Simpson's rule

In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation for n + 1 {displaystyle n+1} equally spaced subdivisions (where n {displaystyle n} is even): (General Form) In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation for n + 1 {displaystyle n+1} equally spaced subdivisions (where n {displaystyle n} is even): (General Form) where Δ x = b − a n {displaystyle Delta x={frac {b-a}{n}}} and x i = a + i Δ x {displaystyle x_{i}=a+iDelta x} . Simpson's rule also corresponds to the three-point Newton-Cotes quadrature rule. In English, the method is credited to the mathematician Thomas Simpson (1710–1761) of Leicestershire, England. However, Johannes Kepler used similar formulas over 100 years prior, and for this reason the method is sometimes called Kepler's rule, or Keplersche Fassregel (Kepler's barrel rule) in German. One derivation replaces the integrand f ( x ) {displaystyle f(x)} by the quadratic polynomial (i.e. parabola) P ( x ) {displaystyle P(x)} which takes the same values as f ( x ) {displaystyle f(x)} at the end points a and b and the midpoint m = (a + b) / 2. One can use Lagrange polynomial interpolation to find an expression for this polynomial, Using integration by substitution one can show that Introducing the step size h = ( b − a ) / 2 {displaystyle h=(b-a)/2} this is also commonly written as Because of the 1 / 3 {displaystyle 1/3} factor Simpson's rule is also referred to as Simpson's 1/3 rule (see below for generalization). The calculation above can be simplified if one observes that (by scaling) there is no loss of generality in assuming that a = − 1 {displaystyle a=-1} 。