LinearSystemSolver.java

package org.drip.numerical.linearalgebra;

/*
 * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
 */

/*!
 * Copyright (C) 2020 Lakshmi Krishnamurthy
 * Copyright (C) 2019 Lakshmi Krishnamurthy
 * Copyright (C) 2018 Lakshmi Krishnamurthy
 * Copyright (C) 2017 Lakshmi Krishnamurthy
 * Copyright (C) 2016 Lakshmi Krishnamurthy
 * Copyright (C) 2015 Lakshmi Krishnamurthy
 * Copyright (C) 2014 Lakshmi Krishnamurthy
 * Copyright (C) 2013 Lakshmi Krishnamurthy
 * 
 *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
 *      asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
 *      analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
 *      equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
 *      numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
 *      and computational support.
 *  
 *      https://lakshmidrip.github.io/DROP/
 *  
 *  DROP is composed of three modules:
 *  
 *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
 *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
 *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
 * 
 *  DROP Product Core implements libraries for the following:
 *  - Fixed Income Analytics
 *  - Loan Analytics
 *  - Transaction Cost Analytics
 * 
 *  DROP Portfolio Core implements libraries for the following:
 *  - Asset Allocation Analytics
 *  - Asset Liability Management Analytics
 *  - Capital Estimation Analytics
 *  - Exposure Analytics
 *  - Margin Analytics
 *  - XVA Analytics
 * 
 *  DROP Computational Core implements libraries for the following:
 *  - Algorithm Support
 *  - Computation Support
 *  - Function Analysis
 *  - Model Validation
 *  - Numerical Analysis
 *  - Numerical Optimizer
 *  - Spline Builder
 *  - Statistical Learning
 * 
 *  Documentation for DROP is Spread Over:
 * 
 *  - Main                     => https://lakshmidrip.github.io/DROP/
 *  - Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
 *  - GitHub                   => https://github.com/lakshmiDRIP/DROP
 *  - Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
 *  - Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
 *  - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
 *  - Release Versions         => https://lakshmidrip.github.io/DROP/version.html
 *  - Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
 *  - Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
 *  - JUnit                    => https://lakshmidrip.github.io/DROP/junit/index.html
 *  - Jacoco                   => https://lakshmidrip.github.io/DROP/jacoco/index.html
 * 
 *  Licensed under the Apache License, Version 2.0 (the "License");
 *      you may not use this file except in compliance with the License.
 *   
 *  You may obtain a copy of the License at
 *      http://www.apache.org/licenses/LICENSE-2.0
 *  
 *  Unless required by applicable law or agreed to in writing, software
 *      distributed under the License is distributed on an "AS IS" BASIS,
 *      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  
 *  See the License for the specific language governing permissions and
 *      limitations under the License.
 */

/**
 * <i>LinearSystemSolver</i> implements the solver for a system of linear equations given by
 * 
 *                                          A * x = B
 * 
 * where A is the matrix, x the set of variables, and B is the result to be solved for. It exports the
 * following functions:
 * 
 * <br><br>
 *  <ul>
 *      <li>
 *          Row Regularization and Diagonal Pivoting
 *      </li>
 *      <li>
 *          Check for Diagonal Dominance
 *      </li>
 *      <li>
 *          Solving the linear system using any one of the following: Gaussian Elimination, Gauss Seidel
 *              reduction, or matrix inversion.
 *      </li>
 *  </ul>
 * 
 * <br><br>
 *  <ul>
 *      <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
 *      <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/NumericalAnalysisLibrary.md">Numerical Analysis Library</a></li>
 *      <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/numerical/README.md">Numerical Quadrature, Differentiation, Eigenization, Linear Algebra, and Utilities</a></li>
 *      <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/numerical/linearalgebra/README.md">Linear Algebra Matrix Transform Library</a></li>
 *  </ul>
 * <br><br>
 *
 * @author Lakshmi Krishnamurthy
 */

public class LinearSystemSolver {

    /**
     * Regularize (i.e., convert the diagonal entries of the given cell to non-zero using suitable linear
     *  transformations)
     * 
     * @param aadblA In/Out Matrix to be regularized
     * @param adblSolution In/out RHS
     * @param iInnerRow Matrix Cell Row that needs to be regularized
     * @param iOuter Matrix Cell Column that needs to be regularized
     * 
     * @return TRUE - Matrix has been successfully regularized
     */

