Fixed spurious "Max iterations exceeded" errors in the stop root-finding problem

The Newton solve in `_calc_rayfunction_stops_only` used the default
perturbation of `na.jacobian`, an absolute 1e-10 in the units of the
variable, which is below the floating-point noise floor of the raytrace
for variables measured in physical units, yielding a Jacobian made of
noise and wild iterates. The default convergence tolerance of
`na.optimize.root_newton` (1e-10 in the units of the residual) had the
same scale-dependence problem: solves whose residuals had converged to
machine precision were reported as "Max iterations exceeded". Both are
now proportional to the size of the target aperture wire.

Failures that remain are re-raised with the names of the stop surfaces
involved and a hint about marking the object surface as the field stop
for dispersive systems, instead of an anonymous "Max iterations
exceeded" from deep inside named_arrays.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

Author: jacobdparker

optika/systems/_sequential.py

@@ -316,24 +316,70 @@ def zfunc(xy: na.AbstractCartesian3dVectorArray):
             else:
                 raise ValueError(f"unrecognized output grid unit, {na.unit(grid_last)}")
 
+            # The residual of the root-finding problem has the same units as
+            # the target grid, so the convergence tolerance must scale with
+            # the size of the target aperture to be achievable in floating
+            # point for systems of any physical scale.
+            scale = np.maximum(
+                np.abs(grid_last.x).max(),
+                np.abs(grid_last.y).max(),
+            )
+            max_abs_error = 1e-9 * np.maximum(
+                scale,
+                1 * na.unit_normalized(scale),
+            )
+
             variables = getattr(result.outputs, component_variable)
 
-            root = na.optimize.root_newton(
-                function=functools.partial(
-                    self._ray_error,
-                    rays=result.outputs,
-                    subsystem=subsystem,
-                    grid_last=grid_last,
-                    component_variable=component_variable,
-                    component_target=component_target,
-                    zfunc=zfunc,
-                ),
-                guess=na.Cartesian2dVectorArray(
-                    x=variables.x,
-                    y=variables.y,
-                ),
+            function = functools.partial(
+                self._ray_error,
+                rays=result.outputs,
+                subsystem=subsystem,
+                grid_last=grid_last,
+                component_variable=component_variable,
+                component_target=component_target,
+                zfunc=zfunc,
             )
 
+            # The default perturbation used by `na.jacobian` is an absolute
+            # 1e-10 in the units of the variable, which is below the
+            # floating-point noise floor of the raytrace for variables
+            # measured in physical units, yielding a Jacobian made of noise.
+            # Use a perturbation proportional to the scale of the problem
+            # instead.
+            if component_variable == "direction":
+                dx = 1e-6
+            else:
+                dx = 1e-6 * np.maximum(
+                    scale,
+                    1 * na.unit_normalized(scale),
+                )
+
+            def jacobian(x, _function=function, _dx=dx):
+                return na.jacobian(function=_function, x=x, dx=_dx)
+
+            try:
+                root = na.optimize.root_newton(
+                    function=function,
+                    guess=na.Cartesian2dVectorArray(
+                        x=variables.x,
+                        y=variables.y,
+                    ),
+                    jacobian=jacobian,
+                    max_abs_error=max_abs_error,
+                )
+            except ValueError as e:
+                raise ValueError(
+                    f"Could not solve for the rays connecting the stop "
+                    f"surfaces {surface_first.name!r} and "
+                    f"{surface_last.name!r}. "
+                    f"If the field stop is only partially reachable at a "
+                    f"single wavelength (e.g. a spectrograph sensor), "
+                    f"consider marking the object surface, with an angular "
+                    f"(dimensionless, sine of the half-angle) aperture, as "
+                    f"the field stop instead."
+                ) from e
+
             variables.x = root.x
             variables.y = root.y
             variables.z = zfunc(root)