Numerical integration: Difference between revisions
Rename Perl 6 -> Raku, alphabetize, minor clean-up
(→{{header|PL/I}}: Rewritten to incorporate all the requirements.) |
Thundergnat (talk | contribs) (Rename Perl 6 -> Raku, alphabetize, minor clean-up) |
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trace(simpson(f1, -1, 2 ,4 ));
</lang>
=={{header|Ada}}==
Specification of a generic package implementing the five specified kinds of numerical integration:
Line 455 ⟶ 456:
fun(x) { ; linear test function
Return x
}</lang>
=={{header|BASIC}}==
Line 866 ⟶ 867:
18000000
18000000</lang>
=={{header|C++}}==
Line 1,095 ⟶ 1,097:
simpsonsIntegration: calculated = 1.8e+07; exact = 1.8e+07; difference = 0.0
</lang>
=={{header|CoffeeScript}}==
{{trans|python}}
Line 1,834 ⟶ 1,837:
end module FunctionHolder</lang>
=={{header|FreeBASIC}}==
Based on the BASIC entry and the BBC BASIC entry
Line 2,977 ⟶ 2,981:
print integrate "i.trapezium "fn2 4 -1 2 ; 2.351014
print integrate "i.simpsons "fn2 4 -1 2 ; 2.447732</lang>
=={{header|Lua}}==
<lang lua>function leftRect( f, a, b, n )
Line 3,558 ⟶ 3,563:
$_
in [0..6000] / 6000000
exact result: 18000000
Line 3,751 ⟶ 3,654:
x 0 - 6000 6000000 17999997 18000000 18000003 18000000 18000000
</pre>
=={{header|PicoLisp}}==
<lang PicoLisp>(scl 6)
(de leftRect (Fun X)
(Fun X) )
(de rightRect (Fun X H)
(Fun (+ X H)) )
(de midRect (Fun X H)
(Fun (+ X (/ H 2))) )
(de trapezium (Fun X H)
(/ (+ (Fun X) (Fun (+ X H))) 2) )
(de simpson (Fun X H)
(*/
(+
(Fun X)
(* 4 (Fun (+ X (/ H 2))))
(Fun (+ X H)) )
6 ) )
(de square (X)
(*/ X X 1.0) )
(de integrate (Fun From To Steps Meth)
(let (H (/ (- To From) Steps) Sum 0)
(for (X From (>= (- To H) X) (+ X H))
(inc 'Sum (Meth Fun X H)) )
(*/ H Sum 1.0) ) )
(prinl (round (integrate square 3.0 7.0 30 simpson)))</lang>
Output:
<pre>105.333</pre>
=={{header|PL/I}}==
Line 3,832 ⟶ 3,771:
1.799999700000000E+0007 1.800000000000000E+0007 1.800000300000000E+0007 1.800000000000000E+0007 1.800000000000000E+0007
</pre>
=={{header|PureBasic}}==
Line 4,268 ⟶ 4,171:
simpson: 17999999.999999993
</lang>
=={{header|Raku}}==
(formerly Perl 6)
The addition of <tt>'''Promise'''</tt>/<tt>'''await'''</tt> allows for concurrent computation, and brings a significant speed-up in running time. Which is not to say that it makes this code fast, but it does make it less slow.
Note that these integrations are done with rationals rather than floats, so should be fairly precise (though of course with so few iterations they are not terribly accurate (except when they are)). Some of the sums do overflow into <tt>Num</tt> (floating point)--currently Rakudo allows 64-bit denominators--but at least all of the interval arithmetic is exact.
