Features:

	- Matrix Util Graham Schmidt Orthogonalization (1, 2, 3)
	- Matrix Util Graham Schmidt Orthonormalization #1 (4, 5)
	- Matrix Util QR Decomposition Vectors in Columns (6, 7)
	- QR Eigen-component Extractor Vectors in Columns (9)
	- Linear Solver Bartels Stewart Scheme (10)
	- Matrix Util Graham Schmidt Orthonormalization #2 (21, 22)
	- Matrix Util Graham Schmidt Orthonormalization #3 (23, 24)


Bug Fixes/Re-organization:

Samples:

	- Matrix Graham Schmidt Process #1 (8)
	- Matrix Graham Schmidt Process #2 (11, 12)
	- Matrix Graham Schmidt Process #3 (13, 14, 15)
	- Matrix Graham Schmidt Process #4 (16, 17, 18)
	- Matrix Graham Schmidt Process #5 (19, 20)
	- Matrix Graham Schmidt Process #6 (25, 26, 27)
	- Matrix Graham Schmidt Process #7 (28, 29, 30)
	- Matrix Graham Schmidt Process #8 (31, 32, 33)


IdeaDRIP:

	- QR Decomposition Computing the - Using Householder Reflections (34-82)
	- QR Decomposition Computing the - Using Householder Reflections - Example (83-108)
	- QR Decomposition Computing the - Using Householder Reflections - Advantages and Disadvantages (109, 110)
	- QR Decomposition Computing the - Using Givens Rotations (111-116)
	- QR Decomposition Computing the - Using Givens Rotations - Example (117-120)

ReleaseNotes/02_19_2024.txt

@@ -0,0 +1,33 @@
+
+Features:
+
+	- Matrix Util Graham Schmidt Orthogonalization (1, 2, 3)
+	- Matrix Util Graham Schmidt Orthonormalization #1 (4, 5)
+	- Matrix Util QR Decomposition Vectors in Columns (6, 7)
+	- QR Eigen-component Extractor Vectors in Columns (9)
+	- Linear Solver Bartels Stewart Scheme (10)
+	- Matrix Util Graham Schmidt Orthonormalization #2 (21, 22)
+	- Matrix Util Graham Schmidt Orthonormalization #3 (23, 24)
+
+
+Bug Fixes/Re-organization:
+
+Samples:
+
+	- Matrix Graham Schmidt Process #1 (8)
+	- Matrix Graham Schmidt Process #2 (11, 12)
+	- Matrix Graham Schmidt Process #3 (13, 14, 15)
+	- Matrix Graham Schmidt Process #4 (16, 17, 18)
+	- Matrix Graham Schmidt Process #5 (19, 20)
+	- Matrix Graham Schmidt Process #6 (25, 26, 27)
+	- Matrix Graham Schmidt Process #7 (28, 29, 30)
+	- Matrix Graham Schmidt Process #8 (31, 32, 33)
+
+
+IdeaDRIP:
+
+	- QR Decomposition Computing the - Using Householder Reflections (34-82)
+	- QR Decomposition Computing the - Using Householder Reflections - Example (83-108)
+	- QR Decomposition Computing the - Using Householder Reflections - Advantages and Disadvantages (109, 110)
+	- QR Decomposition Computing the - Using Givens Rotations (111-116)
+	- QR Decomposition Computing the - Using Givens Rotations - Example (117-120)

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

@@ -157,7 +157,8 @@ public int maxIterations()
 		final double[][] a)
 	{
 		org.drip.numerical.linearalgebra.QR qr = org.drip.numerical.linearalgebra.MatrixUtil.QRDecomposition (
-			a
+			a,
+			false
 		);
 
 		if (null == qr)
@@ -214,7 +215,8 @@ public int maxIterations()
 						v,
 						q
 					)
-				)
+				),
+				false
 			)))
 			{
 				return null;

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

@@ -1036,56 +1036,64 @@ public static final double[] Normalize (
 	/**
 	 * Orthogonalize the Specified Matrix Using the Graham-Schmidt Method
 	 * 
-	 * @param v The Input Matrix
+	 * @param a The Input Matrix
+	 * @param vectorsInColumns TRUE - Vectors in Columns
 	 * 
 	 * @return The Orthogonalized Matrix
 	 */
 
 	public static final double[][] GrahamSchmidtOrthogonalization (
-		final double[][] v)
+		final double[][] a,
+		final boolean vectorsInColumns)
 	{
-		if (null == v || 0 == v.length || v.length != v[0].length) {
+		if (null == a || 0 == a.length || a.length != a[0].length) {
 			return null;
 		}
 
