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Numerical integration: Difference between revisions
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double s = integrate(f, 0.0, 1.0, 10, simpson());</lang>
=={{header|Chapel}}==
<lang chapel>
proc f1(x:real):real {
return x**3;
}
proc f2(x:real):real {
return 1/x;
}
proc f3(x:real):real {
return x;
}
proc leftRectangleIntegration(a: real, b: real, N: int, f): real{
var h: real = (b - a)/N;
var sum: real = 0.0;
var x_n: real;
for n in 0..N-1 {
x_n = a + n * h;
sum = sum + f(x_n);
}
return h * sum;
}
proc rightRectangleIntegration(a: real, b: real, N: int, f): real{
var h: real = (b - a)/N;
var sum: real = 0.0;
var x_n: real;
for n in 0..N-1 {
x_n = a + (n + 1) * h;
sum = sum + f(x_n);
}
return h * sum;
}
proc midpointRectangleIntegration(a: real, b: real, N: int, f): real{
var h: real = (b - a)/N;
var sum: real = 0.0;
var x_n: real;
for n in 0..N-1 {
x_n = a + (n + 0.5) * h;
sum = sum + f(x_n);
}
return h * sum;
}
proc trapezoidIntegration(a: real(64), b: real(64), N: int(64), f): real{
var h: real(64) = (b - a)/N;
var sum: real(64) = f(a) + f(b);
var x_n: real(64);
for n in 1..N-1 {
x_n = a + n * h;
sum = sum + 2.0 * f(x_n);
}
return (h/2.0) * sum;
}
proc simpsonsIntegration(a: real(64), b: real(64), N: int(64), f): real{
var h: real(64) = (b - a)/N;
var sum: real(64) = f(a) + f(b);
var x_n: real(64);
for n in 1..N-1 by 2 {
x_n = a + n * h;
sum = sum + 4.0 * f(x_n);
}
for n in 2..N-2 by 2 {
x_n = a + n * h;
sum = sum + 2.0 * f(x_n);
}
return (h/3.0) * sum;
}
var exact:real;
var calculated:real;
writeln("f(x) = x**3 with 100 steps from 0 to 1");
exact = 0.25;
calculated = leftRectangleIntegration(a = 0.0, b = 1.0, N = 100, f = f1);
writeln("leftRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = rightRectangleIntegration(a = 0.0, b = 1.0, N = 100, f = f1);
writeln("rightRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = midpointRectangleIntegration(a = 0.0, b = 1.0, N = 100, f = f1);
writeln("midpointRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = trapezoidIntegration(a = 0.0, b = 1.0, N = 100, f = f1);
writeln("trapezoidIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = simpsonsIntegration(a = 0.0, b = 1.0, N = 100, f = f1);
writeln("simpsonsIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
writeln();
writeln("f(x) = 1/x with 1000 steps from 1 to 100");
exact = 4.605170;
calculated = leftRectangleIntegration(a = 1.0, b = 100.0, N = 1000, f = f2);
writeln("leftRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = rightRectangleIntegration(a = 1.0, b = 100.0, N = 1000, f = f2);
writeln("rightRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = midpointRectangleIntegration(a = 1.0, b = 100.0, N = 1000, f = f2);
writeln("midpointRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = trapezoidIntegration(a = 1.0, b = 100.0, N = 1000, f = f2);
writeln("trapezoidIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = simpsonsIntegration(a = 1.0, b = 100.0, N = 1000, f = f2);
writeln("simpsonsIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
writeln();
writeln("f(x) = x with 5000000 steps from 0 to 5000");
exact = 12500000;
calculated = leftRectangleIntegration(a = 0.0, b = 5000.0, N = 5000000, f = f3);
writeln("leftRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = rightRectangleIntegration(a = 0.0, b = 5000.0, N = 5000000, f = f3);
writeln("rightRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = midpointRectangleIntegration(a = 0.0, b = 5000.0, N = 5000000, f = f3);
writeln("midpointRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = trapezoidIntegration(a = 0.0, b = 5000.0, N = 5000000, f = f3);
writeln("trapezoidIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = simpsonsIntegration(a = 0.0, b = 5000.0, N = 5000000, f = f3);
writeln("simpsonsIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
writeln();
writeln("f(x) = x with 6000000 steps from 0 to 6000");
exact = 18000000;
calculated = leftRectangleIntegration(a = 0.0, b = 6000.0, N = 6000000, f = f3);
writeln("leftRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = rightRectangleIntegration(a = 0.0, b = 6000.0, N = 6000000, f = f3);
writeln("rightRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = midpointRectangleIntegration(a = 0.0, b = 6000.0, N = 6000000, f = f3);
writeln("midpointRectangleIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = trapezoidIntegration(a = 0.0, b = 6000.0, N = 6000000, f = f3);
writeln("trapezoidIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
calculated = simpsonsIntegration(a = 0.0, b = 6000.0, N = 6000000, f = f3);
writeln("simpsonsIntegration: calculated = ", calculated, "; exact = ", exact, "; difference = ", abs(calculated - exact));
writeln();
</lang>
output
<lang>
f(x) = x**3 with 100 steps from 0 to 1
leftRectangleIntegration: calculated = 0.245025; exact = 0.25; difference = 0.004975
rightRectangleIntegration: calculated = 0.255025; exact = 0.25; difference = 0.005025
midpointRectangleIntegration: calculated = 0.249988; exact = 0.25; difference = 1.25e-05
trapezoidIntegration: calculated = 0.250025; exact = 0.25; difference = 2.5e-05
simpsonsIntegration: calculated = 0.25; exact = 0.25; difference = 5.55112e-17
f(x) = 1/x with 1000 steps from 1 to 100
leftRectangleIntegration: calculated = 4.65499; exact = 4.60517; difference = 0.0498211
rightRectangleIntegration: calculated = 4.55698; exact = 4.60517; difference = 0.0481889
midpointRectangleIntegration: calculated = 4.60476; exact = 4.60517; difference = 0.000407451
trapezoidIntegration: calculated = 4.60599; exact = 4.60517; difference = 0.000816058
simpsonsIntegration: calculated = 4.60517; exact = 4.60517; difference = 3.31627e-06
f(x) = x with 5000000 steps from 0 to 5000
leftRectangleIntegration: calculated = 1.25e+07; exact = 1.25e+07; difference = 2.5
rightRectangleIntegration: calculated = 1.25e+07; exact = 1.25e+07; difference = 2.5
midpointRectangleIntegration: calculated = 1.25e+07; exact = 1.25e+07; difference = 0.0
trapezoidIntegration: calculated = 1.25e+07; exact = 1.25e+07; difference = 1.86265e-09
simpsonsIntegration: calculated = 1.25e+07; exact = 1.25e+07; difference = 3.72529e-09
f(x) = x with 6000000 steps from 0 to 6000
leftRectangleIntegration: calculated = 1.8e+07; exact = 1.8e+07; difference = 3.0
rightRectangleIntegration: calculated = 1.8e+07; exact = 1.8e+07; difference = 3.0
midpointRectangleIntegration: calculated = 1.8e+07; exact = 1.8e+07; difference = 7.45058e-09
trapezoidIntegration: calculated = 1.8e+07; exact = 1.8e+07; difference = 3.72529e-09
simpsonsIntegration: calculated = 1.8e+07; exact = 1.8e+07; difference = 0.0
</lang>
=={{header|CoffeeScript}}==
{{trans|python}}
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