Finite difference — Safekipedia Discoverer

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

The difference operator, shown as Δ (uppercase Delta), is a special tool used to find these differences. For example, it can tell 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 tiny, invisible steps.

Finite differences are important in numerical analysis, where they help us estimate 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 valuable this concept is in understanding how things change.

Basic types

Three basic types of finite differences are forward, backward, and central differences. A forward difference calculates 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. Finally, a central difference averages these two approaches, giving a more balanced view of the change in the function’s value.

Relation with derivatives

Main article: difference quotient

Finite differences help us estimate how fast a function is changing at a point, which is called a derivative. In math, a derivative is usually found using a limit, but with finite differences, we use a small number called "h" instead of getting infinitely close to zero. By doing this, we can approximate the derivative using forward, backward, or central differences.

The central difference method often gives a better approximation than the forward or backward methods because it reduces errors. However, it can sometimes give the wrong answer for certain types of functions that jump back and forth in value.

Higher-order differences

Higher-order differences help us approximate more complex changes in functions. For example, we can use a special method to estimate the rate at which the rate of change itself is changing. This is like figuring out not just how fast something is moving, but also how quickly its speed is increasing or decreasing.

These methods can be used to build better approximations by combining different types of differences. They connect closely to how we understand changes in mathematics and can help solve difficult problems more accurately.

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), applying n pairwise differences results in a constant value. This constant is given by ahⁿn!, where h is the difference between x-values and a is the coefficient of the highest-order term.

This method can help find the polynomial that passes through a set of points where the x-values are evenly spaced. By using a table of differences, we can determine the degree of the polynomial and its coefficients. For example, if it takes three pairwise differences to reach a constant, the polynomial is of degree 3. Using this information, we can solve for each coefficient and build 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 create ways to estimate how a function changes by looking at many points around a central point. This helps us understand the function's behavior, especially when we are near the edge of our data grid and can't look at points on one side. These methods can be shown as patterns on special grids, like hexagons or diamonds, which help us see how the function behaves from different angles.

The rules for these patterns follow certain mathematical principles. For example, when we combine two functions, the way they change together can be broken down into simpler parts, making it easier to study their behavior. This approach helps scientists and engineers solve complex problems by breaking them into smaller, more manageable pieces.

In differential equations

Finite differences are very useful in solving math problems called differential equations. These equations describe how things change, like how heat moves or how fluids flow. Instead of solving these problems with complex math, scientists often use finite differences to approximate the answers. This makes it easier to solve these problems with computers, especially in fields like studying heat and fluid movement.

Newton's series

The Newton series is a way to represent functions using differences between values, named after Isaac Newton. It is similar to how we can describe curves with polynomials in continuous math but works for discrete points instead. This method was first published in Newton's Principia Mathematica in 1687.

This series works well for polynomial functions and many other types, helping to approximate or exactly calculate values at different points. It can also be used in special areas of math and physics, like studying quantum spins or discrete counting.

Calculus of finite differences

The calculus of finite differences studies how to approximate changes in functions using discrete steps. A key idea is the difference operator, often written as Δ, which helps us understand how a function changes from one point to another. For example, given a function f, the forward difference is calculated as f(x + 1) − f(x). This operator acts similarly to the derivative in calculus but uses differences instead of limits.

This approach can be extended to higher orders and various types of differences, such as backward and central differences. These tools are useful in numerical methods, where exact derivatives are hard to compute, and approximations are needed instead. The rules for working with these operators resemble those in regular calculus, making them a valuable part of solving many mathematical problems.

Generalizations

A generalized finite difference is an expression where the difference includes a sum of values of a function at several points, each multiplied by a specific number. This helps in studying how functions change over intervals.

These ideas can be expanded further. For example, the sum can include an endless number of terms, creating an infinite difference. The numbers used in the sum can also change depending on the point at which they are used, leading to weighted finite difference. These generalizations help mathematicians understand how smooth or rough a function’s changes are and are useful in various areas, including the study of structures called difference algebras.

Multivariate finite differences

Finite differences can work with more than one variable, similar to how we think about partial derivatives in several variables.

There are ways to approximate these derivatives using finite differences. For example, the derivative of a function with respect to x and y can be approximated by measuring the function's value at points slightly shifted in both x and y directions. This helps us understand how the function changes when we move in different directions on a grid.

For functions with many variables, special methods can reduce the number of calculations needed to estimate these changes efficiently.