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Module « numpy.linalg »

Fonction lstsq - module numpy.linalg

Signature de la fonction lstsq

def lstsq(a, b, rcond=None) 

Description

help(numpy.linalg.lstsq)

Return the least-squares solution to a linear matrix equation.

Computes the vector `x` that approximately solves the equation
``a @ x = b``. The equation may be under-, well-, or over-determined
(i.e., the number of linearly independent rows of `a` can be less than,
equal to, or greater than its number of linearly independent columns).
If `a` is square and of full rank, then `x` (but for round-off error)
is the "exact" solution of the equation. Else, `x` minimizes the
Euclidean 2-norm :math:`||b - ax||`. If there are multiple minimizing
solutions, the one with the smallest 2-norm :math:`||x||` is returned.

Parameters
----------
a : (M, N) array_like
    "Coefficient" matrix.
b : {(M,), (M, K)} array_like
    Ordinate or "dependent variable" values. If `b` is two-dimensional,
    the least-squares solution is calculated for each of the `K` columns
    of `b`.
rcond : float, optional
    Cut-off ratio for small singular values of `a`.
    For the purposes of rank determination, singular values are treated
    as zero if they are smaller than `rcond` times the largest singular
    value of `a`.
    The default uses the machine precision times ``max(M, N)``.  Passing
    ``-1`` will use machine precision.

    .. versionchanged:: 2.0
        Previously, the default was ``-1``, but a warning was given that
        this would change.

Returns
-------
x : {(N,), (N, K)} ndarray
    Least-squares solution. If `b` is two-dimensional,
    the solutions are in the `K` columns of `x`.
residuals : {(1,), (K,), (0,)} ndarray
    Sums of squared residuals: Squared Euclidean 2-norm for each column in
    ``b - a @ x``.
    If the rank of `a` is < N or M <= N, this is an empty array.
    If `b` is 1-dimensional, this is a (1,) shape array.
    Otherwise the shape is (K,).
rank : int
    Rank of matrix `a`.
s : (min(M, N),) ndarray
    Singular values of `a`.

Raises
------
LinAlgError
    If computation does not converge.

See Also
--------
scipy.linalg.lstsq : Similar function in SciPy.

Notes
-----
If `b` is a matrix, then all array results are returned as matrices.

Examples
--------
Fit a line, ``y = mx + c``, through some noisy data-points:

>>> import numpy as np
>>> x = np.array([0, 1, 2, 3])
>>> y = np.array([-1, 0.2, 0.9, 2.1])

By examining the coefficients, we see that the line should have a
gradient of roughly 1 and cut the y-axis at, more or less, -1.

We can rewrite the line equation as ``y = Ap``, where ``A = [[x 1]]``
and ``p = [[m], [c]]``.  Now use `lstsq` to solve for `p`:

>>> A = np.vstack([x, np.ones(len(x))]).T
>>> A
array([[ 0.,  1.],
       [ 1.,  1.],
       [ 2.,  1.],
       [ 3.,  1.]])

>>> m, c = np.linalg.lstsq(A, y)[0]
>>> m, c
(1.0 -0.95) # may vary

Plot the data along with the fitted line:

>>> import matplotlib.pyplot as plt
>>> _ = plt.plot(x, y, 'o', label='Original data', markersize=10)
>>> _ = plt.plot(x, m*x + c, 'r', label='Fitted line')
>>> _ = plt.legend()
>>> plt.show()

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