mxnet.np.linalg.lstsq

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

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

Solves the equation \(a x = b\) by computing a vector x that minimizes the squared Euclidean 2-norm \(\| b - a x \|^2_2\). 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.

Parameters

  • a ((M, N) ndarray) – “Coefficient” matrix.

  • b ({(M,), (M, K)} ndarray) – 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 of warn or -1 will use the machine precision as rcond parameter. The default of None will use the machine precision times max(M, N).

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 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

MXNetError – If computation does not converge.

Notes

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

Examples

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