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	- R1 Matrix Util Unsafe Power (1, 2, 3)
	- R1 Matrix Util Product Power (4, 5, 6)
	- R1 Square Matrix k Power (37, 38, 39)
	- Numerical Estimation R^k To R^1 Series (40, 41, 42)
	- Implementation of Jordan Normal Form #1 (43, 44, 45)
	- Numerical Jordan Form JSubM Shell (46, 47, 48)
	- Numerical Jordan Form JSubM mSubI (49, 50)
	- Numerical Jordan Form JSubM Lambda (51, 52)
	- Numerical Jordan Form JSubM R1 Grid (53, 54)
	- Numerical Jordan Form JSubM Constructor #1 (55, 56, 57)
	- Numerical Jordan Form JSubM Constructor #2 (58, 59, 60)
	- Numerical Jordan Form JSubM Power #1 (61, 62, 63)
	- Numerical Jordan Form JSubM Power #2 (64, 65, 66)
	- Numerical Jordan Form J Shell (67, 68, 69)
	- Numerical Jordan Form J jSubMArray (70, 71, 72)
	- Numerical Jordan Form J R1 Grid (73, 74, 75)
	- Numerical Jordan Form J Constructor #1 (76, 77, 78)
	- Numerical Jordan Form J Constructor #2 (79, 80, 81)
	- Numerical Jordan Form J Power #1 (82, 83, 84)
	- Numerical Jordan Form J Power #2 (85, 86, 87)
	- Jordan Form VJ Decomposition J (91, 92)
	- Jordan Form VJ Decomposition V (93, 94)
	- Jordan Form VJ Decomposition Constructor (95, 96, 97)
	- Numerical Jordan Form J Diagonal (98, 99)
	- Numerical Jordan Form VJ Decomposition Recover Original (100, 101, 102)
	- Implementation of Matrix Norm Variants (104, 105)
	- R1 Square Matrix Norm #1 (106, 107, 108)
	- R1 Square Matrix Norm #2 (109, 110, 111)
	- R1 Square Matrix Norm #3 (112, 113, 114)
	- R1 Square Evaluator Validator Shell (115, 116, 117)
	- R1 Square Positive Valued Norm (118, 119, 120)


Bug Fixes/Re-organization:

	- Implementation of Jordan Normal Form #2 (103)


Samples:

	- Matrix Power Series #1 (7, 8, 9)
	- Matrix Power Series #2 (10, 11, 12)
	- Matrix Power Series #3 (13, 14, 15)
	- Matrix Power Series #4 (16, 17, 18)
	- Matrix Power Series #5 (19, 20, 21)
	- Matrix Power Series #6 (22, 23, 24)
	- Matrix Power Series #7 (25, 26, 27)
	- Matrix Power Series #8 (28, 29, 30)
	- Matrix Power Series #9 (31, 32, 33)
	- Matrix Power Series #10 (34, 35, 36)


IdeaDRIP:
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Lakshmik committed Aug 31, 2024
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28 changes: 14 additions & 14 deletions IdeaDRIP/NumericalAnalysis/NA_v0.09
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#14 - Spectral Radius
#15 - Spectral Radius
----------------------------
----------------------------
14.1) Definition - Matrix:
15.1) Definition - Matrix:
- Spectral Radius - Max of the Eigenvalues
14.2) Definition - Bounded Complex Linear Operators:
15.2) Definition - Bounded Complex Linear Operators:
- Spectral Radius - Max of the Spectrum
- Gelfand Formula applies to Bounded Complex Linear Operators
14.3) Definition - Graph:
15.3) Definition - Graph:
- Function on Graph Vertex
- Square Integrability of Function across Vertexes
- Graph Function Map - Square Integrability Function Space across Vertexes 1 -> 2
- Graph Adjacency Operator - Function Sum over edges of a vertex
14.4)Upper bounds
15.4)Upper bounds
- Upper bounds on the spectral radius of a matrix <= Power (Norm (A^k), 1/k)
- Upper bounds for spectral radius of a graph
14.5) Jordan Normal Power Sequence
15.5) Jordan Normal Power Sequence
- J (m_i, lambda_i)
- J (m_i, lambda_i) power k
- Jordan Normal J Matrix
- Jordan Normal J Matrix power k
- A = V.J.V^-1
- A^k = V.J^k.V^-1
14.6) Gelfand's formula
15.6) Gelfand's formula
- The Formula
- Gelfand's formula for Commuting Matrices
----------------------------

