Runge-Kutta method: Difference between revisions
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{{
Given the example Differential equation:
:<math>y'(t) = t \times \sqrt {y(t)}</math>
Line 6:
This equation has an exact solution:
:<math>y(t) = \tfrac{1}{16}(t^2 +4)^2</math>
;Task
Demonstrate the commonly used explicit [[wp:Runge–Kutta_methods#Common_fourth-order_Runge.E2.80.93Kutta_method|fourth-order Runge–Kutta method]] to solve the above differential equation.
* Solve the given differential equation over the range <math>t = 0 \ldots 10</math> with a step value of <math>\delta t=0.1</math> (101 total points, the first being given)
* Print the calculated values of <math>y</math> at whole numbered <math>t</math>'s (<math>0.0, 1.0, \ldots 10.0</math>) along with error as compared to the exact solution.
;Method summary
Starting with a given <math>y_n</math> and <math>t_n</math> calculate:
Line 19 ⟶ 23:
:<math>y_{n+1} = y_n + \tfrac{1}{6} (\delta y_1 + 2\delta y_2 + 2\delta y_3 + \delta y_4)</math>
:<math>t_{n+1} = t_n + \delta t\quad</math>
<br><br>
=={{header|11l}}==
{{trans|Python}}
<syntaxhighlight lang="11l">F rk4(f, x0, y0, x1, n)
V vx = [0.0] * (n + 1)
V vy = [0.0] * (n + 1)
V h = (x1 - x0) / Float(n)
V x = x0
V y = y0
vx[0] = x
vy[0] = y
L(i) 1..n
V k1 = h * f(x, y)
V k2 = h * f(x + 0.5 * h, y + 0.5 * k1)
V k3 = h * f(x + 0.5 * h, y + 0.5 * k2)
V k4 = h * f(x + h, y + k3)
vx[i] = x = x0 + i * h
vy[i] = y = y + (k1 + k2 + k2 + k3 + k3 + k4) / 6
R (vx, vy)
F f(Float x, Float y) -> Float
R x * sqrt(y)
V (vx, vy) = rk4(f, 0.0, 1.0, 10.0, 100)
L(x, y) zip(vx, vy)[(0..).step(10)]
print(‘#2.1 #4.5 #2.8’.format(x, y, y - (4 + x * x) ^ 2 / 16))</syntaxhighlight>
{{out}}
<pre>
0.0 1.00000 0.00000000
1.0 1.56250 -1.45721892e-7
2.0 4.00000 -9.194792e-7
3.0 10.56250 -0.00000291
4.0 24.99999 -0.00000623
5.0 52.56249 -0.00001082
6.0 99.99998 -0.00001659
7.0 175.56248 -0.00002352
8.0 288.99997 -0.00003157
9.0 451.56246 -0.00004072
10.0 675.99995 -0.00005098
</pre>
=={{header|Action!}}==
Calculations on a real Atari 8-bit computer take quite long time. It is recommended to use an emulator capable with increasing speed of Atari CPU.
{{libheader|Action! Tool Kit}}
{{libheader|Action! Real Math}}
<syntaxhighlight lang="action!">INCLUDE "D2:PRINTF.ACT" ;from the Action! Tool Kit
INCLUDE "H6:REALMATH.ACT"
DEFINE PTR="CARD"
REAL one,two,four,six
PROC Init()
IntToReal(1,one)
IntToReal(2,two)
IntToReal(4,four)
IntToReal(6,six)
RETURN
PROC Fun=*(REAL POINTER x,y,res)
DEFINE JSR="$20"
DEFINE RTS="$60"
[JSR $00 $00 ;JSR to address set by SetFun
RTS]
PROC SetFun(PTR p)
PTR addr
addr=Fun+1 ;location of address of JSR
PokeC(addr,p)
RETURN
PROC Rate(REAL POINTER x,y,res)
REAL tmp
Sqrt(y,tmp) ;tmp=sqrt(y)
RealMult(x,tmp,res) ;res=x*sqrt(y)
RETURN
PROC RK4(PTR f REAL POINTER dx,x,y,res)
REAL k1,k2,k3,k4,dx2,k12,k22,tmp1,tmp2,tmp3
SetFun(f)
Fun(x,y,tmp1) ;tmp1=f(x,y)
RealMult(dx,tmp1,k1) ;k1=dx*f(x,y)
RealDiv(dx,two,dx2) ;dx2=dx/2
RealDiv(k1,two,k12) ;k12=k1/2
RealAdd(x,dx2,tmp1) ;tmp1=x+dx/2
RealAdd(y,k12,tmp2) ;tmp2=y+k1/2
Fun(tmp1,tmp2,tmp3) ;tmp3=f(x+dx/2,y+k1/2)
RealMult(dx,tmp3,k2) ;k2=dx*f(x+dx/2,y+k1/2)
RealDiv(k2,two,k22) ;k22=k2/2
RealAdd(y,k22,tmp2) ;tmp2=y+k2/2
Fun(tmp1,tmp2,tmp3) ;tmp3=f(x+dx/2,y+k2/2)
RealMult(dx,tmp3,k3) ;k3=dx*f(x+dx/2,y+k2/2)
RealAdd(x,dx,tmp1) ;tmp1=x+dx
RealAdd(y,k3,tmp2) ;tmp2=y+k3
Fun(tmp1,tmp2,tmp3) ;tmp3=f(x+dx,y+k3)
RealMult(dx,tmp3,k4) ;k4=dx*f(x+dx,y+k3)
RealAdd(k2,k3,tmp1) ;tmp1=k2+k3
RealMult(two,tmp1,tmp2) ;tmp2=2*k2+2*k3
RealAdd(k1,tmp2,tmp1) ;tmp3=k1+2*k2+2*k3
RealAdd(tmp1,k4,tmp2) ;tmp2=k1+2*k2+2*k3+k4
RealDiv(tmp2,six,tmp1) ;tmp1=(k1+2*k2+2*k3+k4)/6
RealAdd(y,tmp1,res) ;res=y+(k1+2*k2+2*k3+k4)/6
RETURN
PROC Calc(REAL POINTER x,res)
REAL tmp1,tmp2
RealMult(x,x,tmp1) ;tmp1=x*x
RealDiv(tmp1,four,tmp2) ;tmp2=x*x/4
RealAdd(tmp2,one,tmp1) ;tmp1=x*x/4+1
Power(tmp1,two,res) ;res=(x*x/4+1)^2
RETURN
