1.3 Basic Data Structures and Algorithms

Symbolic computation relies on key data structures like linked lists, trees, and hash tables to represent and manipulate mathematical expressions. These structures enable efficient storage and manipulation of polynomials, matrices, and complex formulas.

Basic algorithms for symbolic manipulation include polynomial operations, differentiation, integration, and simplification. These algorithms often use recursive techniques and pattern matching to handle complex mathematical expressions, trading some efficiency for exact results compared to numerical methods.

Fundamental Data Structures and Algorithms

Fundamental symbolic computation structures

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• Linked lists store polynomials and sparse matrices
  • Nodes contain coefficient and exponent or index enabling efficient insertion, deletion, traversal
• Trees represent expressions and formulas
  • Binary trees have operator or operand nodes with left/right children as subexpressions
  • Syntax trees show formula structure with internal operator nodes and operand leaves for parsing, simplification, evaluation
• Hash tables enable fast lookup of symbolic expressions
  • Key uniquely represents expression mapping to simplified or canonical form value
  • Allows quick comparison and simplification of expressions

Basic symbolic manipulation algorithms

• Polynomial operations performed by traversing linked lists
  • Addition combines like terms by summing coefficients
  • Multiplication uses distributive property to multiply polynomials
• Symbolic differentiation applies rules recursively
  • Chain rule handles composite functions
  • Accounts for special cases (trigonometric, exponential functions)
• Symbolic integration recursively applies rules and techniques
  • Pattern matching identifies known integral forms
  • Utilizes methods like integration by parts and substitution
• Simplification and canonicalization maintain canonical form
  • Applies algebraic identities and rewrite rules
  • Recursively simplifies subexpressions

Efficiency of symbolic algorithms

• Time complexity varies by algorithm
  • Polynomial addition: $O(n)$ for $n$ terms
  • Polynomial multiplication: $O(n^2)$ with naive algorithm
  • Symbolic differentiation: $O(n)$ for expression size $n$
  • Symbolic integration: depends on integrand complexity
• Space complexity depends on data structure
  • Linked lists, trees, hash tables: $O(n)$ for $n$ terms or nodes
• Symbolic algorithms often slower and more memory-intensive than numerical
  • Trade-off exactness for efficiency compared to approximation errors in numerical methods

Symbolic vs numeric data structures

• Symbolic structures handle variable-size expressions
  • Maintain term structure and relationships
  • Linked lists, trees, hash tables common examples
• Numeric structures designed for fixed-size data (integers, floats)
  • Optimized for arithmetic and memory layout
  • Arrays, matrices, vectors typical examples
• Memory differences
  • Symbolic uses pointers and dynamic allocation
  • Numeric uses contiguous memory blocks
• Operational differences
  • Symbolic focuses on algebraic manipulation and rule-based transformations
  • Numeric emphasizes arithmetic and mathematical functions