Features:

	- Matrix Util QR Graham Schmidt Orthonormalization (1, 2, 3)
	- Matrix Util RQ Graham Schmidt Orthonormalization (6, 7, 8)
	- Matrix Util RQ Decomposition Composite (14, 15, 16)
	- Unsafe QR Graham Schmidt Orthogonalization (17, 18)
	- Square Matrix Annotated QR Decomposition (19, 20, 21)
	- Square Matrix Annotated RQ Decomposition (22, 23)


Bug Fixes/Re-organization:

	- Matrix Util QR Decomposition Revamp (10, 11, 12)


Samples:

	- Gram-Schmidt Process Test #1 (4, 5)
	- Gram-Schmidt Process Test #2 (9)
	- Gram-Schmidt Process Test #3 (13)


IdeaDRIP:

	- QR Algorithm (24-26)
	- The Practical QR Algorithm (27-50)
	- QR Algorithm Visualization (51-66)
	- QR Algorithm Visualization - Finding Eigenvalues versus Eigenvectors (67-87)
	- QR Algorithm Visualization - Speeding Up: Shifting and Deflation (88-97)
	- The Implicit QR Algorithm (98-104)
	- The Implicit QR Algorithm - Renaming Proposal (105, 106)
	- QR Algorithm Interpretation and Convergence (107-113)
	- QR Algorithm History (114-120)

ReleaseNotes/02_25_2024.txt

@@ -0,0 +1,34 @@
+
+Features:
+
+	- Matrix Util QR Graham Schmidt Orthonormalization (1, 2, 3)
+	- Matrix Util RQ Graham Schmidt Orthonormalization (6, 7, 8)
+	- Matrix Util RQ Decomposition Composite (14, 15, 16)
+	- Unsafe QR Graham Schmidt Orthogonalization (17, 18)
+	- Square Matrix Annotated QR Decomposition (19, 20, 21)
+	- Square Matrix Annotated RQ Decomposition (22, 23)
+
+
+Bug Fixes/Re-organization:
+
+	- Matrix Util QR Decomposition Revamp (10, 11, 12)
+
+
+Samples:
+
+	- Gram-Schmidt Process Test #1 (4, 5)
+	- Gram-Schmidt Process Test #2 (9)
+	- Gram-Schmidt Process Test #3 (13)
+
+
+IdeaDRIP:
+
+	- QR Algorithm (24-26)
+	- The Practical QR Algorithm (27-50)
+	- QR Algorithm Visualization (51-66)
+	- QR Algorithm Visualization - Finding Eigenvalues versus Eigenvectors (67-87)
+	- QR Algorithm Visualization - Speeding Up: Shifting and Deflation (88-97)
+	- The Implicit QR Algorithm (98-104)
+	- The Implicit QR Algorithm - Renaming Proposal (105, 106)
+	- QR Algorithm Interpretation and Convergence (107-113)
+	- QR Algorithm History (114-120)

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

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

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

@@ -247,6 +247,21 @@ public static final double[][] UnsafeRQGrahamSchmidtOrthogonalization (
 		return u;
 	}
 
+	/**
+	 * Orthogonalize the Specified Matrix Using the QR Graham-Schmidt Method. Unsafe Methods do not validate
+	 * 	the Input Arguments, so <b>use caution</b> in applying these Methods
+	 * 
+	 * @param a The Input Matrix
+	 * 
+	 * @return The QR Orthogonalized Matrix
+	 */
+
+	public static final double[][] UnsafeQRGrahamSchmidtOrthogonalization (
+		final double[][] a)
+	{
+		return Transpose (UnsafeRQGrahamSchmidtOrthogonalization (Transpose (a)));
+	}
+
 	/**
 	 * Indicate if the Cell corresponds to Bottom Left Location in the Matrix
 	 * 
@@ -1135,27 +1150,52 @@ public static final double[][] QRGrahamSchmidtOrthogonalization (
 		return Transpose (RQGrahamSchmidtOrthogonalization (Transpose (a)));
 	}
 
