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Simpson's 1/3 and 3/8 rules

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company name:
Jered Wells
license: Shareware
minimum requirements: No special requirements.
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Simpson's 1/3 and 3/8 rules description


SIMPSON: Simpson's rule for quadratic and cubic numerical integrationRES = 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: %.15fn', a) fprintf('Trapezoid Rule: %.15fn', b) fprintf('Simpson''s 1/3 Rule: %.15fn', c) fprintf('Simpson''s 3/8 Rule: %.15fn', d) fprintf('Actual Integral: %.15fn', e) % http://math.fullerton.edu/mathews/n2003/simpson38rule/Simpson38RuleMod/Links/Simpson38RuleMod_lnk_2.html 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.
Relates:
Integrate - Actual - Rectangle - Trapzxy - Sumy - 002pi - Trapezoid - Simpsonxy - Fprintfrectangle - Cosx Cosxend
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