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RK4 for a vectorial function

by Lucero (Novice)
 on Nov 26, 2021 at 02:38 UTC Need Help??

Lucero has asked for the wisdom of the Perl Monks concerning the following question:

Hello. Im trying to calculate the transition probability for a constant pertubation. What I have to do is apply the RK4 method to a vectorial function
``` :fn(cm, t) =sum over m((-i/h)*<n|V|m>exp(-i(Em -En)t/h))Cm)
The values of |cm|^2 must oscillate around 1 and 0 but I run the code the values I get are bigger and dont have the oscillatory behavior.
```
use warnings;
use strict;
use PDL;
use PDL::Complex;

my \$eprim=0.001;

my \$X =pdl [
[ 0, -sqrt(1),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
+],
[sqrt(1),0, sqrt(2),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
+,0,0],
[0, sqrt(2),0,sqrt(3),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
+,0,0],
[0,0,sqrt(3),0,sqrt(4),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,sqrt(4),0,sqrt(5),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,sqrt(5),0,sqrt(6),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,0,sqrt(6),0,sqrt(7),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,0,0,sqrt(7),0,sqrt(8),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,0,0,0,sqrt(8),0,sqrt(9),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,0,0,0,0,sqrt(9),0,sqrt(10),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
+,0,0],
[0,0,0,0,0,0,0,0,0,sqrt(10),0,sqrt(11),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,sqrt(11),0,sqrt(12),0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,sqrt(12),0,sqrt(13),0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,sqrt(13),0,sqrt(14),0,0,0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(14),0,sqrt(15),0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(15),0,sqrt(16),0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(16),0,sqrt(17),0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(17),0,sqrt(18),0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(18),0,sqrt(19),0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(19),0,sqrt(20),0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(20),0,sqrt(21),0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(21),0,sqrt(22),0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(22),0,sqrt(23),0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(23),0,sqrt(24),0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(24),0,sqrt(25),
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(25),0,sqrt(26
+),0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(26),0,sqrt(
+27),0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(27),0,sqr
+t(27)],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(28),0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(28)]

];

