# Runge-Kutta method

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Runge-Kutta method
You are encouraged to solve this task according to the task description, using any language you may know.

Given the example Differential equation:

${\displaystyle y'(t)=t\times {\sqrt {y(t)}}}$

With initial condition:

${\displaystyle t_{0}=0}$ and ${\displaystyle y_{0}=y(t_{0})=y(0)=1}$

This equation has an exact solution:

${\displaystyle y(t)={\tfrac {1}{16}}(t^{2}+4)^{2}}$

Task

Demonstrate the commonly used explicit   fourth-order Runge–Kutta method   to solve the above differential equation.

• Solve the given differential equation over the range ${\displaystyle t=0\ldots 10}$ with a step value of ${\displaystyle \delta t=0.1}$ (101 total points, the first being given)
• Print the calculated values of ${\displaystyle y}$ at whole numbered ${\displaystyle t}$'s (${\displaystyle 0.0,1.0,\ldots 10.0}$) along with error as compared to the exact solution.

Method summary

Starting with a given ${\displaystyle y_{n}}$ and ${\displaystyle t_{n}}$ calculate:

${\displaystyle \delta y_{1}=\delta t\times y'(t_{n},y_{n})\quad }$
${\displaystyle \delta y_{2}=\delta t\times y'(t_{n}+{\tfrac {1}{2}}\delta t,y_{n}+{\tfrac {1}{2}}\delta y_{1})}$
${\displaystyle \delta y_{3}=\delta t\times y'(t_{n}+{\tfrac {1}{2}}\delta t,y_{n}+{\tfrac {1}{2}}\delta y_{2})}$
${\displaystyle \delta y_{4}=\delta t\times y'(t_{n}+\delta t,y_{n}+\delta y_{3})\quad }$

then:

${\displaystyle y_{n+1}=y_{n}+{\tfrac {1}{6}}(\delta y_{1}+2\delta y_{2}+2\delta y_{3}+\delta y_{4})}$
${\displaystyle t_{n+1}=t_{n}+\delta t\quad }$

## 11l

Translation of: Python
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))
Output:
 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


## 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.

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
Output:
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


## Ada

with Ada.Text_IO; use Ada.Text_IO;
with Ada.Numerics.Generic_Elementary_Functions;
procedure RungeKutta is
type Floaty is digits 15;
type Floaty_Array is array (Natural range <>) of Floaty;
package FIO is new Ada.Text_IO.Float_IO(Floaty); use FIO;
type Derivative is access function(t, y : Floaty) return Floaty;
package Math is new Ada.Numerics.Generic_Elementary_Functions (Floaty);
function calc_err (t, calc : Floaty) return Floaty;

procedure Runge (yp_func : Derivative; t, y : in out Floaty_Array;
dt : Floaty) is
dy1, dy2, dy3, dy4 : Floaty;
begin
for n in t'First .. t'Last-1 loop
dy1 := dt * yp_func(t(n), y(n));
dy2 := dt * yp_func(t(n) + dt / 2.0, y(n) + dy1 / 2.0);
dy3 := dt * yp_func(t(n) + dt / 2.0, y(n) + dy2 / 2.0);
dy4 := dt * yp_func(t(n) + dt, y(n) + dy3);
t(n+1) := t(n) + dt;
y(n+1) := y(n) + (dy1 + 2.0 * (dy2 + dy3) + dy4) / 6.0;
end loop;
end Runge;

procedure Print (t, y : Floaty_Array; modnum : Positive) is begin
for i in t'Range loop
if i mod modnum = 0 then
Put("y(");   Put (t(i), Exp=>0, Fore=>0, Aft=>1);
Put(") = "); Put (y(i), Exp=>0, Fore=>0, Aft=>8);
Put(" Error:"); Put (calc_err(t(i),y(i)), Aft=>5);
New_Line;
end if;
end loop;
end Print;

function yprime (t, y : Floaty) return Floaty is begin
return t * Math.Sqrt (y);
end yprime;
function calc_err (t, calc : Floaty) return Floaty is
actual : constant Floaty := (t**2 + 4.0)**2 / 16.0;
begin return abs(actual-calc);
end calc_err;

dt : constant Floaty := 0.10;
N : constant Positive := 100;
t_arr, y_arr : Floaty_Array(0 .. N);
begin
t_arr(0) := 0.0;
y_arr(0) := 1.0;
Runge (yprime'Access, t_arr, y_arr, dt);
Print (t_arr, y_arr, 10);
end RungeKutta;

Output:
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

## ALGOL 68

BEGIN
PROC rk4 = (PROC (REAL, REAL) REAL f, REAL y, x, dx) REAL :
BEGIN  CO Fourth-order Runge-Kutta method CO
REAL dy1 = dx * f(x, y);
REAL dy2 = dx * f(x + dx / 2.0, y + dy1 / 2.0);
REAL dy3 = dx * f(x + dx / 2.0, y + dy2 / 2.0);
REAL dy4 = dx * f(x + dx, y + dy3);
y + (dy1 + 2.0 * dy2 + 2.0 * dy3 + dy4) / 6.0
END;
REAL x0 = 0, x1 = 10, y0 = 1.0;			CO Boundary conditions. CO
REAL dx = 0.1;					CO Step size. CO
INT num points = ENTIER ((x1 - x0) / dx + 0.5);	CO Add 0.5 for rounding errors. CO
[0:num points]REAL y;   y[0] := y0;			CO Grid and starting point.CO
PROC dy by dx = (REAL x, y) REAL : x * sqrt(y);	CO Differential equation. CO
FOR i TO num points
DO
y[i] := rk4 (dy by dx, y[i-1], x0 + dx * (i - 1), dx)
OD;
print (("   x              true y         calc y       relative error", newline));
FOR i FROM 0 BY 10 TO  num points
DO
REAL x = x0 + dx * i;
REAL true y = (x * x + 4.0) ^ 2 / 16.0;
printf (($3(-zzd.7dxxx), -d.4de-ddl$, x, true y, y[i], y[i] / true y - 1.0))
OD
END
Output:
   x              true y         calc y       relative error
0.0000000      1.0000000      1.0000000    0.0000e 00
1.0000000      1.5625000      1.5624999   -9.3262e-08
2.0000000      4.0000000      3.9999991   -2.2987e-07
3.0000000     10.5625000     10.5624971   -2.7546e-07
4.0000000     25.0000000     24.9999938   -2.4940e-07
5.0000000     52.5625000     52.5624892   -2.0584e-07
6.0000000    100.0000000     99.9999834   -1.6595e-07
7.0000000    175.5625000    175.5624765   -1.3396e-07
8.0000000    289.0000000    288.9999684   -1.0922e-07
9.0000000    451.5625000    451.5624593   -9.0183e-08
10.0000000    676.0000000    675.9999490   -7.5419e-08


## ALGOL W

Translation of: 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.

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.
Output:
   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


## APL

      ∇RK4[⎕]∇
∇
[0]   Z←R(Y¯ RK4)Y;T;YN;TN;∆T;∆Y1;∆Y2;∆Y3;∆Y4
[1]   (T R ∆T)←R
[2]  LOOP:→(R≤TN←¯1↑T)/EXIT
[3]   ∆Y1←∆T×TN Y¯ YN←¯1↑Y
[4]   ∆Y2←∆T×(TN+∆T÷2)Y¯ YN+∆Y1÷2
[5]   ∆Y3←∆T×(TN+∆T÷2)Y¯ YN+∆Y2÷2
[6]   ∆Y4←∆T×(TN+∆T)Y¯ YN+∆Y3
[7]   Y←Y,YN+(∆Y1+(2×∆Y2)+(2×∆Y3)+∆Y4)÷6
[8]   T←T,TN+∆T
[9]   →LOOP
[10] EXIT:Z←T,[⎕IO+.5]Y
∇

∇PRINT[⎕]∇
∇
[0]   PRINT;TABLE
[1]   TABLE←0 10 .1({⍺×⍵*.5}RK4)1
[2]   ⎕←'T' 'RK4 Y' 'ERROR'⍪TABLE,TABLE[;2]-{((4+⍵*2)*2)÷16}TABLE[;1]
∇

Output:
      PRINT
T           RK4 Y              ERROR
0       1               0.000000000E0
0.1     1.005006249    ¯1.303701147E¯9
0.2     1.020099995    ¯5.215366805E¯9
0.3     1.045506238    ¯1.174457109E¯8
0.4     1.081599979    ¯2.093284546E¯8
0.5     1.128906217    ¯3.288601591E¯8
0.6     1.188099952    ¯4.780736740E¯8
0.7     1.260006184    ¯6.602350622E¯8
0.8     1.345599912    ¯8.799725681E¯8
0.9     1.446006136    ¯1.143253423E¯7
. . .


