# scipy.linalg.lstsq

scipy.linalg.lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False, check_finite=True, lapack_driver=None)

  Compute least-squares solution to equation Ax = b.

  Compute a vector x such that the 2-norm `|b - A x|` is minimized.

  Parameters:
    a(M, N) array_like

      Left-hand side array

    b(M,) or (M, K) array_like

      Right hand side array

    cond float, optional

      Cutoff for ‘small’ singular values; used to determine effective rank of a. Singular values smaller than `cond * largest_singular_value` are considered zero.

    overwrite_a bool, optional

      Discard data in a (may enhance performance). Default is False.

    overwrite_b bool, optional

      Discard data in b (may enhance performance). Default is False.

    check_finite bool, optional

      Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

    lapack_driver str, optional

      Which LAPACK driver is used to solve the least-squares problem. Options are `'gelsd'`, `'gelsy'`, `'gelss'`. Default (`'gelsd'`) is a good choice. However, `'gelsy'` can be slightly faster on many problems. `'gelss'` was used historically. It is generally slow but uses less memory.

      New in version 0.17.0.

  Returns:
    x(N,) or (N, K) ndarray

      Least-squares solution.

    residues(K,) ndarray or float

      Square of the 2-norm for each column in `b - a x`, if `M > N` and `ndim(A) == n` (returns a scalar if `b` is 1-D). Otherwise a (0,)-shaped array is returned.

    rank int

      Effective rank of a.

    s(min(M, N),) ndarray or None

      Singular values of a. The condition number of `a` is `s[0] / s[-1]`.

  Raises:
    LinAlgError

      If computation does not converge.

    ValueError

      When parameters are not compatible.

  See also

  `scipy.optimize.nnls`

    linear least squares with non-negativity constraint

  Notes

  When `'gelsy'` is used as a driver, residues is set to a (0,)-shaped array and s is always `None`.

  Examples

      >>> import numpy as np
      >>> from scipy.linalg import lstsq
      >>> import matplotlib.pyplot as plt

  Suppose we have the following data:

      >>> x = np.array([1, 2.5, 3.5, 4, 5, 7, 8.5])
      >>> y = np.array([0.3, 1.1, 1.5, 2.0, 3.2, 6.6, 8.6])

  We want to fit a quadratic polynomial of the form `y = a + b*x**2` to this data. We first form the “design matrix” M, with a constant column of 1s and a column containing `x**2`:

      >>> M = x[:, np.newaxis]**[0, 2]
      >>> M
      array([[  1.  ,   1.  ],
             [  1.  ,   6.25],
             [  1.  ,  12.25],
             [  1.  ,  16.  ],
             [  1.  ,  25.  ],
             [  1.  ,  49.  ],
             [  1.  ,  72.25]])

  We want to find the least-squares solution to `M.dot(p) = y`, where `p` is a vector with length 2 that holds the parameters `a` and `b`.

      >>> p, res, rnk, s = lstsq(M, y)
      >>> p
      array([ 0.20925829,  0.12013861])

  Plot the data and the fitted curve.

      >>> plt.plot(x, y, 'o', label='data')
      >>> xx = np.linspace(0, 9, 101)
      >>> yy = p[0] + p[1]*xx**2
      >>> plt.plot(xx, yy, label='least squares fit, $y = a + bx^2$')
      >>> plt.xlabel('x')
      >>> plt.ylabel('y')
      >>> plt.legend(framealpha=1, shadow=True)
      >>> plt.grid(alpha=0.25)
      >>> plt.show()