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+7,-0.0000000000000002,0.1125603524749182,-0.0000000000000002,-0.34863
+08267332468,0.0000000000000004 ],
[ -0.0000000000000000,-0.0000071522511841,-0.0000000000000000,-0.0001
+531738843782,-0.0000000000000000,-0.0016457996800569,-0.0000000000000
+000,-0.0110351632854802,-0.0000000000000000,-0.0502042359369793,-0.00
+00000000000001,-0.1598827125229744,-0.0000000000000002,-0.35410559958
+16644,-0.0000000000000003,-0.5131002961113091,-0.0000000000000004,-0.
+3827211429896542,-0.0000000000000001,0.0868953534697583,0.00000000000
+00002,0.4060663544932935,0.0000000000000002,0.0906849061585278,0.0000
+000000000001,-0.3565218256839784,0.0000000000000002,-0.09547988946696
+47,-0.0000000000000000,0.3460718461864576 ],
[ -0.0000001292972631,-0.0000000000000001,-0.0000057407224638,0.00000
+00000000000,-0.0000969677496746,0.0000000000000000,-0.000957639115608
+1,-0.0000000000000002,-0.0063336604552633,0.0000000000000000,-0.02981
+94086989161,0.0000000000000001,-0.1025898937974792,0.0000000000000004
+,-0.2581306279126550,0.0000000000000004,-0.4608777058952639,0.0000000
+000000003,-0.5289025115880573,0.0000000000000000,-0.2501979027713430,
+-0.0000000000000003,0.2459229288491326,-0.0000000000000000,0.39768907
+21949536,0.0000000000000001,-0.0441899989974140,-0.0000000000000001,-
+0.3828591066315501,0.0000000000000001 ],
[ 0.0000000000000000,0.0000004836289753,0.0000000000000000,0.00001259
+35691119,0.0000000000000000,0.0001674328686760,0.0000000000000000,0.0
+014204630187498,0.0000000000000000,0.0084233556059741,0.0000000000000
+000,0.0364726872642141,0.0000000000000000,0.1173353239240527,0.000000
+0000000001,0.2791623093846674,0.0000000000000001,0.4743110004921181,0
+.0000000000000002,0.5168194754141317,0.0000000000000002,0.21722128100
+95154,0.0000000000000001,-0.2660675030932704,0.0000000000000000,-0.38
+68615369372250,-0.0000000000000001,0.0582093189941330,0.0000000000000
+002,0.3792058820414961 ],
[ 0.0000000043467313,-0.0000000000000002,0.0000002354338414,0.0000000
+000000001,0.0000049191205605,-0.0000000000000001,0.0000611103982443,0
+.0000000000000001,0.0005191484660093,0.0000000000000002,0.00322513771
+93584,-0.0000000000000002,0.0151840473609755,-0.0000000000000001,0.05
+51258405915315,0.0000000000000001,0.1547931322781031,0.00000000000000
+01,0.3318050095086971,-0.0000000000000004,0.5216362528850688,-0.00000
+00000000001,0.5386522451902931,-0.0000000000000004,0.2190409351878887
+,0.0000000000000002,-0.2724800121518238,0.0000000000000002,-0.4221250
+094454858,0.0000000000000001 ],
[ -0.0000000000000000,-0.0000000167699136,-0.0000000000000000,-0.0000
+005331602679,-0.0000000000000000,-0.0000087760999773,-0.0000000000000
+000,-0.0000937543740401,-0.0000000000000000,-0.0007150017363018,-0.00
+00000000000000,-0.0040915831845336,-0.0000000000000000,-0.01805307191
+06190,-0.0000000000000000,-0.0621905012332527,-0.0000000000000000,-0.
+1672502296997740,-0.0000000000000001,-0.3457819876105404,-0.000000000
+0000002,-0.5268406404622072,-0.0000000000000002,-0.5273801655434678,-
+0.0000000000000002,-0.1998433947965230,0.0000000000000000,0.280500487
+4860440,0.0000000000000003,0.4170641909080779 ],
[ 0.0000000000469012,0.0000000000000001,0.0000000031569888,-0.0000000
+000000001,0.0000000829508766,0.0000000000000000,0.0000013139183774,-0
+.0000000000000001,0.0000144662736142,0.0000000000000001,0.00011879541
+44757,-0.0000000000000001,0.0007576595856397,0.0000000000000002,0.003
+8457059058298,-0.0000000000000001,0.0157544770583618,0.00000000000000
+02,0.0524034914224598,-0.0000000000000002,0.1412730634268711,0.000000
+0000000000,0.3051191205191895,-0.0000000000000000,0.5135323458897670,
+0.0000000000000002,0.6319248579965772,-0.0000000000000001,0.469989404
+2726668,-0.0000000000000000 ],
[ -0.0000000000000000,0.0000000001872384,-0.0000000000000000,0.000000
+0073974097,-0.0000000000000000,0.0000001531161950,-0.0000000000000000
+,0.0000020854332860,-0.0000000000000000,0.0000206106783438,-0.0000000
+000000000,0.0001558996179038,-0.0000000000000000,0.0009318666076831,-
+0.0000000000000000,0.0044889011562547,-0.0000000000000000,0.017620794
+6824076,-0.0000000000000000,0.0565953635430333,-0.0000000000000000,0.
+1482728021860708,-0.0000000000000000,0.3129386149972143,-0.0000000000
+000000,0.5172629611893457,-0.0000000000000001,0.6280792712999990,-0.0
+000000000000001,0.4631473700646182 ]
]; #matrix T de los vectores del doblepozo
my \$vect= transpose(\$vec); #matrix con los vectores en la columna

# <n|V|m>
my \$V= \$vec x \$X x \$vect;
my \$step=0.1;
my \$hbar=1;
#print &Vnm(0,1);
#print&Vnm(1,0);
my @Em=(-3.49090786,-3.48609059,-0.94249804,-0.72223315,0.84486159,1.9
+8334101,3.52168993,5.19562490,7.00900149,8.96758961,11.06144444445427
+60,13.1282317263938650,15.4597663541535190,18.2923955058715430,19.575
+1133106800170,21.3254642902799140,27.3060148356095880,33.615327480770
+2470,48.2864949409823790,58.1657460989797170,83.0782032408711330,97.3
+475523518670660,137.5307666823436800,157.3985331875067300,221.5161226
+004345300,248.6116702653141900,353.3740736965299900,390.0680354316574
+500,577.9792266930371600,628.4681868621187300);

