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richard-churchman
cc0a2bee21
Included open-source Accord.net library code natively to the solution given the deprecated state of that project. Accord.net will be supported inside Jube for own needs (such as upgrading to .Net 7+). Accord.net libraries modified to compile to .Net 6. Modified Accord.net libraries to better support Newtonsoft Json.net serialisation. Included annotations in the Accord.net libraries to house important dependency statistics. Serialised json can be used independently of Jube to recall a Neural Network. Removal of Binary Serializer totally as a precursor to the projects support of .Net 7+. Extensive bug fixing in crucial functions of Neural Network training and recall. While this part of the system remains in Alpha, it is substantially more fit for purpose.
234 lines
8.2 KiB
C#
234 lines
8.2 KiB
C#
// Accord Math Library
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// The Accord.NET Framework
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// http://accord-framework.net
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//
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// Copyright © Diego Catalano, 2014
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// diego.catalano at live.com
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//
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// This library is free software; you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public
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// License as published by the Free Software Foundation; either
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// version 2.1 of the License, or (at your option) any later version.
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//
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// This library is distributed in the hope that it will be useful,
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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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// Lesser General Public License for more details.
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//
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// You should have received a copy of the GNU Lesser General Public
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// License along with this library; if not, write to the Free Software
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// Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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//
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namespace Accord.Math
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{
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/// <summary>
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/// Taylor series expansions for common functions.
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/// </summary>
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///
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/// <remarks>
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/// <para>
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/// In mathematics, a Taylor series is a representation of a function as an infinite sum of terms
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/// that are calculated from the values of the function's derivatives at a single point.</para>
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///
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/// <para>
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/// The concept of a Taylor series was discovered by the Scottish mathematician James Gregory and
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/// formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is
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/// centered at zero, then that series is also called a Maclaurin series, named after the Scottish
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/// mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in
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/// the 18th century.</para>
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///
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/// <para>
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/// It is common practice to approximate a function by using a finite number of terms of its Taylor
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/// series. Taylor's theorem gives quantitative estimates on the error in this approximation. Any
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/// finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
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/// The Taylor series of a function is the limit of that function's Taylor polynomials, provided that
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/// the limit exists. A function may not be equal to its Taylor series, even if its Taylor series
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/// converges at every point. A function that is equal to its Taylor series in an open interval (or
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/// a disc in the complex plane) is known as an analytic function in that interval.</para>
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///
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/// <para>
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/// References:
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/// <list type="bullet">
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/// <item><description><a href="http://en.wikipedia.org/wiki/Taylor_series">
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/// Wikipedia, The Free Encyclopedia. Taylor series. Available at:
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/// http://en.wikipedia.org/wiki/Taylor_series </a></description></item>
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/// <item><description><a href="http://www.haverford.edu/physics/MathAppendices/Taylor_Series.pdf">
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/// Anne Fry, Amy Plofker, Sarah-marie Belcastro, Lyle Roelofs. A Set of Appendices on Mathematical
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/// Methods for Physics Students: Taylor Series Expansions and Approximations. Available at:
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/// http://www.haverford.edu/physics/MathAppendices/Taylor_Series.pdf </a></description></item>
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/// </list></para>
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/// </remarks>
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///
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public static class Taylor
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{
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/// <summary>
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/// Returns the sine of a specified angle by evaluating a Taylor series.
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/// </summary>
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///
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/// <param name="x">An angle, measured in radians.</param>
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/// <param name="nTerms">The number of terms to be evaluated.</param>
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///
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/// <returns>The sine of the angle <paramref name="x"/>.</returns>
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///
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public static double Sin(double x, int nTerms)
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{
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if (nTerms < 2)
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return x;
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if (nTerms == 2)
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return x - (x * x * x) / 6D;
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double mult = x * x * x;
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double fact = 6;
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double sign = 1;
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int factS = 5;
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double result = x - mult / fact;
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for (int i = 3; i <= nTerms; i++)
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{
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mult *= x * x;
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fact *= factS * (factS - 1);
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factS += 2;
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result += sign * (mult / fact);
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sign *= -1;
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}
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return result;
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}
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/// <summary>
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/// Returns the cosine of a specified angle by evaluating a Taylor series.
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/// </summary>
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///
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/// <param name="x">An angle, measured in radians.</param>
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/// <param name="nTerms">The number of terms to be evaluated.</param>
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///
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/// <returns>The cosine of the angle <paramref name="x"/>.</returns>
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///
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public static double Cos(double x, int nTerms)
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{
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if (nTerms < 2)
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return 1;
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if (nTerms == 2)
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return 1 - (x * x) / 2D;
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double mult = x * x;
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double fact = 2;
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double sign = 1;
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int factS = 4;
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double result = 1 - mult / fact;
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for (int i = 3; i <= nTerms; i++)
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{
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mult *= x * x;
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fact *= factS * (factS - 1);
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factS += 2;
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result += sign * (mult / fact);
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sign *= -1;
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}
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return result;
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}
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/// <summary>
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/// Returns the hyperbolic sine of a specified angle by evaluating a Taylor series.
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/// </summary>
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///
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/// <param name="x">An angle, measured in radians.</param>
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/// <param name="nTerms">The number of terms to be evaluated.</param>
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///
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/// <returns>The hyperbolic sine of the angle <paramref name="x"/>.</returns>
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///
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public static double Sinh(double x, int nTerms)
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{
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if (nTerms < 2)
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return x;
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if (nTerms == 2)
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return x + (x * x * x) / 6D;
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double mult = x * x * x;
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double fact = 6;
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int factS = 5;
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double result = x + mult / fact;
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for (int i = 3; i <= nTerms; i++)
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{
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mult *= x * x;
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fact *= factS * (factS - 1);
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factS += 2;
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result += mult / fact;
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}
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return result;
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}
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/// <summary>
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/// Returns the hyperbolic cosine of a specified angle by evaluating a Taylor series.
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/// </summary>
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///
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/// <param name="x">An angle, measured in radians.</param>
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/// <param name="nTerms">The number of terms to be evaluated.</param>
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///
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/// <returns>The hyperbolic cosine of the angle <paramref name="x"/>.</returns>
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///
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public static double Cosh(double x, int nTerms)
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{
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if (nTerms < 2)
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return x;
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if (nTerms == 2)
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return 1 + (x * x) / 2D;
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double mult = x * x;
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double fact = 2;
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int factS = 4;
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double result = 1 + mult / fact;
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for (int i = 3; i <= nTerms; i++)
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{
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mult *= x * x;
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fact *= factS * (factS - 1);
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factS += 2;
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result += mult / fact;
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}
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return result;
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}
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/// <summary>
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/// Returns e raised to the specified power by evaluating a Taylor series.
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/// </summary>
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///
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/// <param name="d">A number specifying a power.</param>
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/// <param name="nTerms">The number of terms to be evaluated.</param>
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///
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/// <returns>Euler's constant raised to the specified power <paramref name="d"/>.</returns>
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///
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public static double Exp(double d, int nTerms)
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{
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if (nTerms < 2)
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return 1 + d;
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if (nTerms == 2)
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return 1 + d + (d * d) / 2;
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double mult = d * d;
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double fact = 2;
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double result = 1 + d + mult / fact;
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for (int i = 3; i <= nTerms; i++)
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{
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mult *= d;
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fact *= i;
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result += mult / fact;
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}
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return result;
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}
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}
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}
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