# Simpson's 1/3 and 3/8 rules

버전 1.2.0.0 (3.11 KB) 작성자:
SIMPSON: Simpson's rule for quadratic and cubic numerical integration
다운로드 수: 4.5K
업데이트 날짜: 2012/3/23

라이선스 보기

RES = SIMPSON(Y) computes an approximation of the integral of Y via
Simpson's 1/3 rule (with unit spacing). Simpson's 1/3 rule uses
quadratic interpolants for numerical integration. To compute the
integral for spacing different from one, multiply RES by the spacing
increment.

For vectors, SIMPSON(Y) is the integral of Y. For matrices, SIMPSON(Y)
is a row vector with the integral over each column. For N-D
arrays, SIMPSON(Y) works across the first non-singleton dimension.

RES = SIMPSON(X,Y) computes the integral of Y with respect to X using
Simpson's 1/3 rule. X and Y must be vectors of the same
length, or X must be a column vector and Y an array whose first
non-singleton dimension is length(X). SIMPSON operates along this
dimension. Note that X must be equally spaced for proper execution of
the 1/3 and 3/8 rules. If X is not equally spaced, the trapezoid rule
(MATLAB's TRAPZ) is recommended.

RES = SIMPSON(X,Y,DIM) or SIMPSON(Y,DIM) integrates across dimension
DIM of Y. The length of X must be the same as size(Y,DIM)).

RES = SIMPSON(X,Y,DIM,RULE) can be used to toggle between Simpson's 1/3
rule and Simpson's 3/8 rule. Simpson's 3/8 rule uses cubic interpolants
to accomplish the numerical integration. If the default value for DIM
is desired, assign an empty matrix.

- RULE options

[DEFAULT] '1/3' Simpson's rule for quadratic interpolants

'3/8' Simpson's rule for cubic interpolants

Examples:
% Integrate Y = SIN(X)
x = 0:0.2:pi;
y = sin(x);
a = sum(y)*0.2; % Rectangle rule
b = trapz(x,y); % Trapezoid rule
c = simpson(x,y,[],'1/3'); % Simpson's 1/3 rule
d = simpson(x,y,[],'3/8'); % Simpson's 3/8 rule
e = cos(x(1))-cos(x(end)); % Actual integral
fprintf('Rectangle Rule: %.15f\n', a)
fprintf('Trapezoid Rule: %.15f\n', b)
fprintf('Simpson''s 1/3 Rule: %.15f\n', c)
fprintf('Simpson''s 3/8 Rule: %.15f\n', d)
fprintf('Actual Integral: %.15f\n', e)

x1 = linspace(0,2,4);
x2 = linspace(0,2,7);
x4 = linspace(0,2,13);
y = @(x) 2+cos(2*sqrt(x));
format long
y1 = y(x1); res1 = simpson(x1,y1,[],'3/8'); disp(res1)
y2 = y(x2); res2 = simpson(x2,y2,[],'3/8'); disp(res2)
y4 = y(x4); res4 = simpson(x4,y4,[],'3/8'); disp(res4)

Class support for inputs X, Y:
float: double, single

### 인용 양식

Jered Wells (2024). Simpson's 1/3 and 3/8 rules (https://www.mathworks.com/matlabcentral/fileexchange/33493-simpson-s-1-3-and-3-8-rules), MATLAB Central File Exchange. 검색됨 .

개발 환경: R2008b
모든 릴리스와 호환
##### 플랫폼 호환성
Windows macOS Linux
##### 카테고리
Help CenterMATLAB Answers에서 Numerical Integration and Differential Equations에 대해 자세히 알아보기

### Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!
버전 게시됨 릴리스 정보
1.2.0.0

Edited example

1.1.0.0

Made inputs and execution congruent with TRAPZ
Help file formatted to MATLAB standard (incl H1 line)
Execution vectorized to accommodate N-dimensional arrays (similar to TRAPZ)
Error checking step included for unequally spaced X

1.0.0.0