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Least squares fit of Laguerre series to data.

Return the coefficients of a Laguerre series of degree `deg` that is the
least squares fit to the data values `y` given at points `x`. If `y` is
1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
fits are done, one for each column of `y`, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form

.. math::  p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),(more...)

        def lagfit(x, y, deg, rcond=None, full=False, w=None):
    """
    Least squares fit of Laguerre series to data.

    Return the coefficients of a Laguerre series of degree `deg` that is the
    least squares fit to the data values `y` given at points `x`. If `y` is
    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
    fits are done, one for each column of `y`, and the resulting
    coefficients are stored in the corresponding columns of a 2-D return.
    The fitted polynomial(s) are in the form

    .. math::  p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),

    where `n` is `deg`.

    Since numpy version 1.7.0, lagfit also supports NA. If any of the
    elements of `x`, `y`, or `w` are NA, then the corresponding rows of the
    linear least squares problem (see Notes) are set to 0. If `y` is 2-D,
    then an NA in any row of `y` invalidates that whole row.

    Parameters
    ----------
    x : array_like, shape (M,)
        x-coordinates of the M sample points ``(x[i], y[i])``.
    y : array_like, shape (M,) or (M, K)
        y-coordinates of the sample points. Several data sets of sample
        points sharing the same x-coordinates can be fitted at once by
        passing in a 2D-array that contains one dataset per column.
    deg : int
        Degree of the fitting polynomial
    rcond : float, optional
        Relative condition number of the fit. Singular values smaller than
        this relative to the largest singular value will be ignored. The
        default value is len(x)*eps, where eps is the relative precision of
        the float type, about 2e-16 in most cases.
    full : bool, optional
        Switch determining nature of return value. When it is False (the
        default) just the coefficients are returned, when True diagnostic
        information from the singular value decomposition is also returned.
    w : array_like, shape (`M`,), optional
        Weights. If not None, the contribution of each point
        ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
        weights are chosen so that the errors of the products ``w[i]*y[i]``
        all have the same variance.  The default value is None.

    Returns
    -------
    coef : ndarray, shape (M,) or (M, K)
        Laguerre coefficients ordered from low to high. If `y` was 2-D,
        the coefficients for the data in column k  of `y` are in column
        `k`.

    [residuals, rank, singular_values, rcond] : present when `full` = True
        Residuals of the least-squares fit, the effective rank of the
        scaled Vandermonde matrix and its singular values, and the
        specified value of `rcond`. For more details, see `linalg.lstsq`.

    Warns
    -----
    RankWarning
        The rank of the coefficient matrix in the least-squares fit is
        deficient. The warning is only raised if `full` = False.  The
        warnings can be turned off by

        >>> import warnings
        >>> warnings.simplefilter('ignore', RankWarning)

    See Also
    --------
    chebfit, legfit, polyfit, hermfit, hermefit
    lagval : Evaluates a Laguerre series.
    lagvander : pseudo Vandermonde matrix of Laguerre series.
    lagweight : Laguerre weight function.
    linalg.lstsq : Computes a least-squares fit from the matrix.
    scipy.interpolate.UnivariateSpline : Computes spline fits.

    Notes
    -----
    The solution is the coefficients of the Laguerre series `p` that
    minimizes the sum of the weighted squared errors

    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,

    where the :math:`w_j` are the weights. This problem is solved by
    setting up as the (typically) overdetermined matrix equation

    .. math:: V(x) * c = w * y,

    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
    coefficients to be solved for, `w` are the weights, and `y` are the
    observed values.  This equation is then solved using the singular value
    decomposition of `V`.

    If some of the singular values of `V` are so small that they are
    neglected, then a `RankWarning` will be issued. This means that the
    coefficient values may be poorly determined. Using a lower order fit
    will usually get rid of the warning.  The `rcond` parameter can also be
    set to a value smaller than its default, but the resulting fit may be
    spurious and have large contributions from roundoff error.

    Fits using Laguerre series are probably most useful when the data can
    be approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Laguerre
    weight. In that case the weight ``sqrt(w(x[i])`` should be used
    together with data values ``y[i]/sqrt(w(x[i])``. The weight function is
    available as `lagweight`.

