Features:

	- Square Matrix is Triangularizable (1, 2)
	- Numerical R^1 -> R^1 Eigenvalue Characteristic Polynomial (3, 4, 5)
	- Square Matrix Eigenvalue Characteristic Polynomial (6, 7, 8)
	- Triangular Matrix Remove R2 Array (9, 10)
	- Triangular Matrix Zero Diagonal Entries (13, 14)
	- Triangular Matrix Is Uni Check (15, 16)
	- Triangular Matrix Is Triangularizable Check (19, 20)
	- Triangular Matrix Determinant and Permanent (21, 22)
	- Square Matrix Triangular Entries #1 (23, 24)
	- Square Matrix Triangular Entries #2 (25, 26)
	- Triangular Matrix Eigenvalue Multiplicity Map (27, 28, 29)
	- Triangular Matrix Eigenvalue Characteristic Polynomial (30, 31, 32)
	- Triangular Matrix Is Unit Check (33, 34)
	- Triangular Matrix Is Normed Check (35, 36)
	- Triangular Matrix Is Upper Unitriangular (37, 38)
	- Triangular Matrix Is Lower Unitriangular (39, 40)
	- Triangular Matrix Atomic Lower #1 (41, 42)
	- Triangular Matrix Atomic Lower #2 (43, 44)
	- Triangular Matrix Atomic Lower #3 (45, 46)
	- Triangular Matrix Atomic Upper #1 (47, 48)
	- Triangular Matrix Atomic Upper #2 (49, 50)
	- Triangular Matrix Is Atomic Unitriangular (51, 52)
	- Triangular Matrix Is Frobenius Unitriangular (53, 54)
	- Triangular Matrix Is Gauss Unitriangular (55, 56)
	- Triangular Matrix Gauss Transformation Unitriangular (57, 58)
	- Linear Algebra Triangular Matrix Constructor (59, 60)
	- Linear Algebra Triangular Matrix Standard (61, 62, 63)
	- Linear Algebra Triangular Matrix Type (64)
	- CRNG Random Matrix Generator Shell (65, 66)
	- Random Matrix Generator Upper Triangular (67, 68, 69)
	- Random Matrix Generator Lower Triangular (70, 71, 72)


Bug Fixes/Re-organization:

	- Linear Algebra Triangular Matrix Revamp (11, 12)
	- Triangular Matrix Determinant/Permanent Revamp (17, 18)
	- Triangular Matrix Variants and Solutions (73, 74)


Samples:

	- Lower Triangular Matrix Shell #1 (75, 76)
	- Lower Triangular Matrix Shell #2 (77, 78)
	- Lower Triangular Matrix Shell #3 (79, 80, 81)
	- Lower Triangular Matrix Shell #4 (82, 83, 84)
	- Lower Triangular Matrix Shell #5 (85, 86, 87)
	- Lower Triangular Matrix Shell #6 (88, 89, 90)
	- Lower Triangular Matrix Shell #7 (91, 92, 93)
	- Lower Triangular Matrix Shell #8 (94, 95, 96)
	- Lower Triangular Matrix Shell #9 (97, 98, 99)
	- Lower Triangular Matrix Shell #10 (100, 101, 102)
	- Lower Triangular Matrix Shell #11 (103, 104, 105)
	- Lower Triangular Matrix Shell #12 (106, 107, 108)
	- Lower Triangular Matrix Shell #13 (109, 110, 111)
	- Lower Triangular Matrix Shell #14 (112, 113, 114)
	- Lower Triangular Matrix Shell #15 (115, 116, 117)
	- Lower Triangular Matrix Shell #16 (118, 119, 120)


IdeaDRIP:

