Features:

	- R^1 -> R^1 Function Definition Condition Number (26, 27, 28)
	- R^1 -> R^1 Addition Operator Function #1 (29, 30, 31)
	- R^1 -> R^1 Addition Operator Function #2 (32, 33, 34)
	- R^1 -> R^1 Addition Operator Function Evaluate (35, 36)
	- R^1 -> R^1 Addition Operator Function a (37, 38)
	- R^1 -> R^1 Addition Operator Function Constructor (39, 40, 41)
	- R^1 -> R^1 Addition Operator Function Condition Number (42, 43, 44)
	- R^1 -> R^1 Addition Operator Function Derivative (45, 46)
	- Built-in R^1 -> R^1 Operator Functions (47)
	- R^1 -> R^1 Scale Operator Function Shell (59, 60)
	- R^1 -> R^1 Scale Operator Function "a" (61, 62)
	- R^1 -> R^1 Scale Operator Function Constructor (63, 64)
	- R^1 -> R^1 Scale Operator Function Evaluate (65, 66)
	- R^1 -> R^1 Scale Operator Function Derivative (67, 68)
	- R^1 -> R^1 Scale Operator Function Condition Number (69, 70)
	- R^1 -> R^1 Reciprocal Operator Function Condition Number (71, 72)
	- R^1 -> R^1 Exponential Operator Function Shell (73, 74)
	- R^1 -> R^1 Exponential Operator Function Constructor (75, 76)
	- R^1 -> R^1 Exponential Operator Function Evaluate (77, 78)
	- R^1 -> R^1 Exponential Operator Function Derivative (79, 80)
	- R^1 -> R^1 Exponential Operator Function Condition Number (81, 82)
	- R^1 -> R^1 Natural Logarithm Operator Function Shell (83, 84)
	- R^1 -> R^1 Natural Logarithm Operator Function Constructor (85, 86)
	- R^1 -> R^1 Natural Logarithm Operator Function Evaluate (87, 88)
	- R^1 -> R^1 Natural Logarithm Operator Function Condition Number (89, 90)
	- Built-in R^1 -> R^1 Trigonometric Functions (91, 92)
	- R^1 -> R^1 Sine Trigonometric Function Shell (93, 94)
	- R^1 -> R^1 Sine Trigonometric Function Constructor (95, 96)
	- R^1 -> R^1 Sine Trigonometric Function Evaluate (97, 98)
	- R^1 -> R^1 Sine Trigonometric Function Condition Number (99, 100)
	- R^1 -> R^1 Cosine Trigonometric Function Shell (101, 102)
	- R^1 -> R^1 Cosine Trigonometric Function Constructor (103, 104)
	- R^1 -> R^1 Cosine Trigonometric Function Evaluate (105, 106)
	- R^1 -> R^1 Cosine Trigonometric Function Condition Number (107, 108)
	- R^1 -> R^1 Tangent Trigonometric Function Shell (109, 110)
	- R^1 -> R^1 Tangent Trigonometric Function Constructor (111, 112)
	- R^1 -> R^1 Tangent Trigonometric Function Evaluate (113, 114)
	- R^1 -> R^1 Tangent Trigonometric Function Condition Number (115, 116)
	- R^1 -> R^1 Inverse Sine Trigonometric Function Shell (117, 118)
	- R^1 -> R^1 Inverse Sine Trigonometric Function Constructor (119, 120)


Bug Fixes/Re-organization:

	- R^1 -> R^1 Addition Operator Function Migration (48, 49)
	- R^1 -> R^1 Flat Operator Function Migration (50, 51)
	- R^1 -> R^1 Offset Idempotent Operator Migration (52)
	- R^1 -> R^1 Convolution Operator Function Migration (53, 54)
	- R^1 -> R^1 Reciprocal Operator Function Migration (55, 56)
	- R^1 -> R^1 Reflection Operator Function Migration (57, 58)


Samples:

IdeaDRIP:

	- Condition Number Matrices (1-5)
	- Condition Number Non-linear (6-7)
	- Condition Number Non-linear - One Variable (8-17)
	- Condition Number Non-linear - Several Variables (18-25)