{{works with|Rakudo|2018.09}}
<lang perl6>use MONKEY-SEE-NO-EVAL;
sub leftrect(&f, $a, $b, $n) {
my $h = ($b - $a) / $n;
$h * [+] do f($_) for $a, $a+$h ... $b-$h;
}
sub rightrect(&f, $a, $b, $n) {
my $h = ($b - $a) / $n;
$h * [+] do f($_) for $a+$h, $a+$h+$h ... $b;
}
sub midrect(&f, $a, $b, $n) {
my $h = ($b - $a) / $n;
$h * [+] do f($_) for $a+$h/2, $a+$h+$h/2 ... $b-$h/2;
}
sub trapez(&f, $a, $b, $n) {
my $h = ($b - $a) / $n;
my $partial-sum += f($_) * 2 for $a+$h, $a+$h+$h ... $b-$h;
$h / 2 * [+] f($a), f($b), $partial-sum;
}
sub simpsons(&f, $a, $b, $n) {
my $h = ($b - $a) / $n;
my $h2 = $h/2;
my $sum1 = f($a + $h2);
my $sum2 = 0;
for $a+$h, *+$h ... $b-$h {
$sum1 += f($_ + $h2);
$sum2 += f($_);
}
($h / 6) * (f($a) + f($b) + 4*$sum1 + 2*$sum2);
}
sub integrate($f, $a, $b, $n, $exact) {
my @r0;
my $e = 0.000001;
@r0.push: "$f\n in [$a..$b] / $n\n";
@r0.push: ' exact result: '~ $exact.round($e);
my (@r1,@r2,@r3,@r4,@r5);
my &f;
EVAL "&f = $f";
my $p1 = Promise.start( { @r1.push: ' rectangle method left: '~ leftrect(&f, $a, $b, $n).round($e) } );
my $p2 = Promise.start( { @r2.push: ' rectangle method right: '~ rightrect(&f, $a, $b, $n).round($e) } );
my $p3 = Promise.start( { @r3.push: ' rectangle method mid: '~ midrect(&f, $a, $b, $n).round($e) } );
my $p4 = Promise.start( { @r4.push: 'composite trapezoidal rule: '~ trapez(&f, $a, $b, $n).round($e) } );
my $p5 = Promise.start( { @r5.push: ' quadratic simpsons rule: '~ simpsons(&f, $a, $b, $n).round($e) } );
await $p1, $p2, $p3, $p4, $p5;
@r0, @r1, @r2, @r3, @r4, @r5;
}
.say for integrate '{ $_ ** 3 }', 0, 1, 100, 0.25; say '';
.say for integrate '1 / *', 1, 100, 1000, log(100); say '';
.say for integrate '*.self', 0, 5_000, 5_000_000, 12_500_000; say '';
.say for integrate '*.self', 0, 6_000, 6_000_000, 18_000_000;</lang>
{{out}}
<pre>{ $_ ** 3 }
in [0..1] / 100
exact result: 0.25
rectangle method left: 0.245025
rectangle method right: 0.255025
rectangle method mid: 0.249988
composite trapezoidal rule: 0.250025
quadratic simpsons rule: 0.25
1 / *
in [1..100] / 1000
exact result: 4.60517
rectangle method left: 4.654991
rectangle method right: 4.556981
rectangle method mid: 4.604763
composite trapezoidal rule: 4.605986
quadratic simpsons rule: 4.60517
*.self
in [0..5000] / 5000000
exact result: 12500000
rectangle method left: 12499997.5
rectangle method right: 12500002.5
rectangle method mid: 12500000
composite trapezoidal rule: 12500000
quadratic simpsons rule: 12500000
*.self
in [0..6000] / 6000000
exact result: 18000000
rectangle method left: 17999997
rectangle method right: 18000003
rectangle method mid: 18000000
composite trapezoidal rule: 18000000
quadratic simpsons rule: 18000000</pre>
=={{header|REXX}}==
Line 4,519 ⟶ 4,525:
trapezium 1.98352353750945 (-0.8%)
simpson 2.0000067844418 (0.0%)</pre>
=={{header|Rust}}==
Line 4,634 ⟶ 4,639:
(define t (integrate square 0 1 10 trapezium))
(define s (integrate square 0 1 10 simpson))</lang>
=={{header|SequenceL}}==
Line 4,762 ⟶ 4,710:
x 0 - 6000 17999997. 18000003. 18000000. 18000000. 18000000."
</pre>
=={{header|Sidef}}==
{{trans|Perl 6}}
<lang ruby>func sum(f, start, from, to) {
var s = 0;
RangeNum(start, to, from-start).each { |i|
s += f(i);
}
return s
}
func leftrect(f, a, b, n) {
var h = ((b - a) / n);
h * sum(f, a, a+h, b-h);
}
func rightrect(f, a, b, n) {
var h = ((b - a) / n);
h * sum(f, a+h, a + 2*h, b);
}
func midrect(f, a, b, n) {
var h = ((b - a) / n);
h * sum(f, a + h/2, a + h + h/2, b - h/2)
}
func trapez(f, a, b, n) {
var h = ((b - a) / n);
h/2 * (f(a) + f(b) + sum({ f(_)*2 }, a+h, a + 2*h, b-h));
}
func simpsons(f, a, b, n) {
var h = ((b - a) / n);
var h2 = h/2;
var sum1 = f(a + h2);
var sum2 = 0;
sum({|i| sum1 += f(i + h2); sum2 += f(i); 0 }, a+h, a+h+h, b-h);
h/6 * (f(a) + f(b) + 4*sum1 + 2*sum2);
}
func tryem(label, f, a, b, n, exact) {
say "\n#{label}\n in [#{a}..#{b}] / #{n}";
say(' exact result: ', exact);
say(' rectangle method left: ', leftrect(f, a, b, n));
say(' rectangle method right: ', rightrect(f, a, b, n));
say(' rectangle method mid: ', midrect(f, a, b, n));
say('composite trapezoidal rule: ', trapez(f, a, b, n));
say(' quadratic simpsons rule: ', simpsons(f, a, b, n));
}
tryem('x^3', { _ ** 3 }, 0, 1, 100, 0.25);
tryem('1/x', { 1 / _ }, 1, 100, 1000, log(100));
tryem('x', { _ }, 0, 5_000, 5_000_000, 12_500_000);
tryem('x', { _ }, 0, 6_000, 6_000_000, 18_000_000);</lang>
=={{header|Standard ML}}==
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