## AWK

# syntax: GAWK -f RUNGE-KUTTA_METHOD.AWK
# converted from BBC BASIC
BEGIN {
print(" t    y         error")
y = 1
for (i=0; i<=100; i++) {
t = i / 10
if (t == int(t)) {
actual = ((t^2+4)^2) / 16
printf("%2d %12.7f %g\n",t,y,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
}
exit(0)
}

Output:
 t    y         error
0    1.0000000 0
1    1.5624999 1.45722e-007
2    3.9999991 9.19479e-007
3   10.5624971 2.90956e-006
4   24.9999938 6.23491e-006
5   52.5624892 1.08197e-005
6   99.9999834 1.65946e-005
7  175.5624765 2.35177e-005
8  288.9999684 3.15652e-005
9  451.5624593 4.07232e-005
10  675.9999490 5.09833e-005


## BASIC

### 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

### BBC BASIC

      y = 1.0
FOR i% = 0 TO 100
t = i% / 10

IF t = INT(t) THEN
actual = ((t^2 + 4)^2) / 16
PRINT "y("; t ") = "; y TAB(20) "Error = ";  actual - y
ENDIF

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 += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
NEXT i%

Output:
y(0) = 1            Error = 0
y(1) = 1.56249985   Error = 1.45721892E-7
y(2) = 3.99999908   Error = 9.19479201E-7
y(3) = 10.5624971   Error = 2.90956245E-6
y(4) = 24.9999938   Error = 6.23490936E-6
y(5) = 52.5624892   Error = 1.08196974E-5
y(6) = 99.9999834   Error = 1.65945964E-5
y(7) = 175.562476   Error = 2.35177287E-5
y(8) = 288.999968   Error = 3.15652015E-5
y(9) = 451.562459   Error = 4.07231605E-5
y(10) = 675.999949  Error = 5.09832905E-5


### 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$(T);") =";Y;TAB(21);"Error =";((T^2+4)^2)/16-Y 140 LET K1=T*SQR(Y) 150 LET K2=(T+.05)*SQR(Y+.05*K1) 160 LET K3=(T+.05)*SQR(Y+.05*K2) 170 LET K4=(T+.1)*SQR(Y+.1*K3) 180 LET Y=Y+.1*(K1+2*(K2+K3)+K4)/6 190 NEXT ### QBasic Works with: QBasic version 1.1 Works with: QuickBasic version 4.5 y! = 1 FOR i = 0 TO 100 t = i / 10 IF t = INT(t) THEN actual! = ((t ^ 2 + 4) ^ 2) / 16 PRINT USING "y(##) = ###.###### Error = "; t; y; PRINT actual - y END IF k1! = t * SQR(y) k2! = (t + .05) * SQR(y + .05 * k1) k3! = (t + .05) * SQR(y + .05 * k2) k4! = (t + .1) * SQR(y + .1 * k3) y = y + .1 * (k1 + 2 * (k2 + k3) + k4) / 6 NEXT i  ### True BASIC Works with: QBasic LET y = 1 FOR i = 0 TO 100 LET t = i / 10 IF t = INT(t) THEN LET actual = ((t ^ 2 + 4) ^ 2) / 16 PRINT "y("; STR$(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


## C

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

double rk4(double(*f)(double, double), double dx, double x, double y)
{
double	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);
return y + (k1 + 2 * k2 + 2 * k3 + k4) / 6;
}

double rate(double x, double y)
{
return x * sqrt(y);
}

int main(void)
{
double *y, x, y2;
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++)
y[i] = rk4(rate, dx, x0 + dx * (i - 1), y[i-1]);

printf("x\ty\trel. err.\n------------\n");
for (i = 0; i < n; i += 10) {
x = x0 + dx * i;
y2 = pow(x * x / 4 + 1, 2);
printf("%g\t%g\t%g\n", x, y[i], y[i]/y2 - 1);
}

return 0;
}

Output:

(errors are relative)

x       y       rel. err.
------------
0       1       0
1       1.5625  -9.3262e-08
2       4       -2.2987e-07
3       10.5625 -2.75462e-07
4       25      -2.49396e-07
5       52.5625 -2.05844e-07
6       100     -1.65946e-07
7       175.562 -1.33956e-07
8       289     -1.09222e-07
9       451.562 -9.01828e-08
10      676     -7.54191e-08


## C#

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;
}
}
}


## C++

Using Lambdas

/*
* 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;
}


## Common Lisp

(defun runge-kutta (f x y x-end n)
(let ((h (float (/ (- x-end x) n) 1d0))
k1 k2 k3 k4)
(setf x (float x 1d0)
y (float y 1d0))
(cons (cons x y)
(loop for i below n do
(setf k1 (* h (funcall f x y))
k2 (* h (funcall f (+ x (* 0.5d0 h)) (+ y (* 0.5d0 k1))))
k3 (* h (funcall f (+ x (* 0.5d0 h)) (+ y (* 0.5d0 k2))))
k4 (* h (funcall f (+ x h) (+ y k3)))
x (+ x h)
y (+ y (/ (+ k1 k2 k2 k3 k3 k4) 6)))
collect (cons x y)))))

(let ((sol (runge-kutta (lambda (x y) (* x (sqrt y))) 0 1 10 100)))
(loop for n from 0
for (x . y) in sol
when (zerop (mod n 10))
collect (list x y (- y (/ (expt (+ 4 (* x x)) 2) 16)))))

((0.0d0 1.0d0 0.0d0)
(0.9999999999999999d0 1.562499854278108d0 -1.4572189210859676d-7)
(2.0000000000000004d0 3.9999990805207988d0 -9.194792029987298d-7)
(3.0000000000000013d0 10.562497090437557d0 -2.9095624576314094d-6)
(4.000000000000002d0 24.999993765090643d0 -6.234909392333066d-6)
(4.999999999999998d0 52.56248918030259d0 -1.081969734428867d-5)
(5.999999999999995d0 99.9999834054036d0 -1.659459609015812d-5)
(6.999999999999991d0 175.56247648227117d0 -2.3517728038768837d-5)
(7.999999999999988d0 288.9999684347983d0 -3.156520000402452d-5)
(8.999999999999984d0 451.56245927683887d0 -4.072315812209126d-5)
(9.99999999999998d0 675.9999490167083d0 -5.0983286655537086d-5))


## Crystal

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

Output:
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


## D

Translation of: Ada
import std.stdio, std.math, std.typecons;

alias FP = real;
alias FPs = Typedef!(FP[101]);

void runge(in FP function(in FP, in FP)
pure nothrow @safe @nogc yp_func,
ref FPs t, ref FPs y, in FP dt) pure nothrow @safe @nogc {
foreach (immutable n; 0 .. t.length - 1) {
immutable FP
dy1 = dt * yp_func(t[n], y[n]),
dy2 = dt * yp_func(t[n] + dt / 2.0, y[n] + dy1 / 2.0),
dy3 = dt * yp_func(t[n] + dt / 2.0, y[n] + dy2 / 2.0),
dy4 = dt * yp_func(t[n] + dt, y[n] + dy3);
t[n + 1] = t[n] + dt;
y[n + 1] = y[n] + (dy1 + 2.0 * (dy2 + dy3) + dy4) / 6.0;
}
}

FP calc_err(in FP t, in FP calc) pure nothrow @safe @nogc {
immutable FP actual = (t ^^ 2 + 4.0) ^^ 2 / 16.0;
return abs(actual - calc);
}

void main() {
enum FP dt = 0.10;
FPs t_arr, y_arr;

t_arr[0] = 0.0;
y_arr[0] = 1.0;
runge((t, y) => t * y.sqrt, t_arr, y_arr, dt);

foreach (immutable i; 0 .. t_arr.length)
if (i % 10 == 0)
writefln("y(%.1f) = %.8f Error: %.6g",
t_arr[i], y_arr[i],
calc_err(t_arr[i], y_arr[i]));
}

Output:
y(0.0) = 1.00000000 Error: 0
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

## Dart

import 'dart:math' as Math;

num RungeKutta4(Function f, num t, num y, num dt){
num k1 = dt * f(t,y);
num k2 = dt * f(t+0.5*dt, y + 0.5*k1);
num k3 = dt * f(t+0.5*dt, y + 0.5*k2);
num k4 = dt * f(t + dt, y + k3);
return y + (1/6) * (k1 + 2*k2 + 2*k3 + k4);
}