PROC RelError(REAL POINTER a,b,res)
REAL tmp
RealDiv(a,b,tmp) ;tmp=a/b
RealSub(tmp,one,res) ;res=a/b-1
RETURN
PROC Main()
REAL x0,x1,x,dx,y,y2,err,tmp1,tmp2
CHAR ARRAY s(20)
INT i,n
Put(125) PutE() ;clear the screen
MathInit()
Init()
PrintF("%-2S %-11S %-8S%E","x","y","rel err")
IntToReal(0,x0)
IntToReal(10,x1)
ValR("0.1",dx)
RealSub(x1,x0,tmp1) ;tmp1=x1-x0
RealDiv(tmp1,dx,tmp2) ;tmp2=(x1-x0)/dx
n=RealToInt(tmp2) ;n=(x1-x0)/dx
i=0
IntToReal(1,y)
DO
IntToReal(i,tmp1) ;tmp1=i
RealMult(dx,tmp1,tmp2) ;tmp2=i*dx
RealAdd(x0,tmp2,x) ;x=x0+i*dx
IF i MOD 10=0 THEN
Calc(x,y2)
RelError(y,y2,err)
StrR(x,s) PrintF("%-2S ",s)
StrR(y,s) PrintF("%-11S ",s)
StrR(err,s) PrintF("%-8S%E",s)
FI
i==+1
IF i>n THEN EXIT FI
RK4(rate,dx,x,y,tmp1) ;tmp1=rk4(rate,dx,x0+dx*(i-1),y)
RealAssign(tmp1,y) ;y=rk4(rate,dx,x0+dx*(i-1),y)
OD
RETURN</syntaxhighlight>
{{out}}
[https://gitlab.com/amarok8bit/action-rosetta-code/-/raw/master/images/Runge-Kutta_method.png Screenshot from Atari 8-bit computer]
<pre>
x y rel err
0 1 0
1 1.56249977 -1.3E-07
2 3.99999882 -2.9E-07
3 10.56249647 -2.9E-07
4 24.99999228 -2.9E-07
5 52.56248607 -2.0E-07
6 99.99997763 -2.1E-07
7 175.562459 -1.8E-07
8 288.999935 -1.9E-07
9 451.562406 0
10 675.999869 -1.4E-07
</pre>
=={{header|Ada}}==
<
with Ada.Numerics.Generic_Elementary_Functions;
procedure RungeKutta is
Line 72 ⟶ 261:
Runge (yprime'Access, t_arr, y_arr, dt);
Print (t_arr, y_arr, 10);
end RungeKutta;</
{{out}}
<pre>y(0.0) = 1.00000000 Error: 0.00000E+00
Line 85 ⟶ 274:
y(9.0) = 451.56245928 Error: 4.07232E-05
y(10.0) = 675.99994902 Error: 5.09833E-05</pre>
=={{header|ALGOL 68}}==
<syntaxhighlight lang="algol68">
BEGIN
PROC rk4 = (PROC (REAL, REAL) REAL f, REAL y, x, dx) REAL :
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
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|ALGOL W}}==
{{Trans|ALGOL 68}}
As originally defined, the signature of a procedure parameter could not be specified in Algol W (as here), modern compilers may require parameter specifications for the "f" parameter of rk4.
<syntaxhighlight lang="algolw">begin
real procedure rk4 ( real procedure f ; real value y, x, dx ) ;
begin % Fourth-order Runge-Kutta method %
real dy1, dy2, dy3, dy4;
dy1 := dx * f(x, y);
dy2 := dx * f(x + dx / 2.0, y + dy1 / 2.0);
dy3 := dx * f(x + dx / 2.0, y + dy2 / 2.0);
dy4 := dx * f(x + dx, y + dy3);
y + (dy1 + 2.0 * dy2 + 2.0 * dy3 + dy4) / 6.0
end rk4;
real x0, x1, y0, dx;
integer numPoints;
x0 := 0; x1 := 10; y0 := 1.0; % Boundary conditions. %
dx := 0.1; % Step size. %
numPoints := entier ((x1 - x0) / dx + 0.5); % Add 0.5 for rounding errors. %
begin
real procedure dyByDx ( real value x, y ) ; x * sqrt(y); % Differential equation. %
real array y ( 0 :: numPoints); y(0) := y0; % Grid and starting point. %
for i := 1 until numPoints do y(i) := rk4 (dyByDx, y(i-1), x0 + dx * (i - 1), dx);
write( " x true y calc y relative error" );
for i := 0 step 10 until numPoints do begin
real x, trueY;
x := x0 + dx * i;
trueY := (x * x + 4.0) ** 2 / 16.0;
write( r_format := "A", r_w := 12, r_d := 7, s_w := 3, x, trueY, y( i )
, r_format := "S", r_w := 12, y( i ) / trueY - 1
)
end for_i
end
end.</syntaxhighlight>
{{out}}
<pre>
x true y calc y relative error
0.0000000 1.0000000 1.0000000 0.0000e+000
1.0000000 1.5625000 1.5624998 -9.3262e-008
2.0000000 4.0000000 3.9999990 -2.2986e-007
3.0000000 10.5625000 10.5624971 -2.7546e-007
4.0000000 25.0000000 24.9999937 -2.4939e-007
5.0000000 52.5625000 52.5624891 -2.0584e-007
6.0000000 100.0000000 99.9999834 -1.6594e-007
7.0000000 175.5625000 175.5624764 -1.3395e-007
8.0000000 289.0000000 288.9999684 -1.0922e-007
9.0000000 451.5625000 451.5624592 -9.0182e-008
10.0000000 676.0000000 675.9999490 -7.5419e-008
</pre>
=={{header|APL}}==
<syntaxhighlight lang="apl">
∇RK4[⎕]∇
∇
[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]
∇
</syntaxhighlight>
{{out}}
<pre>
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
. . .
