"""
This contains a copy of the future version of
scipy.sparse.linalg.eigen.arpack.eigsh
It's an upgraded wrapper of the ARPACK library which
allows the use of shift-invert mode for symmetric matrices.

Find a few eigenvectors and eigenvalues of a matrix.

Uses ARPACK: http://www.caam.rice.edu/software/ARPACK/

"""
# Wrapper implementation notes
#
# ARPACK Entry Points
# -------------------
# The entry points to ARPACK are
# - (s,d)seupd : single and double precision symmetric matrix
# - (s,d,c,z)neupd: single,double,complex,double complex general matrix
# This wrapper puts the *neupd (general matrix) interfaces in eigs()
# and the *seupd (symmetric matrix) in eigsh().
# There is no Hermetian complex/double complex interface.
# To find eigenvalues of a Hermetian matrix you
# must use eigs() and not eigsh()
# It might be desirable to handle the Hermetian case differently
# and, for example, return real eigenvalues.

# Number of eigenvalues returned and complex eigenvalues
# ------------------------------------------------------
# The ARPACK nonsymmetric real and double interface (s,d)naupd return
# eigenvalues and eigenvectors in real (float,double) arrays.
# Since the eigenvalues and eigenvectors are, in general, complex
# ARPACK puts the real and imaginary parts in consecutive entries
# in real-valued arrays.   This wrapper puts the real entries
# into complex data types and attempts to return the requested eigenvalues
# and eigenvectors.

# Solver modes
# ------------
# ARPACK and handle shifted and shift-inverse computations
# for eigenvalues by providing a shift (sigma) and a solver.

__docformat__ = "restructuredtext en"

__all__ = ['eigs', 'eigsh', 'svds', 'ArpackError', 'ArpackNoConvergence']
import warnings

from scipy.sparse.linalg.eigen.arpack import _arpack
import numpy as np
from scipy.sparse.linalg.interface import aslinearoperator, LinearOperator
from scipy.sparse import identity, isspmatrix, isspmatrix_csr
from scipy.linalg import lu_factor, lu_solve
from scipy.sparse.sputils import isdense
from scipy.sparse.linalg import gmres, splu
import scipy
from distutils.version import LooseVersion

_type_conv = {'f': 's', 'd': 'd', 'F': 'c', 'D': 'z'}
_ndigits = {'f': 5, 'd': 12, 'F': 5, 'D': 12}

DNAUPD_ERRORS = {
    0: "Normal exit.",
    1: "Maximum number of iterations taken. "
       "All possible eigenvalues of OP has been found. IPARAM(5) "
       "returns the number of wanted converged Ritz values.",
    2: "No longer an informational error. Deprecated starting "
       "with release 2 of ARPACK.",
    3: "No shifts could be applied during a cycle of the "
       "Implicitly restarted Arnoldi iteration. One possibility "
       "is to increase the size of NCV relative to NEV. ",
    -1: "N must be positive.",
    -2: "NEV must be positive.",
    -3: "NCV-NEV >= 2 and less than or equal to N.",
    -4: "The maximum number of Arnoldi update iterations allowed "
        "must be greater than zero.",
    -5: " WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'",
    -6: "BMAT must be one of 'I' or 'G'.",
    -7: "Length of private work array WORKL is not sufficient.",
    -8: "Error return from LAPACK eigenvalue calculation;",
    -9: "Starting vector is zero.",
    -10: "IPARAM(7) must be 1,2,3,4.",
    -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.",
    -12: "IPARAM(1) must be equal to 0 or 1.",
    -13: "NEV and WHICH = 'BE' are incompatible.",
    -9999: "Could not build an Arnoldi factorization. "
           "IPARAM(5) returns the size of the current Arnoldi "
           "factorization. The user is advised to check that "
           "enough workspace and array storage has been allocated."
}

SNAUPD_ERRORS = DNAUPD_ERRORS

ZNAUPD_ERRORS = DNAUPD_ERRORS.copy()
ZNAUPD_ERRORS[-10] = "IPARAM(7) must be 1,2,3."

CNAUPD_ERRORS = ZNAUPD_ERRORS

DSAUPD_ERRORS = {
    0: "Normal exit.",
    1: "Maximum number of iterations taken. "
       "All possible eigenvalues of OP has been found.",
    2: "No longer an informational error. Deprecated starting with "
       "release 2 of ARPACK.",
    3: "No shifts could be applied during a cycle of the Implicitly "
       "restarted Arnoldi iteration. One possibility is to increase "
       "the size of NCV relative to NEV. ",
    -1: "N must be positive.",
    -2: "NEV must be positive.",
    -3: "NCV must be greater than NEV and less than or equal to N.",
    -4: "The maximum number of Arnoldi update iterations allowed "
        "must be greater than zero.",
    -5: "WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.",
    -6: "BMAT must be one of 'I' or 'G'.",
    -7: "Length of private work array WORKL is not sufficient.",
    -8: "Error return from trid. eigenvalue calculation; "
        "Informational error from LAPACK routine dsteqr .",
    -9: "Starting vector is zero.",
    -10: "IPARAM(7) must be 1,2,3,4,5.",
    -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.",
    -12: "IPARAM(1) must be equal to 0 or 1.",
    -13: "NEV and WHICH = 'BE' are incompatible. ",
    -9999: "Could not build an Arnoldi factorization. "
           "IPARAM(5) returns the size of the current Arnoldi "
           "factorization. The user is advised to check that "
           "enough workspace and array storage has been allocated.",
}

SSAUPD_ERRORS = DSAUPD_ERRORS

DNEUPD_ERRORS = {
    0: "Normal exit.",
    1: "The Schur form computed by LAPACK routine dlahqr "
       "could not be reordered by LAPACK routine dtrsen. "
       "Re-enter subroutine dneupd  with IPARAM(5)NCV and "
       "increase the size of the arrays DR and DI to have "
       "dimension at least dimension NCV and allocate at least NCV "
       "columns for Z. NOTE: Not necessary if Z and V share "
       "the same space. Please notify the authors if this error "
       "occurs.",
    -1: "N must be positive.",
    -2: "NEV must be positive.",
    -3: "NCV-NEV >= 2 and less than or equal to N.",
    -5: "WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'",
    -6: "BMAT must be one of 'I' or 'G'.",
    -7: "Length of private work WORKL array is not sufficient.",
    -8: "Error return from calculation of a real Schur form. "
        "Informational error from LAPACK routine dlahqr .",
    -9: "Error return from calculation of eigenvectors. "
        "Informational error from LAPACK routine dtrevc.",
    -10: "IPARAM(7) must be 1,2,3,4.",
    -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.",
    -12: "HOWMNY = 'S' not yet implemented",
    -13: "HOWMNY must be one of 'A' or 'P' if RVEC = .true.",
    -14: "DNAUPD  did not find any eigenvalues to sufficient "
         "accuracy.",
    -15: "DNEUPD got a different count of the number of converged "
         "Ritz values than DNAUPD got.  This indicates the user "
         "probably made an error in passing data from DNAUPD to "
         "DNEUPD or that the data was modified before entering "
         "DNEUPD",
}

SNEUPD_ERRORS = DNEUPD_ERRORS.copy()
SNEUPD_ERRORS[1] = ("The Schur form computed by LAPACK routine slahqr "
                    "could not be reordered by LAPACK routine strsen . "
                    "Re-enter subroutine dneupd  with IPARAM(5)=NCV and "
                    "increase the size of the arrays DR and DI to have "
                    "dimension at least dimension NCV and allocate at least "
                    "NCV columns for Z. NOTE: Not necessary if Z and V share "
                    "the same space. Please notify the authors if this error "
                    "occurs.")
SNEUPD_ERRORS[-14] = ("SNAUPD did not find any eigenvalues to sufficient "
                      "accuracy.")
SNEUPD_ERRORS[-15] = ("SNEUPD got a different count of the number of "
                      "converged Ritz values than SNAUPD got.  This indicates "
                      "the user probably made an error in passing data from "
                      "SNAUPD to SNEUPD or that the data was modified before "
                      "entering SNEUPD")

ZNEUPD_ERRORS = {0: "Normal exit.",
                 1: "The Schur form computed by LAPACK routine csheqr "
                    "could not be reordered by LAPACK routine ztrsen. "
                    "Re-enter subroutine zneupd with IPARAM(5)=NCV and "
                    "increase the size of the array D to have "
                    "dimension at least dimension NCV and allocate at least "
                    "NCV columns for Z. NOTE: Not necessary if Z and V share "
                    "the same space. Please notify the authors if this error "
                    "occurs.",
                 -1: "N must be positive.",
                 -2: "NEV must be positive.",
                 -3: "NCV-NEV >= 1 and less than or equal to N.",
                 -5: "WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'",
                 -6: "BMAT must be one of 'I' or 'G'.",
                 -7: "Length of private work WORKL array is not sufficient.",
                 -8: "Error return from LAPACK eigenvalue calculation. "
                     "This should never happened.",
                 -9: "Error return from calculation of eigenvectors. "
                     "Informational error from LAPACK routine ztrevc.",
                 -10: "IPARAM(7) must be 1,2,3",
                 -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.",
                 -12: "HOWMNY = 'S' not yet implemented",
                 -13: "HOWMNY must be one of 'A' or 'P' if RVEC = .true.",
                 -14: "ZNAUPD did not find any eigenvalues to sufficient "
                      "accuracy.",
                 -15: "ZNEUPD got a different count of the number of "
                      "converged Ritz values than ZNAUPD got.  This "
                      "indicates the user probably made an error in passing "
                      "data from ZNAUPD to ZNEUPD or that the data was "
                      "modified before entering ZNEUPD"}

CNEUPD_ERRORS = ZNEUPD_ERRORS.copy()
CNEUPD_ERRORS[-14] = ("CNAUPD did not find any eigenvalues to sufficient "
                      "accuracy.")
CNEUPD_ERRORS[-15] = ("CNEUPD got a different count of the number of "
                      "converged Ritz values than CNAUPD got.  This indicates "
                      "the user probably made an error in passing data from "
                      "CNAUPD to CNEUPD or that the data was modified before "
                      "entering CNEUPD")

DSEUPD_ERRORS = {
    0: "Normal exit.",
    -1: "N must be positive.",
    -2: "NEV must be positive.",
    -3: "NCV must be greater than NEV and less than or equal to N.",
    -5: "WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.",
    -6: "BMAT must be one of 'I' or 'G'.",
    -7: "Length of private work WORKL array is not sufficient.",
    -8: ("Error return from trid. eigenvalue calculation; "
         "Information error from LAPACK routine dsteqr."),
    -9: "Starting vector is zero.",
    -10: "IPARAM(7) must be 1,2,3,4,5.",
    -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.",
    -12: "NEV and WHICH = 'BE' are incompatible.",
    -14: "DSAUPD  did not find any eigenvalues to sufficient accuracy.",
    -15: "HOWMNY must be one of 'A' or 'S' if RVEC = .true.",
    -16: "HOWMNY = 'S' not yet implemented",
    -17: ("DSEUPD  got a different count of the number of converged "
          "Ritz values than DSAUPD  got.  This indicates the user "
          "probably made an error in passing data from DSAUPD  to "
          "DSEUPD  or that the data was modified before entering  "
          "DSEUPD.")
}

SSEUPD_ERRORS = DSEUPD_ERRORS.copy()
SSEUPD_ERRORS[-14] = ("SSAUPD  did not find any eigenvalues "
                      "to sufficient accuracy.")
SSEUPD_ERRORS[-17] = ("SSEUPD  got a different count of the number of "
                      "converged "
                      "Ritz values than SSAUPD  got.  This indicates the user "
                      "probably made an error in passing data from SSAUPD  to "
                      "SSEUPD  or that the data was modified before entering  "
                      "SSEUPD.")