void main(){
num t  = 0;
num dt = 0.1;
num tf = 10;
num totalPoints = ((tf-t)/dt).floor()+1;
num y  = 1;
Function f  = (num t, num y) => t * Math.sqrt(y);
Function actual = (num t) => (1/16) * (t*t+4)*(t*t+4);
for (num i = 0; i <= totalPoints; i++){
num relativeError = (actual(t) - y)/actual(t);
if (i%10 == 0){
print('y(${t.round().toStringAsPrecision(3)}) =${y.toStringAsPrecision(11)}  Error = ${relativeError.toStringAsPrecision(11)}'); } y = RungeKutta4(f, t, y, dt); t += dt; } }  Output: y(0.00) = 1.0000000000 Error = 0.0000000000 y(1.00) = 1.5624998543 Error = 9.3262010950e-8 y(2.00) = 3.9999990805 Error = 2.2986980086e-7 y(3.00) = 10.562497090 Error = 2.7546153479e-7 y(4.00) = 24.999993765 Error = 2.4939637555e-7 y(5.00) = 52.562489180 Error = 2.0584442034e-7 y(6.00) = 99.999983405 Error = 1.6594596090e-7 y(7.00) = 175.56247648 Error = 1.3395644308e-7 y(8.00) = 288.99996843 Error = 1.0922214534e-7 y(9.00) = 451.56245928 Error = 9.0182772312e-8 y(10.0) = 675.99994902 Error = 7.5419063100e-8  ## EasyLang Translation of: BASIC256 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 . ## 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.  [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] Output: 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  ## ERRE PROGRAM RUNGE_KUTTA CONST DELTA_T=0.1 FUNCTION Y1(T,Y) Y1=T*SQR(Y) END FUNCTION BEGIN Y=1.0 FOR I%=0 TO 100 DO T=I%*DELTA_T IF T=INT(T) THEN ! print every tenth ACTUAL=((T^2+4)^2)/16 ! exact solution PRINT("Y(";T;")=";Y;TAB(20);"Error=";ACTUAL-Y) END IF K1=Y1(T,Y) K2=Y1(T+DELTA_T/2,Y+DELTA_T/2*K1) K3=Y1(T+DELTA_T/2,Y+DELTA_T/2*K2) K4=Y1(T+DELTA_T,Y+DELTA_T*K3) Y+=DELTA_T*(K1+2*(K2+K3)+K4)/6 END FOR END PROGRAM Output: Y( 0 )= 1 Error= 0 Y( 1 )= 1.5625 Error= 2.384186E-07 Y( 2 )= 3.999999 Error= 7.152558E-07 Y( 3 )= 10.5625 Error= 1.907349E-06 Y( 4 )= 25 Error= 3.814697E-06 Y( 5 )= 52.56249 Error= 7.629395E-06 Y( 6 )= 100 Error= 0 Y( 7 )= 175.5625 Error= 0 Y( 8 )= 289 Error= 0 Y( 9 )= 451.5625 Error= 0 Y( 10 )= 676.0001 Error=-6.103516E-05  ## 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 ) Output:  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  ## F# open System let y'(t,y) = t * sqrt(y) let RungeKutta4 t0 y0 t_max dt = let dy1(t,y) = dt * y'(t,y) let dy2(t,y) = dt * y'(t+dt/2.0, y+dy1(t,y)/2.0) let dy3(t,y) = dt * y'(t+dt/2.0, y+dy2(t,y)/2.0) let dy4(t,y) = dt * y'(t+dt, y+dy3(t,y)) (t0,y0) |> Seq.unfold (fun (t,y) -> if ( t <= t_max) then Some((t,y), (Math.Round(t+dt, 6), y + ( dy1(t,y) + 2.0*dy2(t,y) + 2.0*dy3(t,y) + dy4(t,y))/6.0)) else None ) let y_exact t = (pown (pown t 2 + 4.0) 2)/16.0 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) )  Output: y(0)=1 (relative error:0) y(1)=1.56249985427811 (relative error:-9.32620110027926E-08) y(2)=3.9999990805208 (relative error:-2.29869800194571E-07) y(3)=10.5624970904376 (relative error:-2.75461533583155E-07) y(4)=24.9999937650906 (relative error:-2.49396374552013E-07) y(5)=52.5624891803026 (relative error:-2.05844421730106E-07) y(6)=99.9999834054036 (relative error:-1.65945964192282E-07) y(7)=175.562476482271 (relative error:-1.33956447156969E-07) y(8)=288.999968434799 (relative error:-1.09222150213029E-07) y(9)=451.56245927684 (relative error:-9.01827772459285E-08) y(10)=675.99994901671 (relative error:-7.54190684348899E-08)  ## Fortran program rungekutta implicit none integer, parameter :: dp = kind(1d0) real(dp) :: t, dt, tstart, tstop 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  Output: y( 0.0) = 1.00000000 Error = 0.000000E+00 y( 1.0) = 1.56249985 Error = 1.457219E-07 y( 2.0) = 3.99999908 Error = 9.194792E-07 y( 3.0) = 10.56249709 Error = 2.909562E-06 y( 4.0) = 24.99999377 Error = 6.234909E-06 y( 5.0) = 52.56248918 Error = 1.081970E-05 y( 6.0) = 99.99998341 Error = 1.659460E-05 y( 7.0) = 175.56247648 Error = 2.351773E-05 y( 8.0) = 288.99996843 Error = 3.156520E-05 y( 9.0) = 451.56245928 Error = 4.072316E-05 y(10.0) = 675.99994902 Error = 5.098329E-05  ## FreeBASIC Translation of: BBC BASIC ' version 03-10-2015 ' compile with: fbc -s console ' translation of BBC BASIC Dim As Double y = 1, t, actual, k1, k2, k3, k4 Print For i As Integer = 0 To 100 t = i / 10 If t = Int(t) Then actual = ((t ^ 2 + 4) ^ 2) / 16 Print "y("; Str(t); ") ="; y ; Tab(27); "Error = "; actual - y 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 += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6 Next i ' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End Output: y(0) = 1 Error = 0 y(1) = 1.562499854278108 Error = 1.457218921085968e-007 y(2) = 3.999999080520799 Error = 9.194792012223729e-007 y(3) = 10.56249709043755 Error = 2.909562448749625e-006 y(4) = 24.99999376509064 Error = 6.234909363911356e-006 y(5) = 52.56248918030259 Error = 1.081969741534294e-005 y(6) = 99.99998340540358 Error = 1.659459641700778e-005 y(7) = 175.5624764822713 Error = 2.351772874931157e-005 y(8) = 288.9999684347985 Error = 3.156520148195341e-005 y(9) = 451.5624592768396 Error = 4.072316039582802e-005 y(10) = 675.9999490167097 Error = 5.098329029351589e-005 ## FutureBasic window 1 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 Output: y(0) = 1 Error = 0 y(1) = 1.5624998543 Error = 1.45721892e-7 y(2) = 3.9999990805 Error = 9.19479201e-7 y(3) = 10.5624970904 Error = 2.90956245e-6 y(4) = 24.9999937651 Error = 6.23490936e-6 y(5) = 52.56248918 Error = 1.08196974e-5 y(6) = 99.999983405 Error = 1.65945964e-5 y(7) = 175.562476482 Error = 2.35177287e-5 y(8) = 288.99996843 Error = 3.15652014e-5 y(9) = 451.56245928 Error = 4.07231603e-5 y(10) = 675.99994902 Error = 5.09832903e-5  ## Go Works with: Go1 package main import ( "fmt" "math" ) type ypFunc func(t, y float64) float64 type ypStepFunc func(t, y, dt float64) float64 // newRKStep takes a function representing a differential equation // and returns a function that performs a single step of the forth-order // Runge-Kutta method. func newRK4Step(yp ypFunc) ypStepFunc { return func(t, y, dt float64) float64 { dy1 := dt * yp(t, y) dy2 := dt * yp(t+dt/2, y+dy1/2) dy3 := dt * yp(t+dt/2, y+dy2/2) dy4 := dt * yp(t+dt, y+dy3) return y + (dy1+2*(dy2+dy3)+dy4)/6 } } // example differential equation func yprime(t, y float64) float64 { return t * math.Sqrt(y) } // exact solution of example func actual(t float64) float64 { t = t*t + 4 return t * t / 16 } func main() { t0, tFinal := 0, 10 // task specifies times as integers, dtPrint := 1 // and to print at whole numbers. y0 := 1. // initial y. dtStep := .1 // step value. t, y := float64(t0), y0 ypStep := newRK4Step(yprime) for t1 := t0 + dtPrint; t1 <= tFinal; t1 += dtPrint { printErr(t, y) // print intermediate result for steps := int(float64(dtPrint)/dtStep + .5); steps > 1; steps-- { y = ypStep(t, y, dtStep) t += dtStep } y = ypStep(t, y, float64(t1)-t) // adjust step to integer time t = float64(t1) } printErr(t, y) // print final result } func printErr(t, y float64) { fmt.Printf("y(%.1f) = %f Error: %e\n", t, y, math.Abs(actual(t)-y)) }  Output: 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  ## 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;} }  Output: 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  ## 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); }; Output:  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  ## Haskell Using GHC 7.4.1. dv :: Floating a => a -> a -> a dv = (. sqrt) . (*) fy t = 1 / 16 * (4 + t ^ 2) ^ 2 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)


Example executed in GHCi:

*Main> task
(0,1.0,0.0)
(1,1.5624998542781088,1.4572189122041834e-7)
(2,3.9999990805208006,9.194792029987298e-7)
(3,10.562497090437557,2.909562461184123e-6)
(4,24.999993765090654,6.234909399438493e-6)
(5,52.56248918030265,1.0819697635611192e-5)
(6,99.99998340540378,1.6594596999652822e-5)
(7,175.56247648227165,2.3517730085131916e-5)
(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:

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)

Output:
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

## J

Solution:

NB.*rk4 a Solve function using Runge-Kutta method
NB. y is: y(ta) , ta , tb , tstep
NB. u is: function to solve
NB. eg: fyp rk4 1 0 10 0.1
rk4=: adverb define
'Y0 a b h'=. 4{. y
T=. a + i.@>:&.(%&h) b - a
Y=. Yt=. Y0
for_t. }: T do.
ty=. t,Yt
k1=. h * u ty
k2=. h * u ty + -: h,k1
k3=. h * u ty + -: h,k2
k4=. h * u ty + h,k3
Y=. Y, Yt=. Yt + (%6) * 1 2 2 1 +/@:* k1, k2, k3, k4
end.
T ,. Y
)


Example:

   fy=: (%16) * [: *: 4 + *:             NB. f(t,y)
fyp=: (* %:)/                         NB. f'(t,y)
report_whole=: (10 * i. >:10)&{       NB. report at whole-numbered t values
report_err=: (, {: - [: fy {.)"1      NB. report errors

report_err report_whole fyp rk4 1 0 10 0.1
0       1           0
1  1.5625 _1.45722e_7
2       4 _9.19479e_7
3 10.5625 _2.90956e_6
4      25 _6.23491e_6
5 52.5625 _1.08197e_5
6     100 _1.65946e_5
7 175.562 _2.35177e_5
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.

rk4=: adverb define
'Y0 a b h'=. 4{. y
T=. a + i.@>:&.(%&h) b-a
(,. [: h&(u nextY)@,/\. Y0 ,~ }.)&.|. T
)

NB. nextY a Calculate Yn+1 of a function using Runge-Kutta method
NB. y is: 2-item numeric list of time t and y(t)
NB. u is: function to use
NB. x is: step size
NB. eg: 0.001 fyp nextY 0 1
nextY=: adverb define
:
tableau=. 1 0.5 0.5, x * u y
ks=. (x * [: u y + (* x&,))/\. tableau
({:y) + 6 %~ +/ 1 2 2 1 * ks
)