    public static final boolean RegulariseRow (
        final double[][] aadblA,
        final double[] adblSolution,
        final int iInnerRow,
        final int iOuter)
    {
        double dblInnerScaler = aadblA[iInnerRow][iOuter];

        if (0. != dblInnerScaler) return true;

        int iSize = aadblA.length;
        int iProxyRow = iSize - 1;

        while (0. == aadblA[iProxyRow][iOuter] && iProxyRow >= 0) --iProxyRow;

        if (iProxyRow < 0) return false;

        adblSolution[iInnerRow] += adblSolution[iProxyRow];

        for (int i = 0; i < iSize; ++i)
            aadblA[iInnerRow][i] += aadblA[iProxyRow][i];

        return 0. != aadblA[iInnerRow][iOuter];
    }

    /**
     * Check to see if the matrix is diagonally dominant.
     * 
     * @param aadblA Input Matrix
     * @param bCheckForStrongDominance TRUE - Fail if the matrix is not strongly diagonally dominant.
     * 
     * @return TRUE - Strongly or weakly Diagonally Dominant
     */

    public static final boolean IsDiagonallyDominant (
        final double[][] aadblA,
        final boolean bCheckForStrongDominance)
    {
        if (null == aadblA) return false;

        int iSize = aadblA.length;

        if (0 == iSize || null == aadblA[0] || iSize != aadblA[0].length) return false;

        for (int i = 0; i < iSize; ++i) {
            double dblAbsoluteDiagonalEntry = java.lang.Math.abs (aadblA[i][i]);

            for (int j = 0; j < iSize; ++j) {
                if (i != j) {
                    if ((bCheckForStrongDominance && dblAbsoluteDiagonalEntry <= java.lang.Math.abs
                        (aadblA[i][j])) || (!bCheckForStrongDominance && dblAbsoluteDiagonalEntry <
                            java.lang.Math.abs (aadblA[i][j])))
                        return false;
                }
            }
        }

        return true;
    }

    /**
     * Pivots the matrix A (Refer to wikipedia to find out what "pivot a matrix" means ;))
     * 
     * @param aadblA Input Matrix
     * @param adblB Input RHS
     * 
     * @return The pivoted input matrix and the re-jigged input RHS
     */

    public static final double[] Pivot (
        final double[][] aadblA,
        final double[] adblB)
    {
        if (null == aadblA || null == adblB) return null;

        int iSize = aadblA.length;
        double[] adblSolution = new double[iSize];

        if (0 == iSize || null == aadblA[0] || iSize != aadblA[0].length || iSize != adblB.length)
            return null;

        for (int i = 0; i < iSize; ++i)
            adblSolution[i] = adblB[i];

        for (int iDiagonal = 0; iDiagonal < iSize; ++iDiagonal) {
            if (!RegulariseRow (aadblA, adblSolution, iDiagonal, iDiagonal)) return null;
        }

        return adblSolution;
    }

    /**
     * Solve the Linear System using Matrix Inversion from the Set of Values in the Array
     * 
     * @param aadblAIn Input Matrix
     * @param adblB The Array of Values to be calibrated to
     * 
     * @return The Linear System Solution for the Coefficients
     */

    public static final org.drip.numerical.linearalgebra.LinearizationOutput SolveUsingMatrixInversion (
        final double[][] aadblAIn,
        final double[] adblB)
    {
        if (null == aadblAIn || null == adblB) return null;

        int iSize = aadblAIn.length;
        double[] adblSolution = new double[iSize];

        if (0 == iSize || null == aadblAIn[0] || iSize != aadblAIn[0].length) return null;

        if (adblB.length != iSize) return null;

        double[][] aadblInv = org.drip.numerical.linearalgebra.Matrix.InvertUsingGaussianElimination (aadblAIn);

        if (null == aadblInv) return null;

        double[] adblProduct = org.drip.numerical.linearalgebra.Matrix.Product (aadblInv, adblB);

        if (null == adblProduct || iSize != adblProduct.length) return null;

        for (int i = 0; i < iSize; ++i)
            adblSolution[i] = adblProduct[i];

        try {
            return new LinearizationOutput (adblSolution, aadblInv, "GaussianElimination");
        } catch (java.lang.Exception e) {
            e.printStackTrace();
        }