my \$w=\$Em[1]-\$Em[0];
sub Vnm {
my (\$x1,\$x2)=@_;
return \$V->range([\$x1,\$x2]);
};
my \$n;
my \$m;
my \$f;
my @c = (1, 0, 0, 0); #INITIAL CONDITIONS
#@c=F=(F0,F1,F2,F3)
sub F0 {
my (\$t,@c)=@_;
\$m=0;
\$f=0;
for (\$n=0;\$n<4;\$n++){
my \$a= &Vnm(\$n,\$m)*exp(-(i)*(\$Em[\$m]-\$Em[\$n])*(\$t/\$hbar));
if (\$n!=\$m){
\$f = \$f + \$a*(-(i)*(-\$eprim)*cos(\$w*\$t)*\$c[\$n]);
};
};
return \$f;
};
sub F1 {
my (\$t,@c)=@_;
\$m=1;
\$f=0;
for (\$n=0;\$n<4;\$n++){
my \$a= &Vnm(\$n,\$m)*exp(-(i)*(\$Em[\$m]-\$Em[\$n])*(\$t/\$hbar));
if (\$n!=\$m){
\$f = \$f + \$a*(-(i)*(-\$eprim)*cos(\$w*\$t)*\$c[\$n]);
};
};
return \$f;
};
sub F2 {
my (\$t,@c)=@_;
\$m=2;
\$f=0;
for (\$n=0;\$n<4;\$n++){
my \$a= &Vnm(\$n,\$m)*exp(-(i)*(\$Em[\$m]-\$Em[\$n])*(\$t/\$hbar));
if (\$n!=\$m){
\$f = \$f + \$a*-(i)*(-\$eprim)*cos(\$w*\$t)*\$c[\$n];
};
};
return \$f;
};
sub F3 {
my (\$t,@c)=@_;
\$m=3;
\$f=0;
for (\$n=0;\$n<4;\$n++){
my \$a= &Vnm(\$n,\$m)*exp(-(i)*(\$Em[\$m]-\$Em[\$n])*(\$t/\$hbar));
for (\$n!=\$m){
\$f = \$f +\$a*(-(i)*(-\$eprim)*cos(\$w*\$t)*\$c[\$n]);
};
};
return \$f;
};

#Método  RK4
my \$k1;
my \$k2;
my \$k3;
my \$k4;

my \$k1_1;
my \$k2_1;
my \$k3_1;
my \$k4_1;

my \$k1_2;
my \$k2_2;
my \$k3_2;
my \$k4_2;

my \$k1_3;
my \$k2_3;
my \$k3_3;
my \$k4_3;

sub RK4_c0 {
my (\$t,@c)=@_;
sub k1 { my (\$t1,@c)=@_;return \$step*&F0(\$t1,@c); };
sub k2 { my (\$t2,@c)=@_;return \$step*&F0((\$t2+\$step/2),( @c + (&k1(\$t2
+)/2)));};
sub k3 { my (\$t3,@c)=@_;return \$step*&F0((\$t3+\$step/2),( @c + (&k2(\$t3
+)/2))) ;};
sub k4 { my (\$t4,@c)=@_;return \$step*&F0((\$t4+\$step),( @c + &k3(\$t4)))
+;};
return  \$c[0] + (1/6)*(&k1(\$t,@c)+ 2*(&k2(\$t,@c) + &k3(\$t,@c)) + &k4(\$
+t,@c));
};

sub RK4_c1 {
my (\$t,@c)=@_;
sub k1_1 { my (\$t1,@c)=@_;return \$step*&F1(\$t1,@c); };
sub k2_1 { my (\$t2,@c)=@_;return \$step*&F1((\$t2+\$step/2),( @c + &k1(\$t
+2/2)));};
sub k3_1 { my (\$t3,@c)=@_;return \$step*&F1((\$t3+\$step/2),( @c + &k2(\$t
+3)/2)) ;};
sub k4_1 { my (\$t4,@c)=@_;return \$step*&F1((\$t4+\$step),( @c + &k3(\$t4)
+));};
return  \$c[1] + (1/6)*(&k1_1(\$t,@c)+ 2*(&k2_1(\$t,@c) + &k3_1(\$t,@c)) +
+ &k4_1(\$t,@c));
};