Use:

report_err report_whole fyp rk4 1 0 10 0.1


## Java

Translation of Ada via D

Works with: Java version 8
import static java.lang.Math.*;
import java.util.function.BiFunction;

public class RungeKutta {

static void runge(BiFunction<Double, Double, Double> yp_func, double[] t,
double[] y, double dt) {

for (int n = 0; n < t.length - 1; n++) {
double dy1 = dt * yp_func.apply(t[n], y[n]);
double dy2 = dt * yp_func.apply(t[n] + dt / 2.0, y[n] + dy1 / 2.0);
double dy3 = dt * yp_func.apply(t[n] + dt / 2.0, y[n] + dy2 / 2.0);
double dy4 = dt * yp_func.apply(t[n] + dt, y[n] + dy3);
t[n + 1] = t[n] + dt;
y[n + 1] = y[n] + (dy1 + 2.0 * (dy2 + dy3) + dy4) / 6.0;
}
}

static double calc_err(double t, double calc) {
double actual = pow(pow(t, 2.0) + 4.0, 2) / 16.0;
return abs(actual - calc);
}

public static void main(String[] args) {
double dt = 0.10;
double[] t_arr = new double[101];
double[] y_arr = new double[101];
y_arr[0] = 1.0;

runge((t, y) -> t * sqrt(y), t_arr, y_arr, dt);

for (int i = 0; i < t_arr.length; i++)
if (i % 10 == 0)
System.out.printf("y(%.1f) = %.8f Error: %.6f%n",
t_arr[i], y_arr[i],
calc_err(t_arr[i], y_arr[i]));
}
}

y(0,0) = 1,00000000 Error: 0,000000
y(1,0) = 1,56249985 Error: 0,000000
y(2,0) = 3,99999908 Error: 0,000001
y(3,0) = 10,56249709 Error: 0,000003
y(4,0) = 24,99999377 Error: 0,000006
y(5,0) = 52,56248918 Error: 0,000011
y(6,0) = 99,99998341 Error: 0,000017
y(7,0) = 175,56247648 Error: 0,000024
y(8,0) = 288,99996843 Error: 0,000032
y(9,0) = 451,56245928 Error: 0,000041
y(10,0) = 675,99994902 Error: 0,000051

## JavaScript

### ES5

function rk4(y, x, dx, f) {
var 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;
}

function f(x, y) {
return x * Math.sqrt(y);
}

function actual(x) {
return (1/16) * (x*x+4)*(x*x+4);
}

var y = 1.0,
x = 0.0,
step = 0.1,
steps = 0,
maxSteps = 101,
sampleEveryN = 10;

while (steps < maxSteps) {
if (steps%sampleEveryN === 0) {
console.log("y(" + x + ") =  \t" + y + "\t ± " + (actual(x) - y).toExponential());
}

y = rk4(y, x, step, f);

// using integer math for the step addition
// to prevent floating point errors as 0.2 + 0.1 != 0.3
x = ((x * 10) + (step * 10)) / 10;
steps += 1;
}

Output:
y(0) =  	1	                 ± 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


### ES6

(() => {
'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();
})();

Output:
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

## jq

In this section, two solutions are presented. 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(cond; next):
def _until: if cond then . else (next|_until) end;
_until;

def while(cond; update):
def _while:  if cond then ., (update | _while) else empty end;
_while;

### The Example Differential Equation and its Exact Solution

# yprime maps [t,y] to a number, i.e. t * sqrt(y)
def yprime: .[0] * (.[1] | sqrt);

# The exact solution of yprime:
def actual:
. as $t | (($t*$t) + 4 ) | . * . / 16; ### dy/dt The first solution presented here uses the terminology and style of the Raku version. Generic filters: # n is the number of decimal places of precision def round(n): (if . < 0 then -1 else 1 end) as$s
| $s*10*.*n | if (floor % 10) > 4 then (.+5) else . end | ./10 | floor/n | .*$s;

def abs: if . < 0 then -. else . end;

# Is the input an integer?
def integerq: ((. - ((.+.01) | floor)) | abs) < 0.01;

dy(f)

def dt: 0.1;

# Input: [t, y]; yp is a filter that accepts [t,y] as input
def runge_kutta(yp):
.[0] as $t | .[1] as$y
| (dt * yp) as $a | (dt * ([ ($t + (dt/2)), $y + ($a/2) ] | yp)) as $b | (dt * ([ ($t + (dt/2)), $y + ($b/2) ] | yp)) as $c | (dt * ([ ($t + dt)    , $y +$c     ] | yp)) as $d | ($a + (2*($b +$c)) + $d) / 6 ; # Input: [t,y] def dy(f): runge_kutta(f); Example: # state: [t,y] [0,1] | while( .[0] <= 10; .[0] as$t | .[1] as $y | [$t + dt, $y + dy(yprime) ] ) | .[0] as$t | .[1] as $y | if$t | integerq then
"y(\($t|round(1))) = \($y|round(10000)) ± \( ($t|actual) -$y | abs)"
else empty
end
Output:
$time jq -r -n -f rk4.pl.jq y(0) = 1 ± 0 y(1) = 1.5625 ± 1.4572189210859676e-07 y(2) = 4 ± 9.194792029987298e-07 y(3) = 10.5625 ± 2.9095624576314094e-06 y(4) = 25 ± 6.234909392333066e-06 y(5) = 52.5625 ± 1.081969734428867e-05 y(6) = 100 ± 1.659459609015812e-05 y(7) = 175.5625 ± 2.3517728038768837e-05 y(8) = 289 ± 3.156520000402452e-05 y(9) = 451.5625 ± 4.072315812209126e-05 y(10) = 675.9999 ± 5.0983286655537086e-05 real 0m0.048s user 0m0.013s sys 0m0.006s  ### newRK4Step The second solution follows the nomenclature and style of the Go solution on this page. In the following notes: • ypFunc denotes the type of a jq filter that maps [t, y] to a number; • ypStepFunc denotes the type of a jq filter that maps [t, y, dt] to a number. The heart of the program is the filter newRK4Step(yp), which is of type ypStepFunc and performs a single step of the fourth-order Runge-Kutta method, provided yp is of type ypFunc. # Input: [t, y, dt] def newRK4Step(yp): .[0] as$t | .[1] as $y | .[2] as$dt
| ($dt * ([$t, $y]|yp)) as$dy1
| ($dt * ([$t+$dt/2,$y+$dy1/2]|yp)) as$dy2
| ($dt * ([$t+$dt/2,$y+$dy2/2]|yp)) as$dy3
| ($dt * ([$t+$dt,$y+$dy3] |yp)) as$dy4
| $y + ($dy1+2*($dy2+$dy3)+$dy4)/6 ; def printErr: # input: [t, y] def abs: if . < 0 then -. else . end; .[0] as$t | .[1] as $y | "y(\($t)) = \($y) with error: \( (($t|actual) - $y) | abs )" ; def main(t0; y0; tFinal; dtPrint): def ypStep: newRK4Step(yprime) ; 0.1 as$dtStep     # step value
# [ t, y] is the state vector
| [ t0, y0 ]
| while( .[0] <= tFinal;
.[0] as $t | .[1] as$y
| ($t + dtPrint) as$t1
| (((dtPrint/$dtStep) + 0.5) | floor) as$steps
| [$steps,$t, $y] # state vector | until( .[0] <= 1; .[0] as$steps
| .[1] as $t | .[2] as$y
| [ ($steps - 1), ($t + $dtStep), ([$t, $y,$dtStep]|ypStep) ]
)
| .[1] as $t | .[2] as$y
| [$t1, ([$t, $y, ($t1-$t)] | ypStep)] # adjust step to integer time ) | printErr # print results ; # main(t0; y0; tFinal; dtPrint) main(0; 1; 10; 1) Output: $ time jq -n -r -f runge-kutta.jq
y(0) = 1 with error: 0
y(1) = 1.562499854278108 with error: 1.4572189210859676e-07
y(2) = 3.9999990805207974 with error: 9.194792025546406e-07
y(3) = 10.562497090437544 with error: 2.9095624558550526e-06
y(4) = 24.999993765090615 with error: 6.234909385227638e-06
y(5) = 52.562489180302656 with error: 1.081969734428867e-05
y(6) = 99.99998340540387 with error: 1.6594596132790684e-05
y(7) = 175.56247648227188 with error: 2.3517728124033965e-05
y(8) = 288.9999684347997 with error: 3.156520028824161e-05
y(9) = 451.56245927684154 with error: 4.0723158463151776e-05
y(10) = 675.9999490167129 with error: 5.0983287110284436e-05

real	0m0.023s
user	0m0.014s
sys	0m0.006s


## Julia

Works with: Julia version 0.6

### Using lambda expressions

Translation of: Python
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

δy = rk4(f)
t₀, δt, tmax = 0.0, 0.1, 10.0
y₀ = 1.0

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

Output:
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

### Alternative version

Translation of: Python
function rk4(f::Function, x₀::Float64, y₀::Float64, x₁::Float64, n)
vx = Vector{Float64}(undef, n + 1)
vy = Vector{Float64}(undef, n + 1)
vx[1] = x = x₀
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


