Computer Algebra System Development

Python
class SymbolicExpression:
    def __init__(self, expr):
        self.expr = self._parse(expr)
    
    def differentiate(self, var):
        # Basic symbolic differentiation
        if isinstance(self.expr, Variable) and self.expr.name == var:
            return Constant(1)
        elif isinstance(self.expr, Constant):
            return Constant(0)
        elif isinstance(self.expr, Sum):
            return Sum([term.differentiate(var) for term in self.expr.terms])
        # Add more rules...

Progress:

• Define symbolic data structures (variables, constants, operations)
• Implement expression parsing and normalization
• Build core algebraic operations (expand, factor, simplify)
• Add calculus operations (differentiate, integrate)
• Create equation solving algorithms
• Implement numerical fallback methods
• Add pretty-printing and visualization
• Write comprehensive test suite

Core Architecture:

1. Expression Tree - Represent mathematical expressions as abstract syntax trees
2. Rewrite Rules - Pattern matching system for algebraic transformations
3. Symbolic Engine - Core algorithms for manipulation
4. Numerical Backend - Fallback for when symbolic fails

Example 1: Input: differentiate("x^2 + 3*x + 1", "x") Output: 2*x + 3

Example 2: Input: solve("x^2 - 5*x + 6 = 0", "x") Output: [x = 2, x = 3]

Example 3: Input: expand("(x + 2)*(x - 1)") Output: x^2 + x - 2

• Use immutable expression objects to avoid side effects
• Implement canonical forms for efficient comparison
• Cache expensive computations (factorizations, GCD calculations)
• Separate symbolic and numerical precision handling
• Build modular rule systems for easy extension
• Use visitor pattern for tree traversal operations
• Implement proper operator precedence in parsing
• Support multiple output formats (LaTeX, MathML, plain text)

• Don't mix symbolic zero with numerical zero without careful handling
• Avoid deep recursion in expression evaluation - use iterative approaches
• Don't ignore numerical instability in mixed symbolic-numeric operations
• Never assume expressions are simplified - always normalize first
• Don't implement custom number types without proper arithmetic semantics
• Avoid memory leaks in expression tree construction
• Don't forget edge cases in pattern matching (zero coefficients, identity elements)