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Ingy döt Net
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Task/Numerical-integration/Julia/numerical-integration-2.jl
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24
Task/Numerical-integration/Julia/numerical-integration-2.jl
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@ -0,0 +1,24 @@
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leftrect(f, x, _) = f(x)
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midrect(f, x, h) = f(x + h / 2)
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rightrect(f, x, h) = f(x + h)
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trapezium(f, x, h) = (f(x) + f(x + h)) / 2
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simpson(f, x, h) = (f(x) + 4 * f(x + h / 2) + f(x + h)) / 6
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cube(x) = x * x * x
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reciprocal(x) = inv(x)
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function integrate(f, a, b, steps, meth)
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h = (b - a) / steps
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return h * sum(meth(f, a + i * h, h) for i in 0:steps-1)
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end
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for (a, b, steps, fun) in [
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(0, 1, 100, cube),
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(1, 100, 1000, reciprocal),
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(0, 5000, 5_000_000, identity),
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(0, 6000, 6_000_000, identity)]
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for rule in [leftrect, midrect, rightrect, trapezium, simpson]
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println(string(fun) * " integrated using " * string(rule))
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println(" from $a to $b ($steps steps) = ",
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round(integrate(fun, a, b, steps, rule), sigdigits=14))
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end
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end
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@ -1,48 +1,108 @@
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/*REXX pgm performs numerical integration using 5 different algorithms and show results.*/
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numeric digits 20 /*use twenty decimal digits precision. */
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-- 8 Jun 2025
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include Settings
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arg digs
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if digs = '' then
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digs=9
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numeric digits digs
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do test=1 for 4; say /*perform the 4 different test suites. */
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if test==1 then do; L= 0; H= 1; i= 100; end
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if test==2 then do; L= 1; H= 100; i= 1000; end
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if test==3 then do; L= 0; H= 5000; i= 5000000; end
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if test==4 then do; L= 0; H= 6000; i= 6000000; end
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say center('test' test, 79, "═") /*display a header for the test suite. */
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say ' left rectangular('L", "H', 'i") ──► " left_rect(L, H, i)
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say ' midpoint rectangular('L", "H', 'i") ──► " midpoint_rect(L, H, i)
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say ' right rectangular('L", "H', 'i") ──► " right_rect(L, H, i)
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say ' Simpson('L", "H', 'i") ──► " Simpson(L, H, i)
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say ' trapezium('L", "H', 'i") ──► " trapezium(L, H, i)
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end /*test*/
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exit /*stick a fork in it, we're all done. */
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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f: parse arg y; if test>2 then return y /*choose the "as─is" function. */
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if test==1 then return y**3 /* " " cube function. */
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return 1/y /* " " reciprocal " */
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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left_rect: procedure expose test; parse arg a,b,#; $= 0; h= (b-a)/#
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do x=a by h for #; $= $ + f(x)
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end /*x*/
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return $*h/1
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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midpoint_rect: procedure expose test; parse arg a,b,#; $= 0; h= (b-a)/#
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do x=a+h/2 by h for #; $= $ + f(x)
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end /*x*/
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return $*h/1
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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right_rect: procedure expose test; parse arg a,b,#; $= 0; h= (b-a)/#
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do x=a+h by h for #; $= $ + f(x)
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end /*x*/
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return $*h/1
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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Simpson: procedure expose test; parse arg a,b,#; h= (b-a)/#
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hh= h/2; $= f(a + hh)
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@= 0; do x=1 for #-1; hx=h*x + a; @= @ + f(hx)
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$= $ + f(hx + hh)
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end /*x*/
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say 'NUMERICAL INTEGRATION: COMPARE 5 METHODS'
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say version
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say
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w=Digits()+2
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say Left('Function',11) Left( 'Range',w+4) Left('Method',9) Left( 'Steps',7),
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Left('Result',w) Left('True',w) ' Error'
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say
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call Task 'x**3', 0, 1, 100, 1/4
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call Task '1/x', 1, 100, 1000, Ln(100)/1
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call Task 'x', 0, 5000, 1000000, 12500000
