Runge-Kutta method: Difference between revisions
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{{task}}
Given the example Differential equation:
:
With initial condition:
:
This equation has an exact solution:
:
;Task
Demonstrate the commonly used explicit [[wp:Runge–Kutta_methods#Common_fourth-order_Runge.E2.80.93Kutta_method|fourth-order Runge–Kutta method]] to solve the above differential equation.
* Solve the given differential equation over the range
* Print the calculated values of
;Method summary
Starting with a given
:
:
:
:
then:
:
:
<br><br>
=={{header|11l}}==
{{trans|Python}}
<syntaxhighlight lang="11l">F rk4(f, x0, y0, x1, n)
V vx = [0.0] * (n + 1)
V vy = [0.0] * (n + 1)
V h = (x1 - x0) / Float(n)
V x = x0
V y = y0
vx[0] = x
vy[0] = y
L(i) 1..n
V k1 = h * f(x, y)
V k2 = h * f(x + 0.5 * h, y + 0.5 * k1)
V k3 = h * f(x + 0.5 * h, y + 0.5 * k2)
V k4 = h * f(x + h, y + k3)
vx[i] = x = x0 + i * h
vy[i] = y = y + (k1 + k2 + k2 + k3 + k3 + k4) / 6
R (vx, vy)
F f(Float x, Float y) -> Float
R x * sqrt(y)
V (vx, vy) = rk4(f, 0.0, 1.0, 10.0, 100)
L(x, y) zip(vx, vy)[(0..).step(10)]
print(‘#2.1 #4.5 #2.8’.format(x, y, y - (4 + x * x) ^ 2 / 16))</syntaxhighlight>
{{out}}
<pre>
0.0 1.00000 0.00000000
1.0 1.56250 -1.45721892e-7
2.0 4.00000 -9.194792e-7
3.0 10.56250 -0.00000291
4.0 24.99999 -0.00000623
5.0 52.56249 -0.00001082
6.0 99.99998 -0.00001659
7.0 175.56248 -0.00002352
8.0 288.99997 -0.00003157
9.0 451.56246 -0.00004072
10.0 675.99995 -0.00005098
</pre>
=={{header|Action!}}==
Calculations on a real Atari 8-bit computer take quite long time. It is recommended to use an emulator capable with increasing speed of Atari CPU.
{{libheader|Action! Tool Kit}}
{{libheader|Action! Real Math}}
<syntaxhighlight lang="action!">INCLUDE "D2:PRINTF.ACT" ;from the Action! Tool Kit
INCLUDE "H6:REALMATH.ACT"
DEFINE PTR="CARD"
REAL one,two,four,six
PROC Init()
IntToReal(1,one)
IntToReal(2,two)
IntToReal(4,four)
IntToReal(6,six)
RETURN
PROC Fun=*(REAL POINTER x,y,res)
DEFINE JSR="$20"
DEFINE RTS="$60"
[JSR $00 $00 ;JSR to address set by SetFun
RTS]
PROC SetFun(PTR p)
PTR addr
addr=Fun+1 ;location of address of JSR
PokeC(addr,p)
RETURN
PROC Rate(REAL POINTER x,y,res)
REAL tmp
Sqrt(y,tmp) ;tmp=sqrt(y)
RealMult(x,tmp,res) ;res=x*sqrt(y)
RETURN
PROC RK4(PTR f REAL POINTER dx,x,y,res)
REAL k1,k2,k3,k4,dx2,k12,k22,tmp1,tmp2,tmp3
SetFun(f)
Fun(x,y,tmp1) ;tmp1=f(x,y)
RealMult(dx,tmp1,k1) ;k1=dx*f(x,y)
RealDiv(dx,two,dx2) ;dx2=dx/2
RealDiv(k1,two,k12) ;k12=k1/2
RealAdd(x,dx2,tmp1) ;tmp1=x+dx/2
RealAdd(y,k12,tmp2) ;tmp2=y+k1/2
Fun(tmp1,tmp2,tmp3) ;tmp3=f(x+dx/2,y+k1/2)
RealMult(dx,tmp3,k2) ;k2=dx*f(x+dx/2,y+k1/2)
RealDiv(k2,two,k22) ;k22=k2/2
RealAdd(y,k22,tmp2) ;tmp2=y+k2/2
Fun(tmp1,tmp2,tmp3) ;tmp3=f(x+dx/2,y+k2/2)
RealMult(dx,tmp3,k3) ;k3=dx*f(x+dx/2,y+k2/2)
RealAdd(x,dx,tmp1) ;tmp1=x+dx
RealAdd(y,k3,tmp2) ;tmp2=y+k3
Fun(tmp1,tmp2,tmp3) ;tmp3=f(x+dx,y+k3)
RealMult(dx,tmp3,k4) ;k4=dx*f(x+dx,y+k3)
RealAdd(k2,k3,tmp1) ;tmp1=k2+k3
RealMult(two,tmp1,tmp2) ;tmp2=2*k2+2*k3
RealAdd(k1,tmp2,tmp1) ;tmp3=k1+2*k2+2*k3
RealAdd(tmp1,k4,tmp2) ;tmp2=k1+2*k2+2*k3+k4
RealDiv(tmp2,six,tmp1) ;tmp1=(k1+2*k2+2*k3+k4)/6
RealAdd(y,tmp1,res) ;res=y+(k1+2*k2+2*k3+k4)/6
RETURN
PROC Calc(REAL POINTER x,res)
REAL tmp1,tmp2
RealMult(x,x,tmp1) ;tmp1=x*x
RealDiv(tmp1,four,tmp2) ;tmp2=x*x/4
RealAdd(tmp2,one,tmp1) ;tmp1=x*x/4+1
Power(tmp1,two,res) ;res=(x*x/4+1)^2
RETURN
PROC RelError(REAL POINTER a,b,res)
REAL tmp
RealDiv(a,b,tmp) ;tmp=a/b
RealSub(tmp,one,res) ;res=a/b-1
RETURN
PROC Main()
REAL x0,x1,x,dx,y,y2,err,tmp1,tmp2
CHAR ARRAY s(20)
INT i,n
Put(125) PutE() ;clear the screen
MathInit()
Init()
PrintF("%-2S %-11S %-8S%E","x","y","rel err")
IntToReal(0,x0)
IntToReal(10,x1)
ValR("0.1",dx)
RealSub(x1,x0,tmp1) ;tmp1=x1-x0
RealDiv(tmp1,dx,tmp2) ;tmp2=(x1-x0)/dx
n=RealToInt(tmp2) ;n=(x1-x0)/dx
i=0
IntToReal(1,y)
DO
IntToReal(i,tmp1) ;tmp1=i
RealMult(dx,tmp1,tmp2) ;tmp2=i*dx
RealAdd(x0,tmp2,x) ;x=x0+i*dx
IF i MOD 10=0 THEN
Calc(x,y2)
RelError(y,y2,err)
StrR(x,s) PrintF("%-2S ",s)
StrR(y,s) PrintF("%-11S ",s)
StrR(err,s) PrintF("%-8S%E",s)
FI
i==+1
IF i>n THEN EXIT FI
RK4(rate,dx,x,y,tmp1) ;tmp1=rk4(rate,dx,x0+dx*(i-1),y)
RealAssign(tmp1,y) ;y=rk4(rate,dx,x0+dx*(i-1),y)
OD
RETURN</syntaxhighlight>
{{out}}
[https://gitlab.com/amarok8bit/action-rosetta-code/-/raw/master/images/Runge-Kutta_method.png Screenshot from Atari 8-bit computer]
<pre>
x y rel err
0 1 0
1 1.56249977 -1.3E-07
2 3.99999882 -2.9E-07
3 10.56249647 -2.9E-07
4 24.99999228 -2.9E-07
5 52.56248607 -2.0E-07
6 99.99997763 -2.1E-07
7 175.562459 -1.8E-07
8 288.999935 -1.9E-07
9 451.562406 0
10 675.999869 -1.4E-07
</pre>
=={{header|Ada}}==
<
with Ada.Numerics.Generic_Elementary_Functions;
procedure RungeKutta is
Line 80 ⟶ 261:
Runge (yprime'Access, t_arr, y_arr, dt);
Print (t_arr, y_arr, 10);
end RungeKutta;</
{{out}}
<pre>y(0.0) = 1.00000000 Error: 0.00000E+00
Line 93 ⟶ 274:
y(9.0) = 451.56245928 Error: 4.07232E-05
y(10.0) = 675.99994902 Error: 5.09833E-05</pre>
=={{header|ALGOL 68}}==
<syntaxhighlight lang="algol68">
BEGIN
PROC rk4 = (PROC (REAL, REAL) REAL f, REAL y, x, dx) REAL :
Line 121 ⟶ 303:
OD
END
</syntaxhighlight>
{{out}}
<pre>
Line 136 ⟶ 318:
9.0000000 451.5625000 451.5624593 -9.0183e-08
10.0000000 676.0000000 675.9999490 -7.5419e-08
</pre>
=={{header|ALGOL W}}==
{{Trans|ALGOL 68}}
As originally defined, the signature of a procedure parameter could not be specified in Algol W (as here), modern compilers may require parameter specifications for the "f" parameter of rk4.
