Lay out the whole Newton grid, not one corner of it (#115)

Sources/AngouriMath/Functions/Continuous/Solvers/NumericalSolving/NewtonSolver.cs

@@ -66,11 +66,20 @@ NumericsComplex ChooseGood() =>
             var res = new HashSet<Complex>();
             var df = WithoutConditions(expr.Differentiate(v).Simplify()).Compile(v);
             var f = WithoutConditions(expr.Simplify()).Compile(v);
+            // The shares are worked out by CtxDivide, which rounds to the working precision,
+            // rather than by EDecimal's own operator, which has no context and answers NaN
+            // whenever the quotient does not terminate in base ten. Only step counts of the
+            // form 2^a * 5^b divide exactly, so for 3, 6, 7, 9, 12, 21 and most other values
+            // every share but the first came out NaN, and with it every starting point: the
+            // grid quietly shrank to the single corner where x and y are both zero. The
+            // default of 10 divides exactly, which is why this went unseen -- and why asking
+            // for a finer search made the answer worse rather than better. x^3 - 2x gives all
+            // three of its roots at 10 steps and only one at 21.
             for (int x = 0; x < settings.StepCount.Re; x++)
                 for (int y = 0; y < settings.StepCount.Im; y++)
                 {
-                    var xShare = ((EDecimal)x) / settings.StepCount.Re;
-                    var yShare = ((EDecimal)y) / settings.StepCount.Im;
+                    var xShare = CtxDivide((EDecimal)x, settings.StepCount.Re);
+                    var yShare = CtxDivide((EDecimal)y, settings.StepCount.Im);
                     var value = Complex.Create(
                         settings.From.Re * xShare + settings.To.Re * (1 - xShare),
                         settings.From.Im * yShare + settings.To.Im * (1 - yShare));

Sources/Tests/UnitTests/Algebra/SolveTest/NewtonGridTest.cs

@@ -0,0 +1,80 @@
+//
+// Copyright (c) 2019-2022 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using AngouriMath;
+using AngouriMath.Extensions;
+using PeterO.Numbers;
+using System.Collections.Generic;
+using System.Linq;
+using Xunit;
+
+namespace AngouriMath.Tests.Algebra.PolynomialSolverTests
+{
+    /// <summary>
+    /// The grid <see cref="Entity.SolveNt(Entity.Variable)"/> starts from was laid out with
+    /// EDecimal's contextless division, which answers NaN for a quotient that does not
+    /// terminate in base ten. Every step count outside 2^a * 5^b therefore produced one usable
+    /// starting point and a grid of NaN, so the search covered a single corner of the region
+    /// it was asked for and quietly returned fewer roots the finer it was told to look.
+    /// </summary>
+    public sealed class NewtonGridTest
+    {
+        private static HashSet<Entity.Number.Complex> Roots(string expression, int steps)
+        {
+            using var _ = MathS.Settings.NewtonSolver.Set(new MathS.Settings.NewtonSetting
+            {
+                From = (-10, -10),
+                To = (10, 10),
+                StepCount = (steps, steps),
+                Precision = 30,
+            });
+            return expression.ToEntity().SolveNt("x");
+        }
+
+        private static double[] RealRoots(string expression, int steps) =>
+            Roots(expression, steps)
+                .Where(root => root.ImaginaryPart.EDecimal.Abs().CompareTo(EDecimal.Create(1, -8)) < 0)
+                .Select(root => System.Math.Round(root.RealPart.EDecimal.ToDouble(), 6))
+                .Distinct().OrderBy(value => value).ToArray();
+
+        /// <summary>
+        /// 8, 10, 16, 20 and 40 divide exactly and always worked; 3, 6, 7, 9, 12, 21 and 41 did
+        /// not, and used to give back at most the one root reachable from the corner.
+        /// </summary>
+        [Theory]
+        [InlineData(8)]
+        [InlineData(9)]
+        [InlineData(10)]
+        [InlineData(12)]
+        [InlineData(20)]
+        [InlineData(21)]
+        [InlineData(41)]
+        public void EveryStepCountSearchesTheWholeRegion(int steps) =>
+            Assert.Equal(new[] { -1.414214d, 0d, 1.414214d }, RealRoots("x^3 - 2*x", steps));
+
+        /// <summary>
+        /// Asking for a finer search must not lose roots a coarser one found. This is the
+        /// property the defect broke, stated without naming any particular step count.
+        /// </summary>
+        [Fact]
+        public void AFinerSearchKeepsWhatACoarserOneFound()
+        {
+            var coarse = RealRoots("x^3 - 2*x", 10);
+            foreach (var steps in new[] { 11, 13, 17, 21, 33 })
+                Assert.All(coarse, root => Assert.Contains(root, RealRoots("x^3 - 2*x", steps)));
+        }
+
+        /// <summary>
+        /// A step count that does not divide exactly must lay out the same spread of starting
+        /// points as one that does, rather than collapsing onto a corner: at 21 steps over
+        /// [-10, 10] the search has to reach the roots of sin(x) - x/2 near +-1.8954942.
+        /// </summary>
+        [Fact]
+        public void AStepCountThatDoesNotDivideExactlyStillReachesTheOutlyingRoots() =>
+            Assert.Equal(new[] { -1.895494d, 0d, 1.895494d }, RealRoots("sin(x) - x / 2", 21));
+    }
+}