-		double[][] u = new double[v.length][v.length];
+		double[][] aUsingVectorsInRows = vectorsInColumns ? Transpose (a) : a;
 
-		for (int i = 0; i < v.length; ++i) {
-			for (int j = 0; j < v.length; ++j) {
-				u[i][j] = v[i][j];
+		double[][] u = new double[aUsingVectorsInRows.length][aUsingVectorsInRows.length];
+
+		for (int i = 0; i < aUsingVectorsInRows.length; ++i) {
+			for (int j = 0; j < aUsingVectorsInRows.length; ++j) {
+				u[i][j] = aUsingVectorsInRows[i][j];
 			}
 		}
 
-		for (int i = 1; i < v.length; ++i) {
+		for (int i = 1; i < aUsingVectorsInRows.length; ++i) {
 			for (int j = 0; j < i; ++j) {
-				double[] projectionTrimOff = ProjectVOnUInternal (u[j], v[i]);
+				double[] projectionTrimOff = ProjectVOnUInternal (u[j], aUsingVectorsInRows[i]);
 
 				for (int k = 0; k < projectionTrimOff.length; ++k) {
 					u[i][k] -= projectionTrimOff[k];
 				}
 			}
 		}
 
-		return u;
+		return vectorsInColumns ? Transpose (u) : u;
 	}
 
 	/**
 	 * Orthonormalize the Specified Matrix Using the Graham-Schmidt Method
 	 * 
-	 * @param v The Input Matrix
+	 * @param a The Input Matrix
+	 * @param vectorsInColumns TRUE - Vectors in Columns
 	 * 
 	 * @return The Orthonormalized Matrix
 	 */
 
 	public static final double[][] GrahamSchmidtOrthonormalization (
-		final double[][] v)
+		final double[][] a,
+		final boolean vectorsInColumns)
 	{
-		double[][] u = GrahamSchmidtOrthogonalization (v);
+		double[][] uOrthogonal = GrahamSchmidtOrthogonalization (a, vectorsInColumns);
 
-		if (null == u) {
+		if (null == uOrthogonal) {
 			return null;
 		}
 
+		double[][] u = vectorsInColumns ? Transpose (uOrthogonal) : uOrthogonal;
+
 		for (int i = 0; i < u.length; ++i) {
 			double modulusReciprocal = 1. / ModulusInternal (u[i]);
 
@@ -1094,21 +1102,23 @@ public static final double[][] GrahamSchmidtOrthonormalization (
 			}
 		}
 
-		return u;
+		return vectorsInColumns ? Transpose (u) : u;
 	}
 
 	/**
 	 * Perform a QR Decomposition on the Input Matrix
 	 * 
 	 * @param a The Input Matrix
+	 * @param vectorsInColumns TRUE - Vectors in Columns
 	 * 
 	 * @return The Output of QR Decomposition
 	 */
 
 	public static final QR QRDecomposition (
-		final double[][] a)
+		final double[][] a,
+		final boolean vectorsInColumns)
 	{
-		double[][] q = GrahamSchmidtOrthonormalization (a);
+		double[][] q = GrahamSchmidtOrthonormalization (a, vectorsInColumns);
 
 		try {
 			return null == q ? null : new QR (q, Product (q, a));

src/main/java/org/drip/numerical/linearsolver/BartelsStewartScheme.java

@@ -189,7 +189,7 @@ public double[][] rhsMatrix()
 
 	public void solve()
 	{
-		QR qrA = MatrixUtil.QRDecomposition (_sylvesterEquation.squareMatrixA().r2Array());
+		QR qrA = MatrixUtil.QRDecomposition (_sylvesterEquation.squareMatrixA().r2Array(), false);
 
 		System.out.println();
 
@@ -214,7 +214,8 @@ public void solve()
 		System.out.println();
 