<syntaxhighlight lang="algolw">begin
real procedure rk4 ( real procedure f ; real value y, x, dx ) ;
begin % Fourth-order Runge-Kutta method %
real dy1, dy2, dy3, dy4;
dy1 := dx * f(x, y);
dy2 := dx * f(x + dx / 2.0, y + dy1 / 2.0);
dy3 := dx * f(x + dx / 2.0, y + dy2 / 2.0);
dy4 := dx * f(x + dx, y + dy3);
y + (dy1 + 2.0 * dy2 + 2.0 * dy3 + dy4) / 6.0
end rk4;
real x0, x1, y0, dx;
integer numPoints;
x0 := 0; x1 := 10; y0 := 1.0; % Boundary conditions. %
dx := 0.1; % Step size. %
numPoints := entier ((x1 - x0) / dx + 0.5); % Add 0.5 for rounding errors. %
begin
real procedure dyByDx ( real value x, y ) ; x * sqrt(y); % Differential equation. %
real array y ( 0 :: numPoints); y(0) := y0; % Grid and starting point. %
for i := 1 until numPoints do y(i) := rk4 (dyByDx, y(i-1), x0 + dx * (i - 1), dx);
write( " x true y calc y relative error" );
for i := 0 step 10 until numPoints do begin
real x, trueY;
x := x0 + dx * i;
trueY := (x * x + 4.0) ** 2 / 16.0;
write( r_format := "A", r_w := 12, r_d := 7, s_w := 3, x, trueY, y( i )
, r_format := "S", r_w := 12, y( i ) / trueY - 1
)
end for_i
end
end.</syntaxhighlight>
{{out}}
<pre>
x true y calc y relative error
0.0000000 1.0000000 1.0000000 0.0000e+000
1.0000000 1.5625000 1.5624998 -9.3262e-008
2.0000000 4.0000000 3.9999990 -2.2986e-007
3.0000000 10.5625000 10.5624971 -2.7546e-007
4.0000000 25.0000000 24.9999937 -2.4939e-007
5.0000000 52.5625000 52.5624891 -2.0584e-007
6.0000000 100.0000000 99.9999834 -1.6594e-007
7.0000000 175.5625000 175.5624764 -1.3395e-007
8.0000000 289.0000000 288.9999684 -1.0922e-007
9.0000000 451.5625000 451.5624592 -9.0182e-008
10.0000000 676.0000000 675.9999490 -7.5419e-008
</pre>
=={{header|APL}}==
<syntaxhighlight lang="apl">
∇RK4[⎕]∇
∇
Line 161 ⟶ 392:
[2] ⎕←'T' 'RK4 Y' 'ERROR'⍪TABLE,TABLE[;2]-{((4+⍵*2)*2)÷16}TABLE[;1]
∇
</syntaxhighlight>
{{out}}
<pre>
Line 180 ⟶ 411:
=={{header|AWK}}==
<syntaxhighlight lang="awk">
# syntax: GAWK -f RUNGE-KUTTA_METHOD.AWK
# converted from BBC BASIC
Line 200 ⟶ 431:
exit(0)
}
</syntaxhighlight>
{{out}}
<pre>
Line 218 ⟶ 449:
=={{header|BASIC}}==
==={{header|BASIC256}}===
<syntaxhighlight lang="basic256">y = 1
for i = 0 to 100
t = i / 10
if t = int(t) then
actual = ((t ^ 2 + 4) ^ 2) / 16
print "y("; int(t); ") = "; left(string(y), 13), "Error = "; left(string(actual - y), 13)
end if
k1 = t * sqr(y)
k2 = (t + 0.05) * sqr(y + 0.05 * k1)
k3 = (t + 0.05) * sqr(y + 0.05 * k2)
k4 = (t + 0.10) * sqr(y + 0.10 * k3)
y = y + 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
next i
end</syntaxhighlight>
==={{header|BBC BASIC}}===
<
FOR i% = 0 TO 100
t = i% / 10
Line 233 ⟶ 483:
k4 = (t + 0.10) * SQR(y + 0.10 * k3)
y += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
NEXT i%</
{{out}}
<pre>y(0) = 1 Error = 0
Line 247 ⟶ 497:
y(10) = 675.999949 Error = 5.09832905E-5
</pre>
==={{header|IS-BASIC}}===
<syntaxhighlight lang="is-basic">100 PROGRAM "Runge.bas"
110 LET Y=1
120 FOR T=0 TO 10 STEP .1
130 IF T=INT(T) THEN PRINT "y(";STR$(T);") =";Y;TAB(21);"Error =";((T^2+4)^2)/16-Y
140 LET K1=T*SQR(Y)
150 LET K2=(T+.05)*SQR(Y+.05*K1)
160 LET K3=(T+.05)*SQR(Y+.05*K2)
170 LET K4=(T+.1)*SQR(Y+.1*K3)
180 LET Y=Y+.1*(K1+2*(K2+K3)+K4)/6
190 NEXT</syntaxhighlight>
==={{header|QBasic}}===
{{works with|QBasic|1.1}}
{{works with|QuickBasic|4.5}}
<syntaxhighlight lang="qbasic">y! = 1
FOR i = 0 TO 100
t = i / 10
IF t = INT(t) THEN
actual! = ((t ^ 2 + 4) ^ 2) / 16
PRINT USING "y(##) = ###.###### Error = "; t; y;
PRINT actual - y
END IF
k1! = t * SQR(y)
k2! = (t + .05) * SQR(y + .05 * k1)
k3! = (t + .05) * SQR(y + .05 * k2)
k4! = (t + .1) * SQR(y + .1 * k3)
y = y + .1 * (k1 + 2 * (k2 + k3) + k4) / 6
NEXT i</syntaxhighlight>
==={{header|True BASIC}}===
{{works with|QBasic}}
<syntaxhighlight lang="qbasic">LET y = 1
FOR i = 0 TO 100
LET t = i / 10
IF t = INT(t) THEN
LET actual = ((t ^ 2 + 4) ^ 2) / 16
PRINT "y("; STR$(t); ") ="; y ; TAB(20); "Error = "; actual - y
END IF
LET k1 = t * SQR(y)
LET k2 = (t + 0.05) * SQR(y + 0.05 * k1)
LET k3 = (t + 0.05) * SQR(y + 0.05 * k2)
LET k4 = (t + 0.10) * SQR(y + 0.10 * k3)
LET Y = Y + 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
NEXT i
END</syntaxhighlight>
=={{header|C}}==
<
#include <stdlib.h>
#include <math.h>
Line 272 ⟶ 574:
double x0 = 0, x1 = 10, dx = .1;
int i, n = 1 + (x1 - x0)/dx;
y = (double *)malloc(sizeof(double) * n);
for (y[0] = 1, i = 1; i < n; i++)
Line 285 ⟶ 587:
return 0;
}</
{{out}} (errors are relative)
<pre>
Line 302 ⟶ 604:
10 676 -7.54191e-08
</pre>
=={{header|C sharp|C#}}==
<syntaxhighlight lang="csharp">
using System;
namespace RungeKutta
{
class Program
{
static void Main(string[] args)
{
//Incrementers to pass into the known solution
double t = 0.0;
double T = 10.0;
double dt = 0.1;
// Assign the number of elements needed for the arrays
int n = (int)(((T - t) / dt)) + 1;
// Initialize the arrays for the time index 's' and estimates 'y' at each index 'i'
double[] y = new double[n];
double[] s = new double[n];
// RK4 Variables
double dy1;
double dy2;
double dy3;
double dy4;
// RK4 Initializations
int i = 0;
s[i] = 0.0;
y[i] = 1.0;
Console.WriteLine(" ===================================== ");
Console.WriteLine(" Beging 4th Order Runge Kutta Method ");
Console.WriteLine(" ===================================== ");
Console.WriteLine();
Console.WriteLine(" Given the example Differential equation: \n");
Console.WriteLine(" y' = t*sqrt(y) \n");
Console.WriteLine(" With the initial conditions: \n");
Console.WriteLine(" t0 = 0" + ", y(0) = 1.0 \n");
Console.WriteLine(" Whose exact solution is known to be: \n");
Console.WriteLine(" y(t) = 1/16*(t^2 + 4)^2 \n");
Console.WriteLine(" Solve the given equations over the range t = 0...10 with a step value dt = 0.1 \n");
Console.WriteLine(" Print the calculated values of y at whole numbered t's (0.0,1.0,...10.0) along with the error \n");
Console.WriteLine();
Console.WriteLine(" y(t) " +"RK4" + " ".PadRight(18) + "Absolute Error");
Console.WriteLine(" -------------------------------------------------");
Console.WriteLine(" y(0) " + y[i] + " ".PadRight(20) + (y[i] - solution(s[i])));
// Iterate and implement the Rk4 Algorithm
while (i < y.Length - 1)
{
dy1 = dt * equation(s[i], y[i]);
dy2 = dt * equation(s[i] + dt / 2, y[i] + dy1 / 2);
dy3 = dt * equation(s[i] + dt / 2, y[i] + dy2 / 2);
dy4 = dt * equation(s[i] + dt, y[i] + dy3);
s[i + 1] = s[i] + dt;
y[i + 1] = y[i] + (dy1 + 2 * dy2 + 2 * dy3 + dy4) / 6;
double error = Math.Abs(y[i + 1] - solution(s[i + 1]));
double t_rounded = Math.Round(t + dt, 2);
if (t_rounded % 1 == 0)
{
Console.WriteLine(" y(" + t_rounded + ")" + " " + y[i + 1] + " ".PadRight(5) + (error));
}
i++;
t += dt;
};//End Rk4
Console.ReadLine();
}
// Differential Equation
public static double equation(double t, double y)
{
double y_prime;
return y_prime = t*Math.Sqrt(y);
}
// Exact Solution
public static double solution(double t)
{
double actual;
actual = Math.Pow((Math.Pow(t, 2) + 4), 2)/16;
return actual;
}
}
}</syntaxhighlight>
=={{header|C++}}==
Using Lambdas
<syntaxhighlight lang="cpp">/*
* compiled with:
* g++ (Debian 8.3.0-6) 8.3.0
*
* g++ -std=c++14 -o rk4 %
*
*/
# include <iostream>
# include <math.h>
auto rk4(double f(double, double))
{
return [f](double t, double y, double dt) -> double {
double dy1 { dt * f( t , y ) },
dy2 { dt * f( t+dt/2, y+dy1/2 ) },
dy3 { dt * f( t+dt/2, y+dy2/2 ) },
dy4 { dt * f( t+dt , y+dy3 ) };
return ( dy1 + 2*dy2 + 2*dy3 + dy4 ) / 6;
};
}
int main(void)
{
constexpr
double TIME_MAXIMUM { 10.0 },
T_START { 0.0 },
Y_START { 1.0 },
DT { 0.1 },
WHOLE_TOLERANCE { 1e-12 };
auto dy = rk4( [](double t, double y) -> double { return t*sqrt(y); } ) ;
for (
auto y { Y_START }, t { T_START };
t <= TIME_MAXIMUM;
y += dy(t,y,DT), t += DT
)
if (ceilf(t)-t < WHOLE_TOLERANCE)
printf("y(%4.1f)\t=%12.6f \t error: %12.6e\n", t, y, std::fabs(y - pow(t*t+4,2)/16));
return 0;
}</syntaxhighlight>
=={{header|Common Lisp}}==
<
(let ((h (float (/ (- x-end x) n) 1d0))
k1 k2 k3 k4)
Line 336 ⟶ 780:
(7.999999999999988d0 288.9999684347983d0 -3.156520000402452d-5)
(8.999999999999984d0 451.56245927683887d0 -4.072315812209126d-5)
(9.99999999999998d0 675.9999490167083d0 -5.0983286655537086d-5))</
=={{header|Crystal}}==
{{trans|Run Basic and Ruby output}}
<syntaxhighlight lang="ruby">y, t = 1, 0
while t <= 10
k1 = t * Math.sqrt(y)
k2 = (t + 0.05) * Math.sqrt(y + 0.05 * k1)
k3 = (t + 0.05) * Math.sqrt(y + 0.05 * k2)
k4 = (t + 0.1) * Math.sqrt(y + 0.1 * k3)
printf("y(%4.1f)\t= %12.6f \t error: %12.6e\n", t, y, (((t**2 + 4)**2 / 16) - y )) if (t.round - t).abs < 1.0e-5
y += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
t += 0.1
end</syntaxhighlight>
{{out}}
<pre>
y( 0.0) = 1.000000 error: 0.000000e+00
y( 1.0) = 1.562500 error: 1.457219e-07
y( 2.0) = 3.999999 error: 9.194792e-07
y( 3.0) = 10.562497 error: 2.909562e-06
y( 4.0) = 24.999994 error: 6.234909e-06
y( 5.0) = 52.562489 error: 1.081970e-05
y( 6.0) = 99.999983 error: 1.659460e-05
y( 7.0) = 175.562476 error: 2.351773e-05
y( 8.0) = 288.999968 error: 3.156520e-05
y( 9.0) = 451.562459 error: 4.072316e-05
y(10.0) = 675.999949 error: 5.098329e-05
</pre>
=={{header|D}}==
{{trans|Ada}}
<
alias FP = real;
Line 377 ⟶ 850:
t_arr[i], y_arr[i],
calc_err(t_arr[i], y_arr[i]));
}</
{{out}}
<pre>y(0.0) = 1.00000000 Error: 0
Line 392 ⟶ 865:
=={{header|Dart}}==
<
num RungeKutta4(Function f, num t, num y, num dt){
Line 418 ⟶ 891:
t += dt;
}
}</
{{out}}
<pre>
Line 432 ⟶ 905:
y(9.00) = 451.56245928 Error = 9.0182772312e-8
y(10.0) = 675.99994902 Error = 7.5419063100e-8
</pre>
=={{header|EasyLang}}==
{{trans|BASIC256}}
<syntaxhighlight>
numfmt 6 0
y = 1
for i = 0 to 100
t = i / 10
if t = floor t
h = t * t + 4
actual = h * h / 16
print "y(" & t & ") = " & y & " Error = " & actual - y
.
k1 = t * sqrt y
k2 = (t + 0.05) * sqrt (y + 0.05 * k1)
k3 = (t + 0.05) * sqrt (y + 0.05 * k2)
k4 = (t + 0.10) * sqrt (y + 0.10 * k3)
y += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
.