## Kotlin

// 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)
}

Output:
  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


## Liberty BASIC

'[RC] Runge-Kutta method
'initial conditions
x0 = 0
y0 = 1
'step
h = 0.1
'number of points
N=101

y=y0
FOR i = 0 TO N-1
x = x0+ i*h
IF x = INT(x) THEN
actual = exactY(x)
PRINT "y("; x ;") = "; y; TAB(20); "Error = ";  actual - y
END IF

k1 = h*dydx(x,y)
k2 = h*dydx(x+h/2,y+k1/2)
k3 = h*dydx(x+h/2,y+k2/2)
k4 = h*dydx(x+h,y+k3)
y = y + 1/6 * (k1 + 2*k2 + 2*k3 + k4)
NEXT i

function dydx(x,y)
dydx=x*sqr(y)
end function

function exactY(x)
exactY=(x^2 + 4)^2 / 16
end function
Output:
y(0) = 1           Error = 0
y(1) = 1.56249985  Error = 0.14572189e-6
y(2) = 3.99999908  Error = 0.9194792e-6
y(3) = 10.5624971  Error = 0.29095624e-5
y(4) = 24.9999938  Error = 0.62349094e-5
y(5) = 52.5624892  Error = 0.10819697e-4
y(6) = 99.9999834  Error = 0.16594596e-4
y(7) = 175.562476  Error = 0.23517729e-4
y(8) = 288.999968  Error = 0.31565201e-4
y(9) = 451.562459  Error = 0.4072316e-4
y(10) = 675.999949 Error = 0.5098329e-4


## 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

Output:
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



## Mathematica/Wolfram Language

(* 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}]

(* Numerical solution I (not RK4) *)
Table[{t, y[t], Abs[y[t] - 1/16*(4 + t^2)^2]}, {t, 0, 10}] /.
First@NDSolve[{y'[t] == t*Sqrt[y[t]], y[0] == 1}, y, {t, 0, 10}]

(* Numerical solution II (RK4) *)
f[{t_, y_}] := {1, t Sqrt[y]}
h = 0.1;
phi[y_] := Module[{k1, k2, k3, k4},
k1 = h*f[y];
k2 = h*f[y + 1/2 k1];
k3 = h*f[y + 1/2 k2];
k4 = h*f[y + k3];
y + k1/6 + k2/3 + k3/3 + k4/6]
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]]}]


## 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 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.

function testRK4Programs
figure
hold on
t = 0:0.1:10;
y = 0.0625.*(t.^2+4).^2;
plot(t, y, '-k')
[tode4, yode4] = testODE4(t);
plot(tode4, yode4, '--b')
[trk4, yrk4] = testRK4(t);
plot(trk4, yrk4, ':r')
legend('Exact', 'ODE4', 'RK4')
hold off
fprintf('Time\tExactVal\tODE4Val\tODE4Error\tRK4Val\tRK4Error\n')
for k = 1:10:length(t)
fprintf('%.f\t\t%7.3f\t\t%7.3f\t%7.3g\t%7.3f\t%7.3g\n', t(k), y(k), ...
yode4(k), abs(y(k)-yode4(k)), yrk4(k), abs(y(k)-yrk4(k)))
end
end

function [t, y] = testODE4(t)
y0 = 1;
y = ode4(@(tVal,yVal)tVal*sqrt(yVal), t, y0);
end

function [t, y] = testRK4(t)
dydt = @(tVal,yVal)tVal*sqrt(yVal);
y = zeros(size(t));
y(1) = 1;
for k = 1:length(t)-1
dt = t(k+1)-t(k);
dy1 = dt*dydt(t(k), y(k));
dy2 = dt*dydt(t(k)+0.5*dt, y(k)+0.5*dy1);
dy3 = dt*dydt(t(k)+0.5*dt, y(k)+0.5*dy2);
dy4 = dt*dydt(t(k)+dt, y(k)+dy3);
y(k+1) = y(k)+(dy1+2*dy2+2*dy3+dy4)/6;
end
end

Output:
Time	ExactVal	ODE4Val		ODE4Error	RK4Val		RK4Error
0	  1.000		  1.000		      0		  1.000		      0
1	  1.563		  1.562		1.46e-007	  1.562		1.46e-007
2	  4.000		  4.000		9.19e-007	  4.000		9.19e-007
3	 10.563		 10.562		2.91e-006	 10.562		2.91e-006
4	 25.000		 25.000		6.23e-006	 25.000		6.23e-006
5	 52.563		 52.562		1.08e-005	 52.562		1.08e-005
6	100.000		100.000		1.66e-005	100.000		1.66e-005
7	175.563		175.562		2.35e-005	175.562		2.35e-005
8	289.000		289.000		3.16e-005	289.000		3.16e-005
9	451.563		451.562		4.07e-005	451.562		4.07e-005
10	676.000		676.000		5.10e-005	676.000		5.10e-005


## Maxima

/* Here is how to solve a differential equation */
'diff(y, x) = x * sqrt(y);
ode2(%, y, x);
ic1(%, x = 0, y = 1);
factor(solve(%, y)); /* [y = (x^2 + 4)^2 / 16] */

/* The Runge-Kutta solver is builtin */

load(dynamics)$sol: rk(t * sqrt(y), y, 1, [t, 0, 10, 1.0])$
plot2d([discrete, sol])$/* An implementation of RK4 for one equation */ rk4(f, x0, y0, x1, n) := block([h, x, y, vx, vy, k1, k2, k3, k4], h: bfloat((x1 - x0) / (n - 1)), x: x0, y: y0, vx: makelist(0, n + 1), vy: makelist(0, n + 1), vx[1]: x0, vy[1]: y0, for i from 1 thru n do ( k1: bfloat(h * f(x, y)), k2: bfloat(h * f(x + h / 2, y + k1 / 2)), k3: bfloat(h * f(x + h / 2, y + k2 / 2)), k4: bfloat(h * f(x + h, y + k3)), vy[i + 1]: y: y + (k1 + 2 * k2 + 2 * k3 + k4) / 6, vx[i + 1]: x: x + h ), [vx, vy] )$

[x, y]: rk4(lambda([x, y], x * sqrt(y)), 0, 1, 10, 101)$plot2d([discrete, x, y])$

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]);  ## МК-61/52 ПП 38 П1 ПП 30 П2 ПП 35 П3 2 * ПП 30 ИП2 ИП3 + 2 * + ИП1 + 3 / ИП7 + П7 П8 С/П БП 00 ИП6 ИП5 + П6 <-> ИП7 + П8 ИП8 КвКор ИП6 * ИП5 * В/О Input: 1/2 (h/2) - Р5, 1 (y0) - Р8 and Р7, 0 (t0) - Р6. ## 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)  Output: 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 ## Objeck class RungeKuttaMethod { function : Main(args : String[]) ~ Nil { x0 := 0.0; x1 := 10.0; dx := .1; n := 1 + (x1 - x0)/dx; y := Float->New[n->As(Int)]; y[0] := 1; for(i := 1; i < n; i++;) { y[i] := Rk4(Rate(Float, Float) ~ Float, dx, x0 + dx * (i - 1), y[i-1]); }; for(i := 0; i < n; i += 10;) { x := x0 + dx * i; y2 := (x * x / 4 + 1)->Power(2.0); x_value := x->As(Int); y_value := y[i]; rel_value := y_value/y2 - 1.0; "y({$x_value})={$y_value}; error: {$rel_value}"->PrintLine();
};
}

function : native : Rk4(f : (Float, Float) ~ Float, dx : Float, x : Float, y : Float) ~ Float {
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);

return y + (k1 + 2 * k2 + 2 * k3 + k4) / 6;
}

function : native : Rate(x : Float, y : Float) ~ Float {
return x * y->SquareRoot();
}
}

Output:

y(0)=1.0; error: 0.0
y(1)=1.563; error: -0.0000000933
y(2)=3.1000; error: -0.000000230
y(3)=10.563; error: -0.000000275
y(4)=24.1000; error: -0.000000249
y(5)=52.563; error: -0.000000206
y(6)=99.1000; error: -0.000000166
y(7)=175.563; error: -0.000000134
y(8)=288.1000; error: -0.000000109
y(9)=451.563; error: -0.0000000902
y(10)=675.1000; error: -0.0000000754


## OCaml

let y' t y = t *. sqrt y
let exact t = let u = 0.25*.t*.t +. 1.0 in u*.u

let rk4_step (y,t) h =
let k1 = h *. y' t y in
let k2 = h *. y' (t +. 0.5*.h) (y +. 0.5*.k1) in
let k3 = h *. y' (t +. 0.5*.h) (y +. 0.5*.k2) in
let k4 = h *. y' (t +. h) (y +. k3) in
(y +. (k1+.k4)/.6.0 +. (k2+.k3)/.3.0, t +. h)

let rec loop h n (y,t) =
if n mod 10 = 1 then
Printf.printf "t = %f,\ty = %f,\terr = %g\n" t y (abs_float (y -. exact t));
if n < 102 then loop h (n+1) (rk4_step (y,t) h)

let _ = loop 0.1 1 (1.0, 0.0)

Output:
t = 0.000000,	y = 1.000000,	err = 0
t = 1.000000,	y = 1.562500,	err = 1.45722e-07
t = 2.000000,	y = 3.999999,	err = 9.19479e-07
t = 3.000000,	y = 10.562497,	err = 2.90956e-06
t = 4.000000,	y = 24.999994,	err = 6.23491e-06
t = 5.000000,	y = 52.562489,	err = 1.08197e-05
t = 6.000000,	y = 99.999983,	err = 1.65946e-05
t = 7.000000,	y = 175.562476,	err = 2.35177e-05
t = 8.000000,	y = 288.999968,	err = 3.15652e-05
t = 9.000000,	y = 451.562459,	err = 4.07232e-05
t = 10.000000,	y = 675.999949,	err = 5.09833e-05

## Octave

#Applying the Runge-Kutta method (This code must be implement on a different file than the main one).

function temp = rk4(func,x,pvi,h)
K1 = h*func(x,pvi);
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.

f  = @(t) (1/16)*((t.^2 + 4).^2);
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

