📂Numerical Analysis

Differences in Numerical Analysis

Definition¹

1. Forward difference: $$ \begin{align*} \Delta f(x) =& f(x+h) - f(x) \\ \Delta^{r+1} f(x) =& \Delta^{r} f(x+h) - \Delta^{r} f(x) \end{align*} $$
2. Backward difference: $$ \begin{align*} \nabla f(x) =& f(x) - f(x- h) \\ \nabla^{r+1} f(x) =& \nabla^{r} f(x) - \nabla^{r} f(x- h) \end{align*} $$

Explanation

In general, the term difference is used throughout the study of sequences, but in numerical analysis it specifically refers to the difference between the function values at two node points. In fact, it is an operator that can feel familiar since we have kept seeing it since high school, yet if you look at the frequently appearing formulas in numerical analysis, it tends to be used in quite complicated ways. Although it greatly helps express formulas concisely, it is just as hard to read.

Equations that make heavy use of such operators are harder to manipulate algebraically than one might expect, and require a lot of practice.

See Also

• Differences in Time Series Analysis
• Finite Difference Method FDM
  • Derivation of Finite Differences Using Multiple Points