Numerical integration/Gauss-Legendre Quadrature: Difference between revisions

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<br>visual indication of what's what, &nbsp; but it's the dickens to pay when updating the source code.
<br>visual indication of what's what, &nbsp; but it's the dickens to pay when updating the source code.
<lang rexx>/*REXX program does numerical integration using an N-point Gauss─Legendre quadrature rule. */
<lang rexx>/*REXX program does numerical integration using an N-point Gauss─Legendre quadrature rule. */
pi=pi(); digs=length(pi)-1; numeric digits digs; reps=digs%2
pi= pi(); digs= length(pi)-1; numeric digits digs; reps= digs % 2
!.=.; b=3; a=-b; bma=b-a; bmaH=bma/2; tiny='1e-'digs
!.= .; b= 3; a= -b; bma= b - a; bmaH= bma / 2; tiny= '1e-'digs
trueV=exp(b)-exp(a); bpa=b+a; bpaH=bpa/2
trueV= exp(b)-exp(a); bpa= b + a; bpaH= bpa / 2
say ' step ' center("iterative value",digs+3) ' difference' /*show hdr*/
say ' step ' center("iterative value", digs+3) ' difference' /*show hdr*/
sep='──────' copies("─" ,digs+3) '─────────────'; say sep /* " sep*/
sep='──────' copies("─" , digs+3) '─────────────'; say sep /* " sep*/


do #=1 until dif>0; p0z=1; p0.1=1; p1z=2; p1.1=1; p1.2=0; ##=#+.5; r.=0
do #=1 until dif>0; p0z= 1; p0.1= 1; p1z= 2; p1.1= 1; p1.2= 0; ##= # + .5; r.= 0


/*█*/ do k=2 to #; km=k-1; do y=1 for p1z; T.y=p1.y; end /*y*/
/*█*/ do k=2 to #; km= k - 1; do y=1 for p1z; T.y= p1.y; end /*y*/
/*█*/ T.y=0; TT.=0; do L=1 for p0z; _=L+2; TT._=p0.L; end /*L*/
/*█*/ T.y= 0; TT.= 0; do L=1 for p0z; _= L + 2; TT._= p0.L; end /*L*/
/*█*/
/*█*/
/*█*/ kkm=k+km; do j=1 for p1z+1; T.j=(kkm*T.j-km*TT.j)/k ; end /*j*/
/*█*/ kkm= k + km; do j=1 for p1z +1; T.j= (kkm*T.j -km*TT.j)/k; end /*j*/
/*█*/ p0z=p1z; do n=1 for p0z; p0.n=p1.n ; end /*n*/
/*█*/ p0z= p1z; do n=1 for p0z; p0.n= p1.n ; end /*n*/
/*█*/ p1z=p1z+1; do p=1 for p1z; p1.p= T.p ; end /*p*/
/*█*/ p1z= p1z + 1; do p=1 for p1z; p1.p= T.p ; end /*p*/
/*█*/ end /*k*/
/*█*/ end /*k*/


/*▓*/ do !=1 for #; x=cos( pi * (! - .25) / ## )
/*▓*/ do !=1 for #; x= cos( pi * (! - .25) / ## )
/*▓*/
/*▓*/
/*▓*/ /*░*/ do reps until abs(dx) <= tiny
/*▓*/ /*░*/ do reps until abs(dx) <= tiny
/*▓*/ /*░*/ f=p1.1; df=0; do u=2 to p1z
/*▓*/ /*░*/ f= p1.1; df= 0; do u=2 to p1z; df= f + x*df
/*▓*/ /*░*/ df=f + x*df
/*▓*/ /*░*/ f= p1.u +x*f
/*▓*/ /*░*/ f=p1.u + x*f
/*▓*/ /*░*/ end /*u*/
/*▓*/ /*░*/ end /*u*/
/*▓*/ /*░*/ dx= f / df; x= x - dx
/*▓*/ /*░*/ dx=f/df; x=x-dx
/*▓*/ /*░*/ end /*reps ···*/
/*▓*/ /*░*/ end /*reps ···*/
/*▓*/ r.1.!= x
/*▓*/ r.1.!=x
/*▓*/ r.2.!= 2 / ( (1 - x**2) * df**2)
/*▓*/ r.2.!=2 / ((1 - x**2) * df**2)
/*▓*/ end /*!*/
⚫
$= 0
/*▓*/ end /*!*/
/*▒*/ do m=1 for #; $=$ + r.2.m * exp(bpaH + r.1.m*bmaH); end /*m*/
⚫
$=0
/*▒*/ do m=1 for #; $=$ + r.2.m * exp(bpaH + r.1.m*bmaH); end /*m*/
z= bmaH * $ /*calculate target value (Z)*/
z=bmaH*$ /*calculate target value (Z)*/
dif= z - trueV; z= format(z, 3, digs - 2) /* " difference. */
dif=z-trueV; z=format(z, 3, digs-2) /* " difference. */
Ndif= translate( format(dif, 3, 4, 2, 0), 'e', "E")
⚫
if #\==1 then say center(#, 6) z' ' Ndif /*no display if not computed*/
Ndif=translate( format(dif, 3, 4, 2, 0), 'e', "E")
⚫
if #\==1 then say center(#, 6) z' ' Ndif /*don't display if not computed.*/
end /*#*/
end /*#*/


say sep; xdif=compare(strip(z), trueV); say right("↑", 6+1+xdif)
say sep; xdif= compare( strip(z), trueV); say right("↑", 6 + 1 + xdif)
say left('', 6+1) trueV " {exact value}"; say
say left('', 6 + 1) trueV " {exact value}"; say
say 'Using ' digs " digit precision, the" ,
say 'Using ' digs " digit precision, the" ,
'N-point Gauss─Legendre quadrature (GLQ) had an accuracy of ' xdif-2 " digits."
'N-point Gauss─Legendre quadrature (GLQ) had an accuracy of ' xdif-2 " digits."
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pi: return 3.141592653589793238462643383279502884197169399375105820974944592307816406286286209
pi: return 3.141592653589793238462643383279502884197169399375105820974944592307816406286286209
/*───────────────────────────────────────────────────────────────────────────────────────────*/
/*───────────────────────────────────────────────────────────────────────────────────────────*/
cos: procedure expose !.; parse arg x; if !.x\==. then return !.x; _=1; z=1; y=x*x
cos: procedure expose !.; parse arg x; if !.x\==. then return !.x; _=1; z=1; y=x*x
do k=2 by 2 until p==z; p=z; _=-_*y/(k*(k-1)); z=z+_; end; !.x=z; return z
do k=2 by 2 until p==z; p=z; _= -_*y/(k*(k-1)); z=z+_; end; !.x=z; return z
/*───────────────────────────────────────────────────────────────────────────────────────────*/
/*───────────────────────────────────────────────────────────────────────────────────────────*/
exp: procedure; parse arg x; ix=x % 1; if abs(x-ix) > .5 then ix=ix + sign(x)
exp: procedure; parse arg x; ix= x % 1; if abs(x-ix) > .5 then ix= ix + sign(x)
x=x-ix; z=1; _=1; do j=1 until p==z; p=z; _=_*x/j; z=z+_; end
x= x-ix; z=1; _=1; do j=1 until p==z; p=z; _= _*x/j; z= z+_; end
if z\==0 then z=z * e()**ix; return z</lang>
return z * e()**ix</lang>
{{out|output|text=&nbsp; when using the default inputs:}}
'''output'''
<pre>
<pre>
step iterative value difference
step iterative value difference
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