Polynomial: more integration into FindRoots

MathNet.Numerics.sln.DotSettings

@@ -81,6 +81,7 @@ OTHER DEALINGS IN THE SOFTWARE.
 	<s:Boolean x:Key="/Default/Environment/SettingsMigration/IsMigratorApplied/=JetBrains_002EReSharper_002EPsi_002ECSharp_002ECodeStyle_002ESettingsUpgrade_002ECSharpPlaceAttributeOnSameLineMigration/@EntryIndexedValue">True</s:Boolean>
 	<s:Boolean x:Key="/Default/Environment/SettingsMigration/IsMigratorApplied/=JetBrains_002EReSharper_002EPsi_002ECSharp_002ECodeStyle_002ESettingsUpgrade_002EMigrateBlankLinesAroundFieldToBlankLinesAroundProperty/@EntryIndexedValue">True</s:Boolean>
 	<s:Boolean x:Key="/Default/Environment/SettingsMigration/IsMigratorApplied/=JetBrains_002EReSharper_002EPsi_002ECSharp_002ECodeStyle_002ESettingsUpgrade_002EMigrateThisQualifierSettings/@EntryIndexedValue">True</s:Boolean>
+	<s:Boolean x:Key="/Default/UserDictionary/Words/=Chebychev/@EntryIndexedValue">True</s:Boolean>
 	<s:Boolean x:Key="/Default/UserDictionary/Words/=Numerics/@EntryIndexedValue">True</s:Boolean>
 	
 	</wpf:ResourceDictionary>

src/Numerics.Tests/RootFindingTests/CubicTest.cs

@@ -97,6 +97,19 @@ namespace MathNet.Numerics.UnitTests.RootFindingTests
             Assert.That(roots.Item3.Imaginary, Is.EqualTo(0).Within(1e-14));
         }
 
+        [TestCase(6.0, -5.0, -2.0, 1.0, 3.0, -2.0, 1.0)]
+        public void ComplexRoots_TripleReal_AsPolynomial(double d, double c, double b, double a, double x1, double x2, double x3)
+        {
+            var roots = FindRoots.Polynomial(new[] { d, c, b, a });
+            Assert.That(roots.Length, Is.EqualTo(3));
+            Assert.That(roots[0].Real, Is.EqualTo(x1).Within(1e-14));
+            Assert.That(roots[0].Imaginary, Is.EqualTo(0).Within(1e-14));
+            Assert.That(roots[1].Real, Is.EqualTo(x2).Within(1e-14));
+            Assert.That(roots[1].Imaginary, Is.EqualTo(0).Within(1e-14));
+            Assert.That(roots[2].Real, Is.EqualTo(x3).Within(1e-14));
+            Assert.That(roots[2].Imaginary, Is.EqualTo(0).Within(1e-14));
+        }
+
         [TestCase(-350.0, 162.0, -30.0, 2.0)]
         [TestCase(6.0, -5.0, -2.0, 1.0)]
         [TestCase(1.0, 5.0, 2.0, 1.0)]
@@ -113,5 +126,23 @@ namespace MathNet.Numerics.UnitTests.RootFindingTests
             Assert.That(Polynomial.Evaluate(roots.Item3, d, c, b, a).Real, Is.EqualTo(0).Within(1e-12));
             Assert.That(Polynomial.Evaluate(roots.Item3, d, c, b, a).Imaginary, Is.EqualTo(0).Within(1e-12));
         }
+
+        [TestCase(-350.0, 162.0, -30.0, 2.0)]
+        [TestCase(6.0, -5.0, -2.0, 1.0)]
+        [TestCase(1.0, 5.0, 2.0, 1.0)]
+        [TestCase(1.0, 5.0, 0.0, 1.0)]
+        [TestCase(1.0, 0.0, 2.0, 1.0)]
+        [TestCase(0.0, 0.0, 0.0, 2.0)]
+        public void ComplexRootsAreRoots_AsPolynomial(double d, double c, double b, double a)
+        {
+            var roots = FindRoots.Polynomial(new[] { d, c, b, a });
+            Assert.That(roots.Length, Is.EqualTo(3));
+            Assert.That(Polynomial.Evaluate(roots[0], d, c, b, a).Real, Is.EqualTo(0).Within(1e-12));
+            Assert.That(Polynomial.Evaluate(roots[0], d, c, b, a).Imaginary, Is.EqualTo(0).Within(1e-12));
+            Assert.That(Polynomial.Evaluate(roots[1], d, c, b, a).Real, Is.EqualTo(0).Within(1e-12));
+            Assert.That(Polynomial.Evaluate(roots[1], d, c, b, a).Imaginary, Is.EqualTo(0).Within(1e-12));
+            Assert.That(Polynomial.Evaluate(roots[2], d, c, b, a).Real, Is.EqualTo(0).Within(1e-12));
+            Assert.That(Polynomial.Evaluate(roots[2], d, c, b, a).Imaginary, Is.EqualTo(0).Within(1e-12));
+        }
     }
 }

