Numerical methods are procedures for finding approximate answers when exact algebraic solutions are difficult or impossible. This cheat sheet covers root-finding, numerical integration, differential equation methods, and error analysis. Students need these tools for applied problems in physics, engineering, economics, and data science where real equations often cannot be solved neatly by hand.
The main idea is to turn a hard continuous problem into repeated arithmetic steps. Root-finding methods approximate solutions to f(x) = 0, integration methods approximate area under a curve, and Euler's method approximates a changing quantity from its rate of change. Error estimates, step size, and convergence help students decide whether an approximation is reliable.
Key Facts
- The bisection method starts with an interval [a, b] where f(a) and f(b) have opposite signs, then repeatedly uses the midpoint c = (a + b)/2.
- Newton's method uses the tangent line update x_(n+1) = x_n - f(x_n)/f'(x_n) to approximate a root of f(x) = 0.
- The secant method uses two previous approximations with x_(n+1) = x_n - f(x_n)(x_n - x_(n-1))/(f(x_n) - f(x_(n-1))).
- The trapezoidal rule approximates integral from a to b of f(x) dx by T_n = h/2[f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_(n-1)) + f(x_n)], where h = (b - a)/n.
- Simpson's rule for even n is S_n = h/3[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_(n-2)) + 4f(x_(n-1)) + f(x_n)].
- Euler's method for dy/dx = f(x, y) uses x_(n+1) = x_n + h and y_(n+1) = y_n + h f(x_n, y_n).
- Absolute error is |exact value - approximation|, and relative error is |exact value - approximation|/|exact value|.
- Smaller step sizes usually improve accuracy, but they can increase computation time and may increase round-off error in long calculations.
Vocabulary
- Root
- A root of a function is an input value x where f(x) = 0.
- Iteration
- An iteration is one repeated step in a numerical method that produces a new approximation.
- Step size
- Step size is the spacing h between consecutive x-values used in a numerical approximation.
- Convergence
- Convergence means the approximations produced by a method get closer to the desired value.
- Truncation error
- Truncation error is the error caused by replacing an exact mathematical process with a finite approximation.
- Round-off error
- Round-off error is the error caused by limiting decimal places or computer precision during calculations.
Common Mistakes to Avoid
- Using Newton's method when f'(x_n) = 0 or nearly 0 is wrong because the update x_(n+1) = x_n - f(x_n)/f'(x_n) becomes undefined or unstable.
- Applying the bisection method without checking that f(a) and f(b) have opposite signs is wrong because the interval may not guarantee a root.
- Using Simpson's rule with an odd number of subintervals is wrong because the standard Simpson's rule formula requires n to be even.
- Forgetting to multiply by the step size h in trapezoidal, Simpson's, or Euler's method is wrong because h scales the approximation to the interval length.
- Rounding too early in an iterative method is wrong because small rounding changes can grow and make the final answer less accurate.
Practice Questions
- 1 Use one step of Newton's method to approximate a root of f(x) = x^2 - 10 starting from x_0 = 3.
- 2 Approximate the integral from 0 to 4 of f(x) = x^2 using the trapezoidal rule with n = 4.
- 3 Use Euler's method with h = 0.5 to estimate y at x = 1 for dy/dx = x + y, y(0) = 1.
- 4 Explain why a smaller step size can make a numerical method more accurate but does not always guarantee a perfect answer.