sub RK4_c2 {

my (\$t,@c)=@_;
sub k1_2 { my (\$t1,@c)=@_;return \$step*&F2(\$t1,@c); };
sub k2_2 { my (\$t2,@c)=@_;return \$step*&F2((\$t2+\$step/2),( @c + &k1(\$t
+2)/2));};
sub k3_2 { my (\$t3,@c)=@_;return \$step*&F2((\$t3+\$step/2),( @c + &k2(\$t
+3)/2)) ;};
sub k4_2 { my (\$t4,@c)=@_;return \$step*&F2((\$t4+\$step),( @c + &k3(\$t4)
+));};
return  \$c[2] + (1/6)*(&k1_2(\$t,@c)+ 2*(&k2_2(\$t,@c) + &k3_2(\$t,@c)) +
+ &k4_2(\$t,@c));
};

sub RK4_c3 {
my (\$t,@c)=@_;
sub k1_3 { my (\$t1,@c)=@_;return \$step*&F3(\$t1,@c); };
sub k2_3 { my (\$t2,@c)=@_;return \$step*&F3((\$t2+\$step/2),( @c + &k1(\$t
+2)/2));};
sub k3_3 { my (\$t3,@c)=@_;return \$step*&F3((\$t3+\$step/2),( @c + &k2(\$t
+3)/2)) ;};
sub k4_3 { my (\$t4,@c)=@_;return \$step*&F3((\$t4+\$step),( @c + &k3(\$t4)
+));};
return  \$c[3] + (1/6)*(&k1_3(\$t,@c)+ 2*(&k2_3(\$t,@c) + &k3_3(\$t,@c)) +
+ &k4_3(\$t,@c));
};

my \$t=0;

open (FILE , ">C0_V1W01p2.dat");
for (\$t=0; \$t<2500;\$t+=\$step){
#print  " @c \$t \n";
#print abs(\$c[0])*abs(\$c[0]), "\n";
\$c[0]= &RK4_c0(\$t,@c);
\$c[1]= &RK4_c1(\$t,@c);
\$c[2]= &RK4_c2(\$t,@c);
\$c[3]= &RK4_c3(\$t,@c);
my \$x= abs(\$c[0])**2;
my \$b= abs(\$c[1])**2;
my \$s= abs(\$c[2])**2;
my \$d= abs(\$c[3])**2;

print FILE "\$x \$b \$s \$d \$t\n";

};
close (FILE);

Replies are listed 'Best First'.
Re: RK4 for a vectorial function
by Fletch (Chancellor) on Nov 26, 2021 at 08:51 UTC

Another helping you get help suggestion: you might provide more context what an "RK4 for a vectorial function" is, because I for sure have no clue what you're referring to off hand. If this is some common algorithm for some specific problem domain that's all well and good, but the peanut gallery here is (like myself) probably completely unfamiliar with what you're talking about.

I mean a wild guess and wolfram alpha query turns up Runge-Kutta Method which looks similar if I squint and scroll around. But giving that context out up front rather than making people trying to help do the work first helps us (well, someone here maybe) help you.

The cake is a lie.
The cake is a lie.
The cake is a lie.

Re: RK4 for a vectorial function
by vr (Curate) on Nov 26, 2021 at 07:12 UTC

I definitely understand nothing about what this all is supposed to do. At 1st glance: are you sure about calling

```&F0(\$t1,@c);
...
&F0((\$t2+\$step/2),( @c + (&k1(\$t2)/2)));

? What (how many) arguments do you think F0 receives in each case?

Further, k2_1 calls k1. Just for sake of symmetry, I wonder if it should have been k1_1. Have you tested/debugged each subroutine/section, so they work as expected, instead of typing this huge and complex code to observe final results? And I suspect other monks' advice about code and PBP will follow.

Re: RK4 for a vectorial function
by jwkrahn (Monsignor) on Nov 27, 2021 at 02:24 UTC

You had a lot of duplication in your subroutines so I removed it.

Perhaps this will help?

