Microcomputer Algorithms: Action from Algebra

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
1: General Introduction
	1.1 The Microcomputer Style of Programming
	1.2 Modules and Subroutines
	1.3 Notation and Conventions of This Book
	1.4 The Program Presentation Format
	1.5 Brief Survey of Chapter Contents
2: Root-Finding Methods and Their Application
	2.1 General Introduction
	2.2 The Root-Finding Approach
	2.3 Root-Finding Modules
	2.4 A Note on Cubic Equations
	2.5 Brief Program Descriptions and Programs
		Rootscan
		Newton
		Secant
		Zigzag
	2.6 Extrema and Roots
	2.7 The Programs Search and Maxmin
	2.8 Expectation Values. The Integral Approach
	2.9 The Energy Approach
	2.10 Calculating Local Quantities
3: The Richardson Extrapolation Method
	3.1 General Introduction
	3.2 Microcomputer Calculus
	3.3 An Example. Numerical Differentiation
	3.4 General Applications
	3.5 Richardson Versus Padé
	3.6 The Residual Error Term
	3.7 The Standard Extrapolation Formulae
	3.8 The Non-Standard Case
	3.9 The General Case. The Program Rich
	Romberg. Mathematical Theory
	3.10 MidPoint Integration
	3.11 Finding The AN
	3.12 A Modified MidPoint Rule
	3.13 Relative Merits of Quadrature Rules
	3.14 Three-Point Rules
	Romberg. Programming Notes
	3.15 The Subroutine Structure
	3.16 The Two-Point Methods
	3.17 Three-Point Rules
	3.18 Romberg. Program Analysis and Program
	Romberg. Specimen Results
	3.19 The Error Integral
	3.20 End-Point Singularities
	3.21 Residual Error Examples
	3.22 Final Comments
4: Some Interpolation and Extrapolation Methods
	4.1 General Introduction
	Interp. Mathematical Theory
	4.2 The Lagrange Interpolation Procedure
	4.3 The Divided Difference Approach
	4.4 Computing Derivatives
	Interp. Programming Notes
	4.5 Arrays
	4.6 Variable Indices
	4.7 Derivatives
	4.8 Interp. Program Analysis and Program
	Interp. Specimen Results
	4.9 Runge\'s Example
	4.10 Final Comments
	Spline and Spleen. Mathematical Theory
	4.11 Quadratic Splines
	4.12 The Fundamental Equations
	4.13 The Collocation Method. Spleen
	4.14 Spline. Program Analysis and Program
	Spline. Specimen Results
	4.15 Energy Level Interpolation
	4.16 Spleen. Program Analysis and Program
	Spleen. Specimen Results
	4.17 Energy Level Calculations
	WYNN. Mathematical Theory
	4.18 The Aitken Transformation
	4.19 Multi-Component Sequences
	4.20 Padé Approximants
	4.21 Series of Stieltjes
	4.22 The Algorithm
	WYNN. Programming Notes
	4.23 Setting Array Dimensions
	4.24 Use as a Library Subroutine
	4.25 Labelling The Approximants
	4.26 Storing and Displaying The (M/N)
	4.27 Rounding Errors and Scaling Tests
	4.28 WYNN. Program Analysis and Program
	WYNN. Specimen Results
	4.29 Varying KS
	4.30 Perturbation Series Analysis
	4.31 Comments on The Results
5: The Matrix Inverse and Generalized Inverse
	5.1 General Introduction
	Matin and Genfit. Mathematical Theory
	5.2 Choice of Reduction Method
	5.3 Least-Squares Fitting. The Generalized Inverse
	5.4 An Example. Parabolic Fitting
	5.5 An Iterative Method
	Matin and Genfit. Programming Notes
	5.6 Program Module Structure
	5.7 Forming MT M for Genfit
	5.8 Matin. Program Analysis and Program
	5.9 Genfit. Program Analysis and Program
	Genfit. Specimen Results
	5.10 A Parallel Test
	5.11 Invert. Program Analysis and Program
6: The Matrix Eigenvalue Problem
	6.1 General Introduction
	Jolly. Mathematical Theory
	6.2 The Origin of Jolly
	6.3 Some Simple Transformations
	6.4 The Algorithm
	6.5 The Eigencolumns
	6.6 Eigenvalues by Deflation
	6.7 Rapid Eigenvalue Calculation
	6.8 Non-Symmetric Matrices
	6.9 Preliminary Transformations
	Jolly. Programming Notes
	6.10 The Form of R(N, K)
	6.11 The Matrix Indices
	6.12 The Working Copy, Including .
	6.13 The Subroutine Structure
	6.14 Jolly. Program Analysis and Program
	Jolly. Specimen Results
	6.15 A 5 X 5 Example
	6.16 Matrices with Equal Diagonal Elements
	6.17 Using Aitken Extrapolation
	Folder. Mathematical Theory
	6.18 Introduction
	6.19 The Perturbation Approach
	6.20 An Explicit Example
	6.21 An Alternative Approach
	6.22 Calculating Eigenvectors
	6.23 Solving Linear Equations
	Folder. Programming Notes
	6.24 Array Requirements
	6.25 The Modular Structure
	6.26 Index Permutation
	6.27 The Use of Submatrices
