import unittest
import importlib
import math

class TestTask(unittest.TestCase):
    def setUp(self):
        import user_code
        importlib.reload(user_code)

    def test_accuracy_for_standard_input(self):
        import user_code
        approx, _ = user_code.taylor_exp(1.0, tol=1e-8)
        _dynamic_test(
            self,
            abs(approx - math.exp(1.0)) <= 1e-8,
            "Approximation converges to e^1 within tolerance.",
            f"Approximation error too large: {approx}",
        )

    def test_max_terms_respected(self):
        import user_code
        _, terms = user_code.taylor_exp(5.0, tol=1e-16, max_terms=30)
        _dynamic_test(
            self,
            terms <= 30,
            "Iteration stops when max_terms is reached.",
            f"Exceeded max_terms: {terms}",
        )

    def test_return_structure(self):
        import user_code
        result = user_code.taylor_exp(0.5)
        _dynamic_test(
            self,
            isinstance(result, tuple) and len(result) == 2,
            "Function returns (approximation, number_of_terms).",
            f"Unexpected return value: {result}",
        )

    def test_reasonable_accuracy_for_negative_x(self):
        import user_code
        approx, _ = user_code.taylor_exp(-1.0, tol=1e-8)
        _dynamic_test(
            self,
            abs(approx - math.exp(-1.0)) <= 1e-8,
            "Approximation works correctly for negative x.",
            f"Incorrect approximation for negative x: {approx}",
        )

def _dynamic_test(test_case, condition, success_message, failure_message):
    if condition:
        test_case._testMethodName = success_message
        test_case.assertTrue(True, success_message)
    else:
        test_case._testMethodName = failure_message
        test_case.fail(failure_message)

if __name__ == "__main__":
    unittest.main()


test_main.py

Master the essential numerical methods for approximating mathematical problems on computers. This course blends mathematical intuition, algorithmic reasoning, and hands-on Python implementation to equip you with the tools for scientific computing.

Explore the motivations, limitations, and core concepts underlying numerical computation, including error sources and convergence.

Delve into the implementation and analysis of foundational numerical algorithms for differentiation, integration, and root-finding.

Master numerical approaches for solving ordinary differential equations and analyzing dynamic system stability.

Challenge: Error Analysis and Convergence Behavior

Solution