_SAUPD_ERRORS = {'d': DSAUPD_ERRORS,
                 's': SSAUPD_ERRORS}
_NAUPD_ERRORS = {'d': DNAUPD_ERRORS,
                 's': SNAUPD_ERRORS,
                 'z': ZNAUPD_ERRORS,
                 'c': CNAUPD_ERRORS}
_SEUPD_ERRORS = {'d': DSEUPD_ERRORS,
                 's': SSEUPD_ERRORS}
_NEUPD_ERRORS = {'d': DNEUPD_ERRORS,
                 's': SNEUPD_ERRORS,
                 'z': ZNEUPD_ERRORS,
                 'c': CNEUPD_ERRORS}

# accepted values of parameter WHICH in _SEUPD
_SEUPD_WHICH = ['LM', 'SM', 'LA', 'SA', 'BE']

# accepted values of parameter WHICH in _NAUPD
_NEUPD_WHICH = ['LM', 'SM', 'LR', 'SR', 'LI', 'SI']

class ArpackError(RuntimeError):
    """
    ARPACK error
    """
    def __init__(self, info, infodict=_NAUPD_ERRORS):
        msg = infodict.get(info, "Unknown error")
        RuntimeError.__init__(self, "ARPACK error %d: %s" % (info, msg))

class ArpackNoConvergence(ArpackError):
    """
    ARPACK iteration did not converge

    Attributes
    ----------
    eigenvalues : ndarray
        Partial result. Converged eigenvalues.
    eigenvectors : ndarray
        Partial result. Converged eigenvectors.

    """
    def __init__(self, msg, eigenvalues, eigenvectors):
        ArpackError.__init__(self, -1, {-1: msg})
        self.eigenvalues = eigenvalues
        self.eigenvectors = eigenvectors

class _ArpackParams(object):
    def __init__(self, n, k, tp, mode=1, sigma=None,
                 ncv=None, v0=None, maxiter=None, which="LM", tol=0):
        if k <= 0:
            raise ValueError("k must be positive, k=%d" % k)

        if maxiter is None:
            maxiter = n * 10
        if maxiter <= 0:
            raise ValueError("maxiter must be positive, maxiter=%d" % maxiter)

        if tp not in 'fdFD':
            raise ValueError("matrix type must be 'f', 'd', 'F', or 'D'")

        if v0 is not None:
            # ARPACK overwrites its initial resid,  make a copy
            self.resid = np.array(v0, copy=True)
            info = 1
        else:
            self.resid = np.zeros(n, tp)
            info = 0

        if sigma is None:
            #sigma not used
            self.sigma = 0
        else:
            self.sigma = sigma

        if ncv is None:
            ncv = 2 * k + 1
        ncv = min(ncv, n)

        self.v = np.zeros((n, ncv), tp)  # holds Ritz vectors
        self.iparam = np.zeros(11, "int")

        # set solver mode and parameters
        ishfts = 1
        self.mode = mode
        self.iparam[0] = ishfts
        self.iparam[2] = maxiter
        self.iparam[3] = 1
        self.iparam[6] = mode

        self.n = n
        self.tol = tol
        self.k = k
        self.maxiter = maxiter
        self.ncv = ncv
        self.which = which
        self.tp = tp
        self.info = info

        self.converged = False
        self.ido = 0

    def _raise_no_convergence(self):
        msg = "No convergence (%d iterations, %d/%d eigenvectors converged)"
        k_ok = self.iparam[4]
        num_iter = self.iparam[2]
        try:
            ev, vec = self.extract(True)
        except ArpackError as err:
            msg = "%s [%s]" % (msg, err)
            ev = np.zeros((0,))
            vec = np.zeros((self.n, 0))
            k_ok = 0
        raise ArpackNoConvergence(msg % (num_iter, k_ok, self.k), ev, vec)

class _SymmetricArpackParams(_ArpackParams):
    def __init__(self, n, k, tp, matvec, mode=1, M_matvec=None,
                 Minv_matvec=None, sigma=None,
                 ncv=None, v0=None, maxiter=None, which="LM", tol=0):
        # The following modes are supported:
        #  mode = 1:
        #    Solve the standard eigenvalue problem:
        #      A*x = lambda*x :
        #       A - symmetric
        #    Arguments should be
        #       matvec      = left multiplication by A
        #       M_matvec    = None [not used]
        #       Minv_matvec = None [not used]
        #
        #  mode = 2:
        #    Solve the general eigenvalue problem:
        #      A*x = lambda*M*x
        #       A - symmetric
        #       M - symmetric positive definite
        #    Arguments should be
        #       matvec      = left multiplication by A
        #       M_matvec    = left multiplication by M
        #       Minv_matvec = left multiplication by M^-1
        #
        #  mode = 3:
        #    Solve the general eigenvalue problem in shift-invert mode:
        #      A*x = lambda*M*x
        #       A - symmetric
        #       M - symmetric positive semi-definite
        #    Arguments should be
        #       matvec      = None [not used]
        #       M_matvec    = left multiplication by M
        #                     or None, if M is the identity
        #       Minv_matvec = left multiplication by [A-sigma*M]^-1
        #
        #  mode = 4:
        #    Solve the general eigenvalue problem in Buckling mode:
        #      A*x = lambda*AG*x
        #       A  - symmetric positive semi-definite
        #       AG - symmetric indefinite
        #    Arguments should be
        #       matvec      = left multiplication by A
        #       M_matvec    = None [not used]
        #       Minv_matvec = left multiplication by [A-sigma*AG]^-1
        #
        #  mode = 5:
        #    Solve the general eigenvalue problem in Cayley-transformed mode:
        #      A*x = lambda*M*x
        #       A - symmetric
        #       M - symmetric positive semi-definite
        #    Arguments should be
        #       matvec      = left multiplication by A
        #       M_matvec    = left multiplication by M
        #                     or None, if M is the identity
        #       Minv_matvec = left multiplication by [A-sigma*M]^-1
        if mode == 1:
            if matvec is None:
                raise ValueError("matvec must be specified for mode=1")
            if M_matvec is not None:
                raise ValueError("M_matvec cannot be specified for mode=1")
            if Minv_matvec is not None:
                raise ValueError("Minv_matvec cannot be specified for mode=1")

            self.OP = matvec
            self.B = lambda x: x
            self.bmat = 'I'
        elif mode == 2:
            if matvec is None:
                raise ValueError("matvec must be specified for mode=2")
            if M_matvec is None:
                raise ValueError("M_matvec must be specified for mode=2")
            if Minv_matvec is None:
                raise ValueError("Minv_matvec must be specified for mode=2")

            self.OP = lambda x: Minv_matvec(matvec(x))
            self.OPa = Minv_matvec
            self.OPb = matvec
            self.B = M_matvec
            self.bmat = 'G'
        elif mode == 3:
            if matvec is not None:
                raise ValueError("matvec must not be specified for mode=3")
            if Minv_matvec is None:
                raise ValueError("Minv_matvec must be specified for mode=3")

            if M_matvec is None:
                self.OP = Minv_matvec
                self.OPa = Minv_matvec
                self.B = lambda x: x
                self.bmat = 'I'
            else:
                self.OP = lambda x: Minv_matvec(M_matvec(x))
                self.OPa = Minv_matvec
                self.B = M_matvec
                self.bmat = 'G'
        elif mode == 4:
            if matvec is None:
                raise ValueError("matvec must be specified for mode=4")
            if M_matvec is not None:
                raise ValueError("M_matvec must not be specified for mode=4")
            if Minv_matvec is None:
                raise ValueError("Minv_matvec must be specified for mode=4")
            self.OPa = Minv_matvec
            self.OP = lambda x: self.OPa(matvec(x))
            self.B = matvec
            self.bmat = 'G'
        elif mode == 5:
            if matvec is None:
                raise ValueError("matvec must be specified for mode=5")
            if Minv_matvec is None:
                raise ValueError("Minv_matvec must be specified for mode=5")

            self.OPa = Minv_matvec
            self.A_matvec = matvec

            if M_matvec is None:
                self.OP = lambda x: Minv_matvec(matvec(x) + sigma * x)
                self.B = lambda x: x
                self.bmat = 'I'
            else:
                self.OP = lambda x: Minv_matvec(matvec(x)
                                                + sigma * M_matvec(x))
                self.B = M_matvec
                self.bmat = 'G'
        else:
            raise ValueError("mode=%i not implemented" % mode)

        if which not in _SEUPD_WHICH:
            raise ValueError("which must be one of %s"
                             % ' '.join(_SEUPD_WHICH))
        if k >= n:
            raise ValueError("k must be less than rank(A), k=%d" % k)

        _ArpackParams.__init__(self, n, k, tp, mode, sigma,
                               ncv, v0, maxiter, which, tol)

        if self.ncv > n or self.ncv <= k:
            raise ValueError("ncv must be k<ncv<=n, ncv=%s" % self.ncv)

        self.workd = np.zeros(3 * n, self.tp)
        self.workl = np.zeros(self.ncv * (self.ncv + 8), self.tp)

        ltr = _type_conv[self.tp]
        if ltr not in ["s", "d"]:
            raise ValueError("Input matrix is not real-valued.")

        self._arpack_solver = _arpack.__dict__[ltr + 'saupd']
        self._arpack_extract = _arpack.__dict__[ltr + 'seupd']

        self.iterate_infodict = _SAUPD_ERRORS[ltr]
        self.extract_infodict = _SEUPD_ERRORS[ltr]

        self.ipntr = np.zeros(11, "int")

    def iterate(self):
        self.ido, self.resid, self.v, self.iparam, self.ipntr, self.info = \
            self._arpack_solver(self.ido, self.bmat, self.which, self.k,
                                self.tol, self.resid, self.v, self.iparam,
                                self.ipntr, self.workd, self.workl, self.info)

        xslice = slice(self.ipntr[0] - 1, self.ipntr[0] - 1 + self.n)
        yslice = slice(self.ipntr[1] - 1, self.ipntr[1] - 1 + self.n)
        if self.ido == -1:
            # initialization
            self.workd[yslice] = self.OP(self.workd[xslice])
        elif self.ido == 1:
            # compute y = Op*x
            if self.mode == 1:
                self.workd[yslice] = self.OP(self.workd[xslice])
            elif self.mode == 2:
                self.workd[xslice] = self.OPb(self.workd[xslice])
                self.workd[yslice] = self.OPa(self.workd[xslice])
            elif self.mode == 5:
                Bxslice = slice(self.ipntr[2] - 1, self.ipntr[2] - 1 + self.n)
                Ax = self.A_matvec(self.workd[xslice])
                self.workd[yslice] = self.OPa(Ax + (self.sigma *
                                                    self.workd[Bxslice]))
            else:
                Bxslice = slice(self.ipntr[2] - 1, self.ipntr[2] - 1 + self.n)
                self.workd[yslice] = self.OPa(self.workd[Bxslice])
        elif self.ido == 2:
            self.workd[yslice] = self.B(self.workd[xslice])
        elif self.ido == 3:
            raise ValueError("ARPACK requested user shifts.  Assure ISHIFT==0")
        else:
            self.converged = True

            if self.info == 0:
                pass
            elif self.info == 1:
                self._raise_no_convergence()
            else:
                raise ArpackError(self.info, infodict=self.iterate_infodict)