Output:
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

## PARI/GP

Translation of: C
rk4(f,dx,x,y)={
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
};
rate(x,y)=x*sqrt(y);
go()={
my(x0=0,x1=10,dx=.1,n=1+(x1-x0)\dx,y=vector(n));
y[1]=1;
for(i=2,n,y[i]=rk4(rate, dx, x0 + dx * (i - 1), y[i-1]));
print("x\ty\trel. err.\n------------");
forstep(i=1,n,10,
my(x=x0+dx*i,y2=(x^2/4+1)^2);
print(x "\t" y[i] "\t" y[i]/y2 - 1)
)
};
go()
Output:
x       y       rel. err.
------------
0.100000000     1       -0.00498131231
1.10000000      1.68999982      -0.00383519474
2.10000000      4.40999894      -0.00237694942
3.10000000      11.5599968      -0.00146924588
4.10000000      27.0399933      -0.000961094862
5.10000000      56.2499884      -0.000666538719
6.10000000      106.089982      -0.000485427212
7.10000000      184.959975      -0.000367681962
8.10000000      302.759966      -0.000287408941
9.10000000      470.889955      -0.000230470905

## Pascal

Translation of: Ada

This code has been compiled using Free Pascal 2.6.2.

program RungeKuttaExample;

uses sysutils;

type
TDerivative = function (t, y : Real) : Real;

procedure RungeKutta(yDer : TDerivative;
var t, y : array of Real;
dt   : Real);
var
dy1, dy2, dy3, dy4 : Real;
idx                : Cardinal;

begin
for idx := Low(t) to High(t) - 1 do
begin
dy1 := dt * yDer(t[idx],            y[idx]);
dy2 := dt * yDer(t[idx] + dt / 2.0, y[idx] + dy1 / 2.0);
dy3 := dt * yDer(t[idx] + dt / 2.0, y[idx] + dy2 / 2.0);
dy4 := dt * yDer(t[idx] + dt,       y[idx] + dy3);

t[idx + 1] := t[idx] + dt;
y[idx + 1] := y[idx] + (dy1 + 2.0 * (dy2 + dy3) + dy4) / 6.0;
end;
end;

function CalcError(t, y : Real) : Real;
var
trueVal : Real;

begin
trueVal := sqr(sqr(t) + 4.0) / 16.0;
CalcError := abs(trueVal - y);
end;

procedure Print(t, y : array of Real;
modnum : Integer);
var
idx : Cardinal;

begin
for idx := Low(t) to High(t) do
begin
if idx mod modnum = 0 then
begin
WriteLn(Format('y(%4.1f) = %12.8f  Error: %12.6e',
[t[idx], y[idx], CalcError(t[idx], y[idx])]));
end;
end;
end;

function YPrime(t, y : Real) : Real;
begin
YPrime := t * sqrt(y);
end;

const
dt = 0.10;
N = 100;

var
tArr, yArr : array [0..N] of Real;

begin
tArr[0] := 0.0;
yArr[0] := 1.0;

RungeKutta(@YPrime, tArr, yArr, dt);
Print(tArr, yArr, 10);
end.

Output:
y( 0.0) =   1.00000000  Error: 0.00000E+000
y( 1.0) =   1.56249985  Error: 1.45722E-007
y( 2.0) =   3.99999908  Error: 9.19479E-007
y( 3.0) =  10.56249709  Error: 2.90956E-006
y( 4.0) =  24.99999377  Error: 6.23491E-006
y( 5.0) =  52.56248918  Error: 1.08197E-005
y( 6.0) =  99.99998341  Error: 1.65946E-005
y( 7.0) = 175.56247648  Error: 2.35177E-005
y( 8.0) = 288.99996843  Error: 3.15652E-005
y( 9.0) = 451.56245928  Error: 4.07232E-005
y(10.0) = 675.99994902  Error: 5.09833E-005


## Perl

There are many ways of doing this. Here we define the runge_kutta function as a function of ${\displaystyle y'}$ and ${\displaystyle \delta t}$, returning a closure which itself takes ${\displaystyle (t,y)}$ as argument and returns the next ${\displaystyle (t,y)}$.

Notice how we have to use sprintf to deal with floating point rounding. See perlfaq4.

sub runge_kutta {
my ($yp,$dt) = @_;
sub {
my ($t,$y) = @_;
my @dy =  $dt *$yp->( $t ,$y );
push @dy, $dt *$yp->( $t +$dt/2, $y +$dy[0]/2 );
push @dy, $dt *$yp->( $t +$dt/2, $y +$dy[1]/2 );
push @dy, $dt *$yp->( $t +$dt  , $y +$dy[2] );
return $t +$dt, $y + ($dy[0] + 2*$dy[1] + 2*$dy[2] + $dy[3]) / 6; } } my$RK = runge_kutta sub { $_[0] * sqrt$_[1] }, .1;

for(
my ($t,$y) = (0, 1);
sprintf("%.0f", $t) <= 10; ($t, $y) =$RK->($t,$y)
) {
printf "y(%2.0f) = %12f ± %e\n", $t,$y, abs($y - ($t**2 + 4)**2 / 16)
if sprintf("%.4f", $t) =~ /0000$/;
}

Output:
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

## Phix

Translation of: ERRE
with javascript_semantics
constant dt = 0.1
atom y = 1.0
printf(1,"  x    true/actual y   calculated y    relative error\n")
printf(1," ---   -------------   -------------   --------------\n")
for i=0 to 100 do
atom t = i*dt
if integer(t) then
atom act = power(t*t+4,2)/16
printf(1,"%4.1f  %14.9f  %14.9f   %.9e\n",{t,act,y,abs(y-act)})
end if
atom k1 = t*sqrt(y),
k2 = (t+dt/2)*sqrt(y+dt/2*k1),
k3 = (t+dt/2)*sqrt(y+dt/2*k2),
k4 = (t+dt)*sqrt(y+dt*k3)
y += dt*(k1+2*(k2+k3)+k4)/6
end for

Output:
  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


## PL/I

Runge_Kutta: procedure options (main);           /* 10 March 2014 */
declare (y, dy1, dy2, dy3, dy4) float (18);
declare t fixed decimal (10,1);
declare dt float (18) static initial (0.1);

y = 1;
do t = 0 to 10 by 0.1;
dy1 = dt * ydash(t, y);
dy2 = dt * ydash(t + dt/2, y + dy1/2);
dy3 = dt * ydash(t + dt/2, y + dy2/2);
dy4 = dt * ydash(t + dt,   y + dy3);

if mod(t, 1.0) = 0 then
put skip edit('y(', trim(t), ')=', y, ', error = ', abs(y - (t**2 + 4)**2 / 16 ))
(3 a, column(9), f(16,10), a, f(13,10));
y = y + (dy1 + 2*dy2 + 2*dy3 + dy4)/6;
end;

ydash: procedure (t, y) returns (float(18));
declare (t, y) float (18) nonassignable;
return ( t*sqrt(y) );
end ydash;

end Runge_kutta;
Output:
y(0.0)=     1.0000000000, error =  0.0000000000
y(1.0)=     1.5624998543, error =  0.0000001457
y(2.0)=     3.9999990805, error =  0.0000009195
y(3.0)=    10.5624970904, error =  0.0000029096
y(4.0)=    24.9999937651, error =  0.0000062349
y(5.0)=    52.5624891803, error =  0.0000108197
y(6.0)=    99.9999834054, error =  0.0000165946
y(7.0)=   175.5624764823, error =  0.0000235177
y(8.0)=   288.9999684348, error =  0.0000315652
y(9.0)=   451.5624592768, error =  0.0000407232
y(10.0)=  675.9999490167, error =  0.0000509833


## PowerShell

Works with: PowerShell version 4.0
function Runge-Kutta (${function:F},${function:y}, $y0,$t0, $dt,$tEnd) {
function RK ($tn,$yn)  {
$y1 =$dt*(F -t $tn -y$yn)
$y2 =$dt*(F -t ($tn + (1/2)*$dt) -y ($yn + (1/2)*$y1))
$y3 =$dt*(F -t ($tn + (1/2)*$dt) -y ($yn + (1/2)*$y2))
$y4 =$dt*(F -t ($tn +$dt) -y ($yn +$y3))
$yn + (1/6)*($y1 + 2*$y2 + 2*$y3 + $y4) } function time ($t0, $dt,$tEnd)  {
$end = [MATH]::Floor(($tEnd - $t0)/$dt)
foreach ($_ in 0..$end) { $_*$dt + $t0 } }$time, $yn,$t = (time $t0$dt $tEnd),$y0, 0
foreach ($tn in$time) {
if($t -eq$tn) {
[pscustomobject]@{
t = "$tn" y = "$yn"
error = "$([MATH]::abs($yn - (y $tn)))" }$t += 1
}
$yn = RK$tn $yn } } function F ($t,$y) {$t * [MATH]::Sqrt($y) } function y ($t)  {
(1/16) * [MATH]::Pow($t*$t + 4,2)
}
$y0 = 1$t0 = 0
$dt = 0.1$tEnd = 10
Runge-Kutta  F y $y0$t0  $dt$tEnd


Output:

t                                    y                                    error
-                                    -                                    -----
0                                    1                                    0
1                                    1.56249985427811                     1.45721892108597E-07
2                                    3.9999990805208                      9.19479200778284E-07
3                                    10.5624970904376                     2.90956244874963E-06
4                                    24.9999937650906                     6.23490936391136E-06
5                                    52.5624891803026                     1.08196974153429E-05
6                                    99.9999834054036                     1.65945964170078E-05
7                                    175.562476482271                     2.35177287493116E-05
8                                    288.999968434799                     3.156520142511E-05
9                                    451.56245927684                      4.07231603389846E-05
10                                   675.99994901671                      5.09832902935159E-05


