Numerical Methods Reference Cheat Sheet

Key Facts

• Absolute error is $E_{\text{abs}} = |x - \hat{x}|$, where $x$ is the true value and $\hat{x}$ is the approximation.
• Relative error is $E_{\text{rel}} = \frac{|x - \hat{x}|}{|x|}$ when $x \neq 0$, and percent error is $100E_{\text{rel}}\%$.
• The bisection method requires $f(a)f(b)<0$ and uses the midpoint $c = \frac{a+b}{2}$ to shrink the interval containing a root.
• Newton’s method updates an approximation by $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$ and can converge quickly when $x_0$ is close to a simple root.
• The secant method avoids derivatives by using $x_{n+1}=x_n-f(x_n)\frac{x_n-x_{n-1}}{f(x_n)-f(x_{n-1})}$.
• The Lagrange interpolation polynomial is $P_n(x)=\sum_{i=0}^{n} y_i\prod_{j\neq i}\frac{x-x_j}{x_i-x_j}$ for data points $(x_i,y_i)$.
• The composite trapezoidal rule is $\int_a^b f(x)\,dx \approx \frac{h}{2}\left[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\right]$, where $h=\frac{b-a}{n}$.
• Euler’s method for $y'=f(t,y)$ uses $y_{n+1}=y_n+h f(t_n,y_n)$ to approximate the solution one step at a time.

Vocabulary

Absolute Error

Absolute error is the size of the difference between the true value and an approximation, written as $|x-\hat{x}|$.

Convergence

Convergence means that a sequence of numerical approximations approaches the exact solution or a limiting value.

Root-Finding

Root-finding is the process of approximating values of $x$ that satisfy $f(x)=0$.

Interpolation

Interpolation estimates a function value between known data points using a fitted function such as a polynomial.

Step Size

Step size is the spacing $h$ between consecutive points in a numerical method, often controlling accuracy and cost.

Stability

Stability describes whether small errors from rounding or approximation remain controlled as a numerical method proceeds.