    def extract(self, return_eigenvectors):
        rvec = return_eigenvectors
        ierr = 0
        howmny = 'A'  # return all eigenvectors
        sselect = np.zeros(self.ncv, 'int')  # unused
        d, z, ierr = self._arpack_extract(rvec, howmny, sselect, self.sigma,
                                          self.bmat, self.which, self.k,
                                          self.tol, self.resid, self.v,
                                          self.iparam[0:7], self.ipntr,
                                          self.workd[0:2 * self.n],
                                          self.workl, ierr)
        if ierr != 0:
            raise ArpackError(ierr, infodict=self.extract_infodict)
        k_ok = self.iparam[4]
        d = d[:k_ok]
        z = z[:, :k_ok]

        if return_eigenvectors:
            return d, z
        else:
            return d

class _UnsymmetricArpackParams(_ArpackParams):
    def __init__(self, n, k, tp, matvec, mode=1, M_matvec=None,
                 Minv_matvec=None, sigma=None,
                 ncv=None, v0=None, maxiter=None, which="LM", tol=0):
        # The following modes are supported:
        #  mode = 1:
        #    Solve the standard eigenvalue problem:
        #      A*x = lambda*x
        #       A - square matrix
        #    Arguments should be
        #       matvec      = left multiplication by A
        #       M_matvec    = None [not used]
        #       Minv_matvec = None [not used]
        #
        #  mode = 2:
        #    Solve the generalized eigenvalue problem:
        #      A*x = lambda*M*x
        #       A - square matrix
        #       M - symmetric, positive semi-definite
        #    Arguments should be
        #       matvec      = left multiplication by A
        #       M_matvec    = left multiplication by M
        #       Minv_matvec = left multiplication by M^-1
        #
        #  mode = 3,4:
        #    Solve the general eigenvalue problem in shift-invert mode:
        #      A*x = lambda*M*x
        #       A - square matrix
        #       M - symmetric, positive semi-definite
        #    Arguments should be
        #       matvec      = None [not used]
        #       M_matvec    = left multiplication by M
        #                     or None, if M is the identity
        #       Minv_matvec = left multiplication by [A-sigma*M]^-1
        #    if A is real and mode==3, use the real part of Minv_matvec
        #    if A is real and mode==4, use the imag part of Minv_matvec
        #    if A is complex and mode==3,
        #       use real and imag parts of Minv_matvec
        if mode == 1:
            if matvec is None:
                raise ValueError("matvec must be specified for mode=1")
            if M_matvec is not None:
                raise ValueError("M_matvec cannot be specified for mode=1")
            if Minv_matvec is not None:
                raise ValueError("Minv_matvec cannot be specified for mode=1")

            self.OP = matvec
            self.B = lambda x: x
            self.bmat = 'I'
        elif mode == 2:
            if matvec is None:
                raise ValueError("matvec must be specified for mode=2")
            if M_matvec is None:
                raise ValueError("M_matvec must be specified for mode=2")
            if Minv_matvec is None:
                raise ValueError("Minv_matvec must be specified for mode=2")

            self.OP = lambda x: Minv_matvec(matvec(x))
            self.OPa = Minv_matvec
            self.OPb = matvec
            self.B = M_matvec
            self.bmat = 'G'
        elif mode in (3, 4):
            if matvec is None:
                raise ValueError("matvec must be specified "
                                 "for mode in (3,4)")
            if Minv_matvec is None:
                raise ValueError("Minv_matvec must be specified "
                                 "for mode in (3,4)")

            self.matvec = matvec
            if tp in 'DF':  # complex type
                if mode == 3:
                    self.OPa = Minv_matvec
                else:
                    raise ValueError("mode=4 invalid for complex A")
            else:  # real type
                if mode == 3:
                    self.OPa = lambda x: np.real(Minv_matvec(x))
                else:
                    self.OPa = lambda x: np.imag(Minv_matvec(x))
            if M_matvec is None:
                self.B = lambda x: x
                self.bmat = 'I'
                self.OP = self.OPa
            else:
                self.B = M_matvec
                self.bmat = 'G'
                self.OP = lambda x: self.OPa(M_matvec(x))
        else:
            raise ValueError("mode=%i not implemented" % mode)

        if which not in _NEUPD_WHICH:
            raise ValueError("Parameter which must be one of %s"
                             % ' '.join(_NEUPD_WHICH))
        if k >= n - 1:
            raise ValueError("k must be less than rank(A)-1, k=%d" % k)

        _ArpackParams.__init__(self, n, k, tp, mode, sigma,
                               ncv, v0, maxiter, which, tol)

        if self.ncv > n or self.ncv <= k + 1:
            raise ValueError("ncv must be k+1<ncv<=n, ncv=%s" % self.ncv)

        self.workd = np.zeros(3 * n, self.tp)
        self.workl = np.zeros(3 * self.ncv * (self.ncv + 2), self.tp)

        ltr = _type_conv[self.tp]
        self._arpack_solver = _arpack.__dict__[ltr + 'naupd']
        self._arpack_extract = _arpack.__dict__[ltr + 'neupd']

        self.iterate_infodict = _NAUPD_ERRORS[ltr]
        self.extract_infodict = _NEUPD_ERRORS[ltr]

        self.ipntr = np.zeros(14, "int")

        if self.tp in 'FD':
            self.rwork = np.zeros(self.ncv, self.tp.lower())
        else:
            self.rwork = None

    def iterate(self):
        if self.tp in 'fd':
            self.ido, self.resid, self.v, self.iparam, self.ipntr, self.info =\
                self._arpack_solver(self.ido, self.bmat, self.which, self.k,
                                    self.tol, self.resid, self.v, self.iparam,
                                    self.ipntr,  self.workd, self.workl,
                                    self.info)
        else:
            self.ido, self.resid, self.v, self.iparam, self.ipntr, self.info =\
                self._arpack_solver(self.ido, self.bmat, self.which, self.k,
                                    self.tol, self.resid, self.v, self.iparam,
                                    self.ipntr, self.workd, self.workl,
                                    self.rwork, self.info)

        xslice = slice(self.ipntr[0] - 1, self.ipntr[0] - 1 + self.n)
        yslice = slice(self.ipntr[1] - 1, self.ipntr[1] - 1 + self.n)
        if self.ido == -1:
            # initialization
            self.workd[yslice] = self.OP(self.workd[xslice])
        elif self.ido == 1:
            # compute y = Op*x
            if self.mode in (1, 2):
                self.workd[yslice] = self.OP(self.workd[xslice])
            else:
                Bxslice = slice(self.ipntr[2] - 1, self.ipntr[2] - 1 + self.n)
                self.workd[yslice] = self.OPa(self.workd[Bxslice])
        elif self.ido == 2:
            self.workd[yslice] = self.B(self.workd[xslice])
        elif self.ido == 3:
            raise ValueError("ARPACK requested user shifts.  Assure ISHIFT==0")
        else:
            self.converged = True

            if self.info == 0:
                pass
            elif self.info == 1:
                self._raise_no_convergence()
            else:
                raise ArpackError(self.info, infodict=self.iterate_infodict)

    def extract(self, return_eigenvectors):
        k, n = self.k, self.n

        ierr = 0
        howmny = 'A'  # return all eigenvectors
        sselect = np.zeros(self.ncv, 'int')  # unused
        sigmar = np.real(self.sigma)
        sigmai = np.imag(self.sigma)
        workev = np.zeros(3 * self.ncv, self.tp)

        if self.tp in 'fd':
            dr = np.zeros(k + 1, self.tp)
            di = np.zeros(k + 1, self.tp)
            zr = np.zeros((n, k + 1), self.tp)
            dr, di, zr, ierr = \
                self._arpack_extract(
                    return_eigenvectors, howmny, sselect, sigmar, sigmai,
                    workev, self.bmat, self.which, k, self.tol, self.resid,
                    self.v, self.iparam, self.ipntr, self.workd, self.workl,
                    self.info)
            if ierr != 0:
                raise ArpackError(ierr, infodict=self.extract_infodict)
            nreturned = self.iparam[4]  # number of good eigenvalues returned

            # Build complex eigenvalues from real and imaginary parts
            d = dr + 1.0j * di

            # Arrange the eigenvectors: complex eigenvectors are stored as
            # real,imaginary in consecutive columns
            z = zr.astype(self.tp.upper())

            # The ARPACK nonsymmetric real and double interface (s,d)naupd
            # return eigenvalues and eigenvectors in real (float,double)
            # arrays.

            # Efficiency: this should check that return_eigenvectors == True
            #  before going through this construction.
            if sigmai == 0:
                i = 0
                while i <= k:
                    # check if complex
                    if abs(d[i].imag) != 0:
                        # this is a complex conjugate pair with eigenvalues
                        # in consecutive columns
                        if i < k:
                            z[:, i] = zr[:, i] + 1.0j * zr[:, i + 1]
                            z[:, i + 1] = z[:, i].conjugate()
                            i += 1
                        else:
                            #last eigenvalue is complex: the imaginary part of
                            # the eigenvector has not been returned
                            #this can only happen if nreturned > k, so we'll
                            # throw out this case.
                            nreturned -= 1
                    i += 1

            else:
                # real matrix, mode 3 or 4, imag(sigma) is nonzero:
                # see remark 3 in <s,d>neupd.f
                # Build complex eigenvalues from real and imaginary parts
                i = 0
                while i <= k:
                    if abs(d[i].imag) == 0:
                        d[i] = np.dot(zr[:, i], self.matvec(zr[:, i]))
                    else:
                        if i < k:
                            z[:, i] = zr[:, i] + 1.0j * zr[:, i + 1]
                            z[:, i + 1] = z[:, i].conjugate()
                            d[i] = ((np.dot(zr[:, i],
                                            self.matvec(zr[:, i]))
                                     + np.dot(zr[:, i + 1],
                                              self.matvec(zr[:, i + 1])))
                                    + 1j * (np.dot(zr[:, i],
                                                   self.matvec(zr[:, i + 1]))
                                            - np.dot(zr[:, i + 1],
                                                     self.matvec(zr[:, i]))))
                            d[i + 1] = d[i].conj()
                            i += 1
                        else:
                            #last eigenvalue is complex: the imaginary part of
                            # the eigenvector has not been returned
                            #this can only happen if nreturned > k, so we'll
                            # throw out this case.
                            nreturned -= 1
                    i += 1

