Features:

- RQ Graham Schmidt Orthogonalization (95, 96, 97) - QR Graham Schmidt Orthogonalization #1 (98, 99) - QR Graham Schmidt Orthogonalization #2 (100, 101) - QR Graham Schmidt Orthogonalization #3 (106, 107) - QR Graham Schmidt Orthogonalization #4 (109, 110, 111) - QR Graham Schmidt Orthogonalization #5 (113) - Unsafe RQ Graham Schmidt Orthogonalization (114, 115, 116) - RQ Graham Schmidt Orthogonalization Revamp (118) Bug Fixes/Re-organization: Samples: - Gram-Schmidt Process Test #1 (102, 103) - Gram-Schmidt Process Test #2 (104, 105) - Gram-Schmidt Process Test #3 (108) - Gram-Schmidt Process Test #4 (112) - Gram-Schmidt Process Test #5 (117) - Gram-Schmidt Process Test #6 (119, 120) IdeaDRIP: - Computing the QR Decomposition (1-2) - Computing the QR Decomposition using Gram-Schmidt Process (3-10) - Computing the QR Decomposition using Gram-Schmidt Process – Relation to RQ (11-14) - Computing the QR Decomposition using Gram-Schmidt Process – Advantages and Disadvantages (15-18) - Computing the QR Decomposition using Householder Reflections (19-34) - Computing the QR Decomposition using Householder Reflections – Advantages and Disadvantages (35-34) - Computing the QR Decomposition using Givens' Rotations (39-46) - Computing the QR Decomposition using Givens' Rotations – Advantages and Disadvantages (47-52) - QR Decomposition Connection to a Determinant or a Product of Eigenvalues (53-68) - QR Decomposition Column Pivoting (69-76) - QR Decomposition Use for Solution to Linear Inverse Problems (77-92) - QR Decomposition Generalization (93-94)
lakshmiDRIP · Jul 20, 2024 · 169df7c · 169df7c
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diff --git a/...ricalAnalysis/NumericalAnalysis_v6.42.pdf → ...ricalAnalysis/NumericalAnalysis_v6.43.pdf b/...ricalAnalysis/NumericalAnalysis_v6.42.pdf → ...ricalAnalysis/NumericalAnalysis_v6.43.pdf
diff --git a/Docs/Internal/NumericalAnalysis/Shruti/Source/NumericalAnalysis_v6.43.docx b/Docs/Internal/NumericalAnalysis/Shruti/Source/NumericalAnalysis_v6.43.docx
diff --git a/Docs/Internal/NumericalAnalysis/Shruti/Source/~$mericalAnalysis_v6.42.docx b/Docs/Internal/NumericalAnalysis/Shruti/Source/~$mericalAnalysis_v6.42.docx
diff --git a/Docs/Internal/NumericalAnalysis/Shruti/Source/~$mericalAnalysis_v6.43.docx b/Docs/Internal/NumericalAnalysis/Shruti/Source/~$mericalAnalysis_v6.43.docx
diff --git a/NumericalAnalysisLibrary.md b/NumericalAnalysisLibrary.md
@@ -11,7 +11,7 @@ Numerical Analysis Library contains the supporting Functionality for Numerical M
 
  |        Document         | Link |
  |-------------------------|------|
- | Technical Specification | [*Latest*](https://github.com/lakshmiDRIP/DROP/blob/master/Docs/Internal/NumericalAnalysis/NumericalAnalysis_v6.41.pdf) [*Previous*](https://github.com/lakshmiDRIP/DROP/blob/master/Docs/Internal/NumericalAnalysis) |
+ | Technical Specification | [*Latest*](https://github.com/lakshmiDRIP/DROP/blob/master/Docs/Internal/NumericalAnalysis/NumericalAnalysis_v6.43.pdf) [*Previous*](https://github.com/lakshmiDRIP/DROP/blob/master/Docs/Internal/NumericalAnalysis) |
  | User Guide              |  |
  | API                     | [*Javadoc*](https://lakshmidrip.github.io/DROP/Javadoc/index.html)|
 