</pre>
=={{header|AWK}}==
<syntaxhighlight lang="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)
}
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|BASIC}}==
==={{header|BASIC256}}===
<syntaxhighlight lang="basic256">y = 1
for i = 0 to 100
t = i / 10
if t = int(t) then
actual = ((t ^ 2 + 4) ^ 2) / 16
print "y("; int(t); ") = "; left(string(y), 13), "Error = "; left(string(actual - y), 13)
end if
k1 = t * sqr(y)
k2 = (t + 0.05) * sqr(y + 0.05 * k1)
k3 = (t + 0.05) * sqr(y + 0.05 * k2)
k4 = (t + 0.10) * sqr(y + 0.10 * k3)
y = y + 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
next i
end</syntaxhighlight>
==={{header|BBC BASIC}}===
<syntaxhighlight lang="bbcbasic"> 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%</syntaxhighlight>
{{out}}
<pre>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
</pre>
==={{header|IS-BASIC}}===
<syntaxhighlight lang="is-basic">100 PROGRAM "Runge.bas"
110 LET Y=1
120 FOR T=0 TO 10 STEP .1
130 IF T=INT(T) THEN PRINT "y(";STR$(T);") =";Y;TAB(21);"Error =";((T^2+4)^2)/16-Y
140 LET K1=T*SQR(Y)
150 LET K2=(T+.05)*SQR(Y+.05*K1)
160 LET K3=(T+.05)*SQR(Y+.05*K2)
170 LET K4=(T+.1)*SQR(Y+.1*K3)
180 LET Y=Y+.1*(K1+2*(K2+K3)+K4)/6
190 NEXT</syntaxhighlight>
==={{header|QBasic}}===
{{works with|QBasic|1.1}}
{{works with|QuickBasic|4.5}}
<syntaxhighlight lang="qbasic">y! = 1
FOR i = 0 TO 100
t = i / 10
IF t = INT(t) THEN
actual! = ((t ^ 2 + 4) ^ 2) / 16
PRINT USING "y(##) = ###.###### Error = "; t; y;
PRINT actual - y
END IF
k1! = t * SQR(y)
k2! = (t + .05) * SQR(y + .05 * k1)
k3! = (t + .05) * SQR(y + .05 * k2)
k4! = (t + .1) * SQR(y + .1 * k3)
y = y + .1 * (k1 + 2 * (k2 + k3) + k4) / 6
NEXT i</syntaxhighlight>
==={{header|True BASIC}}===
{{works with|QBasic}}
<syntaxhighlight lang="qbasic">LET y = 1
FOR i = 0 TO 100
LET t = i / 10
IF t = INT(t) THEN
LET actual = ((t ^ 2 + 4) ^ 2) / 16
PRINT "y("; STR$(t); ") ="; y ; TAB(20); "Error = "; actual - y
END IF
LET k1 = t * SQR(y)
LET k2 = (t + 0.05) * SQR(y + 0.05 * k1)
LET k3 = (t + 0.05) * SQR(y + 0.05 * k2)
LET k4 = (t + 0.10) * SQR(y + 0.10 * k3)
LET Y = Y + 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
NEXT i
END</syntaxhighlight>
=={{header|C}}==
<syntaxhighlight lang="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;
}</syntaxhighlight>
{{out}} (errors are relative)
<pre>
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
</pre>
=={{header|C sharp|C#}}==
<syntaxhighlight lang="csharp">
using System;
namespace RungeKutta
{
class Program
{
static void Main(string[] args)
{
//Incrementers to pass into the known solution
double t = 0.0;
double T = 10.0;
double dt = 0.1;
// Assign the number of elements needed for the arrays
int n = (int)(((T - t) / dt)) + 1;
// Initialize the arrays for the time index 's' and estimates 'y' at each index 'i'
double[] y = new double[n];
double[] s = new double[n];
// RK4 Variables
double dy1;
double dy2;
double dy3;
double dy4;
// RK4 Initializations
int i = 0;
s[i] = 0.0;
y[i] = 1.0;
Console.WriteLine(" ===================================== ");
Console.WriteLine(" Beging 4th Order Runge Kutta Method ");
Console.WriteLine(" ===================================== ");
Console.WriteLine();
Console.WriteLine(" Given the example Differential equation: \n");
Console.WriteLine(" y' = t*sqrt(y) \n");
Console.WriteLine(" With the initial conditions: \n");
Console.WriteLine(" t0 = 0" + ", y(0) = 1.0 \n");
Console.WriteLine(" Whose exact solution is known to be: \n");
Console.WriteLine(" y(t) = 1/16*(t^2 + 4)^2 \n");
Console.WriteLine(" Solve the given equations over the range t = 0...10 with a step value dt = 0.1 \n");
Console.WriteLine(" Print the calculated values of y at whole numbered t's (0.0,1.0,...10.0) along with the error \n");
Console.WriteLine();
Console.WriteLine(" y(t) " +"RK4" + " ".PadRight(18) + "Absolute Error");
Console.WriteLine(" -------------------------------------------------");
Console.WriteLine(" y(0) " + y[i] + " ".PadRight(20) + (y[i] - solution(s[i])));
// Iterate and implement the Rk4 Algorithm
while (i < y.Length - 1)
{
dy1 = dt * equation(s[i], y[i]);
dy2 = dt * equation(s[i] + dt / 2, y[i] + dy1 / 2);
dy3 = dt * equation(s[i] + dt / 2, y[i] + dy2 / 2);
dy4 = dt * equation(s[i] + dt, y[i] + dy3);
s[i + 1] = s[i] + dt;
y[i + 1] = y[i] + (dy1 + 2 * dy2 + 2 * dy3 + dy4) / 6;
double error = Math.Abs(y[i + 1] - solution(s[i + 1]));
double t_rounded = Math.Round(t + dt, 2);
if (t_rounded % 1 == 0)
{
Console.WriteLine(" y(" + t_rounded + ")" + " " + y[i + 1] + " ".PadRight(5) + (error));
}
i++;
t += dt;
};//End Rk4
Console.ReadLine();
}
// Differential Equation
public static double equation(double t, double y)
{
double y_prime;
return y_prime = t*Math.Sqrt(y);
}
// Exact Solution
public static double solution(double t)
{
double actual;
actual = Math.Pow((Math.Pow(t, 2) + 4), 2)/16;
return actual;
}
}
}</syntaxhighlight>
=={{header|C++}}==
Using Lambdas
<syntaxhighlight lang="cpp">/*
* compiled with:
* g++ (Debian 8.3.0-6) 8.3.0
*
* g++ -std=c++14 -o rk4 %
*
*/
# include <iostream>
# include <math.h>
auto rk4(double f(double, double))
{
return [f](double t, double y, double dt) -> double {
double dy1 { dt * f( t , y ) },
dy2 { dt * f( t+dt/2, y+dy1/2 ) },
dy3 { dt * f( t+dt/2, y+dy2/2 ) },
dy4 { dt * f( t+dt , y+dy3 ) };
return ( dy1 + 2*dy2 + 2*dy3 + dy4 ) / 6;
};
}
int main(void)
{
constexpr
double TIME_MAXIMUM { 10.0 },
T_START { 0.0 },
Y_START { 1.0 },