```#!/usr/bin/perl
use warnings;
use strict;
use PDL;
use PDL::Complex;

my \$eprim = 0.001;

my \$X = pdl( [
[ 0,         -sqrt( 1 ), 0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ sqrt( 1 ), 0,          sqrt( 2 ), 0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         sqrt( 2 ),  0,         sqrt( 3 ), 0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          sqrt( 3 ), 0,         sqrt( 4 ), 0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         sqrt( 4 ), 0,         sqrt( 5
+), 0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         sqrt( 5 ), 0,
+   sqrt( 6 ), 0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         sqrt( 6
+), 0,         sqrt( 7 ), 0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   sqrt( 7 ), 0,         sqrt( 8 ), 0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         sqrt( 8 ), 0,         sqrt( 9 ),  0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         sqrt( 9 ), 0,          sqrt( 10 ), 0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         sqrt( 10 ), 0,          sqrt( 11
+), 0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          sqrt( 11 ), 0,
+   sqrt( 12 ), 0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          sqrt( 12
+), 0,          sqrt( 13 ), 0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   sqrt( 13 ), 0,          sqrt( 14 ), 0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          sqrt( 14 ), 0,          sqrt( 15 ), 0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          sqrt( 15 ), 0,          sqrt(
+16 ), 0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          sqrt( 16 ), 0,
+      sqrt( 17 ), 0,          0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          sqrt(
+17 ), 0,          sqrt( 18 ), 0,          0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      sqrt( 18 ), 0,          sqrt( 19 ), 0,          0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          sqrt( 19 ), 0,          sqrt( 20 ), 0,          0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          sqrt( 20 ), 0,          sqrt( 21 ), 0,
+         0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          sqrt( 21 ), 0,          sqr
+t( 22 ), 0,          0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          sqrt( 22 ), 0,
+         sqrt( 23 ), 0,          0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          sqr
+t( 23 ), 0,          sqrt( 24 ), 0,          0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         sqrt( 24 ), 0,          sqrt( 25 ), 0,          0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          sqrt( 25 ), 0,          sqrt( 26 ), 0,
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          sqrt( 26 ), 0,          sqrt( 27 ),
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          sqrt( 27 ), 0,
+sqrt( 27 ) ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
+   0,          0,          0,          0,          0,          0,
+      0,          0,          0,          0,          0,          0,
+         0,          0,          0,          0,          sqrt( 28 ),
+0 ],
[ 0,         0,          0,         0,         0,         0,
+   0,         0,         0,         0,          0,          0,
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+0.0000000000000001,  0.0000000829508766,  0.0000000000000000,  0.0000
+013139183774, -0.0000000000000001,  0.0000144662736142,  0.0000000000
+000001,  0.0001187954144757, -0.0000000000000001,  0.0007576595856397
+,  0.0000000000000002,  0.0038457059058298, -0.0000000000000001,  0.0
+157544770583618,  0.0000000000000002,  0.0524034914224598, -0.0000000
+000000002,  0.1412730634268711,  0.0000000000000000,  0.3051191205191
+895, -0.0000000000000000,  0.5135323458897670,  0.0000000000000002,
+0.6319248579965772, -0.0000000000000001,  0.4699894042726668, -0.0000
+000000000000 ],
[ -0.0000000000000000,  0.0000000001872384, -0.0000000000000000,
+0.0000000073974097, -0.0000000000000000,  0.0000001531161950, -0.0000
+000000000000,  0.0000020854332860, -0.0000000000000000,  0.0000206106
+783438, -0.0000000000000000,  0.0001558996179038, -0.0000000000000000
+,  0.0009318666076831, -0.0000000000000000,  0.0044889011562547, -0.0
+000000000000000,  0.0176207946824076, -0.0000000000000000,  0.0565953
+635430333, -0.0000000000000000,  0.1482728021860708, -0.0000000000000
+000,  0.3129386149972143, -0.0000000000000000,  0.5172629611893457, -
+0.0000000000000001,  0.6280792712999990, -0.0000000000000001,  0.4631
+473700646182 ],
] );    # matrix T de los vectores del doblepozo
my \$vect = transpose( \$vec ); # matrix con los vectores en la columna