--------------------------
#15 - Crank–Nicolson method
#16 - Crank–Nicolson method
--------------------------
--------------------------
15.1) von Neumann Stability Validator - Inputs; time-step, diffusivity, space step
16.1) 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
15.2) Set up:
16.2) 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)
15.3) Finite Difference Evolution Scheme:
16.3) 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
15.4) 1D Diffusion:
16.4) 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
15.5) 2D Diffusion:
16.5) 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
15.6) Extension to Iterative Solver Schemes above:
16.6) Extension to Iterative Solver Schemes above:
- Input: State Space "velocity" dF/du
- Input: State "Step Size" delta_u
- Fixed Point Iterative Location Scheme
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--------------------------
#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 - Unitary Matrix
----------------------------
----------------------------
13.1) Properties:
- U.U conjugate = I defines unitary matrix
- U.U conjugate = U conjugate.U = I
- det U = 1
- Eigenvectors are orthogonal
- U conjugate = U inverse
- Norm (U.x) = Norm (x)
- Normal Matrix U
13.2) 2x2 Elementary Matrices
- Wiki Representation #1
- Wiki Representation #2
- Wiki Representation #3
- Wiki Representation #4
----------------------------

----------------------------
#14 - Matrix Norm
----------------------------
----------------------------
14.1) Properties:
- ||A|| >= 0
- ||A|| = 0 only if every element is zero
- ||cA|| = |c|.||A||
- ||A + B|| <= ||A|| + ||B||
- ||A . B|| <= ||A|| . ||B||
14.2) Matrix norms induced by vector p-norms
- p = 1, ∞
- p = 1 => maximum absolute column sum
- p = ∞ => maximum absolute row sum
- Spectral norm (p = 2)
- ‖A‖_2=√(λ_max (A^* A) )=σ_max (A)
- Properties:
- ‖A‖_2=sup⁡{x^* Ay∶x∈K^m; y∈K^n with ‖x‖_2=‖y‖_2=1}
- ‖A^* A‖_2=‖AA^* ‖_2=‖A‖_2^2
- ‖A‖_2=σ_max (A)≤‖A‖_F=√(∑_i▒〖σ_i^2 (A) 〗)
- ‖A‖_2=√(ρ(A^* A) )≤√(‖A^* A‖_∞ )≤√(‖A‖_1∙‖A‖_∞ )
- Matrix Norms Induced by Vector α- and β- Norms
- ‖A‖_(2,∞)=■(max@1≤i≤m) ‖A_(i:) ‖_2
- ‖A‖_(1,2)=■(max@1≤j≤n) ‖A_(:j) ‖_2
- Properties Verification
- Square Matrix Verification
- Consistent and Compatible Norms Verification
- "Entry-wise" matrix norms
- ‖A‖_(p,p)=‖vec (A)‖_p=(∑_(i=1)^n▒∑_(j=1)^n▒|a_ij |^p )^(1/p)
- Entry-wise L_(2,1) and L_(p,q) Matrix Norms
- ‖A‖_(2,1)=∑_(j=1)^n▒‖a_j ‖_2 =∑_(j=1)^n▒√(∑_(i=1)^n▒|a_ij |^2 )
- ‖A‖_(p,q)=[(∑_(i=1)^m▒|a_ij |^p )^(q/p) ]^(1/q)
- Frobenius norm (check the wiki elements)
- Max norm (check the wiki elements)
- Schatten norms
- ‖A‖_p=[∑_(i=1)^min⁡(m,n)▒〖σ_i (A)〗^p ]^(1/p)
- Ky-Fan - ‖A‖_*=trace (√(A^* A))=∑_(i=1)^min⁡(m,n)▒〖σ_i (A) 〗
- Holder's Inequality
- SChatten Inequality
- Monotone norms
- Cut norms (check the wiki elements)
- Equivalence of Norms (check the wiki elements)
- Examples of Norm Equivalence (check the wiki elements)
----------------------------

----------------------------
#15 - Spectral Radius
----------------------------
----------------------------
15.1) Definition - Matrix:
- Spectral Radius - Max of the Eigenvalues
15.2) Definition - Bounded Complex Linear Operators:
- Spectral Radius - Max of the Spectrum
- Gelfand Formula applies to Bounded Complex Linear Operators
15.3) Definition - Graph:
- Function on Graph Vertex
- Square Integrability of Function across Vertexes
- Graph Function Map - Square Integrability Function Space across Vertexes 1 -> 2
- Graph Adjacency Operator - Function Sum over edges of a vertex
15.4)Upper bounds
- Upper bounds on the spectral radius of a matrix <= Power (Norm (A^k), 1/k)
- Upper bounds for spectral radius of a graph
15.5) Jordan Normal Power Sequence
- J (m_i, lambda_i)
- J (m_i, lambda_i) power k
- Jordan Normal J Matrix
- Jordan Normal J Matrix power k
- A = V.J.V^-1
- A^k = V.J^k.V^-1
15.6) Gelfand's formula
- The Formula
- Gelfand's formula for Commuting Matrices
----------------------------

--------------------------
#16 - Crank–Nicolson method
--------------------------
--------------------------
16.1) 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
16.2) 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)
16.3) 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
16.4) 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
16.5) 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
16.6) 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
--------------------------
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