DT { 0.1 },
WHOLE_TOLERANCE { 1e-12 };
auto dy = rk4( [](double t, double y) -> double { return t*sqrt(y); } ) ;
for (
auto y { Y_START }, t { T_START };
t <= TIME_MAXIMUM;
y += dy(t,y,DT), t += DT
)
if (ceilf(t)-t < WHOLE_TOLERANCE)
printf("y(%4.1f)\t=%12.6f \t error: %12.6e\n", t, y, std::fabs(y - pow(t*t+4,2)/16));
return 0;
}</syntaxhighlight>
=={{header|Common Lisp}}==
<syntaxhighlight lang="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))</syntaxhighlight>
=={{header|Crystal}}==
{{trans|Run Basic and Ruby output}}
<syntaxhighlight lang="ruby">y, t = 1, 0
while t <= 10
k1 = t * Math.sqrt(y)
k2 = (t + 0.05) * Math.sqrt(y + 0.05 * k1)
k3 = (t + 0.05) * Math.sqrt(y + 0.05 * k2)
k4 = (t + 0.1) * Math.sqrt(y + 0.1 * k3)
printf("y(%4.1f)\t= %12.6f \t error: %12.6e\n", t, y, (((t**2 + 4)**2 / 16) - y )) if (t.round - t).abs < 1.0e-5
y += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
t += 0.1
end</syntaxhighlight>
{{out}}
<pre>
y( 0.0) = 1.000000 error: 0.000000e+00
y( 1.0) = 1.562500 error: 1.457219e-07
y( 2.0) = 3.999999 error: 9.194792e-07
y( 3.0) = 10.562497 error: 2.909562e-06
y( 4.0) = 24.999994 error: 6.234909e-06
y( 5.0) = 52.562489 error: 1.081970e-05
y( 6.0) = 99.999983 error: 1.659460e-05
y( 7.0) = 175.562476 error: 2.351773e-05
y( 8.0) = 288.999968 error: 3.156520e-05
y( 9.0) = 451.562459 error: 4.072316e-05
y(10.0) = 675.999949 error: 5.098329e-05
</pre>
=={{header|D}}==
{{trans|Ada}}
<
alias
alias FPs = Typedef!(FP[101])
void runge(in FP function(in FP, in FP)
ref FPs t, ref FPs y, in FP dt) pure nothrow @safe @nogc {
foreach (immutable n; 0 .. t.length - 1) {
immutable FP
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;
Line 105 ⟶ 832:
}
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);
Line 116 ⟶ 843:
t_arr[0] = 0.0;
y_arr[0] = 1.0;
runge((t, y) => t *
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]));
}</
{{out}}
<pre>y(0.0) = 1.00000000 Error: 0
Line 136 ⟶ 863:
y(9.0) = 451.56245928 Error: 4.07232e-05
y(10.0) = 675.99994902 Error: 5.09833e-05</pre>
=={{header|Dart}}==
<syntaxhighlight lang="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;
}
}</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|EasyLang}}==
{{trans|BASIC256}}
<syntaxhighlight>
numfmt 6 0
y = 1
for i = 0 to 100
t = i / 10
if t = floor t
h = t * t + 4
actual = h * h / 16
print "y(" & t & ") = " & y & " Error = " & actual - y
.
k1 = t * sqrt y
k2 = (t + 0.05) * sqrt (y + 0.05 * k1)
k3 = (t + 0.05) * sqrt (y + 0.05 * k2)
k4 = (t + 0.10) * sqrt (y + 0.10 * k3)
y += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
.
</syntaxhighlight>
=={{header|EDSAC order code}}==
The EDSAC subroutine library had two Runge-Kutta subroutines: G1 for 35-bit values and G2 for 17-bit values. A demo of G1 is given here. Setting up the parameters is rather complicated, but after that it's just a matter of calling G1 once for every step in the Runge-Kutta process.
Since EDSAC real numbers are restricted to -1 <= x < 1, the values in the Rosetta Code task have to be scaled down. For comparison with other languages it's convenient to divide the y values by 1000. With 100 steps, a convenient time interval is 1/128.
G1 can solve equations in several variables, say y_1, ..., y_n. The user must provide an auxiliary subroutine which calculates dy_1/dt, ..., dy_n/dt from y_1, ..., y_n. If the derivatives also depend on t (as in the Rosetta Code task) it's necessary to add a dummy y variable which is identical with t.
<syntaxhighlight lang="edsac">
[Demo of EDSAC library subroutine G1: Runge-Kutta solution of differential equations.
Full description is in Wilkes, Wheeler & Gill, 1951 edn, pages 32-34, 86-87, 132-134.
Before using G1, we need to fix n, m, a, b, c, d, as defined in WWG pages 86-87:
n = number of equations (2 for the Rosetta Code example).
2^m = multiplier for the hy', as large as possible without causing numeric overflow;
with the scaling chosen here, m = 5.
Variables y are stored in n consecutive long locations, the last of which is aD.
Scaled derivatives (2^m)hy' in n consecutive long locations, the last of which is bD.
G1 uses working variables in n consecutive long locations, the last of which is cD.
d = address of user-supplied auxiliary subroutine, which calculates the (2^m)hy'.
For convenience, keep G1 and its storage together. Start at (say) 400 and place:
variables y at 400D, 402D;
scaled derivatives at 404D, 406D;
workspace for G1 at 408D, 410D;
G1 itself at 412.
If the base address is placed in location 51 at load time, all the above
addresses can be accessed via the G parameter:]
T 51 K
P 400 F
[Now set up the 6 preset parameters specified in WWG:]
T 45 K
P 2#G [H parameter: P a D]
P 4 F [N parameter: P 2n F]
P 4 F [M parameter: P (b-a) F, or V (2048-a+b) F if a > b]
P 4 F [& parameter: P (c-b) F, or V (2048-b+c) F if b > c]
P 8 F [L parameter: P 2^(m-2) F]
P 300 F [X parameter: P d F]
[For other addresses in the program we can optionally use some more parameters:]
T 52 K
P 120 F [A parameter: main routine]
P 56 F [B parameter: print subroutine P1 from EDSAC library]
P 350 F [C parameter: constants for Rosetta code example]
P 78 F [V parameter: square root subroutine]
[Library subroutine to read constants; runs at load time and is then overwritten.
R5, for decimal fractions, seems to be unavailable (lost?), so the values are
here read in as 35-bit integers (i.e. times 2^34) by R2.
