Add always_step option to Newton solver

From updated README.md:

If the optional parameter `always_step = false` (the default), no Newton iteration is performed
if the solution is already converged for the supplied initial `x` values,  and the supplied initial `x` is returned
in the `zero` field of `SolverResults`. If `always_step = true`, a Newton step is performed
before testing for convergence.

README.md

@@ -307,12 +307,17 @@ By default, no linesearch is performed.
 **Note:** it is assumed that a passed linesearch function will at least update the solution
 vector and evaluate the function at the new point.
 
-If `method = :newton` and `linesearch=LineSearches.Static()` (the default), an additional parameter
+If `linesearch=LineSearches.Static()` (the default), an additional parameter
 `apply_step!` (default value `(x, x_old, newton_step)->(x .= x_old .+ newton_step)`) can be used to
 define a problem-specific function to update the `x` value after the Newton iteration. 
 For example, `apply_step! = (x, x_old, newton_step)->(x .= x_old .+ newton_step; x .= max.(x, 1e-80))` 
 will enforce a constraint `x[i] >= 1e-80`.
 
+If the optional parameter `always_step = false` (the default), no Newton iteration is performed
+if the solution is already converged for the supplied initial `x` values,  and the supplied initial `x` is returned
+in the `zero` field of `SolverResults`. If `always_step = true`, a Newton step is performed 
+before testing for convergence.
+
 ## Anderson acceleration
 
 This method is selected with `method = :anderson`.

src/nlsolve/nlsolve.jl

@@ -10,6 +10,7 @@ function nlsolve(df::Union{NonDifferentiable, OnceDifferentiable},
                  linesearch = LineSearches.Static(),
                  linsolve=(x, A, b) -> copyto!(x, A\b),
                  apply_step! = (x, x_old, newton_step)->(x .= x_old .+ newton_step),
+                 always_step = false,
                  factor::Real = one(real(eltype(initial_x))),
                  autoscale::Bool = true,
                  m::Integer = 10,
@@ -22,7 +23,7 @@ function nlsolve(df::Union{NonDifferentiable, OnceDifferentiable},
     end
     if method == :newton
         newton(df, initial_x, xtol, ftol, iterations,
-               store_trace, show_trace, extended_trace, linesearch; linsolve=linsolve, apply_step! =apply_step!)
+               store_trace, show_trace, extended_trace, linesearch; linsolve=linsolve, apply_step! =apply_step!, always_step=always_step)
     elseif method == :trust_region
         trust_region(df, initial_x, xtol, ftol, iterations,
                      store_trace, show_trace, extended_trace, factor,

src/solvers/newton.jl

@@ -43,6 +43,7 @@ function newton_(df::OnceDifferentiable,
                  linesearch,
                  linsolve,
                  apply_step!,
+                 always_step,
                  cache = NewtonCache(df)) where T
     n = length(initial_x)
     copyto!(cache.x, initial_x)
@@ -53,7 +54,12 @@ function newton_(df::OnceDifferentiable,
     x_converged, f_converged = assess_convergence(initial_x, cache.xold, value(df), NaN, ftol)
     stopped = any(isnan, cache.x) || any(isnan, value(df)) ? true : false
 
-    converged = x_converged || f_converged
+    if always_step
+        converged = false
+    else
+        converged = x_converged || f_converged
+    end
+
     x_ls = copy(cache.x)
     tr = SolverTrace()
     tracing = store_trace || show_trace || extended_trace
@@ -145,6 +151,7 @@ function newton(df::OnceDifferentiable,
                 linesearch,
                 cache = NewtonCache(df);
                 linsolve=(x, A, b) -> copyto!(x, A\b),
-                apply_step! = (x, x_old, newton_step)->(x .= x_old .+ newton_step)) where T
-    newton_(df, initial_x, convert(real(T), xtol), convert(real(T), ftol), iterations, store_trace, show_trace, extended_trace, linesearch, linsolve, apply_step!, cache)
+                apply_step! = (x, x_old, newton_step)->(x .= x_old .+ newton_step),
+                always_step::Bool) where T
+    newton_(df, initial_x, convert(real(T), xtol), convert(real(T), ftol), iterations, store_trace, show_trace, extended_trace, linesearch, linsolve, apply_step!, always_step, cache)
 end