## PureBasic

Translation of: BBC Basic
EnableExplicit
Define.i i
Define.d y=1.0, k1=0.0, k2=0.0, k3=0.0, k4=0.0, t=0.0

If OpenConsole()
For i=0 To 100
t=i/10
If Not i%10
PrintN("y("+RSet(StrF(t,0),2," ")+") ="+RSet(StrF(y,4),9," ")+#TAB$+"Error ="+RSet(StrF(Pow(Pow(t,2)+4,2)/16-y,10),14," ")) EndIf 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+0.1*(k1+2*(k2+k3)+k4)/6 Next Print("Press return to exit...") : Input() EndIf End Output: y( 0) = 1.0000 Error = 0.0000000000 y( 1) = 1.5625 Error = 0.0000001457 y( 2) = 4.0000 Error = 0.0000009195 y( 3) = 10.5625 Error = 0.0000029096 y( 4) = 25.0000 Error = 0.0000062349 y( 5) = 52.5625 Error = 0.0000108197 y( 6) = 100.0000 Error = 0.0000165946 y( 7) = 175.5625 Error = 0.0000235177 y( 8) = 289.0000 Error = 0.0000315652 y( 9) = 451.5625 Error = 0.0000407232 y(10) = 675.9999 Error = 0.0000509833 Press return to exit... ## Python from math import sqrt def rk4(f, x0, y0, x1, n): vx = [0] * (n + 1) vy = [0] * (n + 1) h = (x1 - x0) / float(n) vx[0] = x = x0 vy[0] = y = y0 for i in range(1, n + 1): k1 = h * f(x, y) k2 = h * f(x + 0.5 * h, y + 0.5 * k1) k3 = h * f(x + 0.5 * h, y + 0.5 * k2) k4 = h * f(x + h, y + k3) vx[i] = x = x0 + i * h vy[i] = y = y + (k1 + k2 + k2 + k3 + k3 + k4) / 6 return vx, vy def f(x, y): return x * sqrt(y) vx, vy = rk4(f, 0, 1, 10, 100) for x, y in list(zip(vx, vy))[::10]: print("%4.1f %10.5f %+12.4e" % (x, y, y - (4 + x * x)**2 / 16)) 0.0 1.00000 +0.0000e+00 1.0 1.56250 -1.4572e-07 2.0 4.00000 -9.1948e-07 3.0 10.56250 -2.9096e-06 4.0 24.99999 -6.2349e-06 5.0 52.56249 -1.0820e-05 6.0 99.99998 -1.6595e-05 7.0 175.56248 -2.3518e-05 8.0 288.99997 -3.1565e-05 9.0 451.56246 -4.0723e-05 10.0 675.99995 -5.0983e-05  ## R rk4 <- function(f, x0, y0, x1, n) { vx <- double(n + 1) vy <- double(n + 1) vx[1] <- x <- x0 vy[1] <- y <- y0 h <- (x1 - x0)/n for(i in 1:n) { k1 <- h*f(x, y) k2 <- h*f(x + 0.5*h, y + 0.5*k1) k3 <- h*f(x + 0.5*h, y + 0.5*k2) k4 <- h*f(x + h, y + k3) vx[i + 1] <- x <- x0 + i*h vy[i + 1] <- y <- y + (k1 + k2 + k2 + k3 + k3 + k4)/6 } cbind(vx, vy) } sol <- rk4(function(x, y) x*sqrt(y), 0, 1, 10, 100) cbind(sol, sol[, 2] - (4 + sol[, 1]^2)^2/16)[seq(1, 101, 10), ] vx vy [1,] 0 1.000000 0.000000e+00 [2,] 1 1.562500 -1.457219e-07 [3,] 2 3.999999 -9.194792e-07 [4,] 3 10.562497 -2.909562e-06 [5,] 4 24.999994 -6.234909e-06 [6,] 5 52.562489 -1.081970e-05 [7,] 6 99.999983 -1.659460e-05 [8,] 7 175.562476 -2.351773e-05 [9,] 8 288.999968 -3.156520e-05 [10,] 9 451.562459 -4.072316e-05 [11,] 10 675.999949 -5.098329e-05  ## Racket See Euler method#Racket for implementation of simple general ODE-solver. The Runge-Kutta method (define (RK4 F δt) (λ (t y) (define δy1 (* δt (F t y))) (define δy2 (* δt (F (+ t (* 1/2 δt)) (+ y (* 1/2 δy1))))) (define δy3 (* δt (F (+ t (* 1/2 δt)) (+ y (* 1/2 δy2))))) (define δy4 (* δt (F (+ t δt) (+ y δy1)))) (list (+ t δt) (+ y (* 1/6 (+ δy1 (* 2 δy2) (* 2 δy3) δy4))))))  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) #:x-max (+ x h) #:step (/ h n) #:method method))))  Usage: (define (F t y) (* t (sqrt y))) (define (exact-solution t) (* 1/16 (sqr (+ 4 (sqr t))))) (define numeric-solution (ODE-solve F '(0 1) #:x-max 10 #:step 1 #:method (step-subdivision 10 RK4))) (for ([s numeric-solution]) (match-define (list t y) s) (printf "t=~a\ty=~a\terror=~a\n" t y (- y (exact-solution t))))  Output: t=0 y=1 error=0 t=1 y=1.562499854278108 error=-1.4572189210859676e-07 t=2 y=3.999999080520799 error=-9.194792007782837e-07 t=3 y=10.562497090437551 error=-2.9095624487496252e-06 t=4 y=24.999993765090636 error=-6.234909363911356e-06 t=5 y=52.562489180302585 error=-1.0819697415342944e-05 t=6 y=99.99998340540358 error=-1.659459641700778e-05 t=7 y=175.56247648227125 error=-2.3517728749311573e-05 t=8 y=288.9999684347986 error=-3.156520142510999e-05 t=9 y=451.56245927683966 error=-4.07231603389846e-05 t=10 y=675.9999490167097 error=-5.098329029351589e-05  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)")  ## Raku (formerly Perl 6) Works with: rakudo version 2016.03 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;
}

Output:
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

## REXX

    The Runge─Kutta method is used to solve the following differential equation:

y'(t) =  t2  √ y(t)

