Simpson's 1/3 and 3/8 rules

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

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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

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

% 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

See also sum, cumsum, trapz, cumtrapz.

인용 양식

Jered Wells (2024). Simpson's 1/3 and 3/8 rules (, MATLAB Central File Exchange. 검색됨 .

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버전 게시됨 릴리스 정보

Edited example

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