IdeaDRIP/NumericalAnalysis/NA_v0.07

@@ -0,0 +1,192 @@
+
+ --------------------------
+ #1 - Successive Over-Relaxation
+ --------------------------
+ --------------------------
+ 1.1) Successive Over-Relaxation Method for A.x = b; A - n x n square matrix, x - unknown vector, b - RHS vector
+ 1.2) Decompose A into Diagonal, Lower, and upper Triangular Matrices; A = D + L + U
+ 1.3) SOR Scheme uses omega input
+ 1.4) Forward subsitution scheme to iteratively determine the x_i's
+ 1.5) SOR Scheme Linear System Convergence: Inputs A and D, Jacobi Iteration Matrix Spectral Radius, omega
+	- Construct Jacobi Iteration Matrix: C_Jacobian = I - (Inverse D) A
+	- Convergence Verification #1: Ensure that Jacobi Iteration Matrix Spectral Radius is < 1
+	- Convergence Verification #2: Ensure omega between 0 and 2
+	- Optimal Relaxation Parameter Expression in terms of Jacobi Iteration Matrix Spectral Radius
+	- Omega Based Convergence Rate Expression
+	- Gauss-Seidel omega = 1; corresponding Convergence Rate
+	- Optimal Omega Convergence Rate
+ 1.6) Generic Iterative Solver Method:
+	- Inputs: Iterator Function(x) and omega
+	- Unrelaxed Iterated variable: x_n+1 = f(x_n)
+	- SOR Based Iterated variable: x_n+1 = (1-omega).x_n + omega.f(x_n)
+	- SOR Based Iterated variable for Unknown Vector x: x_n+1 = (1-omega).x_n + omega.(L_star inverse)(b - U.x_n)
+ --------------------------
+
+ --------------------------
+ #2 - Successive Over-Relaxation
+ --------------------------
+ --------------------------
+ 2.1) SSOR Algorithm - Inputs; A, omega, and gamma
+	- Decompose into D and L
+	- Pre-conditioner Matrix: Expression from SSOR 
+	- Finally SSOR Iteration Formula
+ --------------------------
+
+ ----------------------------
+ #7 - Tridiagonal matrix algorithm
+ ----------------------------
+ ----------------------------
+ 7.1) Is Tridiagonal Check
+ 7.2) Core Algorithm:
+	- C Prime's and D Prime's Calculation
+	- Back Substitution for the Result
+	- Modified better Book-keeping algorithm
+ 7.3) Sherman-Morrison Algorithm:
+	- Choice of gamma
+	- Construct Tridiagonal B from A and gamma
+	- u Column from gamma and c_n
+	- v Column from a_1 and gamma
+	- Solve for y from By=d
+	- Solve for q from Bq=u
+	- Use Sherman Morrison Formula to extract x
+ 7.4) Alternate Boundary Condition Algorithm:
+	- Solve Au=d for u
+	- Solve Av={-a_2, 0, ..., -c_n} for v
+	- Full solution is x_i = u_i + x_1 * v_i
+	- x_1 if computed using formula
+ ----------------------------
+
+ ----------------------------
+ #8 - Triangular Matrix
+ ----------------------------
+ ----------------------------
+ 8.1) Description:
+	- Lower/Left Triangular Verification
+	- Upper/Right Triangular Verification
+	- Diagonal Matrix Verification
+	- Upper/Lower Trapezoidal Verification
+ 8.2) Forward/Back Substitution:
+	- Inputs => L and b
+		- Forward Substitution
+	- Inputs => U and b
+		- Back Substitution
+ 8.3) Properties:
+	- Is Matrix Normal, i.e.,  A times A transpose = A transpose times A
+	- Characteristic Polynomial
+	- Determinant/Permanent
+ 8.4) Special Forms:
+	- Upper/Lower Unitriangular Matrix Verification
+	- Upper/Lower Strictly Matrix Verification
+	- Upper/Lower Atomic Matrix Verification
+ ----------------------------
+
+ ----------------------------
+ #9 - Sylvester Equation
+ ----------------------------
+ ----------------------------
+ 9.1) Matrix Form:
+	- Inputs: A, B, and C
+	- Size Constraints Verification
+ 9.2) Solution Criteria:
+	- Co-joint EigenSpectrum between A and B
+ 9.3) Numerical Solution:
+	- Decomposition of A/B using Schur Decomposition into Triangular Form
