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Simpson's 1/3 and 3/8 rules

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SIMPSON: Simpson's rule for quadratic and cubic numerical integration



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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.

Comments and Ratings (3)

Evan Provan

eric ross

Andrew Davis

Thanks for putting this up, it works great.

In the first example, I suppose the step size in x should be 0.01, or the multiplier in the a = ... line should be 0.2. Also, the strcat() function in the disp lines is redundant, since the concatenation is already performed by the [] brackets. May I suggest:
>> fprintf('Rectangle Rule: %.15f\n', a);

Thanks again for this submission.



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

MATLAB Release
MATLAB 7.7 (R2008b)

Inspired by: Simpson's Rule Integration

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