Lab 2

Exercise 1: Floating-Point and Symbolic Precision

1. Install and import SymPy
2. Use SymPy to demonstrate the limitations of floating-point arithmetic:
   • Evaluate ((1 / 49) \times 49) using standard Python floating-point arithmetic.
   • Represent the same computation symbolically in SymPy and evaluate.
   • Perform (sin(\pi)) using Python’s math library and compare it with SymPy’s symbolic result.
3. Compare the results and explain why SymPy provides an exact answer.
4. Compute precision and accuracy for some rational estimate.

Exercise 2: Symbolic Manipulations

1. Define symbolic variables (x, y, z) and create these expressions:
   • (expr1 = \frac{x^2 + 2x + 1}{x+1})
   • (expr2 = \sin(x)^2 + \cos(x)^2)
2. Perform the following operations:
   • Simplify (expr1) and (expr2). Explain the simplifications.
   • Substitute (x = \pi/4) in (expr2) and evaluate numerically to verify the trigonometric identity.
3. Create a symbolic expression for (expr3 = (x+y)^3 - (x-y)^3). Factorize and expand it.
4. Substitute (x = 2), (y = 3), and (z = -1) into the factored form of (expr3) and compute the result.

Exercise 3: Symbolic Substitution and Nested Transformations

1. Define (expr = 2x + 3y - z).
2. Perform nested substitutions:
   • Replace (x) with ((z + 1)).
   • Substitute (y) with (2x) in the resulting expression.
   • Replace (z) with (x - 2) and simplify the final expression.
3. Use SymPy to verify whether the substitutions preserve the equivalence of the original and final expressions.
4. Implement a Python function that takes a symbolic expression and a dictionary of substitutions, applies them recursively, and returns the result.

Exercise 4: Exploring Symbolic Trees

1. Define the expression (expr = x^3 + 2x^2 + x + 1).
2. Use sp.srepr(expr) to examine its internal representation.
3. Visualize the structure as a tree using sp.print_tree(expr, assumptions=False).
4. Define a function that:
   • Accepts a symbolic expression.
   • Computes the depth of the tree.
   • Counts the number of unique symbols and operations in the tree.
   • More about expression trees here.
5. Test your function on (expr = (x+y)^5 - (x-y)^5) and compare the results with smaller expressions.

Exercise 5: Numerical Functions and Performance

1. Define the symbolic expression (f(x) = x^4 - 4x^2 + 3).
2. Convert (f(x)) into a numerical function using sp.lambdify:
   • Use Python’s native libraries.
   • Use NumPy and evaluate (f(x)) for (x = [-2, -1, 0, 1, 2]).
3. Plot (f(x)) using NumPy and Matplotlib.