src/Numerics/FindRoots.cs

@@ -114,11 +114,21 @@ namespace MathNet.Numerics
         /// <summary>
         /// Find all roots of a polynomial by calculating the characteristic polynomial of the companion matrix
         /// </summary>
-        /// <param name="poly">the values for the polynomial in ascending order e.G new double[] {5, 0, 2} = "5 + 0 x^1 + 2 x^2"</param>
-        /// <returns>the roots of the polynomial</returns>
-        public static Complex[] Polynomial(double[] poly)
+        /// <param name="coefficients">The coefficients of the polynomial in ascending order, e.g. new double[] {5, 0, 2} = "5 + 0 x^1 + 2 x^2"</param>
+        /// <returns>The roots of the polynomial</returns>
+        public static Complex[] Polynomial(double[] coefficients)
         {
-            return new Polynomial(poly).Roots();
+            return new Polynomial(coefficients).Roots();
+        }
+
+        /// <summary>
+        /// Find all roots of a polynomial by calculating the characteristic polynomial of the companion matrix
+        /// </summary>
+        /// <param name="polynomial">The polynomial.</param>
+        /// <returns>The roots of the polynomial</returns>
+        public static Complex[] Polynomial(Polynomial polynomial)
+        {
+            return polynomial.Roots();
         }
 
         /// <summary>
@@ -139,7 +149,7 @@ namespace MathNet.Numerics
             double location = 0.5*(intervalBegin + intervalEnd);
             double scale = 0.5*(intervalEnd - intervalBegin);
 
-            // evaluate first kind chebyshev nodes
+            // evaluate first kind chebychev nodes
             double angleFactor = Constants.Pi/(2*degree);
 
             var samples = new double[degree];
@@ -168,7 +178,7 @@ namespace MathNet.Numerics
             double location = 0.5*(intervalBegin + intervalEnd);
             double scale = 0.5*(intervalEnd - intervalBegin);
 
-            // evaluate second kind chebyshev nodes
+            // evaluate second kind chebychev nodes
             double angleFactor = Constants.Pi/(degree + 1);
 
             var samples = new double[degree];

src/Numerics/Polynomial.cs

@@ -76,10 +76,21 @@ namespace MathNet.Numerics
         /// Create a constant polynomial.
         /// Example: 3.0 -> "p : x -> 3.0"
         /// </summary>
-        /// <param name="coefficient">just the "x^0" part</param>
+        /// <param name="coefficient">The coefficient of the "x^0" monomial.</param>
         public Polynomial(double coefficient)
         {
-            Coefficients = coefficient == 0.0 ? new double[0] : new[] { coefficient };
+            if (coefficient == 0.0)
+            {
+#if NET40
+                Coefficients = new double[0];
+#else
+                Coefficients = Array.Empty<double>();
+#endif
+            }
+            else
+            {
+                Coefficients = new[] { coefficient };
+            }
         }
 
         /// <summary>
@@ -101,6 +112,17 @@ namespace MathNet.Numerics
         {
         }
 
+        public static Polynomial Zero => new Polynomial();
+
+        /// <summary>
+        /// Least-Squares fitting the points (x,y) to a k-order polynomial y : x -> p0 + p1*x + p2*x^2 + ... + pk*x^k
+        /// </summary>
+        public static Polynomial Fit(double[] x, double[] y, int order, DirectRegressionMethod method = DirectRegressionMethod.QR)
+        {
+            var coefficients = Numerics.Fit.Polynomial(x, y, order, method);
+            return new Polynomial(coefficients);
+        }
+
         static int EvaluateDegree(double[] coefficients)
         {
             for (int i = coefficients.Length - 1; i >= 0; i--)
@@ -114,22 +136,11 @@ namespace MathNet.Numerics
             return -1;
         }
 
-        public static Polynomial Zero => new Polynomial();
-
-        /// <summary>
-        /// Least-Squares fitting the points (x,y) to a k-order polynomial y : x -> p0 + p1*x + p2*x^2 + ... + pk*x^k
-        /// </summary>
-        public static Polynomial Fit(double[] x, double[] y, int order, DirectRegressionMethod method = DirectRegressionMethod.QR)
-        {
-            var coefficients = Numerics.Fit.Polynomial(x, y, order, method);
-            return new Polynomial(coefficients);
-        }
-
         #region Evaluation
 
         /// <summary>
         /// Evaluate a polynomial at point x.
-        /// Coefficients are ordered by power with power k at index k.
+        /// Coefficients are ordered ascending by power with power k at index k.
         /// Example: coefficients [3,-1,2] represent y=2x^2-x+3.
         /// </summary>
         /// <param name="z">The location where to evaluate the polynomial at.</param>
@@ -148,7 +159,7 @@ namespace MathNet.Numerics
 
         /// <summary>
         /// Evaluate a polynomial at point x.
-        /// Coefficients are ordered by power with power k at index k.
+        /// Coefficients are ordered ascending by power with power k at index k.
         /// Example: coefficients [3,-1,2] represent y=2x^2-x+3.
         /// </summary>
         /// <param name="z">The location where to evaluate the polynomial at.</param>
@@ -167,7 +178,7 @@ namespace MathNet.Numerics
 
         /// <summary>
         /// Evaluate a polynomial at point x.
-        /// Coefficients are ordered by power with power k at index k.
+        /// Coefficients are ordered ascending by power with power k at index k.
         /// Example: coefficients [3,-1,2] represent y=2x^2-x+3.
         /// </summary>
         /// <param name="z">The location where to evaluate the polynomial at.</param>