@@ -378,6 +378,28 @@ Numerical Analysis Library contains the supporting Functionality for Numerical M
 		* Borel Subgroups and Borel Subalgebras
 		* Examples
 	* References
+ * QR Decomposition
+	* Introduction
+	* Cases and Definitions
+		* Square Matrix
+		* Rectangular Matrix
+		* QL, RQ and LQ Decompositions
+	* Computing the QR Decomposition
+		* Using the Gram–Schmidt Process
+			* Example
+			* Relation to RQ decomposition
+			* Advantages and Disadvantages
+		* Using Householder Reflections
+			* Example
+			* Advantages and Disadvantages
+		* Using Givens Rotations
+			* Example
+			* Advantages and Disadvantages
+		* Connection to a Determinant or Product of Eigenvalues
+		* Column Pivoting
+		* Using for Solution to Linear Inverse Problems
+		* Generalizations
+		* References
 
 
 ## DROP Specifications

diff --git a/ReleaseNotes/02_22_2024.txt b/ReleaseNotes/02_22_2024.txt
@@ -0,0 +1,39 @@
+
+Features:
+
+	- RQ Graham Schmidt Orthogonalization (95, 96, 97)
+	- QR Graham Schmidt Orthogonalization #1 (98, 99)
+	- QR Graham Schmidt Orthogonalization #2 (100, 101)
+	- QR Graham Schmidt Orthogonalization #3 (106, 107)
+	- QR Graham Schmidt Orthogonalization #4 (109, 110, 111)
+	- QR Graham Schmidt Orthogonalization #5 (113)
+	- Unsafe RQ Graham Schmidt Orthogonalization (114, 115, 116)
+	- RQ Graham Schmidt Orthogonalization Revamp (118)
+
+
+Bug Fixes/Re-organization:
+
+Samples:
+
+	- Gram-Schmidt Process Test #1 (102, 103)
+	- Gram-Schmidt Process Test #2 (104, 105)
+	- Gram-Schmidt Process Test #3 (108)
+	- Gram-Schmidt Process Test #4 (112)
+	- Gram-Schmidt Process Test #5 (117)
+	- Gram-Schmidt Process Test #6 (119, 120)
+
+
+IdeaDRIP:
+
+	- Computing the QR Decomposition (1-2)
+	- Computing the QR Decomposition using Gram-Schmidt Process (3-10)
+	- Computing the QR Decomposition using Gram-Schmidt Process – Relation to RQ (11-14)
+	- Computing the QR Decomposition using Gram-Schmidt Process – Advantages and Disadvantages (15-18)
+	- Computing the QR Decomposition using Householder Reflections (19-34)
+	- Computing the QR Decomposition using Householder Reflections – Advantages and Disadvantages (35-34)
+	- Computing the QR Decomposition using Givens' Rotations (39-46)
+	- Computing the QR Decomposition using Givens' Rotations – Advantages and Disadvantages (47-52)
+	- QR Decomposition Connection to a Determinant or a Product of Eigenvalues (53-68)
+	- QR Decomposition Column Pivoting (69-76)
+	- QR Decomposition Use for Solution to Linear Inverse Problems (77-92)
+	- QR Decomposition Generalization (93-94)
diff --git a/src/main/java/org/drip/numerical/linearalgebra/MatrixUtil.java b/src/main/java/org/drip/numerical/linearalgebra/MatrixUtil.java
@@ -214,6 +214,39 @@ private static final double[][] UnsafeProduct (
 		return product;
 	}
 