</syntaxhighlight>
=={{header|EDSAC order code}}==
The EDSAC subroutine library had two Runge-Kutta subroutines: G1 for 35-bit values and G2 for 17-bit values. A demo of G1 is given here. Setting up the parameters is rather complicated, but after that it's just a matter of calling G1 once for every step in the Runge-Kutta process.
Since EDSAC real numbers are restricted to -1 <= x < 1, the values in the Rosetta Code task have to be scaled down. For comparison with other languages it's convenient to divide the y values by 1000. With 100 steps, a convenient time interval is 1/128.
G1 can solve equations in several variables, say y_1, ..., y_n. The user must provide an auxiliary subroutine which calculates dy_1/dt, ..., dy_n/dt from y_1, ..., y_n. If the derivatives also depend on t (as in the Rosetta Code task) it's necessary to add a dummy y variable which is identical with t.
<syntaxhighlight lang="edsac">
[Demo of EDSAC library subroutine G1: Runge-Kutta solution of differential equations.
Full description is in Wilkes, Wheeler & Gill, 1951 edn, pages 32-34, 86-87, 132-134.
Before using G1, we need to fix n, m, a, b, c, d, as defined in WWG pages 86-87:
n = number of equations (2 for the Rosetta Code example).
2^m = multiplier for the hy', as large as possible without causing numeric overflow;
with the scaling chosen here, m = 5.
Variables y are stored in n consecutive long locations, the last of which is aD.
Scaled derivatives (2^m)hy' in n consecutive long locations, the last of which is bD.
G1 uses working variables in n consecutive long locations, the last of which is cD.
d = address of user-supplied auxiliary subroutine, which calculates the (2^m)hy'.
For convenience, keep G1 and its storage together. Start at (say) 400 and place:
variables y at 400D, 402D;
scaled derivatives at 404D, 406D;
workspace for G1 at 408D, 410D;
G1 itself at 412.
If the base address is placed in location 51 at load time, all the above
addresses can be accessed via the G parameter:]
T 51 K
P 400 F
[Now set up the 6 preset parameters specified in WWG:]
T 45 K
P 2#G [H parameter: P a D]
P 4 F [N parameter: P 2n F]
P 4 F [M parameter: P (b-a) F, or V (2048-a+b) F if a > b]
P 4 F [& parameter: P (c-b) F, or V (2048-b+c) F if b > c]
P 8 F [L parameter: P 2^(m-2) F]
P 300 F [X parameter: P d F]
[For other addresses in the program we can optionally use some more parameters:]
T 52 K
P 120 F [A parameter: main routine]
P 56 F [B parameter: print subroutine P1 from EDSAC library]
P 350 F [C parameter: constants for Rosetta code example]
P 78 F [V parameter: square root subroutine]
[Library subroutine to read constants; runs at load time and is then overwritten.
R5, for decimal fractions, seems to be unavailable (lost?), so the values are
here read in as 35-bit integers (i.e. times 2^34) by R2.
Values are: 0.001, initial value of y
(2^23)/(10^7) and 25/(2^10) for use in calculations
0.5/(10^9) for rounding to 9 d.p. (print routine P1 doesn't do this)]
GKT20FVDL8FA40DUDTFI40FA40FS39FG@S2FG23FA5@T5@E4@E13Z
T#C
17179869F14411518808F419430400F9#
TZ
[Library subroutine M3; prints header at load time and is then overwritten.]
PFGKIFAFRDLFUFOFE@A6FG@E8FEZPF
*SCALED!FOR!EDSAC@&!!TIME!!!!!!!!!Y!VIA!RK!!!!!Y!DIRECT@&
....PK [end text with some blank tape]
[Runge-Kutta: auxiliary subroutine to calculate (2^m)*h*(dy1/dt) and (2^m)*h*(dy2/dt)
from y1, y2, where y1 is the function y in Rosetta Code (but scaled) and y2 = t.
For the Rosetta code example we're using m = 5, h = 2^(-7)]
E25K TX GK
A3F T20@ [set up return as usual]
H2#G V2#G TD [acc := t^2, temp store in 0D]
H#G VD LD YF TD [y1 times t^2, shift left, round, temp store in 0D]
H2#C VD YF T4D [times (2^23)/(10^7), round, to 4D for square root]
[14] A14@ GV A4D T4#G [call square root, result in 4D, copy to (2^m)hy']
A21@ T6#G [1/4, i.e. (2^m)h with m and h as above, to (2^m)ht']
[20] ZF [overwritten by jump back to caller]
[21] RF [constant 1/4]
[Main routine, with two subroutines in the same address block as the main routine.]
E25K TA GK
[0] #F [figures shift on teleprinter]
[1] MF [decimal point (in figures mode)]
[2] !F @F &F [space, carriage return, line feed,]
[5] K4096F [null char]
[6] P100F [constant: nr of Runge-Kutta steps (in address field)]
[7] PF [negative count of Runge-Kutta steps]
[8] P10F [constant: number of steps between printed values]
[9] PF [negative count of steps between printed values]
[Enter with acc = 0]
[10] O@ [set teleprinter to figures]
S6@ T7@ [init negative count of R-K steps]
S8@ T9@ [init negative count of print steps]
[Before using library subroutine G1, clear its working registers (WWG page 33)]
T8#G T10#G
[Set up initial values of y1 and y2 (where y2 = t)]
A#C T#G [load 0.001 from constants section, store in y1]
T2#G [y2 = t = 0]
[20] A20@ G40@ [call subroutine to print initial values]
[Loop round Runge-Kutta steps]
[22] TF A23@ G12G [clear accumulator, call G1 for Runge-Kutta step]
A9@ A2F U9@ [update negative print count]
G33@ [skip printing if not reached 0]
S8@ T9@ [reset negative print count]
A31@ G40@ [call subroutine to print values]
[33] TF [clear accumulator]
A7@ A2F U7@ [increment negative count of Runge-Kutta steps]
G22@ [loop till count = 0]
O5@ ZF [flush teleprinter buffer; stop]
[Subroutine to print y1 as calculated (1) by Runge-Kutta (2) direct from formula]
[40] A3F T71@ [set up return as usual]
A2#G TD [latest t (= y2) from Runge-Kutta, to 0D for printing]
[44] A44@ G72@ [call subroutine to print t]
O2@ O2@ [followed by 2 spaces]
A#G TD [latest y1 from Runge-Kutta, to 0D for printing]
[50] A50@ G72@ [call subroutine to print y1]
O2@ O2@ [followed by 2 spaces]
A 4#C [load constant 25/(2^10)]
H2#G V2#G TD [add t^2, temp store result in 0D]
HD VD LD YF TD [square, shift 1 left, round, result to 0D]
H2#C VD YF TD [times (2^23)/(10^7), round, to 0D for printing]
[67] A67@ G72@ [call subroutine to print y]
O3@ O4@ [print CR, LF]
[71] ZF [overwritten by jump back to caller]
[Second-level subroutine to print number in 0D to 9 decimal places]
[72] A3F T82@ [set up return as usual]
AD A6#C TD [load number, add decimal rounding, to 0D for printing]
O81@ O1@ [print '0.' since P1 doesn't do so]
A79@ GB [call library subroutine P1 for printing]
[81] P9F [parameter for P1, 9 decimals]
[82] ZF [overwritten by jump back to caller]
[Library subroutine G1 for Runge-Kutta process. 66 locations, even address.]
E25K T12G
GKT4#ZH682DT6#ZPNT12#Z!1405DT14#ZTHT16#ZT2HTZA3FT61@A31@G63@&FT6ZPN
T8ZMMO&H4@A20@E23@T14ZAHT16ZA2HT18ZH12#@S12#@T12#@E28@H4#@T4DUFS38@
A25@T38@S6#@A16#@U46#@A8@U37@A9@U55@A24@T39@ZFR1057#@ZFYFU6DV6DRLYF
UDZFZFADLDADLLS6DN4DYFZFA46#@S14#@G29@A65@S11@ZFA35@U65@GXZF
[Replacement for library routine S2 (square root). 38 locations, even address.
Advantages: More accurate for small values of the argument.
Calculates sqrt(0) without going into an infinite loop.
Disadvantages: Longer and slower than S2 (calculates one bit at a time).]
E25K TV
GKA3FT31@A4DG32@A33@T36#@T4DA33@RDU34#@RDS4DS33@A36#@G22@T36#@A4DS34#@
T4DA36#@A33@G25@TFA36#@S33@A36#@T36#@A34#@RDYFG9@ZFZFK4096FPFPFPFPF
[Library subroutine P1 - print a single positive number. 21 locations.
Prints number in 0D to n places of decimals, where
n is specified by 'P n F' pseudo-order after subroutine call.]