ReleaseNotes/02_06_2024.txt

@@ -0,0 +1,64 @@
+
+Features:
+
+ - Square Matrix is Triangularizable (1, 2)
+ - Numerical R^1 -> R^1 Eigenvalue Characteristic Polynomial (3, 4, 5)
+ - Square Matrix Eigenvalue Characteristic Polynomial (6, 7, 8)
+ - Triangular Matrix Remove R2 Array (9, 10)
+ - Triangular Matrix Zero Diagonal Entries (13, 14)
+ - Triangular Matrix Is Uni Check (15, 16)
+ - Triangular Matrix Is Triangularizable Check (19, 20)
+ - Triangular Matrix Determinant and Permanent (21, 22)
+ - Square Matrix Triangular Entries #1 (23, 24)
+ - Square Matrix Triangular Entries #2 (25, 26)
+ - Triangular Matrix Eigenvalue Multiplicity Map (27, 28, 29)
+ - Triangular Matrix Eigenvalue Characteristic Polynomial (30, 31, 32)
+ - Triangular Matrix Is Unit Check (33, 34)
+ - Triangular Matrix Is Normed Check (35, 36)
+ - Triangular Matrix Is Upper Unitriangular (37, 38)
+ - Triangular Matrix Is Lower Unitriangular (39, 40)
+ - Triangular Matrix Atomic Lower #1 (41, 42)
+ - Triangular Matrix Atomic Lower #2 (43, 44)
+ - Triangular Matrix Atomic Lower #3 (45, 46)
+ - Triangular Matrix Atomic Upper #1 (47, 48)
+ - Triangular Matrix Atomic Upper #2 (49, 50)
+ - Triangular Matrix Is Atomic Unitriangular (51, 52)
+ - Triangular Matrix Is Frobenius Unitriangular (53, 54)
+ - Triangular Matrix Is Gauss Unitriangular (55, 56)
+ - Triangular Matrix Gauss Transformation Unitriangular (57, 58)
+ - Linear Algebra Triangular Matrix Constructor (59, 60)
+ - Linear Algebra Triangular Matrix Standard (61, 62, 63)
+ - Linear Algebra Triangular Matrix Type (64)
+ - CRNG Random Matrix Generator Shell (65, 66)
+ - Random Matrix Generator Upper Triangular (67, 68, 69)
+ - Random Matrix Generator Lower Triangular (70, 71, 72)
+
+
+Bug Fixes/Re-organization:
+
+ - Linear Algebra Triangular Matrix Revamp (11, 12)
+ - Triangular Matrix Determinant/Permanent Revamp (17, 18)
+ - Triangular Matrix Variants and Solutions (73, 74)
+
+
+Samples:
+
+ - Lower Triangular Matrix Shell #1 (75, 76)
+ - Lower Triangular Matrix Shell #2 (77, 78)
+ - Lower Triangular Matrix Shell #3 (79, 80, 81)
+ - Lower Triangular Matrix Shell #4 (82, 83, 84)
+ - Lower Triangular Matrix Shell #5 (85, 86, 87)
+ - Lower Triangular Matrix Shell #6 (88, 89, 90)
+ - Lower Triangular Matrix Shell #7 (91, 92, 93)
+ - Lower Triangular Matrix Shell #8 (94, 95, 96)
+ - Lower Triangular Matrix Shell #9 (97, 98, 99)
+ - Lower Triangular Matrix Shell #10 (100, 101, 102)
+ - Lower Triangular Matrix Shell #11 (103, 104, 105)
+ - Lower Triangular Matrix Shell #12 (106, 107, 108)
+ - Lower Triangular Matrix Shell #13 (109, 110, 111)
+ - Lower Triangular Matrix Shell #14 (112, 113, 114)
+ - Lower Triangular Matrix Shell #15 (115, 116, 117)
+ - Lower Triangular Matrix Shell #16 (118, 119, 120)
+
+
+IdeaDRIP:

src/main/java/org/drip/measure/crng/RandomMatrixGenerator.java

@@ -0,0 +1,170 @@
+
+package org.drip.measure.crng;
+
+import org.drip.numerical.linearalgebra.TriangularMatrix;
+
+/*
+ * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
+ */
+
+/*!
+ * Copyright (C) 2025 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,
+ * graph builder/navigator, 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
+ * - Graph Algorithm
+ * - 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
+ * 
+ * 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>RandomMatrixGenerator</i> provides Functionality for generating different Kinds of Random Matrices. The
+ * References are:
+ * 
+ * <br><br>
+ * <ul>
+ * <li>
+ * Axler, S. J. (1997): <i>Linear Algebra Done Right 2<sup>nd</sup> Edition</i> <b>Springer</b>
+ * New York NY
+ * </li>
+ * <li>
+ * Bernstein, D. S. (2009): <i>Matrix Mathematics: Theory, Facts, and Formulas 2<sup>nd</sup>
+ * Edition</i> <b>Princeton University Press</b> Princeton NJ
+ * </li>
+ * <li>
+ * Herstein, I. N. (1975): <i>Topics in Algebra 2<sup>nd</sup> Edition</i> <b>Wiley</b> New York NY
+ * </li>
+ * <li>
+ * Prasolov, V. V. (1994): <i>Topics in Algebra</i> <b>American Mathematical Society</b> Providence
+ * RI
+ * </li>
+ * <li>
+ * Wikipedia (2024): Triangular Matrix https://en.wikipedia.org/wiki/Triangular_matrix
+ * </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 RandomMatrixGenerator
+{
+
+ /**
+ * Construct a Lower Triangular Matrix of Random Elements up to the Maximum Value
+ * 
+ * @param elementCount Number of Elements in the Array
+ * @param maximumElement Maximum Element
+ * @param isEntryInteger TRUE - Entry is an Integer
+ * 
+ * @return Lower Triangular Matrix of Random Elements up to the Maximum Value
+ */
+
+ public static final TriangularMatrix LowerTriangular (
+ final int elementCount,
+ final double maximumElement,
+ final boolean isEntryInteger)
+ {
+ double[][] r2Array = RdRandomSequence.TwoD (elementCount, maximumElement, isEntryInteger);
+
+ for (int i = 0; i < r2Array.length; ++i) {
+ for (int j = i + 1; j < r2Array.length; ++j) {
+ r2Array[i][j] = 0.;
+ }
+ }
+
+ return TriangularMatrix.Standard (r2Array);
+ }
+
+ /**
+ * Construct an Upper Triangular Matrix of Random Elements up to the Maximum Value
+ * 
+ * @param elementCount Number of Elements in the Array
+ * @param maximumElement Maximum Element
+ * @param isEntryInteger TRUE - Entry is an Integer
+ * 
+ * @return Upper Triangular Matrix of Random Elements up to the Maximum Value
+ */
+
+ public static final TriangularMatrix UpperTriangular (
+ final int elementCount,
+ final double maximumElement,
+ final boolean isEntryInteger)
+ {
+ double[][] r2Array = RdRandomSequence.TwoD (elementCount, maximumElement, isEntryInteger);
+
+ for (int i = 0; i < r2Array.length; ++i) {
+ for (int j = 0; j < i; ++j) {
+ r2Array[i][j] = 0.;
+ }
+ }
+
+ return TriangularMatrix.Standard (r2Array);
+ }
+}