            # Now we have k+1 possible eigenvalues and eigenvectors
            # Return the ones specified by the keyword "which"

            if nreturned <= k:
                # we got less or equal as many eigenvalues we wanted
                d = d[:nreturned]
                z = z[:, :nreturned]
            else:
                # we got one extra eigenvalue (likely a cc pair, but which?)
                # cut at approx precision for sorting
                rd = np.round(d, decimals=_ndigits[self.tp])
                if self.which in ['LR', 'SR']:
                    ind = np.argsort(rd.real)
                elif self.which in ['LI', 'SI']:
                    # for LI,SI ARPACK returns largest,smallest
                    # abs(imaginary) why?
                    ind = np.argsort(abs(rd.imag))
                else:
                    ind = np.argsort(abs(rd))
                if self.which in ['LR', 'LM', 'LI']:
                    d = d[ind[-k:]]
                    z = z[:, ind[-k:]]
                if self.which in ['SR', 'SM', 'SI']:
                    d = d[ind[:k]]
                    z = z[:, ind[:k]]
        else:
            # complex is so much simpler...
            d, z, ierr =\
                self._arpack_extract(
                    return_eigenvectors, howmny, sselect, self.sigma, workev,
                    self.bmat, self.which, k, self.tol, self.resid, self.v,
                    self.iparam, self.ipntr, self.workd, self.workl,
                    self.rwork, ierr)

            if ierr != 0:
                raise ArpackError(ierr, infodict=self.extract_infodict)

            k_ok = self.iparam[4]
            d = d[:k_ok]
            z = z[:, :k_ok]

        if return_eigenvectors:
            return d, z
        else:
            return d

def _aslinearoperator_with_dtype(m):
    m = aslinearoperator(m)
    if not hasattr(m, 'dtype'):
        x = np.zeros(m.shape[1])
        m.dtype = (m * x).dtype
    return m

class SpLuInv(LinearOperator):
    """
    SpLuInv:
       helper class to repeatedly solve M*x=b
       using a sparse LU-decopposition of M
    """
    def __init__(self, M):
        self.M_lu = splu(M)
        LinearOperator.__init__(self, M.shape, self._matvec, dtype=M.dtype)
        self.isreal = not np.issubdtype(self.dtype, np.complexfloating)

    def _matvec(self, x):
        # careful here: splu.solve will throw away imaginary
        # part of x if M is real
        if self.isreal and np.issubdtype(x.dtype, np.complexfloating):
            return (self.M_lu.solve(np.real(x))
                    + 1j * self.M_lu.solve(np.imag(x)))
        else:
            return self.M_lu.solve(x)

class LuInv(LinearOperator):
    """
    LuInv:
       helper class to repeatedly solve M*x=b
       using an LU-decomposition of M
    """
    def __init__(self, M):
        self.M_lu = lu_factor(M)
        LinearOperator.__init__(self, M.shape, self._matvec, dtype=M.dtype)

    def _matvec(self, x):
        return lu_solve(self.M_lu, x)

class IterInv(LinearOperator):
    """
    IterInv:
       helper class to repeatedly solve M*x=b
       using an iterative method.
    """
    def __init__(self, M, ifunc=gmres, tol=0):
        if tol <= 0:
            # when tol=0, ARPACK uses machine tolerance as calculated
            # by LAPACK's _LAMCH function.  We should match this
            tol = np.finfo(M.dtype).eps
        self.M = M
        self.ifunc = ifunc
        self.tol = tol
        if hasattr(M, 'dtype'):
            dtype = M.dtype
        else:
            x = np.zeros(M.shape[1])
            dtype = (M * x).dtype
        LinearOperator.__init__(self, M.shape, self._matvec, dtype=dtype)

    def _matvec(self, x):
        b, info = self.ifunc(self.M, x, tol=self.tol)
        if info != 0:
            raise ValueError("Error in inverting M: function "
                             "%s did not converge (info = %i)."
                             % (self.ifunc.__name__, info))
        return b

class IterOpInv(LinearOperator):
    """
    IterOpInv:
       helper class to repeatedly solve [A-sigma*M]*x = b
       using an iterative method
    """
    def __init__(self, A, M, sigma, ifunc=gmres, tol=0):
        if tol <= 0:
            # when tol=0, ARPACK uses machine tolerance as calculated
            # by LAPACK's _LAMCH function.  We should match this
            tol = np.finfo(A.dtype).eps
        self.A = A
        self.M = M
        self.sigma = sigma
        self.ifunc = ifunc
        self.tol = tol

        x = np.zeros(A.shape[1])
        if M is None:
            dtype = self.mult_func_M_None(x).dtype
            self.OP = LinearOperator(self.A.shape,
                                     self.mult_func_M_None,
                                     dtype=dtype)
        else:
            dtype = self.mult_func(x).dtype
            self.OP = LinearOperator(self.A.shape,
                                     self.mult_func,
                                     dtype=dtype)
        LinearOperator.__init__(self, A.shape, self._matvec, dtype=dtype)

    def mult_func(self, x):
        return self.A.matvec(x) - self.sigma * self.M.matvec(x)

    def mult_func_M_None(self, x):
        return self.A.matvec(x) - self.sigma * x

    def _matvec(self, x):
        b, info = self.ifunc(self.OP, x, tol=self.tol)
        if info != 0:
            raise ValueError("Error in inverting [A-sigma*M]: function "
                             "%s did not converge (info = %i)."
                             % (self.ifunc.__name__, info))
        return b

def get_inv_matvec(M, symmetric=False, tol=0):
    if isdense(M):
        return LuInv(M).matvec
    elif isspmatrix(M):
        if isspmatrix_csr(M) and symmetric:
            M = M.T
        return SpLuInv(M).matvec
    else:
        return IterInv(M, tol=tol).matvec

def get_OPinv_matvec(A, M, sigma, symmetric=False, tol=0):
    if sigma == 0:
        return get_inv_matvec(A, symmetric=symmetric, tol=tol)

    if M is None:
        #M is the identity matrix
        if isdense(A):
            if (np.issubdtype(A.dtype, np.complexfloating)
                    or np.imag(sigma) == 0):
                A = np.copy(A)
            else:
                A = A + 0j
            A.flat[::A.shape[1] + 1] -= sigma
            return LuInv(A).matvec
        elif isspmatrix(A):
            A = A - sigma * identity(A.shape[0])
            if symmetric and isspmatrix_csr(A):
                A = A.T
            return SpLuInv(A.tocsc()).matvec
        else:
            return IterOpInv(_aslinearoperator_with_dtype(A), M, sigma,
                             tol=tol).matvec
    else:
        if ((not isdense(A) and not isspmatrix(A)) or
                (not isdense(M) and not isspmatrix(M))):
            return IterOpInv(_aslinearoperator_with_dtype(A),
                             _aslinearoperator_with_dtype(M), sigma,
                             tol=tol).matvec
        elif isdense(A) or isdense(M):
            return LuInv(A - sigma * M).matvec
        else:
            OP = A - sigma * M
            if symmetric and isspmatrix_csr(OP):
                OP = OP.T
            return SpLuInv(OP.tocsc()).matvec

def _eigs(A, k=6, M=None, sigma=None, which='LM', v0=None, ncv=None,
          maxiter=None, tol=0, return_eigenvectors=True, Minv=None, OPinv=None,
          OPpart=None):
    """
    Find k eigenvalues and eigenvectors of the square matrix A.

    Solves ``A * x[i] = w[i] * x[i]``, the standard eigenvalue problem
    for w[i] eigenvalues with corresponding eigenvectors x[i].

    If M is specified, solves ``A * x[i] = w[i] * M * x[i]``, the
    generalized eigenvalue problem for w[i] eigenvalues
    with corresponding eigenvectors x[i]

    Parameters
    ----------
    A : An N x N matrix, array, sparse matrix, or LinearOperator representing \
    the operation A * x, where A is a real or complex square matrix.

    k : int, default 6
        The number of eigenvalues and eigenvectors desired.
        `k` must be smaller than N. It is not possible to compute all
        eigenvectors of a matrix.

    return_eigenvectors : boolean, default True
        Whether to return the eigenvectors along with the eigenvalues.

    M : An N x N matrix, array, sparse matrix, or LinearOperator representing
        the operation M*x for the generalized eigenvalue problem
          ``A * x = w * M * x``
        M must represent a real symmetric matrix.  For best results, M should
        be of the same type as A.  Additionally:
         * If sigma==None, M is positive definite
         * If sigma is specified, M is positive semi-definite
        If sigma==None, eigs requires an operator to compute the solution
        of the linear equation `M * x = b`. This is done internally via a
        (sparse) LU decomposition for an explicit matrix M, or via an
        iterative solver for a general linear operator.  Alternatively,
        the user can supply the matrix or operator Minv, which gives
        x = Minv * b = M^-1 * b

    sigma : real or complex
        Find eigenvalues near sigma using shift-invert mode.  This requires
        an operator to compute the solution of the linear system
        `[A - sigma * M] * x = b`, where M is the identity matrix if
        unspecified. This is computed internally via a (sparse) LU
        decomposition for explicit matrices A & M, or via an iterative
        solver if either A or M is a general linear operator.
        Alternatively, the user can supply the matrix or operator OPinv,
        which gives x = OPinv * b = [A - sigma * M]^-1 * b.
        For a real matrix A, shift-invert can either be done in imaginary
        mode or real mode, specified by the parameter OPpart ('r' or 'i').
        Note that when sigma is specified, the keyword 'which' (below)
        refers to the shifted eigenvalues w'[i] where:
         * If A is real and OPpart == 'r' (default),
            w'[i] = 1/2 * [ 1/(w[i]-sigma) + 1/(w[i]-conj(sigma)) ]
         * If A is real and OPpart == 'i',
            w'[i] = 1/2i * [ 1/(w[i]-sigma) - 1/(w[i]-conj(sigma)) ]
         * If A is complex,
            w'[i] = 1/(w[i]-sigma)

    v0 : array
        Starting vector for iteration.

    ncv : integer
        The number of Lanczos vectors generated
        `ncv` must be greater than `k`; it is recommended that ``ncv > 2*k``.

    which : string ['LM' | 'SM' | 'LR' | 'SR' | 'LI' | 'SI']
        Which `k` eigenvectors and eigenvalues to find:
         - 'LM' : largest magnitude
         - 'SM' : smallest magnitude
         - 'LR' : largest real part
         - 'SR' : smallest real part
         - 'LI' : largest imaginary part
         - 'SI' : smallest imaginary part
        When sigma != None, 'which' refers to the shifted eigenvalues w'[i]
        (see discussion in 'sigma', above).  ARPACK is generally better
        at finding large values than small values.  If small eigenvalues are
        desired, consider using shift-invert mode for better performance.

    maxiter : integer
        Maximum number of Arnoldi update iterations allowed

    tol : float
        Relative accuracy for eigenvalues (stopping criterion)
        The default value of 0 implies machine precision.

    return_eigenvectors : boolean
        Return eigenvectors (True) in addition to eigenvalues

    Minv : N x N matrix, array, sparse matrix, or linear operator
        See notes in M, above.

    OPinv : N x N matrix, array, sparse matrix, or linear operator
        See notes in sigma, above.
    OPpart : 'r' or 'i'.
        See notes in sigma, above

    Returns
    -------
    w : array
        Array of k eigenvalues.

    v : array
        An array of `k` eigenvectors.
        ``v[:, i]`` is the eigenvector corresponding to the eigenvalue w[i].

    Raises
    ------
    ArpackNoConvergence
        When the requested convergence is not obtained.

        The currently converged eigenvalues and eigenvectors can be found
        as ``eigenvalues`` and ``eigenvectors`` attributes of the exception
        object.

    See Also
    --------
    eigsh : eigenvalues and eigenvectors for symmetric matrix A
    svds : singular value decomposition for a matrix A

    Examples
    --------
    Find 6 eigenvectors of the identity matrix:

    >>> from sklearn.utils.arpack import eigs
    >>> id = np.identity(13)
    >>> vals, vecs = eigs(id, k=6)
    >>> vals
    array([ 1.+0.j,  1.+0.j,  1.+0.j,  1.+0.j,  1.+0.j,  1.+0.j])
    >>> vecs.shape
    (13, 6)

    Notes
    -----
    This function is a wrapper to the ARPACK [1]_ SNEUPD, DNEUPD, CNEUPD,
    ZNEUPD, functions which use the Implicitly Restarted Arnoldi Method to
    find the eigenvalues and eigenvectors [2]_.