+	- Forward/Back Substitution
+ ----------------------------
+
+ ----------------------------
+ #10 - Bartels-Stewart Algorithm
+ ----------------------------
+ ----------------------------
+ 10.1) Matrix Form:
+	- Inputs: A, B, and C
+	- Size Constraints Verification
+ 10.2) Schur Decompositions:
+	- R = U^T A U - emits U and R
+	- S = V^T B^T V - emits V and S
+	- F = U^T C V
+	- Y = U^T X V
+	- Solution to R.Y - Y.S^T = F
+	- Finally X = U.Y.V^T
+ 10.3) Computational Costs:
+	- Flops cost for Schur decomposition
+	- Flops cost overall
+ 10.4) Hessenberg-Schur Decompositions:
+	- R = U^T A U becomes H = Q^T A Q - thus emits Q and H (Upper Hessenberg)
+	- Computational Costs
+ ----------------------------
+
+ ----------------------------
+ #11 - Gershgorin Circle Theorem
+ ----------------------------
+ ----------------------------
+ 11.1) Gershgorin Disc:
+	- Diagonal Entry
+	- Radius
+	- One disc per Row/Column in Square Matrix
+	- Optimal Disc based on Row/Column
+ 11.2) Tolerance based Gershgorin Convergence Criterion
+ 11.3) Joint and Disjoint Discs
+ 11.4) Gershgorin Strengthener
+ 11.5) Row/Column Diagonal Dominance
+ ----------------------------
+
+ ----------------------------
+ #12 - Condition Number
+ ----------------------------
+ ----------------------------
+ 12.1) Condition Number Calculation:
+	- Absolute Error
+	- Relative Error
+	- Absolute Condition Number
+	- Relative Condition Number
+ 12.2) Matrix Condition Number Calculation:
+	- Condition Number as a Product of Regular and Inverse Norms
+	- L2 Norm
+	- L2 Norm for Normal Matrix
+	- L2 Norm for Unitary Matrix
+	- Default Condition Number
+	- L Infinity Norm
+	- L Infinity Norm Triangular Matrix
+ 12.3) Non-linear
+	- One Variable
+		- Basic Formula
+		- Condition Numbers for Common Functions
+	- Multi Variables
+ ----------------------------
+
+ --------------------------
+ #13 - Crank–Nicolson method
+ --------------------------
+ --------------------------
+ 13.1) von Neumann Stability Validator - Inputs; time-step, diffusivity, space step
+	- 1D => time step * diffusivity / space step ^2 < 0.5
+	- nD => time step * diffusivity / (space step hypotenuse) ^ 2 < 0.5
+ 13.2) Set up:
+	- Input: Spatial variable x
+	- Input: Time variable t
+	- Inputs: State variable u, du/dx, d2u/dx2 - all at time t
+	- Second Order, 1D => du/dt = F (u, x, t, du/dx, d2u/dx2)
+ 13.3) Finite Difference Evolution Scheme:
+	- Time Step delta_t, space step delta_x
+	- Forward Difference: F calculated at x
+	- Backward Difference: F calculated at x + delta_x
+	- Crank Nicolson: Average of Forward/Backward
+ 13.4) 1D Diffusion:
+	- Inputs: 1D von Neumann Stability Validator, Number of time/space steps
+	- Time Index n, Space Index i
+	- Explicit Tridiagonal form for the discretization - State concentration at n+1 given state concentration at n
+	- Non-linear diffusion coefficient:
+		- Linearization across x_i and x_i+1
+		- Quasi-explicit forms accommodation
+ 13.5) 2D Diffusion:
+	- Inputs: 2D von Neumann Stability Validator, Number of time/space steps
+	- Time Index n, Space Index i, j
+	- Explicit Tridiagonal form for the discretization - State concentration at n+1 given state concentration at n
+	- Explicit Solution using the Alternative Difference Implicit Scheme
+ 13.6) Extension to Iterative Solver Schemes above:
+	- Input: State Space "velocity" dF/du
+	- Input: State "Step Size" delta_u
+	- Fixed Point Iterative Location Scheme
+	- Relaxation Scheme based Robustness => Input: Relaxation Parameter
+ --------------------------