E25K TB
GKA18@U17@S20@T5@H19@PFT5@VDUFOFFFSFL4FTDA5@A2FG6@EFU3FJFM1F
[Define entry point in main routine]
E25K TA GK
E10Z PF [enter at relative address 10 with accumulator = 0]
</syntaxhighlight>
{{out}}
<pre>
SCALED FOR EDSAC
TIME Y VIA RK Y DIRECT
0.000000000 0.001000000 0.001000000
0.078125000 0.001562499 0.001562500
0.156250000 0.003999998 0.004000000
0.234375000 0.010562495 0.010562500
0.312500000 0.024999992 0.025000000
0.390625000 0.052562487 0.052562500
0.468750000 0.099999981 0.100000000
0.546875000 0.175562474 0.175562500
0.625000000 0.288999965 0.289000000
0.703125000 0.451562456 0.451562500
0.781250000 0.675999945 0.676000000
</pre>
=={{header|ERRE}}==
<syntaxhighlight lang="erre">
PROGRAM RUNGE_KUTTA
Line 460 ⟶ 1,122:
Y+=DELTA_T*(K1+2*(K2+K3)+K4)/6
END FOR
END PROGRAM</
{{out}}
<pre>
Line 476 ⟶ 1,138:
</pre>
=={{header|
<syntaxhighlight lang="Excel">
//Worksheet formula to manage looping
=LET(
T₊, SEQUENCE(11, 1, 0, 1),
T, DROP(T₊, -1),
τ, SEQUENCE(1 / δt, 1, 0, δt),
calculated, SCAN(1, T, LAMBDA(y₀, t, REDUCE(y₀, t + τ, RungaKutta4λ(Dλ)))),
calcs, VSTACK(1, calculated),
exact, f(T₊),
HSTACK(T₊, calcs, exact, (exact - calcs) / exact)
)
//Lambda function passed to RungaKutta4λ to evaluate derivatives
Dλ(y,t)
= LAMBDA(y,t, t * SQRT(y))
//Curried Lambda function with derivative function D and y, t as parameters
RungaKutta4λ(Dλ)
= LAMBDA(D,
LAMBDA(yᵣ, tᵣ,
LET(
δy₁, δt * D(yᵣ, tᵣ),
δy₂, δt * D(yᵣ + δy₁ / 2, tᵣ + δt / 2),
δy₃, δt * D(yᵣ + δy₂ / 2, tᵣ + δt / 2),
δy₄, δt * D(yᵣ + δy₃, tᵣ + δt),
yᵣ₊₁, yᵣ + (δy₁ + 2 * δy₂ + 2 * δy₃ + δy₄) / 6,
yᵣ₊₁
)
)
)
//Lambda function returning the exact solution
f(t)
= LAMBDA(t, (1/16) * (t^2 + 4)^2 )
</syntaxhighlight>
{{out}}
<pre>
Time Calculated Exact Rel Error
0.00 1.000000 1.000000 0.00E+00
1.00 1.562500 1.562500 9.33E-08
2.00 3.999999 4.000000 2.30E-07
3.00 10.562497 10.562500 2.75E-07
4.00 24.999994 25.000000 2.49E-07
5.00 52.562489 52.562500 2.06E-07
6.00 99.999983 100.000000 1.66E-07
7.00 175.562476 175.562500 1.34E-07
8.00 288.999968 289.000000 1.09E-07
9.00 451.562459 451.562500 9.02E-08
10.00 675.999949 676.000000 7.54E-08
</pre>
=={{header|F_Sharp|F#}}==
{{works with|F# interactive (fsi.exe)}}
<syntaxhighlight lang="fsharp">
open System
Line 499 ⟶ 1,218:
RungeKutta4 0.0 1.0 10.0 0.1
|> Seq.filter (fun (t,y) -> t % 1.0 = 0.0 )
|> Seq.iter (fun (t,y) -> Console.WriteLine("y({0})={1}\t(relative error:{2})", t, y, (y / y_exact(t))-1.0) )</
{{out}}
Line 517 ⟶ 1,236:
=={{header|Fortran}}==
<
implicit none
integer, parameter :: dp = kind(1d0)
real(
real(dp) :: y, k1, k2, k3, k4
tstart = 0.0d0
tstop = 10.0d0
dt = 0.1d0
y = 1.0d0
t = tstart
write (6, '(a,f4.1,a,f12.8,a,es13.6)') 'y(', t, ') = ', y, ' error = ', &
abs(y-(t**2+4)**2/16)
do while (t < tstop)
k1 = dt*f(t, y)
k2 = dt*f(t+dt/2, y+k1/2)
k3 = dt*f(t+dt/2, y+k2/2)
k4 = dt*f(t+dt, y+k3)
y = y+(k1+2*(k2+k3)+k4)/6
t = t+dt
if (abs(nint(t)-t) <= 1d-12) then
write (6, '(a,f4.1,a,f12.8,a,es13.6)') 'y(', t, ') = ', y, ' error = ', &
abs(y-(t**2+4)**2/16)
end if
end do
contains
function f(t,y)
real(dp), intent(in) :: t, y
real(dp) :: f
f = t*sqrt(y)
end function f
end program rungekutta</syntaxhighlight>
{{out}}
<pre>
Line 561 ⟶ 1,283:
y(10.0) = 675.99994902 Error = 5.098329E-05
</pre>
=={{header|FreeBASIC}}==
{{trans|BBC BASIC}}
<
' compile with: fbc -s console
' translation of BBC BASIC
Line 590 ⟶ 1,313:
' empty keyboard buffer
While Inkey <> ""
Print : Print "hit any key to end program"
Sleep
End</
{{out}}
<pre>y(0) = 1 Error = 0
Line 606 ⟶ 1,329:
y(9) = 451.5624592768396 Error = 4.072316039582802e-005
y(10) = 675.9999490167097 Error = 5.098329029351589e-005</pre>
=={{header|FutureBasic}}==
<
def fn dydx( x as double, y as double ) as double = x * sqr(y)
def fn exactY( x as long ) as double = ( x ^2 + 4 ) ^2 / 16
long i
double h, k1, k2, k3, k4, x, y, result
h = 0.1
y = 1
for i = 0 to 100
x = i * h
if x == int(x)
result = fn exactY( x )
print "y("; mid$( str$(x), 2, len$(str$(x) )); ") = "; y, "Error = "; result - y
end if
k1 = h * fn dydx( x, y )
k2 = h * fn dydx( x + h / 2, y + k1 / 2 )
k3 = h * fn dydx( x + h / 2, y + k2 / 2 )
k4 = h * fn dydx( x + h, y + k3 )
y = y + 1 / 6 * ( k1 + 2 * k2 + 2 * k3 + k4 )
next
HandleEvents</syntaxhighlight>
Output:
<pre>
Line 657 ⟶ 1,374:
=={{header|Go}}==
{{works with|Go1}}
<
import (
Line 713 ⟶ 1,430:
func printErr(t, y float64) {
fmt.Printf("y(%.1f) = %f Error: %e\n", t, y, math.Abs(actual(t)-y))
}</
{{out}}
<pre>
Line 727 ⟶ 1,444:
y(9.0) = 451.562459 Error: 4.072316e-05
y(10.0) = 675.999949 Error: 5.098329e-05
</pre>
=={{header|Groovy}}==
<syntaxhighlight lang="groovy">
class Runge_Kutta{
static void main(String[] args){
def y=1.0,t=0.0,counter=0;
def dy1,dy2,dy3,dy4;
def real;
while(t<=10)
{if(counter%10==0)
{real=(t*t+4)*(t*t+4)/16;
println("y("+t+")="+ y+ " Error:"+ (real-y));
}
dy1=dy(dery(y,t));
dy2=dy(dery(y+dy1/2,t+0.05));
dy3=dy(dery(y+dy2/2,t+0.05));
dy4=dy(dery(y+dy3,t+0.1));
y=y+(dy1+2*dy2+2*dy3+dy4)/6;
t=t+0.1;
counter++;
}
}
static def dery(def y,def t){return t*(Math.sqrt(y));}
static def dy(def x){return x*0.1;}
}
</syntaxhighlight>
{{out}}
<pre>
y(0.0)=1.0 Error:0.0000
y(1.0)=1.562499854278108 Error:1.4572189210859676E-7
y(2.0)=3.999999080520799 Error:9.194792007782837E-7
y(3.0)=10.562497090437551 Error:2.9095624487496252E-6
y(4.0)=24.999993765090636 Error:6.234909363911356E-6
y(5.0)=52.562489180302585 Error:1.0819697415342944E-5
y(6.0)=99.99998340540358 Error:1.659459641700778E-5
y(7.0)=175.56247648227125 Error:2.3517728749311573E-5
y(8.0)=288.9999684347986 Error:3.156520142510999E-5
y(9.0)=451.56245927683966 Error:4.07231603389846E-5
y(10.0)=675.9999490167097 Error:5.098329029351589E-5
</pre>
=={{header|Hare}}==
<syntaxhighlight lang="hare">use fmt;
use math;
export fn main() void = {
rk4_driver(&f, 0.0, 10.0, 1.0, 0.1);
};
fn rk4_driver(func: *fn(_: f64, _: f64) f64, t_init: f64, t_final: f64, y_init: f64, h: f64) void = {
let n = ((t_final - t_init) / h): int;
let tn: f64 = t_init;
let yn: f64 = y_init;
let i: int = 1;
fmt::printfln("{: 2} {: 18} {: 21}", "t", "y(t)", "absolute error")!;
fmt::printfln("{: 2} {: 18} {: 21}", tn, yn, math::absf64(exact(tn) - yn))!;
for (i <= n; i += 1) {
yn = rk4(func, tn, yn, h);
tn = t_init + (i: f64)*h;
if (i % 10 == 0) {
fmt::printfln("{: 2} {: 18} {: 21}\t", tn, yn, math::absf64(exact(tn) - yn))!;
};
};
};
fn rk4(func: *fn(_: f64, _: f64) f64, t: f64, y: f64, h: f64) f64 = {
const k1 = func(t, y);
const k2 = func(t + 0.5*h, y + 0.5*h*k1);
const k3 = func(t + 0.5*h, y + 0.5*h*k2);
const k4 = func(t + h, y + h*k3);
return y + h/6.0 * (k1 + 2.0*k2 + 2.0*k3 + k4);
};
fn f(t: f64, y: f64) f64 = {
return t * math::sqrtf64(y);
};
fn exact(t: f64) f64 = {
return 1.0/16.0 * math::powf64(t*t + 4.0, 2.0);
};</syntaxhighlight>
{{out}}
<pre>
t y(t) absolute error
0 1 0
1 1.562499854278108 1.4572189210859676e-7
2 3.9999990805207997 9.194792003341945e-7
3 10.56249709043755 2.909562450525982e-6
4 24.999993765090633 6.23490936746407e-6
5 52.56248918030258 1.0819697422448371e-5
6 99.99998340540358 1.659459641700778e-5
7 175.56247648227125 2.3517728749311573e-5
8 288.9999684347985 3.156520148195341e-5
9 451.5624592768396 4.072316039582802e-5
10 675.9999490167097 5.098329029351589e-5
</pre>
Line 733 ⟶ 1,550:
Using GHC 7.4.1.
<syntaxhighlight lang
:: Floating a
=> a -> a -> a
dv = (. sqrt) . (*)
rk4
:: (Enum a, Fractional a)
=> (a -> a -> a) -> a -> a -> a -> [(a, a)]
rk4 fd y0 a h = zip ts $ scanl (flip fc) y0 ts
where
ts = [a,h ..]
fc t y =
sum . (y :) . zipWith (*) [1 / 6, 1 / 3, 1 / 3, 1 / 6] $
scanl
(\k f -> h * fd (t + f * h) (y + f * k))
(h * fd t y)
[1 / 2, 1 / 2, 1]
task =
mapM_
(print . (\(x, y) -> (truncate x, y, fy x - y)))
(filter (\(x, _) -> 0 == mod (truncate $ 10 * x) 10) $
take 101 $ rk4 dv 1.0 0 0.1)</syntaxhighlight>
Example executed in GHCi:
<
(0,1.0,0.0)
(1,1.5624998542781088,1.4572189122041834e-7)
Line 763 ⟶ 1,588:
(8,288.99996843479926,3.1565204153594095e-5)
(9,451.562459276841,4.0723166534917254e-5)
(10,675.9999490167125,5.098330132113915e-5)</
(See [[Euler method#Haskell]] for implementation of simple general ODE-solver
Or, disaggregated a little, and expressed in terms of a single scanl:
<syntaxhighlight lang="haskell">rk4 :: Double -> Double -> Double -> Double
rk4 y x dx =
let f x y = x * sqrt y
k1 = dx * f x y
k2 = dx * f (x + dx / 2.0) (y + k1 / 2.0)
k3 = dx * f (x + dx / 2.0) (y + k2 / 2.0)
k4 = dx * f (x + dx) (y + k3)
in y + (k1 + 2.0 * k2 + 2.0 * k3 + k4) / 6.0
actual :: Double -> Double
actual x = (1 / 16) * (x * x + 4) * (x * x + 4)
step :: Double
step = 0.1
ixs :: [Int]
ixs = [0 .. 100]
xys :: [(Double, Double)]
xys =
scanl
(\(x, y) _ -> (((x * 10) + (step * 10)) / 10, rk4 y x step))
(0.0, 1.0)
ixs
samples :: [(Double, Double, Double)]
samples =
zip ixs xys >>=
(\(i, (x, y)) ->
[ (x, y, actual x - y)
| 0 == mod i 10 ])
main :: IO ()
main =
(putStrLn . unlines) $
(\(x, y, v) ->
unwords
[ "y" ++ justifyRight 3 ' ' ('(' : show (round x)) ++ ") = "
, justifyLeft 19 ' ' (show y)
, '±' : show v
]) <$>
samples
where
justifyLeft n c s = take n (s ++ replicate n c)
justifyRight n c s = drop (length s) (replicate n c ++ s)</syntaxhighlight>
{{Out}}
<pre>y (0) = 1.0 ±0.0
y (1) = 1.562499854278108 ±1.4572189210859676e-7
y (2) = 3.999999080520799 ±9.194792007782837e-7
y (3) = 10.562497090437551 ±2.9095624487496252e-6
y (4) = 24.999993765090636 ±6.234909363911356e-6
y (5) = 52.562489180302585 ±1.0819697415342944e-5
y (6) = 99.99998340540358 ±1.659459641700778e-5
y (7) = 175.56247648227125 ±2.3517728749311573e-5
y (8) = 288.9999684347986 ±3.156520142510999e-5
y (9) = 451.56245927683966 ±4.07231603389846e-5
y(10) = 675.9999490167097 ±5.098329029351589e-5</pre>
=={{header|J}}==
'''Solution:'''
<
NB. y is: y(ta) , ta , tb , tstep
NB. u is: function to solve
Line 786 ⟶ 1,671:
end.