+	/**
+	 * Orthonormalize the Specified Matrix Using the RQ Graham-Schmidt Method
+	 * 
+	 * @param a The Input Matrix
+	 * 
+	 * @return The RQ Orthonormalized Matrix
+	 */
+
+	public static final double[][] RQGrahamSchmidtOrthonormalization (
+		final double[][] a)
+	{
+		double[][] u = RQGrahamSchmidtOrthogonalization (a);
+
+		if (null == u) {
+			return null;
+		}
+
+		for (int i = 0; i < u.length; ++i) {
+			double modulusReciprocal = 1. / UnsafeModulus (u[i]);
+
+			for (int j = 0; j < u.length; ++j) {
+				u[i][j] *= modulusReciprocal;
+			}
+		}
+
+		return u;
+	}
+
 	/**
 	 * Orthonormalize the Specified Matrix Using the QR Graham-Schmidt Method
 	 * 
 	 * @param a The Input Matrix
-	 * @param vectorsInColumns TRUE - Vectors in Columns
 	 * 
 	 * @return The QR Orthonormalized Matrix
 	 */
 
-	public static final double[][] GrahamSchmidtOrthonormalization (
-		final double[][] a,
-		final boolean vectorsInColumns)
+	public static final double[][] QRGrahamSchmidtOrthonormalization (
+		final double[][] a)
 	{
-		double[][] uOrthogonal = vectorsInColumns ? QRGrahamSchmidtOrthogonalization (a) :
-			 RQGrahamSchmidtOrthogonalization (a);
+		double[][] uOrthogonal = QRGrahamSchmidtOrthogonalization (a);
 
 		if (null == uOrthogonal) {
 			return null;
 		}
 
-		double[][] u = vectorsInColumns ? Transpose (uOrthogonal) : uOrthogonal;
+		double[][] u = Transpose (uOrthogonal);
 
 		for (int i = 0; i < u.length; ++i) {
 			double modulusReciprocal = 1. / UnsafeModulus (u[i]);
@@ -1165,23 +1205,43 @@ public static final double[][] GrahamSchmidtOrthonormalization (
 			}
 		}
 
-		return vectorsInColumns ? Transpose (u) : u;
+		return Transpose (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 boolean vectorsInColumns)
+		final double[][] a)
+	{
+		double[][] q = QRGrahamSchmidtOrthonormalization (a);
+
+		try {
+			return null == q ? null : new QR (q, Product (q, a));
+		} catch (Exception e) {
+			e.printStackTrace();
+		}
+
+		return null;
+	}
+
+	/**
+	 * Perform a RQ Decomposition on the Input Matrix
+	 * 
+	 * @param a The Input Matrix
+	 * 
+	 * @return The Output of RQ Decomposition
+	 */
+
+	public static final QR RQDecomposition (
+		final double[][] a)
 	{
-		double[][] q = GrahamSchmidtOrthonormalization (a, vectorsInColumns);
+		double[][] q = RQGrahamSchmidtOrthonormalization (a);
 
 		try {
 			return null == q ? null : new QR (q, Product (q, a));

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

@@ -265,6 +265,28 @@ public boolean isTriangularizable()
 		return true;
 	}
 
+	/**
+	 * Generate the QR Decomposition of the Square Matrix
+	 * 
+	 * @return QR Decomposition
+	 */
+
+	public QR qrDecomposition()
+	{
+		return MatrixUtil.QRDecomposition (_r2Array);
+	}
+
+	/**
+	 * Generate the RQ Decomposition of the Square Matrix
+	 * 
+	 * @return RQ Decomposition
+	 */
+
+	public QR rqDecomposition()
+	{
+		return MatrixUtil.RQDecomposition (_r2Array);
+	}
+
 	/**
 	 * Eigenize and Extract the Components of the Specified Matrix
 	 * 

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

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

@@ -133,7 +133,7 @@ public static final void main (
 			FormatUtil.FormatDouble ( MatrixUtil.DotProduct (uTranspose[0], uTranspose[1]), 1, 1, 1.)
 		);
 
-		double[][] q = MatrixUtil.GrahamSchmidtOrthonormalization (a, true);
+		double[][] q = MatrixUtil.QRGrahamSchmidtOrthonormalization (a);
 
 		NumberUtil.PrintMatrix ("ORTHONORMAL", q);
 

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, false);
+		QR qr = MatrixUtil.QRDecomposition (aadblA);
 
 		double[][] aadblR = qr.r();