 		QR qrBTranspose = MatrixUtil.QRDecomposition (
-			_sylvesterEquation.squareMatrixB().transpose().r2Array()
+			_sylvesterEquation.squareMatrixB().transpose().r2Array(),
+			false
 		);
 
 		double[][] v = qrBTranspose.q();

src/main/java/org/drip/sample/matrix/GrahamSchmidtProcess.java

@@ -116,83 +116,43 @@ public static final void main (
 	{
 		EnvManager.InitEnv ("");
 
-		/* double[][] aadblV = new double[][] {
-			{3, 1, 4, 9},
-			{2, 2, 6, 0},
-			{1, 8, 3, 5},
-			{7, 0, 4, 5}
-		}; */
-
-		double[][] aadblV = new double[][] {
-			{ 12,   6,  -4},
-			{-51, 167,  24},
-			{  4, -68, -41}
-		};
-
-		/* double[][] aadblV = new double[][] {
+		double[][] a = new double[][] {
 			{12, -51,   4},
 			{ 6, 167, -68},
 			{-4,  24, -41}
-		}; */
+		};
 
-		double[][] aadblUOrthogonal = MatrixUtil.GrahamSchmidtOrthogonalization (aadblV);
+		double[][] u = MatrixUtil.GrahamSchmidtOrthogonalization (a, true);
 
-		NumberUtil.PrintMatrix (
-			"ORTHOGONAL",
-			aadblUOrthogonal
-		);
+		NumberUtil.PrintMatrix ("ORTHOGONAL", u);
+
+		double[][] uTranspose = MatrixUtil.Transpose (u);
 
 		System.out.println (
 			"ORTHOGONAL TEST: " +
-			FormatUtil.FormatDouble (
-				MatrixUtil.DotProduct (
-					aadblUOrthogonal[0],
-					aadblUOrthogonal[1]
-				),
-				1, 1, 1.
-			)
+			FormatUtil.FormatDouble ( MatrixUtil.DotProduct (uTranspose[0], uTranspose[1]), 1, 1, 1.)
 		);
 
-		double[][] aadblUOrthonormal = MatrixUtil.GrahamSchmidtOrthonormalization (aadblV);
+		double[][] q = MatrixUtil.GrahamSchmidtOrthonormalization (a, true);
 
-		NumberUtil.PrintMatrix (
-			"ORTHONORMAL",
-			aadblUOrthonormal
-		);
+		NumberUtil.PrintMatrix ("ORTHONORMAL", q);
+
+		double[][] qTranspose = MatrixUtil.Transpose (q);
 
 		System.out.println (
 			"ORTHONORMAL TEST: " +
-			FormatUtil.FormatDouble (
-				MatrixUtil.DotProduct (
-					aadblUOrthonormal[0],
-					aadblUOrthonormal[1]
-				),
-				1, 1, 1.
-			)
+			FormatUtil.FormatDouble (MatrixUtil.DotProduct (qTranspose[0], qTranspose[1]), 1, 1, 1.)
 		);
 
 		System.out.println();
 
-		double[][] uvTranspose = MatrixUtil.Product (MatrixUtil.Transpose (aadblUOrthonormal), aadblV);
+		double[][] uvTranspose = MatrixUtil.Product (MatrixUtil.Transpose (q), a);
 
-		NumberUtil.PrintMatrix (
-			"R CHECK #1.1",
-			uvTranspose
-		);
-
-		System.out.println();
-
-		NumberUtil.PrintMatrix (
-			"R CHECK #1.2",
-			MatrixUtil.Product (aadblUOrthonormal, uvTranspose)
-		);
+		NumberUtil.PrintMatrix ("R CHECK #1.1", uvTranspose);
 
 		System.out.println();
 
-		NumberUtil.PrintMatrix (
-			"R CHECK #1.3",
-			MatrixUtil.Product (MatrixUtil.Transpose (aadblV), aadblUOrthonormal)
-		);
+		NumberUtil.PrintMatrix ("R CHECK #1.2", MatrixUtil.Product (q, uvTranspose));
 
 		System.out.println();
 

src/main/java/org/drip/sample/matrix/QRDecomposition.java

@@ -128,7 +128,7 @@ public static final void main (
 			{ 0.0, 0.4, 1.0}
 		};
 
-		QR qr = MatrixUtil.QRDecomposition (aadblA);
+		QR qr = MatrixUtil.QRDecomposition (aadblA, false);
 
 		double[][] aadblR = qr.r();