T ,. Y
)</
'''Example:'''
<
fyp=: (* %:)/ NB. f'(t,y)
report_whole=: (10 * i. >:10)&{ NB. report at whole-numbered t values
Line 804 ⟶ 1,689:
8 289 _3.15652e_5
9 451.562 _4.07232e_5
10 676 _5.09833e_5</
'''Alternative solution:'''
The following solution replaces the for loop as well as the calculation of the increments (ks) with an accumulating suffix.
<
'Y0 a b h'=. 4{. y
T=. a + i.@>:&.(%&h) b-a
Line 825 ⟶ 1,710:
ks=. (x * [: u y + (* x&,))/\. tableau
({:y) + 6 %~ +/ 1 2 2 1 * ks
)</
Use:
Line 833 ⟶ 1,718:
Translation of [[Runge-Kutta_method#Ada|Ada]] via [[Runge-Kutta_method#D|D]]
{{works with|Java|8}}
<
import java.util.function.BiFunction;
Line 870 ⟶ 1,755:
calc_err(t_arr[i], y_arr[i]));
}
}</
<pre>y(0,0) = 1,00000000 Error: 0,000000
Line 885 ⟶ 1,770:
=={{header|JavaScript}}==
===ES5===
<syntaxhighlight lang="javascript">
function rk4(y, x, dx, f) {
var k1 = dx * f(x, y),
Line 922 ⟶ 1,808:
steps += 1;
}
</syntaxhighlight>
{{out}}
<pre>
Line 937 ⟶ 1,823:
y(10) = 675.9999490167097 ± 5.098329029351589e-5
</pre>
===ES6===
<syntaxhighlight lang="javascript">(() => {
'use strict';
// rk4 :: (Double -> Double -> Double) ->
// Double -> Double -> Double -> Double
const rk4 = f => (y, x, dx) => {
const
k1 = dx * f(x, y),
k2 = dx * f(x + dx / 2.0, y + k1 / 2.0),
k3 = dx * f(x + dx / 2.0, y + k2 / 2.0),
k4 = dx * f(x + dx, y + k3);
return y + (k1 + 2.0 * k2 + 2.0 * k3 + k4) / 6.0;
};
// rk :: Double -> Double -> Double -> Double
const rk = rk4((x, y) => x * Math.sqrt(y));
// actual :: Double -> Double
const actual = x => (1 / 16) * ((x * x) + 4) * ((x * x) + 4);
// TEST -------------------------------------------------
// main :: IO ()
const main = () => {
const
step = 0.1,
ixs = enumFromTo(0, 100),
xys = scanl(
xy => Tuple(
((xy[0] * 10) + (step * 10)) / 10, rk(xy[1], xy[0], step)
),
Tuple(0.0, 1.0),
ixs
);
// samples :: [(Double, Double, Double)]
const samples = concatMap(
tpl => 0 === tpl[0] % 10 ? (() => {
const [x, y] = Array.from(tpl[1]);
return [TupleN(x, y, actual(x) - y)];
})() : [],
zip(ixs, xys)
);
console.log(
unlines(map(
tpl => {
const [x, y, v] = Array.from(tpl),
[sn, sm] = splitOn('.', y.toString());
return unwords([
'y' + justifyRight(3, ' ', '(' + Math.round(x).toString()) +
') =',
justifyRight(3, ' ', sn) + '.' + justifyLeft(15, ' ', sm || '0'),
'± ' + v.toExponential()
]);
},
samples
))
);
};
// GENERIC FUNCTIONS ----------------------------
// Tuple (,) :: a -> b -> (a, b)
const Tuple = (a, b) => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});
// TupleN :: a -> b ... -> (a, b ... )
function TupleN() {
const
args = Array.from(arguments),
lng = args.length;
return lng > 1 ? Object.assign(
args.reduce((a, x, i) => Object.assign(a, {
[i]: x
}), {
type: 'Tuple' + (2 < lng ? lng.toString() : ''),
length: lng
})
) : args[0];
};
// concatMap :: (a -> [b]) -> [a] -> [b]
const concatMap = (f, xs) =>
xs.reduce((a, x) => a.concat(f(x)), []);
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = (m, n) =>
Array.from({
length: 1 + n - m
}, (_, i) => m + i)
// justifyLeft :: Int -> Char -> String -> String
const justifyLeft = (n, cFiller, s) =>
n > s.length ? (
s.padEnd(n, cFiller)
) : s;
// justifyRight :: Int -> Char -> String -> String
const justifyRight = (n, cFiller, s) =>
n > s.length ? (
s.padStart(n, cFiller)
) : s;
// Returns Infinity over objects without finite length
// this enables zip and zipWith to choose the shorter
// argument when one is non-finite, like cycle, repeat etc
// length :: [a] -> Int
const length = xs => xs.length || Infinity;
// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) => xs.map(f);
// scanl :: (b -> a -> b) -> b -> [a] -> [b]
const scanl = (f, startValue, xs) =>
xs.reduce((a, x) => {
const v = f(a[0], x);
return Tuple(v, a[1].concat(v));
}, Tuple(startValue, [startValue]))[1];
// splitOn :: String -> String -> [String]
const splitOn = (pat, src) => src.split(pat);
// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = (n, xs) =>
xs.constructor.constructor.name !== 'GeneratorFunction' ? (
xs.slice(0, n)
) : [].concat.apply([], Array.from({
length: n
}, () => {
const x = xs.next();
return x.done ? [] : [x.value];
}));
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// unwords :: [String] -> String
const unwords = xs => xs.join(' ');
// Use of `take` and `length` here allows for zipping with non-finite
// lists - i.e. generators like cycle, repeat, iterate.
// zip :: [a] -> [b] -> [(a, b)]
const zip = (xs, ys) => {
const lng = Math.min(length(xs), length(ys));
return Infinity !== lng ? (() => {
const bs = take(lng, ys);
return take(lng, xs).map((x, i) => Tuple(x, bs[i]));
})() : zipGen(xs, ys);
};
// MAIN ---
return main();
})();</syntaxhighlight>
{{Out}}
<pre>y (0) = 1.0 ± 0e+0
y (1) = 1.562499854278108 ± 1.4572189210859676e-7
y (2) = 3.999999080520799 ± 9.194792007782837e-7
y (3) = 10.562497090437551 ± 2.9095624487496252e-6
y (4) = 24.999993765090636 ± 6.234909363911356e-6
y (5) = 52.562489180302585 ± 1.0819697415342944e-5
y (6) = 99.99998340540358 ± 1.659459641700778e-5
y (7) = 175.56247648227125 ± 2.3517728749311573e-5
y (8) = 288.9999684347986 ± 3.156520142510999e-5
y (9) = 451.56245927683966 ± 4.07231603389846e-5
y(10) = 675.9999490167097 ± 5.098329029351589e-5</pre>
=={{header|jq}}==
Line 942 ⟶ 2,005:
They use "while" and/or "until" as defined in recent versions of jq (after version 1.4).
To use either of the two programs with jq 1.4, simply include the lines in the following block:
<
def _until: if cond then . else (next|_until) end;
_until;
Line 948 ⟶ 2,011:
def while(cond; update):
def _while: if cond then ., (update | _while) else empty end;
_while;</
===The Example Differential Equation and its Exact Solution===
<
def yprime: .[0] * (.[1] | sqrt);
Line 958 ⟶ 2,021:
. as $t
| (( $t*$t) + 4 )
| . * . / 16;</
===dy/dt===
The first solution presented here uses the terminology and style of the
'''Generic filters:'''
<
def round(n):
(if . < 0 then -1 else 1 end) as $s
Line 972 ⟶ 2,035:
# Is the input an integer?
def integerq: ((. - ((.+.01) | floor)) | abs) < 0.01;</
'''dy(f)'''
<
# Input: [t, y]; yp is a filter that accepts [t,y] as input
Line 988 ⟶ 2,051:
# Input: [t,y]
def dy(f): runge_kutta(f);</
''' Example''':
<
[0,1]
| while( .[0] <= 10;
Line 999 ⟶ 2,062:
"y(\($t|round(1))) = \($y|round(10000)) ± \( ($t|actual) - $y | abs)"
else empty
end</
{{out}}
<
y(0) = 1 ± 0
y(1) = 1.5625 ± 1.4572189210859676e-07
Line 1,016 ⟶ 2,079:
real 0m0.048s
user 0m0.013s
sys 0m0.006s</
===newRK4Step===
The second solution follows the nomenclature and style of the Go solution on this page.
Line 1,026 ⟶ 2,090:
The heart of the program is the filter newRK4Step(yp), which is of type ypStepFunc and performs a single
step of the fourth-order Runge-Kutta method, provided yp is of type ypFunc.