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_v6.44.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_v6.45.pdf) [*Previous*](https://github.com/lakshmiDRIP/DROP/blob/master/Docs/Internal/NumericalAnalysis) |
  | User Guide              |  |
  | API                     | [*Javadoc*](https://lakshmidrip.github.io/DROP/Javadoc/index.html)|
 
@@ -408,6 +408,14 @@ Numerical Analysis Library contains the supporting Functionality for Numerical M
 	* Application
 	* Example
 	* References
+ * Condition Number
+	* Introduction
+	* General Definition in the Context of Error Analysis
+	* Matrices
+	* Non-linear
+		* One variable
+		* Several variables
+	* References
 
 
 ## DROP Specifications

ReleaseNotes/03_04_2024.txt

@@ -0,0 +1,16 @@
+
+Features:
+
+Bug Fixes/Re-organization:
+
+Samples:
+
+IdeaDRIP:
+
+	- Condition Number Matrices #1 (1-33)
+	- Condition Number Non-linear (34-35)
+	- Condition Number Non-linear - One Variable (36-45)
+	- Condition Number Non-linear - Several Variables (46-53)
+	- Condition Number Matrices - Introduction (54-77)
+	- Condition Number General Definition in the Context of Error Analysis (78-81)
+	- Condition Number Matrices #2 (82-120)

ReleaseNotes/03_05_2024.txt

@@ -0,0 +1,63 @@
+
+Features:
+
+	- R^1 -> R^1 Function Definition Condition Number (26, 27, 28)
+	- R^1 -> R^1 Addition Operator Function #1 (29, 30, 31)
+	- R^1 -> R^1 Addition Operator Function #2 (32, 33, 34)
+	- R^1 -> R^1 Addition Operator Function Evaluate (35, 36)
+	- R^1 -> R^1 Addition Operator Function a (37, 38)
+	- R^1 -> R^1 Addition Operator Function Constructor (39, 40, 41)
+	- R^1 -> R^1 Addition Operator Function Condition Number (42, 43, 44)
+	- R^1 -> R^1 Addition Operator Function Derivative (45, 46)
+	- Built-in R^1 -> R^1 Operator Functions (47)
+	- R^1 -> R^1 Scale Operator Function Shell (59, 60)
+	- R^1 -> R^1 Scale Operator Function "a" (61, 62)
+	- R^1 -> R^1 Scale Operator Function Constructor (63, 64)
+	- R^1 -> R^1 Scale Operator Function Evaluate (65, 66)
+	- R^1 -> R^1 Scale Operator Function Derivative (67, 68)
+	- R^1 -> R^1 Scale Operator Function Condition Number (69, 70)
+	- R^1 -> R^1 Reciprocal Operator Function Condition Number (71, 72)
+	- R^1 -> R^1 Exponential Operator Function Shell (73, 74)
+	- R^1 -> R^1 Exponential Operator Function Constructor (75, 76)
+	- R^1 -> R^1 Exponential Operator Function Evaluate (77, 78)
+	- R^1 -> R^1 Exponential Operator Function Derivative (79, 80)
+	- R^1 -> R^1 Exponential Operator Function Condition Number (81, 82)
+	- R^1 -> R^1 Natural Logarithm Operator Function Shell (83, 84)
+	- R^1 -> R^1 Natural Logarithm Operator Function Constructor (85, 86)
+	- R^1 -> R^1 Natural Logarithm Operator Function Evaluate (87, 88)
+	- R^1 -> R^1 Natural Logarithm Operator Function Condition Number (89, 90)
+	- Built-in R^1 -> R^1 Trigonometric Functions (91, 92)
+	- R^1 -> R^1 Sine Trigonometric Function Shell (93, 94)
+	- R^1 -> R^1 Sine Trigonometric Function Constructor (95, 96)
+	- R^1 -> R^1 Sine Trigonometric Function Evaluate (97, 98)
+	- R^1 -> R^1 Sine Trigonometric Function Condition Number (99, 100)
+	- R^1 -> R^1 Cosine Trigonometric Function Shell (101, 102)
+	- R^1 -> R^1 Cosine Trigonometric Function Constructor (103, 104)
+	- R^1 -> R^1 Cosine Trigonometric Function Evaluate (105, 106)
+	- R^1 -> R^1 Cosine Trigonometric Function Condition Number (107, 108)
+	- R^1 -> R^1 Tangent Trigonometric Function Shell (109, 110)
+	- R^1 -> R^1 Tangent Trigonometric Function Constructor (111, 112)
+	- R^1 -> R^1 Tangent Trigonometric Function Evaluate (113, 114)
+	- R^1 -> R^1 Tangent Trigonometric Function Condition Number (115, 116)
+	- R^1 -> R^1 Inverse Sine Trigonometric Function Shell (117, 118)
+	- R^1 -> R^1 Inverse Sine Trigonometric Function Constructor (119, 120)
+
+
+Bug Fixes/Re-organization:
+
+	- R^1 -> R^1 Addition Operator Function Migration (48, 49)
+	- R^1 -> R^1 Flat Operator Function Migration (50, 51)
+	- R^1 -> R^1 Offset Idempotent Operator Migration (52)
+	- R^1 -> R^1 Convolution Operator Function Migration (53, 54)
+	- R^1 -> R^1 Reciprocal Operator Function Migration (55, 56)
+	- R^1 -> R^1 Reflection Operator Function Migration (57, 58)
+
+
+Samples:
+
+IdeaDRIP:
+
+	- Condition Number Matrices (1-5)
+	- Condition Number Non-linear (6-7)
+	- Condition Number Non-linear - One Variable (8-17)
+	- Condition Number Non-linear - Several Variables (18-25)