# <n|V|m>
my \$V    = \$vec x \$X x \$vect;
my \$step = 0.1;
my \$hbar = 1;
#print Vnm( 0,1 );
#print Vnm( 1,0 );
my @Em = (
-3.49090786,          -3.48609059,          -0.94249804,
+ -0.72223315,           0.84486159,
1.98334101,           3.52168993,           5.19562490,
+  7.00900149,           8.96758961,
11.0614444444542760,  13.1282317263938650,  15.4597663541535190,
+ 18.2923955058715430,  19.5751133106800170,
21.3254642902799140,  27.3060148356095880,  33.6153274807702470,
+ 48.2864949409823790,  58.1657460989797170,
83.0782032408711330,  97.3475523518670660, 137.5307666823436800,
+157.3985331875067300, 221.5161226004345300,
248.6116702653141900, 353.3740736965299900, 390.0680354316574500,
+577.9792266930371600, 628.4681868621187300,
);

my \$w = \$Em[ 1 ] - \$Em[ 0 ];

sub Vnm {
my ( \$x1, \$x2 ) = @_;
return \$V->range( [ \$x1, \$x2 ] );
}

my @c = ( 1, 0, 0, 0 );    #INITIAL CONDITIONS
#@c = F = ( F0, F1, F2, F3 )

sub F {
my ( \$m, \$t, @c ) = @_;

my \$f = 0;
for my \$n ( 0 .. 3 ) {
my \$a = Vnm( \$n, \$m ) * exp( -( i ) * ( \$Em[ \$m ] - \$Em[ \$n ]
+) * ( \$t / \$hbar ) );
if ( \$n != \$m ) {
\$f += \$a * ( -( i ) * ( -\$eprim ) * cos( \$w * \$t ) * \$c[ \$
+n ] );
}
}

return \$f;
}

sub RK4_c {
my ( \$m, \$t, @c ) = @_;

my \$k1 = sub {
my ( \$t, @c ) = @_;
return \$step * F( \$m,   \$t,                 @c );
};
my \$k2 = sub {
my ( \$t, @c ) = @_;
return \$step * F( \$m, ( \$t + \$step / 2 ), ( @c + ( \$k1->( \$t )
+ / 2 ) ) );
};
my \$k3 = sub {
my ( \$t, @c ) = @_;
return \$step * F( \$m, ( \$t + \$step / 2 ), ( @c + ( \$k2->( \$t )
+ / 2 ) ) );
};
my \$k4 = sub {
my ( \$t, @c ) = @_;
return \$step * F( \$m, ( \$t + \$step ),     ( @c +   \$k3->( \$t )
+ ) );
};

return \$c[ \$m ] + ( 1 / 6 ) * ( \$k1->( \$t, @c ) + 2 * ( \$k2->( \$t,
+ @c ) + \$k3->( \$t, @c ) ) + \$k4->( \$t, @c ) );
}

open my \$FILE, '>', 'C0_V1W01p2.dat';

for ( my \$t = 0; \$t < 2500; \$t += \$step ) {
#print  " @c \$t \n";
#print abs( \$c[ 0 ] ) * abs( \$c[ 0 ] ), "\n";

my ( \$x, \$b, \$s, \$d ) = map abs( RK4_c( \$_, \$t, @c ) ) ** 2, 0 ..
+3;

print \$FILE "\$x \$b \$s \$d \$t\n";
}

close \$FILE;
Re: RK4 for a vectorial function
by bliako (Monsignor) on Nov 26, 2021 at 08:33 UTC

for 0<t<25 I get beautiful oscillations in columns 3 and 4 of the output file

side remark: You don't need to declare subs k1_1 etc. as variables (e.g. my \$k1_1;) (and you could also implement them outside the RK4_... subs, EDIT: did you mean to use them as closures? you don't seem to because you pass them the important params and step is "global" in their scope). EDIT: also you can stop calling subs using the &mysub() notation, instead mysub() is valid.

Re: RK4 for a vectorial function
by etj (Pilgrim) on Dec 18, 2021 at 15:18 UTC
Sorry to have only just noticed this; I see you're using PDL::Complex, which is now officially deprecated. If you delete that line, then with a recent (2.047+) PDL "i" will return a "cdouble" (a "native complex" type) with the obvious value and all the type stuff will do the right thing.

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