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lstsq(a, b, rcond='warn')

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 $||b - ax||$ . If there are multiple minimizing solutions, the one with the smallest 2-norm $||x||$ is returned.

Notes

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

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 :None:None:`rcond` times the largest singular value of a.

versionchanged

If not set, a FutureWarning is given. The previous default of -1 will use the machine precision as :None:None:`rcond` parameter, the new default will use the machine precision times :None:None:`max(M, N)`. To silence the warning and use the new default, use rcond=None , to keep using the old behavior, use rcond=-1 .

Raises

LinAlgError: If computation does not converge.

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.

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

Examples

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

>>> 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 :None:None:`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, rcond=None)[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()

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

numpy.polynomial.polynomial.polyfit numpy.ma.extras.polyfit numpy.polynomial.hermite_e.hermefit numpy.linalg.qr numpy.polyfit numpy.polynomial.chebyshev.chebfit numpy.linalg.solve numpy.polynomial.hermite.hermfit numpy.polynomial.legendre.legfit numpy.polynomial.laguerre.lagfit numpy.linalg.lstsq

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GitHub : /numpy/linalg/linalg.py#2157
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