linalg.lstsq
Description
Return the least-squares solution to a linear matrix equation.
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. Otherwise, x minimizes the Euclidean 2-norm. If there are multiple minimizing solutions, the one with the smallest 2-norm is returned.
Mandatory Input Parameters
Parameter |
Type |
Description |
|---|---|---|
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. |
Optional Input Parameters
Parameter |
Type |
Default Value |
Description |
|---|---|---|---|
rcond |
float |
None |
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. |
Return Value
Parameter |
Type |
Description |
|---|---|---|
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 value of a |
Examples
>>> import numpy as np
>>> # y = mx + c
>>> x = np.array([0,1,2,3])
>>> y = np.array([-1,0.2,0.9,2.1])
>>>
>>> # Write the line equation as y = Ap, where A = [[x 1]] and p = [[m], [c]].
>>> 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
(1.0000000000000002, -0.9499999999999996)
>>>