Values are: 0.001, initial value of y
(2^23)/(10^7) and 25/(2^10) for use in calculations
0.5/(10^9) for rounding to 9 d.p. (print routine P1 doesn't do this)]
GKT20FVDL8FA40DUDTFI40FA40FS39FG@S2FG23FA5@T5@E4@E13Z
T#C
17179869F14411518808F419430400F9#
TZ
[Library subroutine M3; prints header at load time and is then overwritten.]
PFGKIFAFRDLFUFOFE@A6FG@E8FEZPF
*SCALED!FOR!EDSAC@&!!TIME!!!!!!!!!Y!VIA!RK!!!!!Y!DIRECT@&
....PK [end text with some blank tape]
[Runge-Kutta: auxiliary subroutine to calculate (2^m)*h*(dy1/dt) and (2^m)*h*(dy2/dt)
from y1, y2, where y1 is the function y in Rosetta Code (but scaled) and y2 = t.
For the Rosetta code example we're using m = 5, h = 2^(-7)]
E25K TX GK
A3F T20@ [set up return as usual]
H2#G V2#G TD [acc := t^2, temp store in 0D]
H#G VD LD YF TD [y1 times t^2, shift left, round, temp store in 0D]
H2#C VD YF T4D [times (2^23)/(10^7), round, to 4D for square root]
[14] A14@ GV A4D T4#G [call square root, result in 4D, copy to (2^m)hy']
A21@ T6#G [1/4, i.e. (2^m)h with m and h as above, to (2^m)ht']
[20] ZF [overwritten by jump back to caller]
[21] RF [constant 1/4]
[Main routine, with two subroutines in the same address block as the main routine.]
E25K TA GK
[0] #F [figures shift on teleprinter]
[1] MF [decimal point (in figures mode)]
[2] !F @F &F [space, carriage return, line feed,]
[5] K4096F [null char]
[6] P100F [constant: nr of Runge-Kutta steps (in address field)]
[7] PF [negative count of Runge-Kutta steps]
[8] P10F [constant: number of steps between printed values]
[9] PF [negative count of steps between printed values]
[Enter with acc = 0]
[10] O@ [set teleprinter to figures]
S6@ T7@ [init negative count of R-K steps]
S8@ T9@ [init negative count of print steps]
[Before using library subroutine G1, clear its working registers (WWG page 33)]
T8#G T10#G
[Set up initial values of y1 and y2 (where y2 = t)]
A#C T#G [load 0.001 from constants section, store in y1]
T2#G [y2 = t = 0]
[20] A20@ G40@ [call subroutine to print initial values]
[Loop round Runge-Kutta steps]
[22] TF A23@ G12G [clear accumulator, call G1 for Runge-Kutta step]
A9@ A2F U9@ [update negative print count]
G33@ [skip printing if not reached 0]
S8@ T9@ [reset negative print count]
A31@ G40@ [call subroutine to print values]
[33] TF [clear accumulator]
A7@ A2F U7@ [increment negative count of Runge-Kutta steps]
G22@ [loop till count = 0]
O5@ ZF [flush teleprinter buffer; stop]
[Subroutine to print y1 as calculated (1) by Runge-Kutta (2) direct from formula]
[40] A3F T71@ [set up return as usual]
A2#G TD [latest t (= y2) from Runge-Kutta, to 0D for printing]
[44] A44@ G72@ [call subroutine to print t]
O2@ O2@ [followed by 2 spaces]
A#G TD [latest y1 from Runge-Kutta, to 0D for printing]
[50] A50@ G72@ [call subroutine to print y1]
O2@ O2@ [followed by 2 spaces]
A 4#C [load constant 25/(2^10)]
H2#G V2#G TD [add t^2, temp store result in 0D]
HD VD LD YF TD [square, shift 1 left, round, result to 0D]
H2#C VD YF TD [times (2^23)/(10^7), round, to 0D for printing]
[67] A67@ G72@ [call subroutine to print y]
O3@ O4@ [print CR, LF]
[71] ZF [overwritten by jump back to caller]
[Second-level subroutine to print number in 0D to 9 decimal places]
[72] A3F T82@ [set up return as usual]
AD A6#C TD [load number, add decimal rounding, to 0D for printing]
O81@ O1@ [print '0.' since P1 doesn't do so]
A79@ GB [call library subroutine P1 for printing]
[81] P9F [parameter for P1, 9 decimals]
[82] ZF [overwritten by jump back to caller]
[Library subroutine G1 for Runge-Kutta process. 66 locations, even address.]
E25K T12G
GKT4#ZH682DT6#ZPNT12#Z!1405DT14#ZTHT16#ZT2HTZA3FT61@A31@G63@&FT6ZPN
T8ZMMO&H4@A20@E23@T14ZAHT16ZA2HT18ZH12#@S12#@T12#@E28@H4#@T4DUFS38@
A25@T38@S6#@A16#@U46#@A8@U37@A9@U55@A24@T39@ZFR1057#@ZFYFU6DV6DRLYF
UDZFZFADLDADLLS6DN4DYFZFA46#@S14#@G29@A65@S11@ZFA35@U65@GXZF
[Replacement for library routine S2 (square root). 38 locations, even address.
Advantages: More accurate for small values of the argument.
Calculates sqrt(0) without going into an infinite loop.
Disadvantages: Longer and slower than S2 (calculates one bit at a time).]
E25K TV
GKA3FT31@A4DG32@A33@T36#@T4DA33@RDU34#@RDS4DS33@A36#@G22@T36#@A4DS34#@
T4DA36#@A33@G25@TFA36#@S33@A36#@T36#@A34#@RDYFG9@ZFZFK4096FPFPFPFPF
[Library subroutine P1 - print a single positive number. 21 locations.
Prints number in 0D to n places of decimals, where
n is specified by 'P n F' pseudo-order after subroutine call.]
