Docs, scaling factor

Author: DaniGlez

lib/SimpleNonlinearSolve/src/bracketing/itp.jl

@@ -15,16 +15,22 @@ I. F. D. Oliveira and R. H. C. Takahashi.
 The following keyword parameters are accepted.
 
   - `n₀::Int = 10`, the 'slack'. Must not be negative. When n₀ = 0 the worst-case is
-    identical to that of bisection, but increacing n₀ provides greater oppotunity for
+    identical to that of bisection, but increasing n₀ provides greater opportunity for
     superlinearity.
-  - `scaled_κ₁::Float64 = 0.2`. Must not be negative. The recomended value is `0.2`.
+  - `scaled_κ₁::Float64 = 0.2`. Must not be negative. The recommended value is `0.2`.
     Lower values produce tighter asymptotic behaviour, while higher values improve the
     steady-state behaviour when truncation is not helpful.
   - `κ₂::Real = 2`. Must lie in [1, 1+ϕ ≈ 2.62). Higher values allow for a greater
     convergence rate, but also make the method more succeptable to worst-case performance.
-    In practice, κ=1, 2 seems to work well due to the computational simplicity, as κ₂ is
+    In practice, κ₂=1, 2 seems to work well due to the computational simplicity, as κ₂ is
     used as an exponent in the method.
 
+### Computation of κ₁
+
+In the current implementation, we compute κ₁ = scaled_κ₁·|Δx₀|^(1 - κ₂); this allows κ₁ to
+adapt to the dimension of the problem in order to keep the proposed initial step
+proportional to Δx₀.
+
 ### Worst Case Performance
 
 n½ + `n₀` iterations, where n½ is the number of iterations using bisection
@@ -72,8 +78,8 @@ function SciMLBase.solve(prob::IntervalNonlinearProblem, alg::ITP, args...;
     end
     ϵ = abstol
     #defining variables/cache
-    k1 = alg.scaled_k1 / abs(right - left)
     k2 = alg.k2
+    k1 = alg.scaled_k1 * abs(right - left)^(1 - k2)
     n0 = alg.n0
     n_h = ceil(log2(abs(right - left) / (2 * ϵ)))
     mid = (left + right) / 2