    References
    ----------
    .. [1] ARPACK Software, http://www.caam.rice.edu/software/ARPACK/
    .. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang,  ARPACK USERS GUIDE:
       Solution of Large Scale Eigenvalue Problems by Implicitly Restarted
       Arnoldi Methods. SIAM, Philadelphia, PA, 1998.
    """
    if A.shape[0] != A.shape[1]:
        raise ValueError('expected square matrix (shape=%s)' % (A.shape,))
    if M is not None:
        if M.shape != A.shape:
            raise ValueError('wrong M dimensions %s, should be %s'
                             % (M.shape, A.shape))
        if np.dtype(M.dtype).char.lower() != np.dtype(A.dtype).char.lower():
            warnings.warn('M does not have the same type precision as A. '
                          'This may adversely affect ARPACK convergence')
    n = A.shape[0]

    if k <= 0 or k >= n:
        raise ValueError("k must be between 1 and rank(A)-1")

    if sigma is None:
        matvec = _aslinearoperator_with_dtype(A).matvec

        if OPinv is not None:
            raise ValueError("OPinv should not be specified "
                             "with sigma = None.")
        if OPpart is not None:
            raise ValueError("OPpart should not be specified with "
                             "sigma = None or complex A")

        if M is None:
            #standard eigenvalue problem
            mode = 1
            M_matvec = None
            Minv_matvec = None
            if Minv is not None:
                raise ValueError("Minv should not be "
                                 "specified with M = None.")
        else:
            #general eigenvalue problem
            mode = 2
            if Minv is None:
                Minv_matvec = get_inv_matvec(M, symmetric=True, tol=tol)
            else:
                Minv = _aslinearoperator_with_dtype(Minv)
                Minv_matvec = Minv.matvec
            M_matvec = _aslinearoperator_with_dtype(M).matvec
    else:
        #sigma is not None: shift-invert mode
        if np.issubdtype(A.dtype, np.complexfloating):
            if OPpart is not None:
                raise ValueError("OPpart should not be specified "
                                 "with sigma=None or complex A")
            mode = 3
        elif OPpart is None or OPpart.lower() == 'r':
            mode = 3
        elif OPpart.lower() == 'i':
            if np.imag(sigma) == 0:
                raise ValueError("OPpart cannot be 'i' if sigma is real")
            mode = 4
        else:
            raise ValueError("OPpart must be one of ('r','i')")

        matvec = _aslinearoperator_with_dtype(A).matvec
        if Minv is not None:
            raise ValueError("Minv should not be specified when sigma is")
        if OPinv is None:
            Minv_matvec = get_OPinv_matvec(A, M, sigma,
                                           symmetric=False, tol=tol)
        else:
            OPinv = _aslinearoperator_with_dtype(OPinv)
            Minv_matvec = OPinv.matvec
        if M is None:
            M_matvec = None
        else:
            M_matvec = _aslinearoperator_with_dtype(M).matvec

    params = _UnsymmetricArpackParams(n, k, A.dtype.char, matvec, mode,
                                      M_matvec, Minv_matvec, sigma,
                                      ncv, v0, maxiter, which, tol)

    while not params.converged:
        params.iterate()

    return params.extract(return_eigenvectors)

def _eigsh(A, k=6, M=None, sigma=None, which='LM', v0=None, ncv=None,
           maxiter=None, tol=0, return_eigenvectors=True, Minv=None,
           OPinv=None, mode='normal'):
    """
    Find k eigenvalues and eigenvectors of the real symmetric square matrix
    or complex hermitian matrix A.

    Solves ``A * x[i] = w[i] * x[i]``, the standard eigenvalue problem for
    w[i] eigenvalues with corresponding eigenvectors x[i].

    If M is specified, solves ``A * x[i] = w[i] * M * x[i]``, the
    generalized eigenvalue problem for w[i] eigenvalues
    with corresponding eigenvectors x[i]

    Parameters
    ----------
    A : An N x N matrix, array, sparse matrix, or LinearOperator representing
        the operation A * x, where A is a real symmetric matrix
        For buckling mode (see below) A must additionally be positive-definite
    k : integer
        The number of eigenvalues and eigenvectors desired.
        `k` must be smaller than N. It is not possible to compute all
        eigenvectors of a matrix.

    M : An N x N matrix, array, sparse matrix, or linear operator representing
        the operation M * x for the generalized eigenvalue problem
          ``A * x = w * M * x``.
        M must represent a real, symmetric matrix.  For best results, M should
        be of the same type as A.  Additionally:
         * If sigma == None, M is symmetric positive definite
         * If sigma is specified, M is symmetric positive semi-definite
         * In buckling mode, M is symmetric indefinite.
        If sigma == None, eigsh requires an operator to compute the solution
        of the linear equation `M * x = b`. This is done internally via a
        (sparse) LU decomposition for an explicit matrix M, or via an
        iterative solver for a general linear operator.  Alternatively,
        the user can supply the matrix or operator Minv, which gives
        x = Minv * b = M^-1 * b
    sigma : real
        Find eigenvalues near sigma using shift-invert mode.  This requires
        an operator to compute the solution of the linear system
        `[A - sigma * M] x = b`, where M is the identity matrix if
        unspecified.  This is computed internally via a (sparse) LU
        decomposition for explicit matrices A & M, or via an iterative
        solver if either A or M is a general linear operator.
        Alternatively, the user can supply the matrix or operator OPinv,
        which gives x = OPinv * b = [A - sigma * M]^-1 * b.
        Note that when sigma is specified, the keyword 'which' refers to
        the shifted eigenvalues w'[i] where:
         - if mode == 'normal',
             w'[i] = 1 / (w[i] - sigma)
         - if mode == 'cayley',
             w'[i] = (w[i] + sigma) / (w[i] - sigma)
         - if mode == 'buckling',
             w'[i] = w[i] / (w[i] - sigma)
        (see further discussion in 'mode' below)
    v0 : array
        Starting vector for iteration.
    ncv : integer
        The number of Lanczos vectors generated
        ncv must be greater than k and smaller than n;
        it is recommended that ncv > 2*k
    which : string ['LM' | 'SM' | 'LA' | 'SA' | 'BE']
        If A is a complex hermitian matrix, 'BE' is invalid.
        Which `k` eigenvectors and eigenvalues to find
         - 'LM' : Largest (in magnitude) eigenvalues
         - 'SM' : Smallest (in magnitude) eigenvalues
         - 'LA' : Largest (algebraic) eigenvalues
         - 'SA' : Smallest (algebraic) eigenvalues
         - 'BE' : Half (k/2) from each end of the spectrum
                  When k is odd, return one more (k/2+1) from the high end
        When sigma != None, 'which' refers to the shifted eigenvalues w'[i]
        (see discussion in 'sigma', above).  ARPACK is generally better
        at finding large values than small values.  If small eigenvalues are
        desired, consider using shift-invert mode for better performance.
    maxiter : integer
        Maximum number of Arnoldi update iterations allowed
    tol : float
        Relative accuracy for eigenvalues (stopping criterion).
        The default value of 0 implies machine precision.
    Minv : N x N matrix, array, sparse matrix, or LinearOperator
        See notes in M, above
    OPinv : N x N matrix, array, sparse matrix, or LinearOperator
        See notes in sigma, above.
    return_eigenvectors : boolean
        Return eigenvectors (True) in addition to eigenvalues
    mode : string ['normal' | 'buckling' | 'cayley']
        Specify strategy to use for shift-invert mode.  This argument applies
        only for real-valued A and sigma != None.  For shift-invert mode,
        ARPACK internally solves the eigenvalue problem
        ``OP * x'[i] = w'[i] * B * x'[i]``
        and transforms the resulting Ritz vectors x'[i] and Ritz values w'[i]
        into the desired eigenvectors and eigenvalues of the problem
        ``A * x[i] = w[i] * M * x[i]``.
        The modes are as follows:
          - 'normal'   : OP = [A - sigma * M]^-1 * M
                         B = M
                         w'[i] = 1 / (w[i] - sigma)
          - 'buckling' : OP = [A - sigma * M]^-1 * A
                         B = A
                         w'[i] = w[i] / (w[i] - sigma)
          - 'cayley'   : OP = [A - sigma * M]^-1 * [A + sigma * M]
                         B = M
                         w'[i] = (w[i] + sigma) / (w[i] - sigma)
        The choice of mode will affect which eigenvalues are selected by
        the keyword 'which', and can also impact the stability of
        convergence (see [2] for a discussion)

    Returns
    -------
    w : array
        Array of k eigenvalues
    v : array
        An array of k eigenvectors
        The v[i] is the eigenvector corresponding to the eigenvector w[i]

    Raises
    ------
    ArpackNoConvergence
        When the requested convergence is not obtained.

        The currently converged eigenvalues and eigenvectors can be found
        as ``eigenvalues`` and ``eigenvectors`` attributes of the exception
        object.

    See Also
    --------
    eigs : eigenvalues and eigenvectors for a general (nonsymmetric) matrix A
    svds : singular value decomposition for a matrix A

    Notes
    -----
    This function is a wrapper to the ARPACK [1]_ SSEUPD and DSEUPD
    functions which use the Implicitly Restarted Lanczos Method to
    find the eigenvalues and eigenvectors [2]_.

    Examples
    --------
    >>> from sklearn.utils.arpack import eigsh
    >>> id = np.identity(13)
    >>> vals, vecs = eigsh(id, k=6)
    >>> vals # doctest: +SKIP
    array([ 1.+0.j,  1.+0.j,  1.+0.j,  1.+0.j,  1.+0.j,  1.+0.j])
    >>> print(vecs.shape)
    (13, 6)

    References
    ----------
    .. [1] ARPACK Software, http://www.caam.rice.edu/software/ARPACK/
    .. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang,  ARPACK USERS GUIDE:
       Solution of Large Scale Eigenvalue Problems by Implicitly Restarted
       Arnoldi Methods. SIAM, Philadelphia, PA, 1998.
    """
    # complex hermitian matrices should be solved with eigs
    if np.issubdtype(A.dtype, np.complexfloating):
        if mode != 'normal':
            raise ValueError("mode=%s cannot be used with "
                             "complex matrix A" % mode)
        if which == 'BE':
            raise ValueError("which='BE' cannot be used with complex matrix A")
        elif which == 'LA':
            which = 'LR'
        elif which == 'SA':
            which = 'SR'
        ret = eigs(A, k, M=M, sigma=sigma, which=which, v0=v0,
                   ncv=ncv, maxiter=maxiter, tol=tol,
                   return_eigenvectors=return_eigenvectors, Minv=Minv,
                   OPinv=OPinv)

        if return_eigenvectors:
            return ret[0].real, ret[1]
        else:
            return ret.real

    if A.shape[0] != A.shape[1]:
        raise ValueError('expected square matrix (shape=%s)' % (A.shape,))
    if M is not None:
        if M.shape != A.shape:
            raise ValueError('wrong M dimensions %s, should be %s'
                             % (M.shape, A.shape))
        if np.dtype(M.dtype).char.lower() != np.dtype(A.dtype).char.lower():
            warnings.warn('M does not have the same type precision as A. '
                          'This may adversely affect ARPACK convergence')
    n = A.shape[0]

    if k <= 0 or k >= n:
        raise ValueError("k must be between 1 and rank(A)-1")

    if sigma is None:
        A = _aslinearoperator_with_dtype(A)
        matvec = A.matvec

        if OPinv is not None:
            raise ValueError("OPinv should not be specified "
                             "with sigma = None.")
        if M is None:
            #standard eigenvalue problem
            mode = 1
            M_matvec = None
            Minv_matvec = None
            if Minv is not None:
                raise ValueError("Minv should not be "
                                 "specified with M = None.")
        else:
            #general eigenvalue problem
            mode = 2
            if Minv is None:
                Minv_matvec = get_inv_matvec(M, symmetric=True, tol=tol)
            else:
                Minv = _aslinearoperator_with_dtype(Minv)
                Minv_matvec = Minv.matvec
            M_matvec = _aslinearoperator_with_dtype(M).matvec
    else:
        # sigma is not None: shift-invert mode
        if Minv is not None:
            raise ValueError("Minv should not be specified when sigma is")

