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.
Understanding Numerical Methods
A numerical answer is only useful if the procedure is behaving well. This is called convergence. A sequence converges when later estimates settle toward one value instead of jumping around.
Bisection is dependable because it keeps a root trapped inside a shrinking interval, provided the function is continuous there. Its weakness is speed. Newton's method can reach many correct digits very quickly near a well behaved root.
However, a poor starting value can send it toward a different root or make the calculation fail near a horizontal tangent. The secant method avoids calculating a derivative, which helps when a derivative is difficult to obtain, but it has less built in protection than bisection.
Root finding appears whenever a quantity must meet a target. An engineer may find the time when a falling object reaches the ground. An economist may find an interest rate that makes a loan balance match a payment plan.
In physics, roots can identify an equilibrium position where the total force is zero. A graph gives valuable evidence before any calculation starts.
Look for crossings of the horizontal axis, steep or flat sections, gaps, and turning points. A graph cannot usually give enough precision by itself, but it can reveal whether a chosen starting interval or estimate makes sense.
Numerical integration has a different kind of uncertainty. It replaces a curved boundary with simple shapes. Trapezoids tend to overestimate or underestimate depending on the curve's bending.
A curve that bends upward lies below many trapezoid tops, so the estimate is often too large. Simpson's rule uses curved pieces indirectly through weighted sample values. It is usually more accurate for smooth data, though it requires an even number of subintervals.
Neither rule can repair missing information. If a graph has a sharp corner, a vertical-looking change, or rapidly changing values, use more sample points near that feature rather than spreading points evenly without thought.
Euler's method is useful for modelling motion, cooling, population change, and circuits because many real systems specify a rate instead of a complete formula for the future. Each Euler step assumes the current rate remains constant for one short interval. That assumption creates accumulated error.
If the rate changes quickly, a large step can drift far from the real path. Some differential equations are especially sensitive, so reducing the step size is important but not always sufficient. Students should record units at every stage, keep extra decimal places during work, and round only at the end.
Compare results from two step sizes when possible. If halving the step changes the answer greatly, the first result was not yet trustworthy. Relative error is especially useful when comparing quantities of very different sizes, while absolute error shows the actual size of the difference.