<
def newRK4Step(yp):
.[0] as $t | .[1] as $y | .[2] as $dt
Line 1,068 ⟶ 2,132:
# main(t0; y0; tFinal; dtPrint)
main(0; 1; 10; 1)</
{{out}}
<
y(0) = 1 with error: 0
y(1) = 1.562499854278108 with error: 1.4572189210859676e-07
Line 1,085 ⟶ 2,149:
real 0m0.023s
user 0m0.014s
sys 0m0.006s</
=={{header|Julia}}==
=== Using lambda expressions ===
{{trans|Python}}
<syntaxhighlight lang="julia">f(x, y) = x * sqrt(y)
theoric(t) = (t ^ 2 + 4.0) ^ 2 / 16.0
rk4(f) = (t, y, δt) -> # 1st (result) lambda
((δy1) -> # 2nd lambda
((δy2) -> # 3rd lambda
((δy3) -> # 4th lambda
((δy4) -> ( δy1 + 2δy2 + 2δy3 + δy4 ) / 6 # 5th and deepest lambda: calc y_{n+1}
)(δt * f(t + δt, y + δy3)) # calc δy₄
)(δt * f(t + δt / 2, y + δy2 / 2)) # calc δy₃
)(δt * f(t + δt / 2, y + δy1 / 2)) # calc δy₂
)(δt * f(t, y)) # calc δy₁
t, y = t₀, y₀
while t ≤ tmax
if t ≈ round(t) @printf("y(%4.1f) = %10.6f\terror: %12.6e\n", t, y, abs(y - theoric(t))) end
y += δy(t, y, δt)
t += δt
end</syntaxhighlight>
{{out}}
<pre>y( 0.0) = 1.000000 error: 0.000000e+00
y( 1.0) = 1.562500 error: 1.457219e-07
y( 2.0) = 3.999999 error: 9.194792e-07
y( 3.0) = 10.562497 error: 2.909562e-06
y( 4.0) = 24.999994 error: 6.234909e-06
y( 5.0) = 52.562489 error: 1.081970e-05
y( 6.0) = 99.999983 error: 1.659460e-05
y( 7.0) = 175.562476 error: 2.351773e-05
y( 8.0) = 288.999968 error: 3.156520e-05
y( 9.0) = 451.562459 error: 4.072316e-05
y(10.0) = 675.999949 error: 5.098329e-05</pre>
=== Alternative version ===
{{trans|Python}}
<syntaxhighlight lang="julia">function rk4(f::Function, x₀::Float64, y₀::Float64, x₁::Float64, n)
vx = Vector{Float64}(undef, n
vy = Vector{Float64}(undef,
vx[1] =
vy[1] = y = y₀
h = (x₁ - x₀) / n
for i in 1:n
k₁ = h * f(x, y)
k₂ = h * f(x + 0.5h, y + 0.5k₁)
k₃ = h * f(x + 0.5h, y + 0.5k₂)
k₄ = h * f(x + h, y + k₃)
vx[i + 1] = x = x₀ + i * h
vy[i + 1] = y = y + (k₁ + 2k₂ + 2k₃ + k₄) / 6
end
return vx, vy
end
vx, vy = rk4(f, 0.0, 1.0, 10.0, 100)
for (x, y) in Iterators.take(zip(vx, vy), 10)
@printf("%4.1f %10.5f %+12.4e\n", x, y, y - theoric(x))
end</syntaxhighlight>
=={{header|Kotlin}}==
<syntaxhighlight lang="scala">// version 1.1.2
typealias Y = (Double) -> Double
typealias Yd = (Double, Double) -> Double
fun rungeKutta4(t0: Double, tz: Double, dt: Double, y: Y, yd: Yd) {
var tn = t0
var yn = y(tn)
val z = ((tz - t0) / dt).toInt()
for (i in 0..z) {
if (i % 10 == 0) {
val exact = y(tn)
val error = yn - exact
println("%4.1f %10f %10f %9f".format(tn, yn, exact, error))
}
if (i == z) break
val dy1 = dt * yd(tn, yn)
val dy2 = dt * yd(tn + 0.5 * dt, yn + 0.5 * dy1)
val dy3 = dt * yd(tn + 0.5 * dt, yn + 0.5 * dy2)
val dy4 = dt * yd(tn + dt, yn + dy3)
yn += (dy1 + 2.0 * dy2 + 2.0 * dy3 + dy4) / 6.0
tn += dt
}
}
fun main(args: Array<String>) {
println(" T RK4 Exact Error")
println("---- ---------- ---------- ---------")
val y = fun(t: Double): Double {
val x = t * t + 4.0
return x * x / 16.0
}
val yd = fun(t: Double, yt: Double) = t * Math.sqrt(yt)
rungeKutta4(0.0, 10.0, 0.1, y, yd)
}</syntaxhighlight>
{{out}}
<pre>
T RK4 Exact Error
---- ---------- ---------- ---------
6.0 99.999983 100.000000 -0.000017
7.0 175.562476 175.562500 -0.000024
8.0 288.999968 289.000000 -0.000032
9.0 451.562459 451.562500 -0.000041
10.0 675.999949 676.000000 -0.000051
</pre>
=={{header|Liberty BASIC}}==
<syntaxhighlight lang="lb">
'[RC] Runge-Kutta method
'initial conditions
Line 1,171 ⟶ 2,304:
exactY=(x^2 + 4)^2 / 16
end function
</syntaxhighlight>
{{Out}}
<pre>
Line 1,187 ⟶ 2,320:
</pre>
=={{header|
<syntaxhighlight lang="lua">local df = function (t, y)
-- derivative of function by value y at time t
return t*y^0.5
end
local dt = 0.1
local y = 1
print ("t", "realY"..' ', "y", ' '.."error")
print ("---", "-------"..' ', "---------------", ' '.."--------------------")
for i = 0, 100 do
local t = i*dt
if t%1 == 0 then
local realY = (t*t+4)^2/16
print (t, realY..' ', y, ' '..realY-y)
end
local dy1 = df(t, y)
local dy2 = df(t+dt/2, y+dt/2*dy1)
local dy3 = df(t+dt/2, y+dt/2*dy2)
local dy4 = df(t+dt, y+dt*dy3)
y = y + dt*(dy1+2*dy2+2*dy3+dy4)/6
end</syntaxhighlight>
{{Out}}
<pre>t realY y error
--- ------- --------------- --------------------
0.0 1.0 1 0.0
1.0 1.5625 1.5624998542781 1.457218921086e-007
2.0 4.0 3.9999990805208 9.1947919989011e-007
3.0 10.5625 10.562497090438 2.9095624469733e-006
4.0 25.0 24.999993765091 6.2349093639114e-006
5.0 52.5625 52.562489180303 1.0819697415343e-005
6.0 100.0 99.999983405404 1.6594596417008e-005
7.0 175.5625 175.56247648227 2.3517728749312e-005
8.0 289.0 288.9999684348 3.156520142511e-005
9.0 451.5625 451.56245927684 4.0723160338985e-005
10.0 676.0 675.99994901671 5.0983290293516e-005
</pre>
=={{header|Mathematica}}/{{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">(* Symbolic solution *)
DSolve[{y'[t] == t*Sqrt[y[t]], y[0] == 1}, y, t]
Table[{t, 1/16 (4 + t^2)^2}, {t, 0, 10}]
Line 1,207 ⟶ 2,381:
solution = NestList[phi, {0, 1}, 101];
Table[{y[[1]], y[[2]], Abs[y[[2]] - 1/16 (y[[1]]^2 + 4)^2]},
{y, solution[[1 ;; 101 ;; 10]]}]
=={{header|MATLAB}}==
The normally-used built-in solver is the ode45 function, which uses a non-fixed-step solver with 4th/5th order Runge-Kutta methods. The MathWorks Support Team released a [http://www.mathworks.com/matlabcentral/answers/98293-is-there-a-fixed-step-ordinary-differential-equation-ode-solver-in-matlab-8-0-r2012b#answer_107643 package of fixed-step RK method ODE solvers] on MATLABCentral. The ode4 function contained within uses a 4th-order Runge-Kutta method. Here is code that tests both ode4 and my own function, shows that they are the same, and compares them to the exact solution.
<
figure
hold on
Line 1,248 ⟶ 2,421:
y(k+1) = y(k)+(dy1+2*dy2+2*dy3+dy4)/6;
end
end</
{{out}}
<pre>
Line 1,266 ⟶ 2,439:
=={{header|Maxima}}==
<
'diff(y, x) = x * sqrt(y);
ode2(%, y, x);
Line 1,305 ⟶ 2,478:
s: map(lambda([x], (x^2 + 4)^2 / 16), x)$
for i from 1 step 10 thru 101 do print(x[i], " ", y[i], " ", y[i] - s[i]);</
=={{header|МК-61/52}}==
Line 1,318 ⟶ 2,491:
''Input:'' 1/2 (h/2) - Р5, 1 (y<sub>0</sub>) - Р8 and Р7, 0 (t<sub>0</sub>) - Р6.
=={{header|Nim}}==
<syntaxhighlight lang="nim">import math
proc fn(t, y: float): float =
result = t * math.sqrt(y)
proc solution(t: float): float =
result = (t^2 + 4)^2 / 16
proc rk(start, stop, step: float) =
let nsteps = int(round((stop - start) / step)) + 1
let delta = (stop - start) / float(nsteps - 1)
var cur_y = 1.0
for i in 0..(nsteps - 1):
let cur_t = start + delta * float(i)
if abs(cur_t - math.round(cur_t)) < 1e-5:
echo "y(", cur_t, ") = ", cur_y, ", error = ", solution(cur_t) - cur_y
let dy1 = step * fn(cur_t, cur_y)
let dy2 = step * fn(cur_t + 0.5 * step, cur_y + 0.5 * dy1)
let dy3 = step * fn(cur_t + 0.5 * step, cur_y + 0.5 * dy2)
let dy4 = step * fn(cur_t + step, cur_y + dy3)
import math, strformat
proc fn(t, y: float): float =
result = t * math.sqrt(y)
proc solution(t: float): float =
result = (t^2 + 4)^2 / 16
proc rk(start, stop, step: float) =
let nsteps = int(round((stop - start) / step)) + 1
let delta = (stop - start) / float(nsteps - 1)
var cur_y = 1.0
for i in 0..<nsteps:
let cur_t = start + delta * float(i)
if abs(cur_t - math.round(cur_t)) < 1e-5:
echo &"y({cur_t}) = {cur_y}, error = {solution(cur_t) - cur_y}"
let dy1 = step * fn(cur_t, cur_y)
let dy2 = step * fn(cur_t + 0.5 * step, cur_y + 0.5 * dy1)
let dy3 = step * fn(cur_t + 0.5 * step, cur_y + 0.5 * dy2)
let dy4 = step * fn(cur_t + step, cur_y + dy3)
cur_y += (dy1 + 2 * (dy2 + dy3) + dy4) / 6
rk(start = 0, stop = 10, step = 0.1)
cur_y += (dy1 + 2.0 * (dy2 + dy3) + dy4) </syntaxhighlight>
{{out}}
<pre>y(0.0) = 1.0, error = 0.0
y(1.0) = 1.562499854278108, error = 1.457218921085968e-07
y(2.0) = 3.9999990805208, error = 9.194792003341945e-07
y(3.0) = 10.56249709043755, error = 2.909562448749625e-06
y(4.0) = 24.99999376509064, error = 6.234909363911356e-06
y(5.0) = 52.56248918030258, error = 1.081969741534294e-05
y(6.0) = 99.99998340540358, error = 1.659459641700778e-05
y(7.0) = 175.5624764822713, error = 2.351772874931157e-05
y(8.0) = 288.9999684347986, error = 3.156520142510999e-05
y(9.0) = 451.5624592768397, error = 4.07231603389846e-05
y(10.0) = 675.9999490167097, error = 5.098329029351589e-05</pre>
=={{header|Objeck}}==
<
function : Main(args : String[]) ~ Nil {
x0 := 0.0; x1 := 10.0; dx := .1;
Line 1,355 ⟶ 2,591:
return x * y->SquareRoot();
}
}</
Output:
Line 1,373 ⟶ 2,609:
=={{header|OCaml}}==
<
let exact t = let u = 0.25*.t*.t +. 1.0 in u*.u
Line 1,388 ⟶ 2,624:
if n < 102 then loop h (n+1) (rk4_step (y,t) h)
let _ = loop 0.1 1 (1.0, 0.0)</
{{out}}
<pre>t = 0.000000, y = 1.000000, err = 0
Line 1,403 ⟶ 2,639:
=={{header|Octave}}==
<syntaxhighlight lang="octave">
#Applying the Runge-Kutta method (This code must be implement on a different file than the main one).
K2 = h*func(x+0.5*h,pvi+0.5*K1);
K3 = h*func(x+0.5*h,pvi+0.5*K2);
K4 = h*func(x+h,pvi+K3);
temp = pvi + (K1 + 2*K2 + 2*K3 + K4)/6;
endfunction
#Main Program.
df = @(t,y) t*sqrt(y);
pvi = 1.0;
h = 0.1;
Yn = pvi;
for x = 0:h:10-h
pvi = rk4(df,x,pvi,h);
Yn = [Yn pvi];
endfor
fprintf('Time \t Exact Value \t ODE4 Value \t Num. Error\n');
for i=0:10
fprintf('%d \t %.5f \t %.5f \t %.4g \n',i,f(i),Yn(1+i*10),f(i)-Yn(1+i*10));
endfor
</syntaxhighlight>
{{out}}
<pre>
Time Exact Value ODE4 Value Num. Error
0 1.00000 1.00000 0
1 1.56250 1.56250 1.457e-007
2 4.00000 4.00000 9.195e-007
3 10.56250 10.56250 2.91e-006
4 25.00000 24.99999 6.235e-006
5 52.56250 52.56249 1.082e-005
6 100.00000 99.99998 1.659e-005
7 175.56250 175.56248 2.352e-005
8 289.00000 288.99997 3.157e-005
9 451.56250 451.56246 4.072e-005
10 676.00000 675.99995 5.098e-005</pre>
=={{header|PARI/GP}}==
{{trans|C}}
<
my(k1=dx*f(x,y), k2=dx*f(x+dx/2,y+k1/2), k3=dx*f(x+dx/2,y+k2/2), k4=dx*f(x+dx,y+k3));
y + (k1 + 2*k2 + 2*k3 + k4) / 6
Line 1,446 ⟶ 2,702:
)
};
go()</
{{out}}
<pre>x y rel. err.
