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();