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call Task 'x', 0, 5000, 4, 12500000
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call Task '4/(x**2+1)', 0, 1, 100, Pi()/1
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call Task '4/(x**2+1)', 0, 1, 1000, Pi()/1
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call Task 'Sin(x)', 0, Pi()/1, 100, 2
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call Task 'Sin(x)', 0, Pi()/1, 10000, 2
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call Task 'Cos(x)', 0, Pi()/1, 100, 0
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call Task 'Cos(x)', 0, Pi()/1, 10000, 0
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call Task 'Tan(x)', 0, Pi()/1, 100, 0
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call Task 'Tan(x)', 0, Pi()/1, 10000, 0
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call Task 'Exp(x)', -3, 3, 100, Exp(3)-Exp(-3)
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call Task 'Exp(x)', -3, 3, 10000, Exp(3)-Exp(-3)
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call Task 'Gamma(x)', 1, 8, 100, 2603.238829328642145
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call Task 'Gamma(x)', 1, 8, 10000, 2603.238829328642145
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call Timer
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exit
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return h * (f(a) + f(b) + 4*$ + 2*@) / 6
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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trapezium: procedure expose test; parse arg a,b,#; $= 0; h= (b-a)/#
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do x=a by h for #; $= $ + (f(x) + f(x+h))
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end /*x*/
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return $*h/2
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Task:
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procedure expose glob.
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arg ff,aa,bb,steps,true
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w=Digits()+2
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res=LeftRect(ff,aa,bb,steps); diff=res-true
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say Left(ff,11) Left(aa '-' bb,w+4) Left('LeftRect',9) Left(steps,7),
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Left(Std(res),w) Left(true,w) Format(diff,2,4,,0)
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res=MidRect(ff,aa,bb,steps); diff=res-true
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say Left(ff,11) Left(aa '-' bb,w+4) Left('MidRect',9) Left(steps,7),
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Left(Std(res),w) Left(true,w) Format(diff,2,4,,0)
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res=RightRect(ff,aa,bb,steps); diff=res-true
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say Left(ff,11) Left(aa '-' bb,w+4) Left('RightRect',9) Left(steps,7),
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Left(Std(res),w) Left(true,w) Format(diff,2,4,,0)
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res=Trapezoid(ff,aa,bb,steps); diff=res-true
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say Left(ff,11) Left(aa '-' bb,w+4) Left('Trapezoid',9) Left(steps,7),
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Left(Std(res),w) Left(true,w) Format(diff,2,4,,0)
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res=Simpson(ff,aa,bb,steps); diff=res-true
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say Left(ff,11) Left(aa '-' bb,w+4) Left('Simpson',9) Left(steps,7),
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Left(Std(res),w) Left(true,w) Format(diff,2,4,,0)
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say
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return
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LeftRect:
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procedure expose glob.
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arg ff,aa,bb,steps
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h=(bb-aa)/steps; s=0
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do n = 0 to steps-1
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s=s+Eval(ff,aa+n*h)
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end
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return s*h
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MidRect:
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procedure expose glob.
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arg ff,aa,bb,steps
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h=(bb-aa)/steps; s=0; aa=aa-h/2
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do n = 1 to steps
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s=s+Eval(ff,aa+n*h)
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end
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return s*h
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RightRect:
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procedure expose glob.
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arg ff,aa,bb,steps
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h=(bb-aa)/steps; s=0
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do n = 1 to steps
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s=s+Eval(ff,aa+n*h)
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end
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return s*h
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Trapezoid:
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procedure expose glob.
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arg ff,aa,bb,steps
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h=(bb-aa)/steps; s=0.5*(Eval(ff,aa)+Eval(ff,bb))
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do n = 1 to steps-1
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s=s+Eval(ff,aa+n*h)
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end
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return s*h
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Simpson:
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procedure expose glob.
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arg ff,aa,bb,steps
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h=(bb-aa)/steps; s=Eval(ff,aa)+Eval(ff,bb)
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do n = 1 by 2 to steps-1
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s=s+4*Eval(ff,aa+n*h)
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end
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do n = 2 by 2 to steps-1
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s=s+2*Eval(ff,aa+n*h)
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end
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return s*h/3
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include Functions
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include Special
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include Constants
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include Helper
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include Abend
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