Line 1,466 ⟶ 2,722:
This code has been compiled using Free Pascal 2.6.2.
<
uses sysutils;
Line 1,536 ⟶ 2,792:
RungeKutta(@YPrime, tArr, yArr, dt);
Print(tArr, yArr, 10);
end.</
{{out}}
<pre>y( 0.0) = 1.00000000 Error: 0.00000E+000
Line 1,558 ⟶ 2,814:
Notice how we have to use sprintf to deal with floating point rounding. See perlfaq4.
<
my ($yp, $dt) = @_;
sub {
Line 1,579 ⟶ 2,835:
printf "y(%2.0f) = %12f ± %e\n", $t, $y, abs($y - ($t**2 + 4)**2 / 16)
if sprintf("%.4f", $t) =~ /0000$/;
}</
{{out}}
Line 1,594 ⟶ 2,850:
y(10) = 675.999949 ± 5.098329e-05</pre>
=={{header|
{{trans|ERRE}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">dt</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0.1</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1.0</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" x true/actual y calculated y relative error\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" --- ------------- ------------- --------------\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">100</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">*</span><span style="color: #000000;">dt</span>
<span style="color: #008080;">if</span> <span style="color: #004080;">integer</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">act</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">16</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%4.1f %14.9f %14.9f %.9e\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">act</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">-</span><span style="color: #000000;">act</span><span style="color: #0000FF;">)})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">k1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">k2</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">k1</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">k3</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">k2</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">k4</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">*</span><span style="color: #000000;">k3</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">y</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">dt</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">k1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">k2</span><span style="color: #0000FF;">+</span><span style="color: #000000;">k3</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">k4</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">6</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
x true/actual y calculated y relative error
--- ------------- ------------- --------------
0.0 1.000000000 1.000000000 0.000000000e+0
1.0 1.562500000 1.562499854 1.457218921e-7
2.0 4.000000000 3.999999081 9.194791999e-7
3.0 10.562500000 10.562497090 2.909562447e-6
4.0 25.000000000 24.999993765 6.234909363e-6
5.0 52.562500000 52.562489180 1.081969741e-5
6.0 100.000000000 99.999983405 1.659459641e-5
7.0 175.562500000 175.562476482 2.351772874e-5
8.0 289.000000000 288.999968435 3.156520142e-5
9.0 451.562500000 451.562459277 4.072316033e-5
10.0 676.000000000 675.999949017 5.098329030e-5
</pre>
=={{header|PL/I}}==
<syntaxhighlight lang="pl/i">
Runge_Kutta: procedure options (main); /* 10 March 2014 */
declare (y, dy1, dy2, dy3, dy4) float (18);
Line 1,657 ⟶ 2,915:
end Runge_kutta;
</syntaxhighlight>
{{out}}
<pre>
Line 1,677 ⟶ 2,935:
{{works with|PowerShell|4.0}}
<syntaxhighlight lang="powershell">
function Runge-Kutta (${function:F}, ${function:y}, $y0, $t0, $dt, $tEnd) {
function RK ($tn,$yn) {
Line 1,714 ⟶ 2,972:
$tEnd = 10
Runge-Kutta F y $y0 $t0 $dt $tEnd
</syntaxhighlight>
<b>Output:</b>
<pre>
Line 1,731 ⟶ 2,989:
10 675.99994901671 5.09832902935159E-05
</pre>
=={{header|PureBasic}}==
{{trans|BBC Basic}}
<
Define.i i
Define.d y=1.0, k1=0.0, k2=0.0, k3=0.0, k4=0.0, t=0.0
Line 1,751 ⟶ 3,010:
Print("Press return to exit...") : Input()
EndIf
End</
{{out}}
<pre>y( 0) = 1.0000 Error = 0.0000000000
Line 1,767 ⟶ 3,026:
=={{header|Python}}==
<syntaxhighlight lang="python">from math import sqrt
def rk4(f, x0, y0, x1, n):
Line 1,840 ⟶ 3,060:
8.0 288.99997 -3.1565e-05
9.0 451.56246 -4.0723e-05
10.0 675.99995 -5.0983e-05</
=={{header|R}}==
<
vx <- double(n + 1)
vy <- double(n + 1)
Line 1,875 ⟶ 3,095:
[9,] 8 288.999968 -3.156520e-05
[10,] 9 451.562459 -4.072316e-05
[11,] 10 675.999949 -5.098329e-05</
=={{header|Racket}}==
Line 1,882 ⟶ 3,102:
The Runge-Kutta method
<
(define (RK4 F δt)
(λ (t y)
Line 1,891 ⟶ 3,111:
(list (+ t δt)
(+ y (* 1/6 (+ δy1 (* 2 δy2) (* 2 δy3) δy4))))))
</syntaxhighlight>
The method modifier which divides each time-step into ''n'' sub-steps:
<
(define ((step-subdivision n method) F h)
(λ (x . y) (last (ODE-solve F (cons x y)
Line 1,900 ⟶ 3,120:
#:step (/ h n)
#:method method))))
</syntaxhighlight>
Usage:
<
(define (F t y) (* t (sqrt y)))
Line 1,914 ⟶ 3,134:
(match-define (list t y) s)
(printf "t=~a\ty=~a\terror=~a\n" t y (- y (exact-solution t))))
</syntaxhighlight>
{{out}}
<pre>
Line 1,932 ⟶ 3,152:
Graphical representation:
<
> (require plot)
> (plot (list (function exact-solution 0 10 #:label "Exact solution")
(points numeric-solution #:label "Runge-Kutta method"))
#:x-label "t" #:y-label "y(t)")
</syntaxhighlight>
[[File:runge-kutta.png]]
=={{header|Raku}}==
(formerly Perl 6)
{{Works with|rakudo|2016.03}}
<syntaxhighlight lang="raku" line>sub runge-kutta(&yp) {
return -> \t, \y, \δt {
my $a = δt * yp( t, y );
my $b = δt * yp( t + δt/2, y + $a/2 );
my $c = δt * yp( t + δt/2, y + $b/2 );
my $d = δt * yp( t + δt, y + $c );
($a + 2*($b + $c) + $d) / 6;
}
}
constant δt = .1;
my &δy = runge-kutta { $^t * sqrt($^y) };
loop (
my ($t, $y) = (0, 1);
$t <= 10;
($t, $y) »+=« (δt, δy($t, $y, δt))
) {
printf "y(%2d) = %12f ± %e\n", $t, $y, abs($y - ($t**2 + 4)**2 / 16)
if $t %% 1;
}</syntaxhighlight>
{{out}}
<pre>y( 0) = 1.000000 ± 0.000000e+00
y( 1) = 1.562500 ± 1.457219e-07
y( 2) = 3.999999 ± 9.194792e-07
y( 3) = 10.562497 ± 2.909562e-06
y( 4) = 24.999994 ± 6.234909e-06
y( 5) = 52.562489 ± 1.081970e-05
y( 6) = 99.999983 ± 1.659460e-05
y( 7) = 175.562476 ± 2.351773e-05
y( 8) = 288.999968 ± 3.156520e-05
y( 9) = 451.562459 ± 4.072316e-05
y(10) = 675.999949 ± 5.098329e-05</pre>
=={{header|REXX}}==
<big><big> y'(t) = t<sup>2</sup> √<span style="text-decoration: overline"> y(t) </span></big></big>
The exact solution: <big><big> y(t) = (t<sup>2</sup>+4)<sup>2</sup> ÷ 16 </big></big>
<syntaxhighlight lang="rexx">/*REXX program uses the Runge─Kutta method to solve the equation: y'(t) = t² √[y(t)] */
numeric digits 40; f= digits() % 4 /*use 40 decimal digs, but only show 10*/
x0= 0; x1= 10; dx= .1 /*define variables: X0 X1 DX */
n=1 + (x1-x0) / dx
y.=1;
end /*m*/
w= digits() % 2
say center('X', f, "═") center('Y', w+2, "═") center("relative error", w+8, '═') /*hdr*/
do i=0 to n-1 by 10; x= (x0 + dx*i) / 1; $= y.i / (x*x/4+1)**2 - 1
say center(x, f) fmt(y.i) left('', 2 + ($>=0) ) fmt($)
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
fmt: parse arg z; z= right( format(
return translate( right(z, (z>=0) + w + 5*hasE + 2*(jus & (z<0) ) ), 'e', "E")
/*──────────────────────────────────────────────────────────────────────────────────────*/
k2= dx * (x + dxH) * sqrt(y + k1/2)
return y + (k1 + k2*2
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); m.=9; numeric form; h=d+6
numeric digits; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g * .5'e'_ % 2
Programming note: the '''fmt''' function is used to
align the output with attention paid to the different ways some
<br>REXXes format numbers that are in floating point representation.
{{out|output|text= when using Regina REXX:}}
<pre>
════X═════ ══════════Y═══════════ ═══════relative error═══════
Line 1,994 ⟶ 3,254:
10 675.9999490167 -7.5419068846e-8
</pre>
<pre>
════X═════ ══════════Y═══════════ ═══════relative error═══════
0 1 0
1 1.5624998543 -0.0000000933
2 3.9999990805 -0.0000002299
3 10.5624970904 -0.0000002755
4 24.9999937651 -0.0000002494
5 52.5624891803 -0.0000002058
6 99.9999834054 -0.0000001659
7 175.5624764823 -0.000000134
8 288.9999684348 -0.0000001092
9 451.5624592768 -0.0000000902
10 675.9999490167 -0.0000000754
</pre>
=={{header|Ring}}==
<
decimals(8)
y = 1.0
Line 2,025 ⟶ 3,285:
y += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
next
</syntaxhighlight>
Output:
Line 2,043 ⟶ 3,303:
=={{header|Ruby}}==
<
return ->(t,y,dt){
->(dy1 ){
Line 2,069 ⟶ 3,329:
printf("y(%4.1f)\t= %12.6f \t error: %12.6e\n",t,y,find_error(t,y)) if is_whole?(t)
t, y = t + DT, y + dy.call(t,y,DT)
end</
{{Out}}
<pre>
Line 2,086 ⟶ 3,346:
=={{header|Run BASIC}}==
<
while t <= 10
k1 = t * sqr(y)
Line 2,097 ⟶ 3,357:
t = t + .1
wend
end</
{{out}}
<pre>y( 0) = 1.0000000 Error =0
Line 2,110 ⟶ 3,370:
y( 9) = 451.5624593 Error =4.07231581e-5
y(10) = 675.9999490 Error =5.09832864e-5
</pre>
=={{header|Rust}}==
This is a translation of the javascript solution with some minor differences.