    References
    ----------
    .. [1] Wikipedia, "Curve fitting",
           http://en.wikipedia.org/wiki/Curve_fitting

    Examples
    --------
    >>> from numpy.polynomial.laguerre import lagfit, lagval
    >>> x = np.linspace(0, 10)
    >>> err = np.random.randn(len(x))/10
    >>> y = lagval(x, [1, 2, 3]) + err
    >>> lagfit(x, y, 2)
    array([ 0.96971004,  2.00193749,  3.00288744])

    """
    order = int(deg) + 1
    x = np.asarray(x) + 0.0
    y = np.asarray(y) + 0.0

    # check arguments.
    if deg < 0 :
        raise ValueError("expected deg >= 0")
    if x.ndim != 1:
        raise TypeError("expected 1D vector for x")
    if x.size == 0:
        raise TypeError("expected non-empty vector for x")
    if y.ndim < 1 or y.ndim > 2 :
        raise TypeError("expected 1D or 2D array for y")
    if len(x) != len(y):
        raise TypeError("expected x and y to have same length")

    # set up the least squares matrices in transposed form
    lhs = lagvander(x, deg).T
    rhs = y.T
    if w is not None:
        w = np.asarray(w) + 0.0
        if w.ndim != 1:
            raise TypeError("expected 1D vector for w")
        if len(x) != len(w):
            raise TypeError("expected x and w to have same length")
        # apply weights. Don't use inplace operations as they
        # can cause problems with NA.
        lhs = lhs * w
        rhs = rhs * w

    # set rcond
    if rcond is None :
        rcond = len(x)*np.finfo(x.dtype).eps

    # Determine the norms of the design matrix columns.
    if issubclass(lhs.dtype.type, np.complexfloating):
        scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1))
    else:
        scl = np.sqrt(np.square(lhs).sum(1))
    scl[scl == 0] = 1

    # Solve the least squares problem.
    c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond)
    c = (c.T/scl).T

    # warn on rank reduction
    if rank != order and not full:
        msg = "The fit may be poorly conditioned"
        warnings.warn(msg, pu.RankWarning)

    if full :
        return c, [resids, rank, s, rcond]
    else :
        return c
        


src/n/u/nupic-linux64-HEAD/lib64/python2.6/site-packages/numpy/polynomial/tests/test_laguerre.py   nupic-linux64(Download)
        y = f(x)
        #
        coef3 = lag.lagfit(x, y, 3)
        assert_equal(len(coef3), 4)
        assert_almost_equal(lag.lagval(x, coef3), y)
        #
        coef4 = lag.lagfit(x, y, 4)
        assert_almost_equal(lag.lagval(x, coef4), y)
        #
        coef2d = lag.lagfit(x, np.array([y,y]).T, 3)
        assert_almost_equal(coef2d, np.array([coef3,coef3]).T)
        # test weighting
        w = np.zeros_like(x)
        yw = y.copy()
        w[1::2] = 1
        y[0::2] = 0
        wcoef3 = lag.lagfit(x, yw, 3, w=w)
        assert_almost_equal(wcoef3, coef3)
        #
        wcoef2d = lag.lagfit(x, np.array([yw,yw]).T, 3, w=w)
        assert_almost_equal(wcoef2d, np.array([coef3,coef3]).T)
        # test scaling with complex values x points whose square

src/n/u/numpy-1.8.1/numpy/polynomial/tests/test_laguerre.py   numpy(Download)
        y = f(x)
        #
        coef3 = lag.lagfit(x, y, 3)
        assert_equal(len(coef3), 4)
        assert_almost_equal(lag.lagval(x, coef3), y)
        #
        coef4 = lag.lagfit(x, y, 4)
        assert_almost_equal(lag.lagval(x, coef4), y)
        #
        coef2d = lag.lagfit(x, np.array([y, y]).T, 3)
        assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
        # test weighting
        w = np.zeros_like(x)
        yw = y.copy()
        w[1::2] = 1
        y[0::2] = 0
        wcoef3 = lag.lagfit(x, yw, 3, w=w)
        assert_almost_equal(wcoef3, coef3)
        #
        wcoef2d = lag.lagfit(x, np.array([yw, yw]).T, 3, w=w)
        assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
        # test scaling with complex values x points whose square