E25K TB
GKA18@U17@S20@T5@H19@PFT5@VDUFOFFFSFL4FTDA5@A2FG6@EFU3FJFM1F
[Define entry point in main routine]
E25K TA GK
E10Z PF [enter at relative address 10 with accumulator = 0]
</syntaxhighlight>
{{out}}
<pre>
SCALED FOR EDSAC
TIME Y VIA RK Y DIRECT
0.000000000 0.001000000 0.001000000
0.078125000 0.001562499 0.001562500
0.156250000 0.003999998 0.004000000
0.234375000 0.010562495 0.010562500
0.312500000 0.024999992 0.025000000
0.390625000 0.052562487 0.052562500
0.468750000 0.099999981 0.100000000
0.546875000 0.175562474 0.175562500
0.625000000 0.288999965 0.289000000
0.703125000 0.451562456 0.451562500
0.781250000 0.675999945 0.676000000
</pre>
=={{header|ERRE}}==
<syntaxhighlight lang="erre">
PROGRAM RUNGE_KUTTA
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</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|Excel}}==
<syntaxhighlight lang="Excel">
//Worksheet formula to manage looping
=LET(
T₊, SEQUENCE(11, 1, 0, 1),
T, DROP(T₊, -1),
τ, SEQUENCE(1 / δt, 1, 0, δt),
calculated, SCAN(1, T, LAMBDA(y₀, t, REDUCE(y₀, t + τ, RungaKutta4λ(Dλ)))),
calcs, VSTACK(1, calculated),
exact, f(T₊),
HSTACK(T₊, calcs, exact, (exact - calcs) / exact)
)
//Lambda function passed to RungaKutta4λ to evaluate derivatives
Dλ(y,t)
= LAMBDA(y,t, t * SQRT(y))
//Curried Lambda function with derivative function D and y, t as parameters
RungaKutta4λ(Dλ)
= LAMBDA(D,
LAMBDA(yᵣ, tᵣ,
LET(
δy₁, δt * D(yᵣ, tᵣ),
δy₂, δt * D(yᵣ + δy₁ / 2, tᵣ + δt / 2),
δy₃, δt * D(yᵣ + δy₂ / 2, tᵣ + δt / 2),
δy₄, δt * D(yᵣ + δy₃, tᵣ + δt),
yᵣ₊₁, yᵣ + (δy₁ + 2 * δy₂ + 2 * δy₃ + δy₄) / 6,
yᵣ₊₁
)
)
)
//Lambda function returning the exact solution
f(t)
= LAMBDA(t, (1/16) * (t^2 + 4)^2 )
</syntaxhighlight>
{{out}}
<pre>
Time Calculated Exact Rel Error
0.00 1.000000 1.000000 0.00E+00
1.00 1.562500 1.562500 9.33E-08
2.00 3.999999 4.000000 2.30E-07
3.00 10.562497 10.562500 2.75E-07
4.00 24.999994 25.000000 2.49E-07
5.00 52.562489 52.562500 2.06E-07
6.00 99.999983 100.000000 1.66E-07
7.00 175.562476 175.562500 1.34E-07
8.00 288.999968 289.000000 1.09E-07
9.00 451.562459 451.562500 9.02E-08
10.00 675.999949 676.000000 7.54E-08
</pre>
=={{header|F_Sharp|F#}}==
{{works with|F# interactive (fsi.exe)}}
<syntaxhighlight lang="fsharp">
open System
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) )</syntaxhighlight>
{{out}}
<pre>
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)
</pre>
=={{header|Fortran}}==
<syntaxhighlight lang="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</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|FreeBASIC}}==
{{trans|BBC BASIC}}
<syntaxhighlight lang="freebasic">' 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</syntaxhighlight>
{{out}}
<pre>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</pre>
=={{header|FutureBasic}}==
<syntaxhighlight lang="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</syntaxhighlight>
Output:
<pre>
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
</pre>
=={{header|Go}}==
{{works with|Go1}}
<
import (
Line 195 ⟶ 1,430:
func printErr(t, y float64) {
fmt.Printf("y(%.1f) = %f Error: %e\n", t, y, math.Abs(actual(t)-y))
}</
{{out}}
<pre>
Line 210 ⟶ 1,445:
y(10.0) = 675.999949 Error: 5.098329e-05
</pre>
=={{header|Groovy}}==
<syntaxhighlight lang="groovy">
class Runge_Kutta{
static void main(String[] args){
def y=1.0,t=0.0,counter=0;
def dy1,dy2,dy3,dy4;
def real;
while(t<=10)
{if(counter%10==0)
{real=(t*t+4)*(t*t+4)/16;
println("y("+t+")="+ y+ " Error:"+ (real-y));
}
dy1=dy(dery(y,t));
dy2=dy(dery(y+dy1/2,t+0.05));
dy3=dy(dery(y+dy2/2,t+0.05));
dy4=dy(dery(y+dy3,t+0.1));
y=y+(dy1+2*dy2+2*dy3+dy4)/6;
t=t+0.1;
counter++;
}
}
static def dery(def y,def t){return t*(Math.sqrt(y));}
static def dy(def x){return x*0.1;}
}
</syntaxhighlight>
{{out}}
<pre>
y(0.0)=1.0 Error:0.0000
y(1.0)=1.562499854278108 Error:1.4572189210859676E-7
y(2.0)=3.999999080520799 Error:9.194792007782837E-7
y(3.0)=10.562497090437551 Error:2.9095624487496252E-6
y(4.0)=24.999993765090636 Error:6.234909363911356E-6
y(5.0)=52.562489180302585 Error:1.0819697415342944E-5
y(6.0)=99.99998340540358 Error:1.659459641700778E-5
y(7.0)=175.56247648227125 Error:2.3517728749311573E-5
y(8.0)=288.9999684347986 Error:3.156520142510999E-5
y(9.0)=451.56245927683966 Error:4.07231603389846E-5
y(10.0)=675.9999490167097 Error:5.098329029351589E-5
</pre>
=={{header|Hare}}==
<syntaxhighlight lang="hare">use fmt;
use math;
export fn main() void = {
rk4_driver(&f, 0.0, 10.0, 1.0, 0.1);
};
fn rk4_driver(func: *fn(_: f64, _: f64) f64, t_init: f64, t_final: f64, y_init: f64, h: f64) void = {
let n = ((t_final - t_init) / h): int;
let tn: f64 = t_init;
let yn: f64 = y_init;
let i: int = 1;
fmt::printfln("{: 2} {: 18} {: 21}", "t", "y(t)", "absolute error")!;
fmt::printfln("{: 2} {: 18} {: 21}", tn, yn, math::absf64(exact(tn) - yn))!;
for (i <= n; i += 1) {
yn = rk4(func, tn, yn, h);
tn = t_init + (i: f64)*h;
if (i % 10 == 0) {
fmt::printfln("{: 2} {: 18} {: 21}\t", tn, yn, math::absf64(exact(tn) - yn))!;
};
};
};
fn rk4(func: *fn(_: f64, _: f64) f64, t: f64, y: f64, h: f64) f64 = {
const k1 = func(t, y);
const k2 = func(t + 0.5*h, y + 0.5*h*k1);
const k3 = func(t + 0.5*h, y + 0.5*h*k2);
const k4 = func(t + h, y + h*k3);
return y + h/6.0 * (k1 + 2.0*k2 + 2.0*k3 + k4);
};
fn f(t: f64, y: f64) f64 = {
return t * math::sqrtf64(y);
};
fn exact(t: f64) f64 = {
return 1.0/16.0 * math::powf64(t*t + 4.0, 2.0);
};</syntaxhighlight>
{{out}}
<pre>
t y(t) absolute error
0 1 0
1 1.562499854278108 1.4572189210859676e-7
2 3.9999990805207997 9.194792003341945e-7
3 10.56249709043755 2.909562450525982e-6
4 24.999993765090633 6.23490936746407e-6
5 52.56248918030258 1.0819697422448371e-5
6 99.99998340540358 1.659459641700778e-5
7 175.56247648227125 2.3517728749311573e-5
8 288.9999684347985 3.156520148195341e-5
9 451.5624592768396 4.072316039582802e-5
10 675.9999490167097 5.098329029351589e-5
</pre>
=={{header|Haskell}}==
Using GHC 7.4.1.