+	/**
+	 * Orthogonalize the Specified Matrix Using the RQ 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 RQ Orthogonalized Matrix
+	 */
+
+	public static final double[][] UnsafeRQGrahamSchmidtOrthogonalization (
+		final double[][] a)
+	{
+		double[][] u = new double[a.length][a.length];
+
+		for (int i = 0; i < a.length; ++i) {
+			for (int j = 0; j < a.length; ++j) {
+				u[i][j] = a[i][j];
+			}
+		}
+
+		for (int i = 1; i < a.length; ++i) {
+			for (int j = 0; j < i; ++j) {
+				double[] projectionTrimOff = UnsafeProjectVOnU (u[j], a[i]);
+
+				for (int k = 0; k < projectionTrimOff.length; ++k) {
+					u[i][k] -= projectionTrimOff[k];
+				}
+			}
+		}
+
+		return u;
+	}
+
 	/**
 	 * Indicate if the Cell corresponds to Bottom Left Location in the Matrix
 	 * 
@@ -1074,59 +1107,49 @@ public static final double[] Normalize (
 	}
 
 	/**
-	 * Orthogonalize the Specified Matrix Using the Graham-Schmidt Method
+	 * Orthogonalize the Specified Matrix Using the RQ Graham-Schmidt Method
 	 * 
 	 * @param a The Input Matrix
-	 * @param vectorsInColumns TRUE - Vectors in Columns
 	 * 
-	 * @return The Orthogonalized Matrix
+	 * @return The RQ Orthogonalized Matrix
 	 */
 
-	public static final double[][] GrahamSchmidtOrthogonalization (
-		final double[][] a,
-		final boolean vectorsInColumns)
+	public static final double[][] RQGrahamSchmidtOrthogonalization (
+		final double[][] a)
 	{
-		if (null == a || 0 == a.length || a.length != a[0].length) {
-			return null;
-		}
-
-		double[][] aUsingVectorsInRows = vectorsInColumns ? Transpose (a) : a;
-
-		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 < aUsingVectorsInRows.length; ++i) {
-			for (int j = 0; j < i; ++j) {
-				double[] projectionTrimOff = UnsafeProjectVOnU (u[j], aUsingVectorsInRows[i]);
+		return null == a || 0 == a.length || a.length != a[0].length ?
+			null : UnsafeRQGrahamSchmidtOrthogonalization (a);
+	}
 
-				for (int k = 0; k < projectionTrimOff.length; ++k) {
-					u[i][k] -= projectionTrimOff[k];
-				}
-			}
-		}
+	/**
+	 * Orthogonalize the Specified Matrix Using the QR Graham-Schmidt Method
+	 * 
+	 * @param a The Input Matrix
+	 * 
+	 * @return The QR Orthogonalized Matrix
+	 */
 
-		return vectorsInColumns ? Transpose (u) : u;
+	public static final double[][] QRGrahamSchmidtOrthogonalization (
+		final double[][] a)
+	{
+		return Transpose (RQGrahamSchmidtOrthogonalization (Transpose (a)));
 	}
 
 	/**
-	 * Orthonormalize the Specified Matrix Using the Graham-Schmidt Method
+	 * Orthonormalize the Specified Matrix Using the QR Graham-Schmidt Method
 	 * 
 	 * @param a The Input Matrix
 	 * @param vectorsInColumns TRUE - Vectors in Columns
 	 * 
-	 * @return The Orthonormalized Matrix
+	 * @return The QR Orthonormalized Matrix
 	 */
 
 	public static final double[][] GrahamSchmidtOrthonormalization (
 		final double[][] a,
 		final boolean vectorsInColumns)
 	{
-		double[][] uOrthogonal = GrahamSchmidtOrthogonalization (a, vectorsInColumns);
+		double[][] uOrthogonal = vectorsInColumns ? QRGrahamSchmidtOrthogonalization (a) :
+			 RQGrahamSchmidtOrthogonalization (a);
 
 		if (null == uOrthogonal) {
 			return null;

diff --git a/src/main/java/org/drip/sample/matrix/GrahamSchmidtProcess.java b/src/main/java/org/drip/sample/matrix/GrahamSchmidtProcess.java
@@ -122,7 +122,7 @@ public static final void main (
 			{-4,  24, -41}
 		};
 
-		double[][] u = MatrixUtil.GrahamSchmidtOrthogonalization (a, true);
+		double[][] u = MatrixUtil.QRGrahamSchmidtOrthogonalization (a);
 
 		NumberUtil.PrintMatrix ("ORTHOGONAL", u);