Merge pull request scipy#5784 from nmayorov/ls_tol

MAINT: Change default tolerances of least_squares to 1e-8

doc/source/tutorial/optimize.rst

@@ -458,9 +458,9 @@ finally plots the original data and the fitted model function:
     >>> x0 = np.array([2.5, 3.9, 4.15, 3.9])
     >>> res = least_squares(fun, x0, jac=jac, bounds=(0, 100), args=(u, y), verbose=1)
     `ftol` termination condition is satisfied.
-    Function evaluations 129, initial cost 4.4383e+00, final cost 1.5375e-04, first-order optimality 1.41e-05.
+    Function evaluations 130, initial cost 4.4383e+00, final cost 1.5375e-04, first-order optimality 4.92e-08.
     >>> res.x
-    array([ 0.19280849,  0.19124746,  0.12304994,  0.13604619])
+    array([ 0.19280596,  0.19130423,  0.12306063,  0.13607247])
 
     >>> import matplotlib.pyplot as plt
     >>> u_test = np.linspace(0, 5)

scipy/optimize/_lsq/least_squares.py

@@ -232,10 +232,9 @@ def loss_function(f, cost_only=False):
 
 def least_squares(
         fun, x0, jac='2-point', bounds=(-np.inf, np.inf), method='trf',
-        ftol=EPS**0.5, xtol=EPS**0.5, gtol=EPS**0.5, x_scale=1.0,
-        loss='linear', f_scale=1.0, diff_step=None, tr_solver=None,
-        tr_options={}, jac_sparsity=None, max_nfev=None, verbose=0, args=(),
-        kwargs={}):
+        ftol=1e-8, xtol=1e-8, gtol=1e-8, x_scale=1.0, loss='linear',
+        f_scale=1.0, diff_step=None, tr_solver=None, tr_options={},
+        jac_sparsity=None, max_nfev=None, verbose=0, args=(), kwargs={}):
     """Solve a nonlinear least-squares problem with bounds on the variables.
 
     Given the residuals f(x) (an m-dimensional function of n variables) and
@@ -292,24 +291,22 @@ def least_squares(
 
         Default is 'trf'. See Notes for more information.
     ftol : float, optional
-        Tolerance for termination by the change of the cost function.
-        Default is the square root of machine epsilon. The optimization process
-        is stopped when ``dF < ftol * F``, and there was an adequate agreement
-        between a local quadratic model and the true model in the last step.
+        Tolerance for termination by the change of the cost function. Default
+        is 1e-8. The optimization process is stopped when  ``dF < ftol * F``,
+        and there was an adequate agreement between a local quadratic model and
+        the true model in the last step.
     xtol : float, optional
         Tolerance for termination by the change of the independent variables.
-        Default is the square root of machine epsilon. The exact condition
-        checked depends on the `method` used:
+        Default is 1e-8. The exact condition depends on the `method` used:
 
             * For 'trf' and 'dogbox' : ``norm(dx) < xtol * (xtol + norm(x))``
             * For 'lm' : ``Delta < xtol * norm(xs)``, where ``Delta`` is
               a trust-region radius and ``xs`` is the value of ``x``
               scaled according to `x_scale` parameter (see below).
 
     gtol : float, optional
-        Tolerance for termination by the norm of the gradient. Default is
-        the square root of machine epsilon. The exact condition depends
-        on a `method` used:
+        Tolerance for termination by the norm of the gradient. Default is 1e-8.
+        The exact condition depends on a `method` used:
 
             * For 'trf' : ``norm(g_scaled, ord=np.inf) < gtol``, where
               ``g_scaled`` is the value of the gradient scaled to account for
@@ -578,9 +575,9 @@ def least_squares(
     >>> res_1.x
     array([ 1.,  1.])
     >>> res_1.cost
-    2.4651903288156619e-30
+    9.8669242910846867e-30
     >>> res_1.optimality
-    4.4408921315878507e-14
+    8.8928864934219529e-14
 
     We now constrain the variables, in such a way that the previous solution
     becomes infeasible. Specifically, we require that ``x[1] >= 1.5``, and
@@ -636,7 +633,7 @@ def least_squares(
     >>> res_3 = least_squares(fun_broyden, x0_broyden,
     ...                       jac_sparsity=sparsity_broyden(n))
     >>> res_3.cost
-    4.5687161966109073e-23
+    4.5687069299604613e-23
     >>> res_3.optimality
     1.1650454296851518e-11