<syntaxhighlight lang="haskell">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)</syntaxhighlight>
Example executed in GHCi:
<syntaxhighlight lang="haskell">*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)</syntaxhighlight>
(See [[Euler method#Haskell]] for implementation of simple general ODE-solver)
Or, disaggregated a little, and expressed in terms of a single scanl:
<syntaxhighlight lang="haskell">rk4 :: Double -> Double -> Double -> Double
rk4 y x dx =
let f x y = x * sqrt y
k1 = dx * f x y
k2 = dx * f (x + dx / 2.0) (y + k1 / 2.0)
k3 = dx * f (x + dx / 2.0) (y + k2 / 2.0)
k4 = dx * f (x + dx) (y + k3)
in y + (k1 + 2.0 * k2 + 2.0 * k3 + k4) / 6.0
actual :: Double -> Double
actual x = (1 / 16) * (x * x + 4) * (x * x + 4)
step :: Double
step = 0.1
ixs :: [Int]
ixs = [0 .. 100]
xys :: [(Double, Double)]
xys =
scanl
(\(x, y) _ -> (((x * 10) + (step * 10)) / 10, rk4 y x step))
(0.0, 1.0)
ixs
samples :: [(Double, Double, Double)]
samples =
zip ixs xys >>=
(\(i, (x, y)) ->
[ (x, y, actual x - y)
| 0 == mod i 10 ])
main :: IO ()
main =
(putStrLn . unlines) $
(\(x, y, v) ->
unwords
[ "y" ++ justifyRight 3 ' ' ('(' : show (round x)) ++ ") = "
, justifyLeft 19 ' ' (show y)
, '±' : show v
]) <$>
samples
where
justifyLeft n c s = take n (s ++ replicate n c)
justifyRight n c s = drop (length s) (replicate n c ++ s)</syntaxhighlight>
{{Out}}
<pre>y (0) = 1.0 ±0.0
y (1) = 1.562499854278108 ±1.4572189210859676e-7
y (2) = 3.999999080520799 ±9.194792007782837e-7
y (3) = 10.562497090437551 ±2.9095624487496252e-6
y (4) = 24.999993765090636 ±6.234909363911356e-6
y (5) = 52.562489180302585 ±1.0819697415342944e-5
y (6) = 99.99998340540358 ±1.659459641700778e-5
y (7) = 175.56247648227125 ±2.3517728749311573e-5
y (8) = 288.9999684347986 ±3.156520142510999e-5
y (9) = 451.56245927683966 ±4.07231603389846e-5
y(10) = 675.9999490167097 ±5.098329029351589e-5</pre>
=={{header|J}}==
'''Solution:'''
<
NB. y is: y(ta) , ta , tb , tstep
NB. u is: function to solve
NB. eg: fyp rk4 1 0 10 0.1
rk4=:
'Y0 a b h'=. 4{. y
T=. a + i.@>:&.(%&h) b - a
Line 229 ⟶ 1,671:
end.
T ,. Y
)</
'''Example:'''
<
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
0 1 0
1 1.5625 _1.45722e_7
Line 247 ⟶ 1,689:
8 289 _3.15652e_5
9 451.562 _4.07232e_5
10 676 _5.09833e_5</
'''Alternative solution:'''
The following solution replaces the for loop as well as the calculation of the increments (ks) with an accumulating suffix.
<syntaxhighlight lang="j">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
)</syntaxhighlight>
Use:
report_err report_whole fyp rk4 1 0 10 0.1
=={{header|Java}}==
Translation of [[Runge-Kutta_method#Ada|Ada]] via [[Runge-Kutta_method#D|D]]
{{works with|Java|8}}
<syntaxhighlight lang="java">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]));
}
}</syntaxhighlight>
<pre>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</pre>
=={{header|JavaScript}}==
===ES5===
<syntaxhighlight lang="javascript">
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;
}
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
===ES6===
<syntaxhighlight lang="javascript">(() => {
'use strict';
// rk4 :: (Double -> Double -> Double) ->
// Double -> Double -> Double -> Double
const rk4 = f => (y, x, dx) => {
const
k1 = dx * f(x, y),
k2 = dx * f(x + dx / 2.0, y + k1 / 2.0),
k3 = dx * f(x + dx / 2.0, y + k2 / 2.0),
k4 = dx * f(x + dx, y + k3);
return y + (k1 + 2.0 * k2 + 2.0 * k3 + k4) / 6.0;
};
// rk :: Double -> Double -> Double -> Double
const rk = rk4((x, y) => x * Math.sqrt(y));
// actual :: Double -> Double
const actual = x => (1 / 16) * ((x * x) + 4) * ((x * x) + 4);
// TEST -------------------------------------------------
// main :: IO ()
const main = () => {
const
step = 0.1,
ixs = enumFromTo(0, 100),
xys = scanl(
xy => Tuple(
((xy[0] * 10) + (step * 10)) / 10, rk(xy[1], xy[0], step)
),
Tuple(0.0, 1.0),
ixs
);
// samples :: [(Double, Double, Double)]
const samples = concatMap(
tpl => 0 === tpl[0] % 10 ? (() => {
const [x, y] = Array.from(tpl[1]);
return [TupleN(x, y, actual(x) - y)];
})() : [],
zip(ixs, xys)
);
console.log(
unlines(map(
tpl => {
const [x, y, v] = Array.from(tpl),
[sn, sm] = splitOn('.', y.toString());
return unwords([
'y' + justifyRight(3, ' ', '(' + Math.round(x).toString()) +
') =',
justifyRight(3, ' ', sn) + '.' + justifyLeft(15, ' ', sm || '0'),
'± ' + v.toExponential()
]);
},
samples
))
);
};
// GENERIC FUNCTIONS ----------------------------
// Tuple (,) :: a -> b -> (a, b)
const Tuple = (a, b) => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});
// TupleN :: a -> b ... -> (a, b ... )
function TupleN() {
const
args = Array.from(arguments),
lng = args.length;
return lng > 1 ? Object.assign(
args.reduce((a, x, i) => Object.assign(a, {
[i]: x
}), {
type: 'Tuple' + (2 < lng ? lng.toString() : ''),
length: lng
})
) : args[0];
};
// concatMap :: (a -> [b]) -> [a] -> [b]
const concatMap = (f, xs) =>
xs.reduce((a, x) => a.concat(f(x)), []);