	6.28 Folder. Program Analysis and Program
	Folder. Specimen Results
	6.29 Simple Test Example
	6.30 A Complex Matrix Example
	6.31 Finding Double Roots
	6.32 An Iterative Method
	6.33 Band Matrices
	Hitter. Mathematical Theory
	6.34 The Self-Consistency Problem
	6.35 The Brillouin-Wigner Method
	6.36 The Direct Iteration Procedure
	6.37 Relaxation Parameters and Rayleigh Quotients
	6.38 The Matrix H
	6.39 Algebraic Matrix Elements
	Hitter. Programming Notes
	6.40 The Relaxation Parameter
	6.41 Displaying The Eigencolumn
	6.42 The λ Parameter
	6.43 Hitter. Program Analysis and Program
	Hitter. Specimen Results
	6.44 Eigenvalues for BETOSC
	6.45 BETOSC. Program Analysis and Program
	6.46 Using Folder with BETOSC
7: Two Perturbation Methods
	7.1 General Introduction
	Hyposc. Mathematical Theory
	7.2 Hypervirial Theorems
	7.3 The Perturbation Recurrence Relation
	Hyposc. Programming Notes
	7.4 Use of The Value λ = 1
	7.5 Use of The F Factor
	7.6 Calculation of Array Indices
	7.7 Calculation of (xN)
	7.8 The Calculation of WKB Results
	7.9 Variable α and ß
	7.10 The T Array and The WYNN Algorithm
	7.11 Hyposc. Program Analysis and Program
	Hyposc. Specimen Results
	7.12 Perturbed Oscillator Energies
	7.13 Recent Developments
	Rally. Mathematical Theory
	7.14 Convergence Criteria
	7.15 The Recurrence Relations
	Rally. Programming Notes
	7.16 The Array Problem
	7.17 The Use of λ = 1
	7.18 Rally. Program Analysis and Program
	Rally. Specimen Results
	7.19 Rayleigh-Schrödinger Mode
	7.20 Brillouin-Wigner Mode
	7.21 Rayleigh Iteration Mode
8: Finite Difference Eigenvalue Calculations
	8.1 General Introduction
	Radial. Mathematical Theory
	8.2 The Finite Difference Approximation
	8.3 The Shooting Approach
	8.4 Expectation Values
	Radial. Programming Notes
	8.5 Initial Conditions
	8.6 Storage Requirements
	8.7 Node Counting
	8.8 The Choice of L
	8.9 The Index I
	8.10 Radial. Program Analysis and Program
	Radial. Specimen Results
	8.11 The Coulomb Potential
	8.12 A Comment on Singular Potentials
	Fidif. Mathematical Theory
	8.13 The Three-Term Recurrence Relation
	8.14 Obtaining H4 Error
	8.15 Initial and Boundary Conditions
	8.16 Forwards and Backwards Shooting
	8.17 The Numerov Method
	8.18 The Three Methods of Fidif
	8.19 The Propagator Approach
	8.20 Excited State Calculations
	Fidif. Programming Notes
	8.21 The Range of x
	8.22 An Economical Starting Procedure
	8.23 The Boundary Conditions
	8.24 Wavefunction Scaling
	8.25 Fidif. Program Analysis and Program
	Fidif. Specimen Results
	8.26 Comparison of Methods 2 and 3
	8.27 Extrapolation for Method 1
	8.28 Expectation Values
	8.29 Final Comments
9: Recurrence Relation Methods
	9.1 General Introduction
	Serat and Serosc. Mathematical Theory
	9.2 The Perturbed Coulomb Potential
	9.3 The Recurrence Relation
	9.4 The Zero-Coefficient Test
	9.5 Forwards Nested Multiplication
	9.6 The Perturbed Oscillator Potential
	9.7 The \'False Eigenvalue\' Effect
	Serat and Serosc. Programming Notes
	9.8 Array Dimensions
	9.9 Modular Structure
	9.10 Precomputation. Reduction in Strength
	9.11 The Scaling Factor
	9.12 Serat. Program Analysis and Program
	Serat and Serosc. Specimen Results
	9.13 Published Applications for Serat
	9.14 Some Specimen Results for Ssaosc
	9.15 The CT Parameter
	Momosc. Mathematical Theory
	9.16 Inner Product Recurrence Relations
	9.17 The Computational Procedure
	9.18 A Link with Matrix Theory
	9.19 The Angular Momentum Variable
	9.20 A Perturbation Approach
	Momosc. Programming Notes
	9.21 The Singularity Problem
	9.22 Momosc. Program Analysis and Program
	Momosc. Specimen Results
	9.23 Comparison with Serosc
	9.24 Some Specimen Expectation Values
10: Two Research Problems
	10.1 General Introduction
	Twodosc. Mathematical Theory
	10.2 The Recurrence Relations
	Twodosc. Programming Notes
	10.3 Array Requirements
	10.4 Exploiting The Symmetry
	10.5 Twodosc. Program Analysis and Program
	Twodosc. Specimen Results
	10.6 Results at Small λ
	Zeeman. Mathematical Theory
	10.7 Introduction
	10.8 Implicit Basis Methods
	10.9 Using Explicit Basis Functions
	Zeeman. Programming Notes
	10.10 Array Requirements
	10.11 Subroutine Structure
	10.12 Zeeman. Program Analysis and Program
	Zeeman. Speciman Results
	10.13 Shooting-Relaxation Mode
Bibliography
Index