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Applied Mathematics · Quick Reference
Numerical Methods Cheat Sheet
In one line: numerical methods find approximate solutions to problems that are too difficult or impossible to solve exactly — essential tools whenever an equation resists a clean algebraic answer.
Key Ideas
1Why Numerical Methods Matter. Many real-world equations (especially nonlinear ones) have no exact algebraic solution, so we approximate the answer instead, often using a computer.
2Bisection Method. Repeatedly halves an interval known to contain a root, narrowing in on the answer by checking which half still contains a sign change.
3Newton's Method. Uses a function's derivative to rapidly converge on a root: x_(n+1) = x_n - f(x_n)/f'(x_n), starting from an initial guess.
4Numerical Integration (Trapezoidal Rule). Approximates the area under a curve by dividing it into trapezoids instead of computing an exact integral.
5Error and Convergence. Numerical methods produce approximations, not exact answers; the 'error' shrinks as you do more iterations or use smaller step sizes, until the method 'converges' close enough to the true answer.
Worked Examples
Using the bisection method, a root is known to lie between x=1 and x=2. What is the first midpoint tested?
x = 1.5
Apply one step of Newton's method to f(x)=x^2-2, starting at x_0=1.5 (f'(x)=2x).
x_1 ≈ 1.4167 (getting closer to sqrt(2) ≈ 1.4142)
Why can't every equation be solved with simple algebra, requiring numerical methods instead?
Because such equations lack a closed-form algebraic solution, numerical approximation becomes necessary
Formulas
Newton's method: x_(n+1) = x_n - f(x_n)/f'(x_n)
Practice Yourself
A root lies between x=0 and x=4. Find the first bisection midpoint.
2
What information does Newton's method require besides the function itself?
Its derivative
What does 'convergence' mean in a numerical method?
The approximations get closer and closer to the true answer with more iterations