The exact solution:   y(t)  =  (t2+4)2 ÷ 16


/*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;                    do m=1  for n-1;    p= m - 1;      y.m= RK4(dx,  x0 + dx*p,  y.p)
end   /*m*/             /*   [↑]  use 4th order Runge─Kutta.   */
w= digits() % 2                                  /*W: width used for displaying numbers.*/
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() end /*i*/ /*└┴┴┴───◄─────── aligns positive #'s. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ fmt: parse arg z; z= right( format(z, w, f), w); hasE= pos('E', z)>0; has.= pos(., z)>0 jus= has. & \hasE; T= 'T'; if jus then z= left( strip( strip(z, T, 0), T, .), w) return translate( right(z, (z>=0) + w + 5*hasE + 2*(jus & (z<0) ) ), 'e', "E") /*──────────────────────────────────────────────────────────────────────────────────────*/ RK4: procedure; parse arg dx,x,y; dxH= dx/2; k1= dx * (x ) * sqrt(y ) k2= dx * (x + dxH) * sqrt(y + k1/2) k3= dx * (x + dxH) * sqrt(y + k2/2) k4= dx * (x + dx ) * sqrt(y + k3 ) return y + (k1 + k2*2 + k3*2 + k4) / 6 /*──────────────────────────────────────────────────────────────────────────────────────*/ 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 do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/ do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/; return g  Programming note: the fmt function is used to align the output with attention paid to the different ways some REXXes format numbers that are in floating point representation. output when using Regina REXX: ════X═════ ══════════Y═══════════ ═══════relative error═══════ 0 1 0 1 1.5624998543 -9.3262010935e-8 2 3.9999990805 -2.2986980019e-7 3 10.5624970904 -2.7546153356e-7 4 24.9999937651 -2.4939637459e-7 5 52.5624891803 -2.0584442174e-7 6 99.9999834054 -1.6594596403e-7 7 175.5624764823 -1.3395644713e-7 8 288.9999684348 -1.0922215040e-7 9 451.5624592768 -9.0182777476e-8 10 675.9999490167 -7.5419068846e-8  output when using PC/REXX, Personal REXX, ROO, or R4 REXX: ════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  ## Ring decimals(8) y = 1.0 for i = 0 to 100 t = i / 10 if t = floor(t) actual = (pow((pow(t,2) + 4),2)) / 16 see "y(" + t + ") = " + y + " error = " + (actual - y) + nl ok 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 next Output: y(0) = 1 error = 0 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  ## Ruby def calc_rk4(f) return ->(t,y,dt){ ->(dy1 ){ ->(dy2 ){ ->(dy3 ){ ->(dy4 ){ ( dy1 + 2*dy2 + 2*dy3 + dy4 ) / 6 }.call( dt * f.call( t + dt , y + dy3 ))}.call( dt * f.call( t + dt/2, y + dy2/2 ))}.call( dt * f.call( t + dt/2, y + dy1/2 ))}.call( dt * f.call( t , y ))} end TIME_MAXIMUM, WHOLE_TOLERANCE = 10.0, 1.0e-5 T_START, Y_START, DT = 0.0, 1.0, 0.10 def my_diff_eqn(t,y) ; t * Math.sqrt(y) ; end def my_solution(t ) ; (t**2 + 4)**2 / 16 ; end def find_error(t,y) ; (y - my_solution(t)).abs ; end def is_whole?(t ) ; (t.round - t).abs < WHOLE_TOLERANCE ; end dy = calc_rk4( ->(t,y){my_diff_eqn(t,y)} ) t, y = T_START, Y_START while t <= TIME_MAXIMUM 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  Output: 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  ## Run BASIC y = 1 while t <= 10 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) if right(using("##.#",t),1) = "0" then  print "y(";using("##",t);") ="; using("####.#######", y);chr$(9);"Error ="; (((t^2 + 4)^2) /16) -y y = y + .1 *(k1 + 2 * (k2 + k3) + k4) / 6 t = t + .1 wend end Output: y( 0) = 1.0000000 Error =0 y( 1) = 1.5624999 Error =1.45721892e-7 y( 2) = 3.9999991 Error =9.19479203e-7 y( 3) = 10.5624971 Error =2.90956246e-6 y( 4) = 24.9999938 Error =6.23490939e-6 y( 5) = 52.5624892 Error =1.08196973e-5 y( 6) = 99.9999834 Error =1.65945961e-5 y( 7) = 175.5624765 Error =2.3517728e-5 y( 8) = 288.9999684 Error =3.15652e-5 y( 9) = 451.5624593 Error =4.07231581e-5 y(10) = 675.9999490 Error =5.09832864e-5  ## Rust This is a translation of the javascript solution with some minor differences. 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; } }  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  ## 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) } } }  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  ## Sidef Translation of: Raku func runge_kutta(yp) { func (t, y, δt) { var a = (δt * yp(t, y)); var b = (δt * yp(t + δt/2, y + a/2)); var c = (δt * yp(t + δt/2, y + b/2)); var d = (δt * yp(t + δt, y + c)); (a + 2*(b + c) + d) / 6; } } define δt = 0.1; var δy = runge_kutta(func(t, y) { t * y.sqrt }); var(t, y) = (0, 1); loop { t.is_int && printf("y(%2d) = %12f ± %e\n", t, y, abs(y - ((t**2 + 4)**2 / 16))); t <= 10 || break; y += δy(t, y, δt); t += δt; }  Output: 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  ## Standard ML fun step y' (tn,yn) dt = let val dy1 = dt * y'(tn,yn) val dy2 = dt * y'(tn + 0.5 * dt, yn + 0.5 * dy1) val dy3 = dt * y'(tn + 0.5 * dt, yn + 0.5 * dy2) val dy4 = dt * y'(tn + dt, yn + dy3) in (tn + dt, yn + (1.0 / 6.0) * (dy1 + 2.0*dy2 + 2.0*dy3 + dy4)) end (* Suggested test case *) fun testy' (t,y) = t * Math.sqrt y fun testy t = (1.0 / 16.0) * Math.pow(Math.pow(t,2.0) + 4.0, 2.0) (* Test-runner that iterates the step function and prints the results. *) fun test t0 y0 dt steps print_freq y y' = let fun loop i (tn,yn) = if i = steps then () else let val (t1,y1) = step y' (tn,yn) dt val y1' = y tn val () = if i mod print_freq = 0 then (print ("Time: " ^ Real.toString tn ^ "\n"); print ("Exact: " ^ Real.toString y1' ^ "\n"); print ("Approx: " ^ Real.toString yn ^ "\n"); print ("Error: " ^ Real.toString (y1' - yn) ^ "\n\n")) else () in loop (i+1) (t1,y1) end in loop 0 (t0,y0) end (* Run the suggested test case *) val () = test 0.0 1.0 0.1 101 10 testy testy'  Output: Time: 0.0 Exact: 1.0 Approx: 1.0 Error: ~1.11022302463E~16 Time: 1.0 Exact: 1.5625 Approx: 1.56249985428 Error: 1.45722452549E~07 Time: 2.0 Exact: 4.0 Approx: 3.99999908052 Error: 9.19479203443E~07 Time: 3.0 Exact: 10.5625 Approx: 10.5624970904 Error: 2.90956245586E~06 Time: 4.0 Exact: 25.0 Approx: 24.9999937651 Error: 6.23490938878E~06 Time: 5.0 Exact: 52.5625 Approx: 52.5624891803 Error: 1.08196973727E~05 Time: 6.0 Exact: 100.0 Approx: 99.9999834054 Error: 1.65945961186E~05 Time: 7.0 Exact: 175.5625 Approx: 175.562476482 Error: 2.35177280956E~05 Time: 8.0 Exact: 289.0 Approx: 288.999968435 Error: 3.15651997767E~05 Time: 9.0 Exact: 451.5625 Approx: 451.562459277 Error: 4.07231581221E~05 Time: 10.0 Exact: 676.0 Approx: 675.999949017 Error: 5.09832866555E~05 ## 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 | +----------------------------------------------+  ## Swift Translation of: C import Foundation func rk4(dx: Double, x: Double, y: Double, f: (Double, Double) -> Double) -> Double { let k1 = dx * f(x, y) let k2 = dx * f(x + dx / 2, y + k1 / 2) let k3 = dx * f(x + dx / 2, y + k2 / 2) let k4 = dx * f(x + dx, y + k3) return y + (k1 + 2 * k2 + 2 * k3 + k4) / 6 } var y = [Double]() var x: Double = 0.0 var y2: Double = 0.0 var x0: Double = 0.0 var x1: Double = 10.0 var dx: Double = 0.1 var i = 0 var n = Int(1 + (x1 - x0) / dx) y.append(1) for i in 1..<n { y.append(rk4(dx, x: x0 + dx * (Double(i) - 1), y: y[i - 1]) { (x: Double, y: Double) -> Double in return x * sqrt(y) }) } print(" x y rel. err.") print("------------------------------") for (var i = 0; i < n; i += 10) { x = x0 + dx * Double(i) y2 = pow(x * x / 4 + 1, 2) print(String(format: "%2g %11.6g %11.5g", x, y[i], y[i]/y2 - 1)) }  Output:  x y rel. err. ------------------------------ 0 1 0 1 1.5625 -9.3262e-08 2 4 -2.2987e-07 3 10.5625 -2.7546e-07 4 25 -2.494e-07 5 52.5625 -2.0584e-07 6 100 -1.6595e-07 7 175.562 -1.3396e-07 8 289 -1.0922e-07 9 451.562 -9.0183e-08 10 676 -7.5419e-08 ## Tcl package require Tcl 8.5 # Hack to bring argument function into expression proc tcl::mathfunc::dy {t y} {upvar 1 dyFn dyFn;$dyFn $t$y}

proc rk4step {dyFn y* t* dt} {
upvar 1 ${y*} y${t*} t
set dy1 [expr {$dt * dy($t,       $y)}] set dy2 [expr {$dt * dy($t+$dt/2, $y+$dy1/2)}]
set dy3 [expr {$dt * dy($t+$dt/2,$y+$dy2/2)}] set dy4 [expr {$dt * dy($t+$dt,   $y+$dy3)}]
set y [expr {$y + ($dy1 + 2*$dy2 + 2*$dy3 + $dy4)/6.0}] set t [expr {$t + $dt}] } proc y {t} {expr {($t**2 + 4)**2 / 16}}
proc δy {t y} {expr {$t * sqrt($y)}}

proc printvals {t y} {
set err [expr {abs($y - [y$t])}]
puts [format "y(%.1f) = %.8f\tError: %.8e" $t$y $err] } set t 0.0 set y 1.0 set dt 0.1 printvals$t $y for {set i 1} {$i <= 101} {incr i} {
rk4step  δy  y t  $dt if {$i%10 == 0} {
printvals $t$y
}
}

Output:
y(0.0) = 1.00000000	Error: 0.00000000e+00
y(1.0) = 1.56249985	Error: 1.45721892e-07
y(2.0) = 3.99999908	Error: 9.19479203e-07
y(3.0) = 10.56249709	Error: 2.90956245e-06
y(4.0) = 24.99999377	Error: 6.23490939e-06
y(5.0) = 52.56248918	Error: 1.08196973e-05
y(6.0) = 99.99998341	Error: 1.65945961e-05
y(7.0) = 175.56247648	Error: 2.35177280e-05
y(8.0) = 288.99996843	Error: 3.15652000e-05
y(9.0) = 451.56245928	Error: 4.07231581e-05
y(10.0) = 675.99994902	Error: 5.09832864e-05

## V (Vlang)

Translation of: Ring
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 } }  Output: 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  ## Wren Translation of: Kotlin Library: Wren-fmt 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)

Output:
  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


## 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.;
]
Output:
  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


## zkl

Translation of: OCaml
fcn yp(t,y) { t * y.sqrt() }
fcn exact(t){ u:=0.25*t*t + 1.0; u*u }

fcn rk4_step([(y,t)],h){
k1:=h * yp(t,y);
k2:=h * yp(t + 0.5*h, y + 0.5*k1);
k3:=h * yp(t + 0.5*h, y + 0.5*k2);
k4:=h * yp(t + h, y + k3);
T(y + (k1+k4)/6.0 + (k2+k3)/3.0, t + h);
}

fcn loop(h,n,[(y,t)]){
if(n % 10 == 1)
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
}
Output:
loop(0.1,1,T(1.0, 0.0))
t = 0.000000,	y = 1.000000,	err = 0
t = 1.000000,	y = 1.562500,	err = 1.45722e-07
t = 2.000000,	y = 3.999999,	err = 9.19479e-07
t = 3.000000,	y = 10.562497,	err = 2.90956e-06
t = 4.000000,	y = 24.999994,	err = 6.23491e-06
t = 5.000000,	y = 52.562489,	err = 1.08197e-05
t = 6.000000,	y = 99.999983,	err = 1.65946e-05
t = 7.000000,	y = 175.562476,	err = 2.35177e-05
t = 8.000000,	y = 288.999968,	err = 3.15652e-05
t = 9.000000,	y = 451.562459,	err = 4.07232e-05
t = 10.000000,	y = 675.999949,	err = 5.09833e-05