<syntaxhighlight lang="rust">fn runge_kutta4(fx: &dyn Fn(f64, f64) -> f64, x: f64, y: f64, dx: f64) -> f64 {
let k1 = dx * fx(x, y);
let k2 = dx * fx(x + dx / 2.0, y + k1 / 2.0);
let k3 = dx * fx(x + dx / 2.0, y + k2 / 2.0);
let k4 = dx * fx(x + dx, y + k3);
y + (k1 + 2.0 * k2 + 2.0 * k3 + k4) / 6.0
}
fn f(x: f64, y: f64) -> f64 {
x * y.sqrt()
}
fn actual(x: f64) -> f64 {
(1.0 / 16.0) * (x * x + 4.0).powi(2)
}
fn main() {
let mut y = 1.0;
let mut x = 0.0;
let step = 0.1;
let max_steps = 101;
let sample_every_n = 10;
for steps in 0..max_steps {
if steps % sample_every_n == 0 {
println!("y({}):\t{:.10}\t\t {:E}", x, y, actual(x) - y)
}
y = runge_kutta4(&f, x, y, step);
x = ((x * 10.0) + (step * 10.0)) / 10.0;
}
}</syntaxhighlight>
<pre>
y(0): 1.0000000000 0E0
y(1): 1.5624998543 1.4572189210859676E-7
y(2): 3.9999990805 9.194792007782837E-7
y(3): 10.5624970904 2.9095624487496252E-6
y(4): 24.9999937651 6.234909363911356E-6
y(5): 52.5624891803 1.0819697415342944E-5
y(6): 99.9999834054 1.659459641700778E-5
y(7): 175.5624764823 2.3517728749311573E-5
y(8): 288.9999684348 3.156520142510999E-5
y(9): 451.5624592768 4.07231603389846E-5
y(10): 675.9999490167 5.098329029351589E-5
</pre>
=={{header|Scala}}==
<syntaxhighlight lang="scala">object Main extends App {
val f = (t: Double, y: Double) => t * Math.sqrt(y) // Runge-Kutta solution
val g = (t: Double) => Math.pow(t * t + 4, 2) / 16 // Exact solution
new Calculator(f, Some(g)).compute(100, 0, .1, 1)
}
class Calculator(f: (Double, Double) => Double, g: Option[Double => Double] = None) {
def compute(counter: Int, tn: Double, dt: Double, yn: Double): Unit = {
if (counter % 10 == 0) {
val c = (x: Double => Double) => (t: Double) => {
val err = Math.abs(x(t) - yn)
f" Error: $err%7.5e"
}
val s = g.map(c(_)).getOrElse((x: Double) => "") // If we don't have exact solution, just print nothing
println(f"y($tn%4.1f) = $yn%12.8f${s(tn)}") // Else, print Error estimation here
}
if (counter > 0) {
val dy1 = dt * f(tn, yn)
val dy2 = dt * f(tn + dt / 2, yn + dy1 / 2)
val dy3 = dt * f(tn + dt / 2, yn + dy2 / 2)
val dy4 = dt * f(tn + dt, yn + dy3)
val y = yn + (dy1 + 2 * dy2 + 2 * dy3 + dy4) / 6
val t = tn + dt
compute(counter - 1, t, dt, y)
}
}
}</syntaxhighlight>
<pre>
y( 0.0) = 1.00000000 Error: 0.00000e+00
y( 1.0) = 1.56249985 Error: 1.45722e-07
y( 2.0) = 3.99999908 Error: 9.19479e-07
y( 3.0) = 10.56249709 Error: 2.90956e-06
y( 4.0) = 24.99999377 Error: 6.23491e-06
y( 5.0) = 52.56248918 Error: 1.08197e-05
y( 6.0) = 99.99998341 Error: 1.65946e-05
y( 7.0) = 175.56247648 Error: 2.35177e-05
y( 8.0) = 288.99996843 Error: 3.15652e-05
y( 9.0) = 451.56245928 Error: 4.07232e-05
y(10.0) = 675.99994902 Error: 5.09833e-05
</pre>
=={{header|Sidef}}==
{{trans|
<
func (t, y, δt) {
var a = (δt * yp(t, y));
Line 2,134 ⟶ 3,486:
y += δy(t, y, δt);
t += δt;
}</
{{out}}
<pre>
Line 2,151 ⟶ 3,503:
=={{header|Standard ML}}==
<
let
val dy1 = dt * y'(tn,yn)
Line 2,191 ⟶ 3,543:
(* Run the suggested test case *)
val () = test 0.0 1.0 0.1 101 10 testy testy'</
{{out}}
<pre>Time: 0.0
Line 2,247 ⟶ 3,599:
Approx: 675.999949017
Error: 5.09832866555E~05</pre>
=={{header|Stata}}==
<syntaxhighlight lang="stata">function rk4(f, t0, y0, t1, n) {
h = (t1-t0)/(n-1)
a = J(n, 2, 0)
a[1, 1] = t = t0
a[1, 2] = y = y0
for (i=2; i<=n; i++) {
k1 = h*(*f)(t, y)
k2 = h*(*f)(t+0.5*h, y+0.5*k1)
k3 = h*(*f)(t+0.5*h, y+0.5*k2)
k4 = h*(*f)(t+h, y+k3)
t = t+h
y = y+(k1+2*k2+2*k3+k4)/6
a[i, 1] = t
a[i, 2] = y
}
return(a)
}
function f(t, y) {
return(t*sqrt(y))
}
a = rk4(&f(), 0, 1, 10, 101)
t = a[., 1]
a = a, a[., 2]:-(t:^2:+4):^2:/16
a[range(1,101,10), .]
1 2 3
+----------------------------------------------+
1 | 0 1 0 |
2 | 1 1.562499854 -1.45722e-07 |
3 | 2 3.999999081 -9.19479e-07 |
4 | 3 10.56249709 -2.90956e-06 |
5 | 4 24.99999377 -6.23491e-06 |
6 | 5 52.56248918 -.0000108197 |
7 | 6 99.99998341 -.0000165946 |
8 | 7 175.5624765 -.0000235177 |
9 | 8 288.9999684 -.0000315652 |
10 | 9 451.5624593 -.0000407232 |
11 | 10 675.999949 -.0000509833 |
+----------------------------------------------+</syntaxhighlight>
=={{header|Swift}}==
{{trans|C}}
<
func rk4(dx: Double, x: Double, y: Double, f: (Double, Double) -> Double) -> Double {
Line 2,287 ⟶ 3,682:
print(String(format: "%2g %11.6g %11.5g", x, y[i], y[i]/y2 - 1))
}</
{{out}}
Line 2,305 ⟶ 3,700:
=={{header|Tcl}}==
<
# Hack to bring argument function into expression
Line 2,337 ⟶ 3,732:
printvals $t $y
}
}</
{{out}}
<pre>y(0.0) = 1.00000000 Error: 0.00000000e+00
Line 2,350 ⟶ 3,745:
y(9.0) = 451.56245928 Error: 4.07231581e-05
y(10.0) = 675.99994902 Error: 5.09832864e-05</pre>
=={{header|V (Vlang)}}==
{{trans|Ring}}
<syntaxhighlight lang="Zig">
import math
fn main() {
mut t, mut k1, mut k2, mut k3, mut k4, mut y := 0.0, 0.0, 0.0, 0.0, 0.0, 1.0
for i in 0..101 {
t = i / 10.0
if t == math.floor(t) {
actual := math.pow((math.pow(t, 2) + 4), 2)/16
println("y(${t:.0}) = ${y:.8f} error = ${(actual - y):.8f}")
}
k1 = t * math.sqrt(y)
k2 = (t + 0.05) * math.sqrt(y + 0.05 * k1)
k3 = (t + 0.05) * math.sqrt(y + 0.05 * k2)
k4 = (t + 0.10) * math.sqrt(y + 0.10 * k3)
y += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
}
}
</syntaxhighlight>
{{out}}
<pre>
y(0) = 1.00000000 error = 0.00000000
y(1) = 1.56249985 error = 0.00000015
y(2) = 3.99999908 error = 0.00000092
y(3) = 10.56249709 error = 0.00000291
y(4) = 24.99999377 error = 0.00000623
y(5) = 52.56248918 error = 0.00001082
y(6) = 99.99998341 error = 0.00001659
y(7) = 175.56247648 error = 0.00002352
y(8) = 288.99996843 error = 0.00003157
y(9) = 451.56245928 error = 0.00004072
y(10) = 675.99994902 error = 0.00005098
</pre>
=={{header|Wren}}==
{{trans|Kotlin}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="wren">import "./fmt" for Fmt
var rungeKutta4 = Fn.new { |t0, tz, dt, y, yd|
var tn = t0
var yn = y.call(tn)
var z = ((tz - t0)/dt).truncate
for (i in 0..z) {
if (i % 10 == 0) {
var exact = y.call(tn)
var error = yn - exact
Fmt.print("$4.1f $10f $10f $9f", tn, yn, exact, error)
}
if (i == z) break
var dy1 = dt * yd.call(tn, yn)
var dy2 = dt * yd.call(tn + 0.5 * dt, yn + 0.5 * dy1)
var dy3 = dt * yd.call(tn + 0.5 * dt, yn + 0.5 * dy2)
var dy4 = dt * yd.call(tn + dt, yn + dy3)
yn = yn + (dy1 + 2.0 * dy2 + 2.0 * dy3 + dy4) / 6.0
tn = tn + dt
}
}
System.print(" T RK4 Exact Error")
System.print("---- --------- ---------- ---------")
var y = Fn.new { |t|
var x = t * t + 4.0
return x * x / 16.0
}
var yd = Fn.new { |t, yt| t * yt.sqrt }
rungeKutta4.call(0, 10, 0.1, y, yd)</syntaxhighlight>
{{out}}
<pre>
T RK4 Exact Error
---- --------- ---------- ---------
0.0 1.000000 1.000000 0.000000
1.0 1.562500 1.562500 -0.000000
2.0 3.999999 4.000000 -0.000001
3.0 10.562497 10.562500 -0.000003
4.0 24.999994 25.000000 -0.000006
5.0 52.562489 52.562500 -0.000011
6.0 99.999983 100.000000 -0.000017
7.0 175.562476 175.562500 -0.000024
8.0 288.999968 289.000000 -0.000032
9.0 451.562459 451.562500 -0.000041
10.0 675.999949 676.000000 -0.000051
</pre>
=={{header|XPL0}}==
<syntaxhighlight lang "XPL0">func real Y_(T, Y);
real T, Y;
return T*sqrt(Y);
def DT = 0.1;
real T, Y, Exact, DY1, DY2, DY3, DY4;
[Text(0, " T RK Exact Error^m^j");
T:= 0.; Y:= 1.;
repeat if Mod(T+.001, 1.) < .01 then
[Format(2, 1);
RlOut(0, T);
Format(5, 7);
RlOut(0, Y);
Exact:= sq(T*T+4.)/16.;
RlOut(0, Exact);
RlOut(0, Y-Exact);
CrLf(0);
];
DY1:= DT * Y_(T, Y);
DY2:= DT * Y_(T+DT/2., Y+DY1/2.);
DY3:= DT * Y_(T+DT/2., Y+DY2/2.);
DY4:= DT * Y_(T+DT, Y+DY3);
Y:= Y + (DY1 + 2.*DY2 + 2.*DY3 + DY4) / 6.;
T:= T + DT;
until T > 10.;
]</syntaxhighlight>
{{out}}
<pre>
T RK Exact Error
0.0 1.0000000 1.0000000 0.0000000
1.0 1.5624999 1.5625000 -0.0000001
2.0 3.9999991 4.0000000 -0.0000009
3.0 10.5624971 10.5625000 -0.0000029
4.0 24.9999938 25.0000000 -0.0000062
5.0 52.5624892 52.5625000 -0.0000108
6.0 99.9999834 100.0000000 -0.0000166
7.0 175.5624765 175.5625000 -0.0000235
8.0 288.9999684 289.0000000 -0.0000316
9.0 451.5624593 451.5625000 -0.0000407
10.0 675.9999490 676.0000000 -0.0000510
</pre>
=={{header|zkl}}==
{{trans|OCaml}}
<
fcn exact(t){ u:=0.25*t*t + 1.0; u*u }
Line 2,368 ⟶ 3,894:
print("t = %f,\ty = %f,\terr = %g\n".fmt(t,y,(y - exact(t)).abs()));
if(n < 102) return(loop(h,(n+1),rk4_step(T(y,t),h))) //tail recursion
}</
{{out}}
<pre>
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