Numerical integration: Difference between revisions

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(→‎{{header|MATLAB}}: Changed the solution to conform with the task specs and added a more detailed example of how to use the solution functions)
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=={{header|MATLAB}}==
 
For all of the examples given, the function that is passed to the method as parameter f is a function handle.
f.m (The funtion you are wanting to approximate an integral too)
<lang MATLAB>
function f=f(x)
f=x;
</lang>▼
 
Function for performing left rectangular integration: leftRectIntegration.m
Numerical_Integration.m
<lang MATLAB>function integral = leftRectIntegration(f,a,b,n)
function Numerical_Integration(a,b,n)
format long;▼
h=(b-a)/n;
x1=linspace(a,b,n+1); ▼
LftRecRl=h.*sum(f(x1(1:n)))
 
▲ format long;
x2=linspace(a,b,n+1); ▼
width = (b-a)/n; %calculate the width of each devision
RgtRecRl=h.*sum(f(x2(2:n+1)))
▲x1 x = linspace(a,b,n+1); %define x-axis
integral = width * sum( f(x(1:n-1)) );
▲end</lang>
 
Function for performing right rectangular integration: rightRectIntegration.m
x3=linspace(a,b,n+1);▼
<lang MATLAB>function integral = rightRectIntegration(f,a,b,n)
MdPtRl=h.*sum(f((x3(1:n)+x3(2:n+1))/2))▼
 
format long;
x4=linspace(a,b,n+1); ▼
width = (b-a)/n; %calculate the width of each devision
TrpRl=h.*sum((f(x3(1:n))+f(x4(2:n+1)))/2)
▲x2 x = linspace(a,b,n+1); %define x-axis
integral = width * sum( f(x(2:n)) );
end</lang>▼
 
Function for performing mid-point rectangular integration: midPointRectIntegration.m
x5=linspace(a,b,n+1);
<lang MATLAB>function integral = midPointRectIntegration(f,a,b,n)
SmpRl=h/6.*sum(f(x5(1:n))+4*f((x5(1:n)+x5(2:n+1))/2)+f(x5(2:n+1)))
 
▲</lang>
format long;
width = (b-a)/n; %calculate the width of each devision
▲x3 x = linspace(a,b,n+1); %define x-axis
▲MdPtRl integral =h. width * sum( f( (x3x(1:n-1)+x3x(2:n+1))/2 ) );
end</lang>
 
Function for performing trapezoidal integration: trapezoidalIntegration.m
<lang MATLAB>function integral = trapezoidalIntegration(f,a,b,n)
 
format long;
▲x4 x = linspace(a,b,n+1); %define x-axis
integral = trapz( x,f(x) );
end</lang>
 
Simpson's rule for numerical integration is already included in MATLAB as "quad()". It is not the same as the above examples, instead of specifying the amount of points to divide the x-axis into, the programmer passes the acceptable error tolerance for the calculation (parameter "tol").
<lang MATLAB>integral = quad(f,a,b,tol)</lang>
 
{{header|Sample inputs}}
 
Using anonymous functions
 
<lang MATLAB>trapezoidalIntegration(@(x)( exp(-(x.^2)) ),0,10,100000)
 
ans =
 
0.886226925452753</lang>
 
Using predefined functions
 
Built-in MATLAB function sin(x):
<lang MATLAB>quad(@sin,0,pi,1/1000000000000)
 
ans =
 
2.000000000000000</lang>
 
User defined scripts and functions:
fermiDirac.m
<lang MATLAB>function answer = fermiDirac(x)
k = 8.617343e-5; %Boltazmann's Constant in eV/K
answer = 1./( 1+exp( (x)/(k*2000) ) ); %Fermi-Dirac distribution with mu = 0 and T = 2000K
end</lang>
 
<lang MATLAB> rightRectIntegration(@fermiDirac,-1,1,1000000)
 
ans =
 
0.999998006023282</lang>
 
=={{header|OCaml}}==
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