Finite difference

A finite difference is a way to see how a function or idea changes between two points. It looks at the change between the value of a function at one point and its value at another point, often very close. This helps us guess how fast something is changing, which is useful in many parts of math and science.

The difference operator, shown as Δ (uppercase Delta), is a special tool used to find these changes. For example, it can show us how much a function changes when we move one step forward. This idea is similar to how we study changes in calculus, but it uses whole numbers instead of very small steps.

Finite differences are important in numerical analysis, where they help us guess rates of change without needing exact formulas. They were first introduced by Brook Taylor in 1715 and have been studied by many famous mathematicians like George Boole, L. M. Milne-Thomson, and Károly Jordan. Their ideas go back even further, to the time of Jost Bürgi and Isaac Newton, showing how old and useful this idea is in understanding how things change.

Basic types

There are three basic types of finite differences: forward, backward, and central differences.

A forward difference finds the change in a function’s value by subtracting the value at one point from the value at a later point.

A backward difference does the opposite, subtracting the value at an earlier point from the value at a later point.

A central difference averages these two methods, giving a more balanced view of how the function’s value changes.

Relation with derivatives

Main article: difference quotient

Finite differences help us guess how fast a function is changing at a point. This change is called a derivative. In math, a derivative is usually found using a special math rule. But with finite differences, we use a small number called "h" instead. This lets us guess the derivative using forward, backward, or central differences.

The central difference method often gives a better guess than the forward or backward methods. But it can sometimes give the wrong answer for certain functions.

Higher-order differences

Higher-order differences help us understand more complex changes in functions. For example, they can show us how the rate of change itself is changing. This is like learning not just how fast something is moving, but also how quickly its speed is changing.

These methods improve our guesses by using different types of differences together. They are important in mathematics for solving hard problems more precisely.

Polynomials

For a polynomial of degree n, which looks like P(x) = axⁿ + bxⁿ⁻¹ + l.o.t. (where "l.o.t." means lower order terms), doing n pairwise differences gives a constant value. This constant is ahⁿn!, where h is the space between x-values and a is the biggest number in the polynomial.

This way helps us find the polynomial that goes through a group of points where the x-values are evenly spaced. By using a table of differences, we can learn the degree of the polynomial and its numbers. For example, if it takes three pairwise differences to get a constant, the polynomial is of degree 3. With this knowledge, we can find each number and make the polynomial.

x	y
1	4
4	109
7	772
10	2641
13	6364

x	y	Δy	Δ²y	Δ³y
1	4
4	109	105
7	772	663	558
10	2641	1869	1206	648
13	6364	3723	1854	648

x	y	Δy	Δ²y
1	4 − 4(1)³ = 4 − 4 = 0
4	109 − 4(4)³ = 109 − 256 = −147	−147
7	772 − 4(7)³ = 772 − 1372 = −600	−453	−306
10	2641 − 4(10)³ = 2641 − 4000 = −1359	−759	−306
13	6364 − 4(13)³ = 6364 − 8788 = −2424	−1065	−306

x	y	Δy
1	0 − (−17(1)²) = 0 + 17 = 17
4	−147 − (−17(4)²) = −147 + 272 = 125	108
7	−600 − (−17(7)²) = −600 + 833 = 233	108
10	−1359 − (−17(10)²) = −1359 + 1700 = 341	108
13	−2424 − (−17(13)²) = −2424 + 2873 = 449	108

x	y
1	17 − 36(1) = 17 − 36 = −19
4	125 − 36(4) = 125 − 144 = −19
7	233 − 36(7) = 233 − 252 = −19
10	341 − 36(10) = 341 − 360 = −19
13	449 − 36(13) = 449 − 468 = −19

Arbitrarily sized kernels

Further information: Five-point stencil

Using linear algebra, we can find ways to guess how a function changes by checking many points around a central point. This helps us learn about the function, especially when we are close to the edge of our data and can't see points on one side. These methods can be shown as patterns on special grids, like hexagons or diamonds. These patterns help us see how the function behaves from different directions.

The rules for these patterns follow certain math ideas. For example, when we combine two functions, we can break down how they change together into simpler parts. This makes it easier to study their behavior. This way of thinking helps scientists and engineers solve hard problems by splitting them into smaller, easier pieces.

In differential equations

Finite differences help us solve math problems called differential equations. These equations describe how things change, like how heat moves or how fluids flow. Instead of using hard math, scientists use finite differences to find answers that are close enough. This makes it easier to solve these problems with computers, especially when studying heat and how fluids move.

Newton's series

The Newton series is a way to show functions by looking at how values change. It is named after Isaac Newton. This idea is like describing curves with polynomial math, but it works for points that are not smooth.

This series was first shared in Newton's Principia Mathematica in 1687.

It works well for polynomial functions and many other types. It helps to guess or find exact values at different points. It can also be used in special parts of math and physics.

Calculus of finite differences

The calculus of finite differences looks at how to estimate changes in functions using steps. A main idea is the difference operator, written as Δ. It helps us see how a function changes from one point to the next. For a function f, the forward difference is found by doing f(x + 1) − f(x). This works like a derivative in calculus but uses differences.

We can use this idea for more complex cases and different types of differences, like backward and central differences. These tools are helpful in numerical methods when exact derivatives are difficult to find, and we need approximations. The rules for using these operators are similar to those in regular calculus, which makes them useful for solving many math problems.

Generalizations

A generalized finite difference is a way to look at how a function changes. It adds up values of the function at different points, each multiplied by a special number.

These ideas can be made even broader. For instance, the sum can go on forever, making an infinite difference. The numbers can also change based on where they are used, resulting in a weighted finite difference. These ideas help mathematicians study how smoothly or roughly a function changes. They are useful in many areas, like studying structures known as difference algebras.

Multivariate finite differences

Finite differences can work with more than one variable, just like partial derivatives in several variables.

We can approximate these derivatives by using finite differences. For example, we can estimate how a function changes when we move slightly in both the x and y directions. This helps us see how the function changes in different directions on a grid.

For functions with many variables, there are special methods to make these calculations faster and easier.