src/main/java/org/drip/execution/adaptive/CoordinatedVariationTrajectoryGenerator.java

@@ -146,7 +146,7 @@ private org.drip.execution.dynamics.LinearPermanentExpectationParameters realize
 		try {
 			return new org.drip.execution.dynamics.LinearPermanentExpectationParameters (new
 				org.drip.execution.parameters.ArithmeticPriceDynamicsSettings (0., new
-					org.drip.function.r1tor1.FlatUnivariate (_cv.referenceVolatility() * java.lang.Math.exp
+					org.drip.function.r1tor1operator.Flat (_cv.referenceVolatility() * java.lang.Math.exp
 						(-0.5 * dblMarketState)), 0.), new
 							org.drip.execution.profiletime.UniformParticipationRateLinear
 								(org.drip.execution.impact.ParticipationRateLinear.NoImpact()), new

src/main/java/org/drip/execution/athl/DynamicsParameters.java

@@ -158,7 +158,7 @@ public org.drip.execution.dynamics.LinearPermanentExpectationParameters almgren2
 		try {
 			return org.drip.execution.dynamics.ArithmeticPriceEvolutionParametersBuilder.Almgren2003 (new
 				org.drip.execution.parameters.ArithmeticPriceDynamicsSettings (0., new
-					org.drip.function.r1tor1.FlatUnivariate (_afp.dailyVolatility()), 0.), new
+					org.drip.function.r1tor1operator.Flat (_afp.dailyVolatility()), 0.), new
 						org.drip.execution.profiletime.UniformParticipationRateLinear (new
 							org.drip.execution.athl.PermanentImpactNoArbitrage (_afp)), new
 								org.drip.execution.profiletime.UniformParticipationRate (new

src/main/java/org/drip/execution/dynamics/ArithmeticPriceEvolutionParametersBuilder.java

