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Chapter8
Chapter 8: Advanced Problem Solving
8.0 Advanced Problem Solving
8.1 Numeric Functions
8.2 Polynomial Factorization
8.2.1 Integer and Rational Number Coefficients
8.2.2 Finite Field Coefficients
8.2.3 Simple Algebraic Extension Field Coefficients
8.2.4 Factoring Rational Functions
8.3 Manipulating Symbolic Roots of a Polynomial
8.3.1 Using a Single Root of a Polynomial
8.3.2 Using All Roots of a Polynomial
8.4 Computation of Eigenvalues and Eigenvectors
8.5 Solution of Linear and Polynomial Equations
8.5.1 Solution of Systems of Linear Equations
8.5.2 Solution of a Single Polynomial Equation
8.5.3 Solution of Systems of Polynomial Equations
8.6 Limits
8.7 Laplace Transforms
8.8 Integration
8.9 Working with Power Series
8.9.1 Creation of Power Series
8.9.2 Coefficients of Power Series
8.9.3 Power Series Arithmetic
8.9.4 Functions on Power Series
8.9.5 Converting to Power Series
8.9.6 Power Series from Formulas
8.9.7 Substituting Numerical Values in Power Series
8.9.8 Example: Bernoulli Polynomials and Sums of Powers
8.10 Solution of Differential Equations
8.10.1 Closed-Form Solutions of Linear Differential Equations
8.10.2 Closed-Form Solutions of Non-Linear Differential Equations
8.10.3 Power Series Solutions of Differential Equations
8.11 Finite Fields
8.11.1 Modular Arithmetic and Prime Fields
8.11.2 Extensions of Finite Fields
8.11.3 Irreducible Modulus Polynomial Representations
8.11.4 Cyclic Group Representations
8.11.5 Normal Basis Representations
8.11.6 Conversion Operations for Finite Fields
8.11.7 Utility Operations for Finite Fields
8.12 Primary Decomposition of Ideals
8.13 Computation of Galois Groups
8.14 Non-Associative Algebras and Modelling Genetic Laws