        # normal mode
        if mode == 'normal':
            mode = 3
            matvec = None
            if OPinv is None:
                Minv_matvec = get_OPinv_matvec(A, M, sigma,
                                               symmetric=True, tol=tol)
            else:
                OPinv = _aslinearoperator_with_dtype(OPinv)
                Minv_matvec = OPinv.matvec
            if M is None:
                M_matvec = None
            else:
                M = _aslinearoperator_with_dtype(M)
                M_matvec = M.matvec

        # buckling mode
        elif mode == 'buckling':
            mode = 4
            if OPinv is None:
                Minv_matvec = get_OPinv_matvec(A, M, sigma,
                                               symmetric=True, tol=tol)
            else:
                Minv_matvec = _aslinearoperator_with_dtype(OPinv).matvec
            matvec = _aslinearoperator_with_dtype(A).matvec
            M_matvec = None

        # cayley-transform mode
        elif mode == 'cayley':
            mode = 5
            matvec = _aslinearoperator_with_dtype(A).matvec
            if OPinv is None:
                Minv_matvec = get_OPinv_matvec(A, M, sigma,
                                               symmetric=True, tol=tol)
            else:
                Minv_matvec = _aslinearoperator_with_dtype(OPinv).matvec
            if M is None:
                M_matvec = None
            else:
                M_matvec = _aslinearoperator_with_dtype(M).matvec

        # unrecognized mode
        else:
            raise ValueError("unrecognized mode '%s'" % mode)

    params = _SymmetricArpackParams(n, k, A.dtype.char, matvec, mode,
                                    M_matvec, Minv_matvec, sigma,
                                    ncv, v0, maxiter, which, tol)

    while not params.converged:
        params.iterate()

    return params.extract(return_eigenvectors)

def _svds(A, k=6, ncv=None, tol=0):
    """Compute k singular values/vectors for a sparse matrix using ARPACK.

    Parameters
    ----------
    A : sparse matrix
        Array to compute the SVD on
    k : int, optional
        Number of singular values and vectors to compute.
    ncv : integer
        The number of Lanczos vectors generated
        ncv must be greater than k+1 and smaller than n;
        it is recommended that ncv > 2*k
    tol : float, optional
        Tolerance for singular values. Zero (default) means machine precision.

    Notes
    -----
    This is a naive implementation using an eigensolver on A.H * A or
    A * A.H, depending on which one is more efficient.

    """
    if not (isinstance(A, np.ndarray) or isspmatrix(A)):
        A = np.asarray(A)

    n, m = A.shape

    if np.issubdtype(A.dtype, np.complexfloating):
        herm = lambda x: x.T.conjugate()
        eigensolver = eigs
    else:
        herm = lambda x: x.T
        eigensolver = eigsh

    if n > m:
        X = A
        XH = herm(A)
    else:
        XH = A
        X = herm(A)

    if hasattr(XH, 'dot'):
        def matvec_XH_X(x):
            return XH.dot(X.dot(x))
    else:
        def matvec_XH_X(x):
            return np.dot(XH, np.dot(X, x))

    XH_X = LinearOperator(matvec=matvec_XH_X, dtype=X.dtype,
                          shape=(X.shape[1], X.shape[1]))

    # Ignore deprecation warnings here: dot on matrices is deprecated,
    # but this code is a backport anyhow
    with warnings.catch_warnings():
        warnings.simplefilter('ignore', DeprecationWarning)
        eigvals, eigvec = eigensolver(XH_X, k=k, tol=tol ** 2)
    s = np.sqrt(eigvals)

    if n > m:
        v = eigvec
        if hasattr(X, 'dot'):
            u = X.dot(v) / s
        else:
            u = np.dot(X, v) / s
        vh = herm(v)
    else:
        u = eigvec
        if hasattr(X, 'dot'):
            vh = herm(X.dot(u) / s)
        else:
            vh = herm(np.dot(X, u) / s)

    return u, s, vh

# check if backport is actually needed:
if scipy.version.version >= LooseVersion('0.10'):
    from scipy.sparse.linalg import eigs, eigsh, svds
else:
    eigs, eigsh, svds = _eigs, _eigsh, _svds


from __future__ import division
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import paraBEM

from paraBEM.liftingline import LiftingLine
from paraBEM.utils import check_path

# WingGeometry
spw = 2
numpos = 50
z_fac_1 = -0.3
z_fac_2 = -0.7
y = np.sin(np.linspace(0, np.pi/2, numpos)) * spw/2
x = [0. for _ in y]
z = [i**2 * z_fac_1 + i**6 * z_fac_2 for i in y]

mirror = lambda xyz: [xyz[0], -xyz[1], xyz[2]]
wing = list(zip(x, y, z))
wing = list(map(mirror, wing))[::-1] + list(wing)[1:]
wing = [paraBEM.Vector3(*i) for i in wing]

# LiftingLine
lifting_line = LiftingLine(wing)
lifting_line.v_inf = paraBEM.Vector3(1, 0, 0)
lifting_line.solve_for_best_gamma(1)
gamma = [i.best_gamma for i in lifting_line.segments]
gamma_max = max(gamma)

# Plot
gamma_el = lambda y: gamma_max * (1 - (y / spw * 2)**2)**(1 / 2)
mids = [[i.mids.x, i.mids.y, i.mids.z] for i in lifting_line.segments]
x, y, z = zip(*mids)

fig = plt.figure()
ax1 = fig.add_subplot(3, 1, 1)
ax1.plot(y, z)

ax2 = fig.add_subplot(3, 1, 2)
ax2.plot(y, x, marker="x")

ax3 = fig.add_subplot(3, 1, 3)
y_el = np.linspace(-1, 1, 400)
ax3.plot([-spw/2] + list(y) + [spw/2], [0] + gamma + [0], marker="x")
ax3.plot(y_el, list(map(gamma_el, y_el)))
plt.savefig(check_path("results/2d/liftingline.png"))

total = 0
for i in lifting_line.segments:
    total += i.lift_factor * i.best_gamma
print(total)


"""
==============================================
Scaling the regularization parameter for SVCs
==============================================

The following example illustrates the effect of scaling the
regularization parameter when using :ref:`svm` for
:ref:`classification <svm_classification>`.
For SVC classification, we are interested in a risk minimization for the
equation:

.. math::

    C \sum_{i=1, n} \mathcal{L} (f(x_i), y_i) + \Omega (w)

where

    - :math:`C` is used to set the amount of regularization
    - :math:`\mathcal{L}` is a `loss` function of our samples
      and our model parameters.
    - :math:`\Omega` is a `penalty` function of our model parameters

If we consider the loss function to be the individual error per
sample, then the data-fit term, or the sum of the error for each sample, will
increase as we add more samples. The penalization term, however, will not
increase.

When using, for example, :ref:`cross validation <cross_validation>`, to
set the amount of regularization with `C`, there will be a
different amount of samples between the main problem and the smaller problems
within the folds of the cross validation.

Since our loss function is dependent on the amount of samples, the latter
will influence the selected value of `C`.
The question that arises is `How do we optimally adjust C to
account for the different amount of training samples?`

The figures below are used to illustrate the effect of scaling our
`C` to compensate for the change in the number of samples, in the
case of using an `l1` penalty, as well as the `l2` penalty.

l1-penalty case
-----------------
In the `l1` case, theory says that prediction consistency
(i.e. that under given hypothesis, the estimator
learned predicts as well as a model knowing the true distribution)
is not possible because of the bias of the `l1`. It does say, however,
that model consistency, in terms of finding the right set of non-zero
parameters as well as their signs, can be achieved by scaling
`C1`.

l2-penalty case
-----------------
The theory says that in order to achieve prediction consistency, the
penalty parameter should be kept constant
as the number of samples grow.

Simulations
------------

The two figures below plot the values of `C` on the `x-axis` and the
corresponding cross-validation scores on the `y-axis`, for several different
fractions of a generated data-set.

In the `l1` penalty case, the cross-validation-error correlates best with
the test-error, when scaling our `C` with the number of samples, `n`,
which can be seen in the first figure.

For the `l2` penalty case, the best result comes from the case where `C`
is not scaled.

.. topic:: Note:

    Two separate datasets are used for the two different plots. The reason
    behind this is the `l1` case works better on sparse data, while `l2`
    is better suited to the non-sparse case.
"""
print(__doc__)

# Author: Andreas Mueller <amueller@ais.uni-bonn.de>
#         Jaques Grobler <jaques.grobler@inria.fr>
# License: BSD 3 clause

import numpy as np
import matplotlib.pyplot as plt

from sklearn.svm import LinearSVC
from sklearn.cross_validation import ShuffleSplit
from sklearn.grid_search import GridSearchCV
from sklearn.utils import check_random_state
from sklearn import datasets

rnd = check_random_state(1)

# set up dataset
n_samples = 100
n_features = 300

# l1 data (only 5 informative features)
X_1, y_1 = datasets.make_classification(n_samples=n_samples,
                                        n_features=n_features, n_informative=5,
                                        random_state=1)

# l2 data: non sparse, but less features
y_2 = np.sign(.5 - rnd.rand(n_samples))
X_2 = rnd.randn(n_samples, n_features / 5) + y_2[:, np.newaxis]
X_2 += 5 * rnd.randn(n_samples, n_features / 5)

clf_sets = [(LinearSVC(penalty='l1', loss='squared_hinge', dual=False,
                       tol=1e-3),
             np.logspace(-2.3, -1.3, 10), X_1, y_1),
            (LinearSVC(penalty='l2', loss='squared_hinge', dual=True,
                       tol=1e-4),
             np.logspace(-4.5, -2, 10), X_2, y_2)]

colors = ['b', 'g', 'r', 'c']

for fignum, (clf, cs, X, y) in enumerate(clf_sets):
    # set up the plot for each regressor
    plt.figure(fignum, figsize=(9, 10))

    for k, train_size in enumerate(np.linspace(0.3, 0.7, 3)[::-1]):
        param_grid = dict(C=cs)
        # To get nice curve, we need a large number of iterations to
        # reduce the variance
        grid = GridSearchCV(clf, refit=False, param_grid=param_grid,
                            cv=ShuffleSplit(n=n_samples, train_size=train_size,
                                            n_iter=250, random_state=1))
        grid.fit(X, y)
        scores = [x[1] for x in grid.grid_scores_]

        scales = [(1, 'No scaling'),
                  ((n_samples * train_size), '1/n_samples'),
                  ]

        for subplotnum, (scaler, name) in enumerate(scales):
            plt.subplot(2, 1, subplotnum + 1)
            plt.xlabel('C')
            plt.ylabel('CV Score')
            grid_cs = cs * float(scaler)  # scale the C's
            plt.semilogx(grid_cs, scores, label="fraction %.2f" %
                         train_size)
            plt.title('scaling=%s, penalty=%s, loss=%s' %
                      (name, clf.penalty, clf.loss))

    plt.legend(loc="best")
plt.show()


# -*- coding: utf-8 -*- <nbformat>3.0</nbformat>

# <headingcell level=1>

# Plotting with Matplotlib

# <headingcell level=2>

# Prepare for action

# <codecell>

import numpy as np
import scipy as sp
import sympy

# Pylab combines the pyplot functionality (for plotting) with the numpy
# functionality (for mathematics and for working with arrays) in a single namespace
# aims to provide a closer MATLAB feel (the easy way). Note that his approach
# should only be used when doing some interactive quick and dirty data inspection.
# DO NOT USE THIS FOR SCRIPTS
#from pylab import *