@@ -195,7 +195,7 @@ public static final org.drip.execution.dynamics.LinearPermanentExpectationParame
 		try {
 			return new org.drip.execution.dynamics.ArithmeticPriceEvolutionParameters (new
 				org.drip.execution.parameters.ArithmeticPriceDynamicsSettings (0., new
-					org.drip.function.r1tor1.FlatUnivariate (dblPriceVolatility), 0.), new
+					org.drip.function.r1tor1operator.Flat (dblPriceVolatility), 0.), new
 						org.drip.execution.profiletime.UniformParticipationRate
 							(org.drip.execution.impact.ParticipationRateLinear.NoImpact()),
 								bprlTemporaryExpectation, new
@@ -255,7 +255,7 @@ public static final org.drip.execution.dynamics.ArithmeticPriceEvolutionParamete
 			final org.drip.execution.tradingtime.CoordinatedVariation cv)
 	{
 		try {
-			return CoordinatedVariation (new org.drip.function.r1tor1.FlatUnivariate (dblPriceVolatility),
+			return CoordinatedVariation (new org.drip.function.r1tor1operator.Flat (dblPriceVolatility),
 				cv);
 		} catch (java.lang.Exception e) {
 			e.printStackTrace();
@@ -281,7 +281,7 @@ public static final org.drip.execution.dynamics.ArithmeticPriceEvolutionParamete
 		try {
 			return new org.drip.execution.dynamics.LinearPermanentExpectationParameters (new
 				org.drip.execution.parameters.ArithmeticPriceDynamicsSettings (0., new
-					org.drip.function.r1tor1.FlatUnivariate (cv.referenceVolatility()), 0.), new
+					org.drip.function.r1tor1operator.Flat (cv.referenceVolatility()), 0.), new
 						org.drip.execution.profiletime.UniformParticipationRateLinear
 							(org.drip.execution.impact.ParticipationRateLinear.NoImpact()), new
 								org.drip.execution.profiletime.UniformParticipationRateLinear

src/main/java/org/drip/execution/parameters/ArithmeticPriceDynamicsSettings.java

@@ -147,7 +147,7 @@ public static final ArithmeticPriceDynamicsSettings FromAnnualReturnsSettings (
 
 		try {
 			return new ArithmeticPriceDynamicsSettings (dblPrice * dblAnnualReturnsExpectation / 250., new
-				org.drip.function.r1tor1.FlatUnivariate (dblPrice * dblAnnualReturnsVolatility /
+				org.drip.function.r1tor1operator.Flat (dblPrice * dblAnnualReturnsVolatility /
 					java.lang.Math.sqrt (250.)), dblSerialCorrelation);
 		} catch (java.lang.Exception e) {
 			e.printStackTrace();

src/main/java/org/drip/function/definition/R1ToR1.java

@@ -414,4 +414,21 @@ public org.drip.function.definition.PoleResidue poleResidue (
 	{
 		return null;
 	}
+
+	/**
+	 * Compute the Condition Number at the specified Variate
+	 * 
+	 * @param x Variate
+	 * 
+	 * @return The Condition Number
+	 * 
+	 * @throws Exception Thrown if the Condition Number cannor be computed
+	 */
+
+	public double conditionNumber (
+		final double x)
+		throws Exception
+	{
+		return x * derivative (x, 1) / evaluate (x);
+	}
 }

src/main/java/org/drip/function/r1tor1/BernsteinPolynomial.java

@@ -97,7 +97,7 @@
  * @author Lakshmi Krishnamurthy
  */
 
-public class BernsteinPolynomial extends org.drip.function.r1tor1.UnivariateConvolution {
+public class BernsteinPolynomial extends org.drip.function.r1tor1operator.Convolution {
 
 	/**
 	 * Construct a BernsteinPolynomial instance
@@ -114,7 +114,7 @@ public BernsteinPolynomial (
 		throws java.lang.Exception
 	{
 		super (new org.drip.function.r1tor1.NaturalLogSeriesElement (iBaseExponent), new
-			org.drip.function.r1tor1.UnivariateReflection (new org.drip.function.r1tor1.NaturalLogSeriesElement
+			org.drip.function.r1tor1operator.Reflection (new org.drip.function.r1tor1.NaturalLogSeriesElement
 				(iComplementExponent)));
 	}