src/main/java/org/drip/numerical/eigen/EigenOutput.java

@@ -4,6 +4,8 @@
 import java.util.HashMap;
 import java.util.Map;
 
+import org.drip.function.definition.R1ToR1;
+
 /*
  * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
  */
@@ -197,4 +199,29 @@ public double determinant()
 
  return determinant;
  }
+
+ /**
+ * Retrieve the Characteristic Polynomial of the Eigenvalues
+ * 
+ * @return Characteristic Polynomial of the Eigenvalues
+ */
+
+ public R1ToR1 characteristicPolynomial()
+ {
+ return new R1ToR1 (null)
+ {
+ @Override public double evaluate (
+ final double x)
+ throws Exception
+ {
+ double value = 1.;
+
+ for (double eigenValue : _eigenValueArray) {
+ value *= (x - eigenValue);
+ }
+
+ return value;
+ }
+ };
+ }
 }

src/main/java/org/drip/numerical/linearalgebra/SquareMatrix.java

@@ -3,6 +3,7 @@
 
 import java.util.Map;
 
+import org.drip.function.definition.R1ToR1;
 import org.drip.numerical.common.NumberUtil;
 import org.drip.numerical.eigen.EigenOutput;
 import org.drip.numerical.eigen.QREigenComponentExtractor;
@@ -129,7 +130,7 @@ public class SquareMatrix
  * @return Standard Instance of <i>SquareMatrix</i>
  */
 
- public static final SquareMatrix Standard (
+ public static SquareMatrix Standard (
  final double[][] r2Array)
  {
  if (!MatrixUtil.IsSquare (r2Array)) {
@@ -253,6 +254,17 @@ public boolean isNormal()
  return true;
  }
 
+ /**
+ * Calculate whether the Matrix is "Triangularizable"
+ * 
+ * @return TRUE - Matrix is "Triangularizable"
+ */
+
+ public boolean isTriangularizable()
+ {
+ return true;
+ }
+
  /**
  * Eigenize and Extract the Components of the Specified Matrix
  * 
@@ -264,6 +276,23 @@ public EigenOutput eigenize()
  return QREigenComponentExtractor.Standard().eigenize (_r2Array);
  }
 
+ /**
+ * Retrieve the Array of Diagonal Entries
+ * 
+ * @return Array of Diagonal Entries
+ */
+
+ public double[] diagonalEntryArray()
+ {
+ double[] diagonalEntryArray = new double[_r2Array.length];
+
+ for (int i = 0; i < _r2Array.length; ++i) {
+ diagonalEntryArray[i] = _r2Array[i][i];
+ }
+
+ return diagonalEntryArray;
+ }
+
  /**
  * Retrieve the Eigenvalue Multiplicity Map
  * 
@@ -296,4 +325,17 @@ public double determinant()
 
  return eigenOutput.determinant();
  }
+
+ /**
+ * Retrieve the Characteristic Polynomial of the Eigenvalues
+ * 
+ * @return Characteristic Polynomial of the Eigenvalues
+ */
+
+ public R1ToR1 characteristicPolynomial()
+ {
+ EigenOutput eigenOutput = QREigenComponentExtractor.Standard().eigenize (_r2Array);
+
+ return null == eigenOutput ? null : eigenOutput.characteristicPolynomial();
+ }
 }