, p, i, p
                        dsk[prev] = (np.dot, (name, j, p), (name_upper, p, i))
                        prevs.append(prev)
                    target = (operator.sub, target, (sum, prevs))
                dsk[name_upper, j, i] = (_solve_triangular_lower,(name, j, j), target)
                dsk[name, i, j] = (np.transpose, (name_upper, j, i))

    dsk = sharedict.merge(a.dask, (name, dsk))
    cho = scipy.linalg.cholesky(np.array([[1, 2], [2, 5]], dtype=a.dtype))

    lower = Array(dsk, name, shape=a.shape, chunks=a.chunks, dtype=cho.dtype)
    # do not use .T, because part of transposed blocks are already calculated
    upper = Array(dsk, name_upper, shape=a.shape, chunks=a.chunks, dtype=cho.dtype)
    return lower, upper


def _sort_decreasing(x):
    x[::-1].sort()
    return x


def lstsq(a, b):
    """
    Return the least-squares solution to a linear matrix equation using
    QR decomposition.

    Solves the equation `a x = b` by computing a vector `x` that
    minimizes the Euclidean 2-norm `|| b - a x ||^2`.  The equation may
    be under-, well-, or over- determined (i.e., the number of
    linearly independent rows of `a` can be less than, equal to, or
    greater than its number of linearly independent columns).  If `a`
    is square and of full rank, then `x` (but for round-off error) is
    the "exact" solution of the equation.

    Parameters
    ----------
    a : (M, N) array_like
        "Coefficient" matrix.
    b : (M,) array_like
        Ordinate or "dependent variable" values.

    Returns
    -------
    x : (N,) Array
        Least-squares solution. If `b` is two-dimensional,
        the solutions are in the `K` columns of `x`.
    residuals : (1,) Array
        Sums of residuals; squared Euclidean 2-norm for each column in
        ``b - a*x``.
    rank : Array
        Rank of matrix `a`.
    s : (min(M, N),) Array
        Singular values of `a`.
    """
    q, r = qr(a)
    x = solve_triangular(r, q.T.dot(b))
    residuals = b - a.dot(x)
    residuals = (residuals ** 2).sum(keepdims=True)

    token = tokenize(a, b)

    # r must be a triangular with single block

    # rank
    rname = 'lstsq-rank-' + token
    rdsk = {(rname, ): (np.linalg.matrix_rank, (r.name, 0, 0))}
    rdsk = sharedict.merge(r.dask, (rname, rdsk))
    # rank must be an integer
    rank = Array(rdsk, rname, shape=(), chunks=(), dtype=int)

    # singular
    sname = 'lstsq-singular-' + token
    rt = r.T
    sdsk = {(sname, 0): (_sort_decreasing,
                         (np.sqrt,
                          (np.linalg.eigvals,
                           (np.dot, (rt.name, 0, 0), (r.name, 0, 0)))))}
    sdsk = sharedict.merge(rt.dask, (sname, sdsk))
    _, _, _, ss = np.linalg.lstsq(np.array([[1, 0], [1, 2]], dtype=a.dtype),
                                  np.array([0, 1], dtype=b.dtype))
    s = Array(sdsk, sname, shape=(r.shape[0], ),
              chunks=r.shape[0], dtype=ss.dtype)

    return x, residuals, rank, s


@wraps(np.linalg.norm)
def norm(x, ord=None, axis=None, keepdims=False):
    if x.ndim > 2:
        raise ValueError("Improper number of dimensions to norm.")

    if axis is None:
        axis = tuple(range(x.ndim))
    elif isinstance(axis, Number):
        axis = (int(axis),)
    else:
        axis = tuple(axis)

    if ord == "fro":
        ord = None
        if len(axis) == 1:
            raise ValueError("Invalid norm order for vectors.")

    # Coerce to double precision.
    r = x.astype(np.promote_types(x.dtype, float))

    if ord is None:
        r = (abs(r) ** 2).sum(axis=axis, keepdims=keepdims) ** 0.5
    elif ord == "nuc":
        if len(axis) == 1:
            raise ValueError("Invalid norm order for vectors.")

        r = svd(x)[1][None].sum(keepdims=keepdims)
    elif ord == np.inf:
        r = abs(r)
        if len(axis) == 1:
            r = r.max(axis=axis, keepdims=keepdims)
        else:
            r = r.sum(axis=axis[1], keepdims=keepdims).max(keepdims=keepdims)
    elif ord == -np.inf:
        r = abs(r)
        if len(axis) == 1:
            r = r.min(axis=axis, keepdims=keepdims)
        else:
            r = r.sum(axis=axis[1], keepdims=keepdims).min(keepdims=keepdims)
    elif ord == 0:
        if len(axis) == 2:
            raise ValueError("Invalid norm order for matrices.")

        r = (r != 0).astype(r.dtype).sum(axis=0, keepdims=keepdims)
    elif ord == 1:
        r = abs(r)
        if len(axis) == 1:
            r = r.sum(axis=axis, keepdims=keepdims)
        else:
            r = r.sum(axis=axis[0], keepdims=keepdims).max(keepdims=keepdims)
    elif len(axis) == 2 and ord == -1:
        r = abs(r).sum(axis=axis[0], keepdims=keepdims).min(keepdims=keepdims)
    elif len(axis) == 2 and ord == 2:
        r = svd(x)[1][None].max(keepdims=keepdims)
    elif len(axis) == 2 and ord == -2:
        r = svd(x)[1][None].min(keepdims=keepdims)
    else:
        if len(axis) == 2:
            raise ValueError("Invalid norm order for matrices.")

        r = (abs(r) ** ord).sum(axis=axis, keepdims=keepdims) ** (1.0 / ord)

    return r