# the convienient Matplotib plotting interface pyplot (the tidy/right way)
# use this for building scripts. The examples here will all use pyplot.
import matplotlib.pyplot as plt

# for using the matplotlib API directly (the hard and verbose way)
# use this when building applications, and/or backends
import matplotlib as mpl

# <markdowncell>

# How would you like the IPython notebook show your plots? In order to use the
# matplotlib IPython magic youre IPython notebook should be launched as
#
#     ipython notebook --matplotlib=inline
#
# Make plots appear as a pop up window, chose the backend: 'gtk', 'inline', 'osx', 'qt', 'qt4', 'tk', 'wx'
#
#     %matplotlib qt
#
# or inline the notebook (no panning, zooming through the plot). Not working in IPython 0.x
#
#     %matplotib inline
#

# <codecell>

# activate pop up plots
#%matplotlib qt
# or change to inline plots
# %matplotlib inline

# <headingcell level=3>

# Matplotlib documentation

# <markdowncell>

# Finding your own way (aka RTFM). Hint: there is search box available!
#
# * http://matplotlib.org/contents.html
#
# The Matplotlib API docs:
#
# * http://matplotlib.org/api/index.html
#
# Pyplot, object oriented plotting:
#
# * http://matplotlib.org/api/pyplot_api.html
# * http://matplotlib.org/api/pyplot_summary.html
#
# Extensive gallery with examples:
#
# * http://matplotlib.org/gallery.html

# <headingcell level=3>

# Tutorials for those who want to start playing

# <markdowncell>

# If reading manuals is too much for you, there is a very good tutorial available here:
#
# * http://nbviewer.ipython.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-4-Matplotlib.ipynb
#
# Note that this tutorial uses
#
#     from pylab import *
#
# which is usually not adviced in more advanced script environments. When using
#
#     import matplotlib.pyplot as plt
#
# you need to preceed all plotting commands as used in the above tutorial with
#
#     plt.

# <markdowncell>

# Give me more!
#
# [EuroScipy 2012 Matlotlib tutorial](http://www.loria.fr/~rougier/teaching/matplotlib/). Note that here the author uses ```from pylab import * ```. When using ```import matplotliblib.pyplot as plt``` the plotting commands need to be proceeded with ```plt.```

# <headingcell level=2>

# Plotting template starting point

# <codecell>

# some sample data
x = np.arange(-10,10,0.1)

# <markdowncell>

# To change the default plot configuration values.

# <codecell>

page_width_cm = 13
dpi = 200
inch = 2.54 # inch in cm
# setting global plot configuration using the RC configuration style
plt.rc('font', family='serif')
plt.rc('xtick', labelsize=12) # tick labels
plt.rc('ytick', labelsize=20) # tick labels
plt.rc('axes', labelsize=20)  # axes labels
# If you don’t need LaTeX, don’t use it. It is slower to plot, and text
# looks just fine without. If you need it, e.g. for symbols, then use it.
#plt.rc('text', usetex=True) #<- P-E: Doesn't work on my Mac

# <codecell>

# create a figure instance, note that figure size is given in inches!
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(8,6))
# set the big title (note aligment relative to figure)
fig.suptitle("suptitle 16, figure alignment", fontsize=16)

# actual plotting
ax.plot(x, x**2, label="label 12")

# set axes title (note aligment relative to axes)
ax.set_title("title 14, axes alignment", fontsize=14)

# axes labels
ax.set_xlabel('xlabel 12')
ax.set_ylabel(r'$y_{\alpha}$ 12', fontsize=8)

# legend
ax.legend(fontsize=12, loc="best")

# saving the figure in different formats
# fig.savefig('figure-%03i.png' % dpi, dpi=dpi)
# fig.savefig('figure.svg')
# fig.savefig('figure.eps')

# <codecell>

# following steps are only relevant when using figures as pop up windows (with %matplotlib qt)
# to update a figure with has been modified
fig.canvas.draw()
# show a figure
fig.show()

# <headingcell level=2>

# Exercise

# <markdowncell>

# The current section is about you trying to figure out how to do several plotting features. You should use the previously mentioned resources to find how to do that. In many cases, google is your friend!

# <markdowncell>

# * add a grid to the plot

# <codecell>

plt.plot(x,x**2)
plt.grid('on')

# <markdowncell>

# * change the location of the legend to different places

# <codecell>

plt.plot(x,x**2, label="label 12")
plt.legend(fontsize=12, loc="upper right")

# <markdowncell>

# * find a way to control the line type and color, marker type and color, control the frequency of the marks (`markevery`). See plot options at: http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot

# <codecell>
stride = max( int(len(x) / 20), 1)
plt.plot(x,x**2, 'ko-',color='forestgreen', markevery=stride,label="label 12")
plt.legend(fontsize=12, loc="upper center")
# <markdowncell>

# * add different sub-plots

# <codecell>

fig, axes = plt.subplots(nrows=2, ncols=1,sharex=True)
axes[0].plot(x,x**2)
axes[1].plot(x,-x**2)

# <markdowncell>

# * size the figure such that when included on an A4 page the fonts are given in their true size

# <codecell>
# matplotlib.rcParams.update({'font.size': 22})
fig, axes = plt.subplots(nrows=2, ncols=1,sharex=True)
axes[0].plot(x,x**2)
axes[1].plot(x,-x**2)
fig.set_size_inches(8.2,3) # using A4 width in inches?
fig.set_dpi(100)
for ax in axes:
    for item in ([ax.title, ax.xaxis.label, ax.yaxis.label] +
                 ax.get_xticklabels() + ax.get_yticklabels()):
        item.set_fontsize(12)
# ax[0].set('xtick', labelsize=12) # tick labels
# .rc('ytick', labelsize=20) # tick labels
# .rc('axes', labelsize=20)  # axes labels
# fig.savefig('figure.pdf')
# <markdowncell>

# * make a contour plot

# <codecell>
X, Y = np.meshgrid(x,x)
plt.figure()
plt.contourf(X,Y,X*Y,linewidth=0.3,cmap=plt.get_cmap('hsv'),levels=np.arange(-1,1,0.1))
plt.show

# im=ax.contourf(x,y,ui,levels=np.arange(Umean-5*Ustd,Umean+5*Ustd,Ustd/30),cmap=plt.get_cmap('hsv'),linewidth=0.1)
# <markdowncell>

# * use twinx() to create a second axis on the right for the second plot

# <codecell>
plt.figure()
ax=plt.gca()
ax.plot(x,x**2)
ax2 = ax.twinx()
ax2.plot(x,x**4, 'r')
# <markdowncell>

# * add horizontal and vertical lines using axvline(), axhline()

# <codecell>

plt.figure()
plt.plot(x,x**2)
plt.axvline(2)
plt.axhline(10)

# <markdowncell>

# * autoformat dates for nice printing on the x-axis using fig.autofmt_xdate()

# <codecell>

import datetime
dates = np.array([datetime.datetime.now() + datetime.timedelta(days=i) for i in xrange(24)])
fig, ax = plt.subplots(nrows=1, ncols=1)
ax.plot(dates,xrange(24))
fig.autofmt_xdate()
# <headingcell level=2>

# Advanced exercises

# <markdowncell>

# We are going to play a bit with regression

# <markdowncell>

# * Create a vector x of equally spaced number between $x \in [0, 5\pi]$ of 1000 points (keyword: linspace)

# <codecell>
n=1000
x=np.linspace(0,5*np.pi,n)

# <markdowncell>

# * create a vector y, so that y=sin(x) with some random noise

# <codecell>
y   = np.sin(x) +np.random.rand(n)-0.5
yth = np.sin(x)

# <markdowncell>

# * plot it like this: ![test](files/plt1.png)

# <codecell>
fig=plt.figure()
ax=plt.gca()
ax.plot(x,y,'b.')
ax.plot(x,yth,'k--',label=r'$y=sin(x)$')

# <markdowncell>

# Try to do a polynomial fit on y(x) with different polynomial degree (Use numpy.polyfit to obtain coefficients)
#
# Plot it like this (use np.poly1d(coef)(x) to plot polynomials) ![test](files/plt2.png)

# <codecell>
for order in xrange(9):
    coeff=np.polyfit(x,y,order)
    ax.plot(x,np.poly1d(coeff)(x),label='deg %d'%order)

# shrink current axis by 20%
box = ax.get_position()
ax.set_position([box.x0, box.y0, box.width * 0.8, box.height])

# Put a legend to the right of the current axis
ax.legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.show()
# <codecell>


"""Testing utilities."""

# Copyright (c) 2011, 2012
# Authors: Pietro Berkes,
#          Andreas Muller
#          Mathieu Blondel
#          Olivier Grisel
#          Arnaud Joly
# License: BSD 3 clause
import inspect
import pkgutil
import warnings

import scipy as sp
from functools import wraps
try:
    # Python 2
    from urllib2 import urlopen
    from urllib2 import HTTPError
except ImportError:
    # Python 3+
    from urllib.request import urlopen
    from urllib.error import HTTPError

import sklearn
from sklearn.base import BaseEstimator
from .fixes import savemat

# Conveniently import all assertions in one place.
from nose.tools import assert_equal
from nose.tools import assert_not_equal
from nose.tools import assert_true
from nose.tools import assert_false
from nose.tools import assert_raises
from nose.tools import raises
from nose import SkipTest
from nose import with_setup

from numpy.testing import assert_almost_equal
from numpy.testing import assert_array_equal
from numpy.testing import assert_array_almost_equal
from numpy.testing import assert_array_less
import numpy as np

from sklearn.base import (ClassifierMixin, RegressorMixin, TransformerMixin,
                          ClusterMixin)

__all__ = ["assert_equal", "assert_not_equal", "assert_raises", "raises",
           "with_setup", "assert_true", "assert_false", "assert_almost_equal",
           "assert_array_equal", "assert_array_almost_equal",
           "assert_array_less"]

try:
    from nose.tools import assert_in, assert_not_in
except ImportError:
    # Nose < 1.0.0

    def assert_in(x, container):
        assert_true(x in container, msg="%r in %r" % (x, container))

    def assert_not_in(x, container):
        assert_false(x in container, msg="%r in %r" % (x, container))

def _assert_less(a, b, msg=None):
    message = "%r is not lower than %r" % (a, b)
    if msg is not None:
        message += ": " + msg
    assert a < b, message

def _assert_greater(a, b, msg=None):
    message = "%r is not greater than %r" % (a, b)
    if msg is not None:
        message += ": " + msg
    assert a > b, message

# To remove when we support numpy 1.7
def assert_warns(warning_class, func, *args, **kw):
    with warnings.catch_warnings(record=True) as w:
        # Cause all warnings to always be triggered.
        warnings.simplefilter("always")

        # Trigger a warning.
        result = func(*args, **kw)

        # Verify some things
        if not len(w) > 0:
            raise AssertionError("No warning raised when calling %s"
                                 % func.__name__)

        if not w[0].category is warning_class:
            raise AssertionError("First warning for %s is not a "
                                 "%s( is %s)"
                                 % (func.__name__, warning_class, w[0]))

    return result

# To remove when we support numpy 1.7
def assert_no_warnings(func, *args, **kw):
    # XXX: once we may depend on python >= 2.6, this can be replaced by the
    # warnings module context manager.
    with warnings.catch_warnings(record=True) as w:
        warnings.simplefilter('always')

        result = func(*args, **kw)
        if len(w) > 0:
            raise AssertionError("Got warnings when calling %s: %s"
                                 % (func.__name__, w))
    return result

def ignore_warnings(fn):
    """Decorator to catch and hide warnings without visual nesting"""
    @wraps(fn)
    def wrapper(*args, **kwargs):
        with warnings.catch_warnings(record=True) as w:
            warnings.simplefilter('always')
            return fn(*args, **kwargs)
            w[:] = []
    return wrapper

try:
    from nose.tools import assert_less
except ImportError:
    assert_less = _assert_less

try:
    from nose.tools import assert_greater
except ImportError:
    assert_greater = _assert_greater

def _assert_allclose(actual, desired, rtol=1e-7, atol=0,
                     err_msg='', verbose=True):
    actual, desired = np.asanyarray(actual), np.asanyarray(desired)
    if np.allclose(actual, desired, rtol=rtol, atol=atol):
        return
    msg = ('Array not equal to tolerance rtol=%g, atol=%g: '
           'actual %s, desired %s') % (rtol, atol, actual, desired)
    raise AssertionError(msg)

if hasattr(np.testing, 'assert_allclose'):
    assert_allclose = np.testing.assert_allclose
else:
    assert_allclose = _assert_allclose

def assert_raise_message(exception, message, function, *args, **kwargs):
    """Helper function to test error messages in exceptions"""

    try:
        function(*args, **kwargs)
        raise AssertionError("Should have raised %r" % exception(message))
    except exception as e:
        error_message = str(e)
        assert_in(message, error_message)

def fake_mldata(columns_dict, dataname, matfile, ordering=None):
    """Create a fake mldata data set.