        return null;
    }

    /**
     * Solve the Linear System using Gaussian Elimination from the Set of Values in the Array
     * 
     * @param aadblAIn Input Matrix
     * @param adblB The Array of Values to be calibrated to
     * 
     * @return The Linear System Solution for the Coefficients
     */

    public static final org.drip.numerical.linearalgebra.LinearizationOutput SolveUsingGaussianElimination (
        final double[][] aadblAIn,
        final double[] adblB)
    {
        if (null == aadblAIn || null == adblB) return null;

        int iSize = aadblAIn.length;
        double[][] aadblA = new double[iSize][iSize];

        if (0 == iSize || null == aadblAIn[0] || iSize != aadblAIn[0].length) return null;

        if (adblB.length != iSize) return null;

        for (int i = 0; i < iSize; ++i) {
            for (int j = 0; j < iSize; ++j)
                aadblA[i][j] = aadblAIn[i][j];
        }

        double[] adblSolution = Pivot (aadblA, adblB);

        if (null == adblSolution || adblSolution.length != iSize) return null;

        for (int iEliminationDiagonalPivot = iSize - 1; iEliminationDiagonalPivot >= 0;
            --iEliminationDiagonalPivot) {
            for (int iRow = 0; iRow < iSize; ++iRow) {
                if (iRow == iEliminationDiagonalPivot) continue;

                if (0. == aadblA[iRow][iEliminationDiagonalPivot]) continue;

                double dblEliminationRatio = aadblA[iEliminationDiagonalPivot][iEliminationDiagonalPivot] /
                    aadblA[iRow][iEliminationDiagonalPivot];
                adblSolution[iRow] = adblSolution[iRow] * dblEliminationRatio -
                    adblSolution[iEliminationDiagonalPivot];

                for (int iCol = 0; iCol < iSize; ++iCol)
                    aadblA[iRow][iCol] = aadblA[iRow][iCol] * dblEliminationRatio -
                        aadblA[iEliminationDiagonalPivot][iCol];
            }
        }

        for (int i = iSize - 1; i >= 0; --i) {
            for (int j = iSize - 1; j > i; --j)
                adblSolution[i] -= adblSolution[j] * aadblA[i][j];

            adblSolution[i] /= aadblA[i][i];
        }

        try {
            return new LinearizationOutput (adblSolution, aadblA, "GaussianElimination");
        } catch (java.lang.Exception e) {
            e.printStackTrace();
        }

        return null;
    }

    /**
     * Solve the Linear System using the Gauss-Seidel algorithm from the Set of Values in the Array
     * 
     * @param aadblAIn Input Matrix
     * @param adblB The Array of Values to be calibrated to
     * 
     * @return The Linear System Solution for the Coefficients
     */

    public static final org.drip.numerical.linearalgebra.LinearizationOutput SolveUsingGaussSeidel (
        final double[][] aadblAIn,
        final double[] adblB)
    {
        if (null == aadblAIn || null == adblB) return null;

        int NUM_SIM = 5;
        int iSize = aadblAIn.length;
        double[] adblSolution = new double[iSize];
        double[][] aadblA = new double[iSize][iSize];

        if (0 == iSize || null == aadblAIn[0] || iSize != aadblAIn[0].length || iSize != adblB.length)
            return null;

        for (int i = 0; i < iSize; ++i) {
            for (int j = 0; j < iSize; ++j)
                aadblA[i][j] = aadblAIn[i][j];
        }

        double[] adblRHS = Pivot (aadblA, adblB);

        if (null == adblRHS || iSize != adblRHS.length ||
            !org.drip.numerical.linearalgebra.LinearSystemSolver.IsDiagonallyDominant (aadblA, true))
            return null;

        for (int i = 0; i < iSize; ++i)
            adblSolution[i] = 0.;

        for (int k = 0; k < NUM_SIM; ++k) {
            for (int i = 0; i < iSize; ++i) {
                adblSolution[i] = adblRHS[i];

                for (int j = 0; j < iSize; ++j) {
                    if (j != i) adblSolution[i] -= aadblA[i][j] * adblSolution[j];
                }

                adblSolution[i] /= aadblA[i][i];
            }
        }

        try {
            return new LinearizationOutput (adblSolution, aadblA, "GaussianSeidel");
        } catch (java.lang.Exception e) {
            e.printStackTrace();
        }

        return null;
    }
}