    Parameters
    ----------
    columns_dict: contains data as
                  columns_dict[column_name] = array of data
    dataname: name of data set
    matfile: file-like object or file name
    ordering: list of column_names, determines the ordering in the data set

    Note: this function transposes all arrays, while fetch_mldata only
    transposes 'data', keep that into account in the tests.
    """
    datasets = dict(columns_dict)

    # transpose all variables
    for name in datasets:
        datasets[name] = datasets[name].T

    if ordering is None:
        ordering = sorted(list(datasets.keys()))
    # NOTE: setting up this array is tricky, because of the way Matlab
    # re-packages 1D arrays
    datasets['mldata_descr_ordering'] = sp.empty((1, len(ordering)),
                                                 dtype='object')
    for i, name in enumerate(ordering):
        datasets['mldata_descr_ordering'][0, i] = name

    savemat(matfile, datasets, oned_as='column')

class mock_mldata_urlopen(object):

    def __init__(self, mock_datasets):
        """Object that mocks the urlopen function to fake requests to mldata.

        `mock_datasets` is a dictionary of {dataset_name: data_dict}, or
        {dataset_name: (data_dict, ordering).
        `data_dict` itself is a dictionary of {column_name: data_array},
        and `ordering` is a list of column_names to determine the ordering
        in the data set (see `fake_mldata` for details).

        When requesting a dataset with a name that is in mock_datasets,
        this object creates a fake dataset in a StringIO object and
        returns it. Otherwise, it raises an HTTPError.
        """
        self.mock_datasets = mock_datasets

    def __call__(self, urlname):
        dataset_name = urlname.split('/')[-1]
        if dataset_name in self.mock_datasets:
            resource_name = '_' + dataset_name
            from io import BytesIO
            matfile = BytesIO()

            dataset = self.mock_datasets[dataset_name]
            ordering = None
            if isinstance(dataset, tuple):
                dataset, ordering = dataset
            fake_mldata(dataset, resource_name, matfile, ordering)

            matfile.seek(0)
            return matfile
        else:
            raise HTTPError(urlname, 404, dataset_name + " is not available",
                            [], None)

def install_mldata_mock(mock_datasets):
    # Lazy import to avoid mutually recursive imports
    from sklearn import datasets
    datasets.mldata.urlopen = mock_mldata_urlopen(mock_datasets)

def uninstall_mldata_mock():
    # Lazy import to avoid mutually recursive imports
    from sklearn import datasets
    datasets.mldata.urlopen = urlopen

# Meta estimators need another estimator to be instantiated.
meta_estimators = ["OneVsOneClassifier",
                   "OutputCodeClassifier", "OneVsRestClassifier", "RFE",
                   "RFECV", "BaseEnsemble"]
# estimators that there is no way to default-construct sensibly
other = ["Pipeline", "FeatureUnion", "GridSearchCV", "RandomizedSearchCV"]

def all_estimators(include_meta_estimators=False, include_other=False,
                   type_filter=None):
    """Get a list of all estimators from sklearn.

    This function crawls the module and gets all classes that inherit
    from BaseEstimator. Classes that are defined in test-modules are not
    included.
    By default meta_estimators such as GridSearchCV are also not included.

    Parameters
    ----------
    include_meta_estimators : boolean, default=False
        Whether to include meta-estimators that can be constructed using
        an estimator as their first argument. These are currently
        BaseEnsemble, OneVsOneClassifier, OutputCodeClassifier,
        OneVsRestClassifier, RFE, RFECV.

    include_others : boolean, default=False
        Wether to include meta-estimators that are somehow special and can
        not be default-constructed sensibly. These are currently
        Pipeline, FeatureUnion and GridSearchCV

    type_filter : string or None, default=None
        Which kind of estimators should be returned. If None, no filter is
        applied and all estimators are returned.  Possible values are
        'classifier', 'regressor', 'cluster' and 'transformer' to get
        estimators only of these specific types.

    Returns
    -------
    estimators : list of tuples
        List of (name, class), where ``name`` is the class name as string
        and ``class`` is the actuall type of the class.
    """
    def is_abstract(c):
        if not(hasattr(c, '__abstractmethods__')):
            return False
        if not len(c.__abstractmethods__):
            return False
        return True

    all_classes = []
    # get parent folder
    path = sklearn.__path__
    for importer, modname, ispkg in pkgutil.walk_packages(
            path=path, prefix='sklearn.', onerror=lambda x: None):
        module = __import__(modname, fromlist="dummy")
        if ".tests." in modname:
            continue
        classes = inspect.getmembers(module, inspect.isclass)
        all_classes.extend(classes)

    all_classes = set(all_classes)

    estimators = [c for c in all_classes
                  if (issubclass(c[1], BaseEstimator)
                      and c[0] != 'BaseEstimator')]
    # get rid of abstract base classes
    estimators = [c for c in estimators if not is_abstract(c[1])]

    if not include_other:
        estimators = [c for c in estimators if not c[0] in other]
    # possibly get rid of meta estimators
    if not include_meta_estimators:
        estimators = [c for c in estimators if not c[0] in meta_estimators]

    if type_filter == 'classifier':
        estimators = [est for est in estimators
                      if issubclass(est[1], ClassifierMixin)]
    elif type_filter == 'regressor':
        estimators = [est for est in estimators
                      if issubclass(est[1], RegressorMixin)]
    elif type_filter == 'transformer':
        estimators = [est for est in estimators
                      if issubclass(est[1], TransformerMixin)]
    elif type_filter == 'cluster':
        estimators = [est for est in estimators
                      if issubclass(est[1], ClusterMixin)]
    elif type_filter is not None:
        raise ValueError("Parameter type_filter must be 'classifier', "
                         "'regressor', 'transformer', 'cluster' or None, got"
                         " %s." % repr(type_filter))

    # We sort in order to have reproducible test failures
    return sorted(estimators)

def set_random_state(estimator, random_state=0):
    if "random_state" in estimator.get_params().keys():
        estimator.set_params(random_state=random_state)

def if_matplotlib(func):
    """Test decorator that skips test if matplotlib not installed. """

    @wraps(func)
    def run_test(*args, **kwargs):
        try:
            import matplotlib
            matplotlib.use('Agg', warn=False)
            # this fails if no $DISPLAY specified
            matplotlib.pylab.figure()
        except:
            raise SkipTest('Matplotlib not available.')
        else:
            return func(*args, **kwargs)
    return run_test


# Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
#         Gael Varoquaux <gael.varoquaux@normalesup.org>
#         Virgile Fritsch <virgile.fritsch@inria.fr>
#
# License: BSD 3 clause

import numpy as np

from sklearn.utils.testing import assert_almost_equal
from sklearn.utils.testing import assert_array_almost_equal
from sklearn.utils.testing import assert_raises
from sklearn.utils.validation import NotFittedError

from sklearn import datasets
from sklearn.covariance import empirical_covariance, MinCovDet, \
    EllipticEnvelope

X = datasets.load_iris().data
X_1d = X[:, 0]
n_samples, n_features = X.shape

def test_mcd():
    # Tests the FastMCD algorithm implementation
    # Small data set
    # test without outliers (random independent normal data)
    launch_mcd_on_dataset(100, 5, 0, 0.01, 0.1, 80)
    # test with a contaminated data set (medium contamination)
    launch_mcd_on_dataset(100, 5, 20, 0.01, 0.01, 70)
    # test with a contaminated data set (strong contamination)
    launch_mcd_on_dataset(100, 5, 40, 0.1, 0.1, 50)

    # Medium data set
    launch_mcd_on_dataset(1000, 5, 450, 0.1, 0.1, 540)

    # Large data set
    launch_mcd_on_dataset(1700, 5, 800, 0.1, 0.1, 870)

    # 1D data set
    launch_mcd_on_dataset(500, 1, 100, 0.001, 0.001, 350)

def launch_mcd_on_dataset(n_samples, n_features, n_outliers, tol_loc, tol_cov,
                          tol_support):

    rand_gen = np.random.RandomState(0)
    data = rand_gen.randn(n_samples, n_features)
    # add some outliers
    outliers_index = rand_gen.permutation(n_samples)[:n_outliers]
    outliers_offset = 10. * \
        (rand_gen.randint(2, size=(n_outliers, n_features)) - 0.5)
    data[outliers_index] += outliers_offset
    inliers_mask = np.ones(n_samples).astype(bool)
    inliers_mask[outliers_index] = False

    pure_data = data[inliers_mask]
    # compute MCD by fitting an object
    mcd_fit = MinCovDet(random_state=rand_gen).fit(data)
    T = mcd_fit.location_
    S = mcd_fit.covariance_
    H = mcd_fit.support_
    # compare with the estimates learnt from the inliers
    error_location = np.mean((pure_data.mean(0) - T) ** 2)
    assert(error_location < tol_loc)
    error_cov = np.mean((empirical_covariance(pure_data) - S) ** 2)
    assert(error_cov < tol_cov)
    assert(np.sum(H) >= tol_support)
    assert_array_almost_equal(mcd_fit.mahalanobis(data), mcd_fit.dist_)

def test_mcd_issue1127():
    # Check that the code does not break with X.shape = (3, 1)
    # (i.e. n_support = n_samples)
    rnd = np.random.RandomState(0)
    X = rnd.normal(size=(3, 1))
    mcd = MinCovDet()
    mcd.fit(X)

def test_outlier_detection():
    rnd = np.random.RandomState(0)
    X = rnd.randn(100, 10)
    clf = EllipticEnvelope(contamination=0.1)
    assert_raises(NotFittedError, clf.predict, X)
    assert_raises(NotFittedError, clf.decision_function, X)
    clf.fit(X)
    y_pred = clf.predict(X)
    decision = clf.decision_function(X, raw_values=True)
    decision_transformed = clf.decision_function(X, raw_values=False)

    assert_array_almost_equal(
        decision, clf.mahalanobis(X))
    assert_array_almost_equal(clf.mahalanobis(X), clf.dist_)
    assert_almost_equal(clf.score(X, np.ones(100)),
                        (100 - y_pred[y_pred == -1].size) / 100.)
    assert(sum(y_pred == -1) == sum(decision_transformed < 0))