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = (m, n) =>
Array.from({
length: 1 + n - m
}, (_, i) => m + i)
// justifyLeft :: Int -> Char -> String -> String
const justifyLeft = (n, cFiller, s) =>
n > s.length ? (
s.padEnd(n, cFiller)
) : s;
// justifyRight :: Int -> Char -> String -> String
const justifyRight = (n, cFiller, s) =>
n > s.length ? (
s.padStart(n, cFiller)
) : s;
// Returns Infinity over objects without finite length
// this enables zip and zipWith to choose the shorter
// argument when one is non-finite, like cycle, repeat etc
// length :: [a] -> Int
const length = xs => xs.length || Infinity;
// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) => xs.map(f);
// scanl :: (b -> a -> b) -> b -> [a] -> [b]
const scanl = (f, startValue, xs) =>
xs.reduce((a, x) => {
const v = f(a[0], x);
return Tuple(v, a[1].concat(v));
}, Tuple(startValue, [startValue]))[1];
// splitOn :: String -> String -> [String]
const splitOn = (pat, src) => src.split(pat);
// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = (n, xs) =>
xs.constructor.constructor.name !== 'GeneratorFunction' ? (
xs.slice(0, n)
) : [].concat.apply([], Array.from({
length: n
}, () => {
const x = xs.next();
return x.done ? [] : [x.value];
}));
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// unwords :: [String] -> String
const unwords = xs => xs.join(' ');
// Use of `take` and `length` here allows for zipping with non-finite
// lists - i.e. generators like cycle, repeat, iterate.
// zip :: [a] -> [b] -> [(a, b)]
const zip = (xs, ys) => {
const lng = Math.min(length(xs), length(ys));
return Infinity !== lng ? (() => {
const bs = take(lng, ys);
return take(lng, xs).map((x, i) => Tuple(x, bs[i]));
})() : zipGen(xs, ys);
};
// MAIN ---
return main();
})();</syntaxhighlight>
{{Out}}
<pre>y (0) = 1.0 ± 0e+0
y (1) = 1.562499854278108 ± 1.4572189210859676e-7
y (2) = 3.999999080520799 ± 9.194792007782837e-7
y (3) = 10.562497090437551 ± 2.9095624487496252e-6
y (4) = 24.999993765090636 ± 6.234909363911356e-6
y (5) = 52.562489180302585 ± 1.0819697415342944e-5
y (6) = 99.99998340540358 ± 1.659459641700778e-5
y (7) = 175.56247648227125 ± 2.3517728749311573e-5
y (8) = 288.9999684347986 ± 3.156520142510999e-5
y (9) = 451.56245927683966 ± 4.07231603389846e-5
y(10) = 675.9999490167097 ± 5.098329029351589e-5</pre>
=={{header|jq}}==
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:
<syntaxhighlight lang="jq">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;</syntaxhighlight>
===The Example Differential Equation and its Exact Solution===
<syntaxhighlight lang="jq"># 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;</syntaxhighlight>
===dy/dt===
The first solution presented here uses the terminology and style of the Raku version.
'''Generic filters:'''
<syntaxhighlight lang="jq"># 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;</syntaxhighlight>
'''dy(f)'''
<syntaxhighlight lang="jq">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);</syntaxhighlight>
''' Example''':
<syntaxhighlight lang="jq"># 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) - $y | abs)"
else empty
end</syntaxhighlight>
{{out}}
<syntaxhighlight lang="sh">$ 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</syntaxhighlight>
===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.
<syntaxhighlight lang="jq"># 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-$t)] | ypStep)] # adjust step to integer time
)
| printErr # print results
;
# main(t0; y0; tFinal; dtPrint)
main(0; 1; 10; 1)</syntaxhighlight>
{{out}}
<syntaxhighlight lang="sh">$ 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</syntaxhighlight>
=={{header|Julia}}==
{{works with|Julia|0.6}}
=== Using lambda expressions ===
{{trans|Python}}
<syntaxhighlight lang="julia">f(x, y) = x * sqrt(y)
theoric(t) = (t ^ 2 + 4.0) ^ 2 / 16.0
rk4(f) = (t, y, δt) -> # 1st (result) lambda
((δy1) -> # 2nd lambda
((δy2) -> # 3rd lambda
((δy3) -> # 4th lambda
((δy4) -> ( δy1 + 2δy2 + 2δy3 + δy4 ) / 6 # 5th and deepest lambda: calc y_{n+1}
)(δt * f(t + δt, y + δy3)) # calc δy₄
)(δt * f(t + δt / 2, y + δy2 / 2)) # calc δy₃
)(δt * f(t + δt / 2, y + δy1 / 2)) # calc δy₂
)(δt * f(t, y)) # calc δy₁
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</syntaxhighlight>
{{out}}
<pre>y( 0.0) = 1.000000 error: 0.000000e+00
y( 1.0) = 1.562500 error: 1.457219e-07
y( 2.0) = 3.999999 error: 9.194792e-07
y( 3.0) = 10.562497 error: 2.909562e-06
y( 4.0) = 24.999994 error: 6.234909e-06
y( 5.0) = 52.562489 error: 1.081970e-05
y( 6.0) = 99.999983 error: 1.659460e-05
y( 7.0) = 175.562476 error: 2.351773e-05
y( 8.0) = 288.999968 error: 3.156520e-05
y( 9.0) = 451.562459 error: 4.072316e-05
y(10.0) = 675.999949 error: 5.098329e-05</pre>
=== Alternative version ===
{{trans|Python}}
<syntaxhighlight lang="julia">function rk4(f::Function, x₀::Float64, y₀::Float64, x₁::Float64, n)
vx = Vector{Float64}(undef, n + 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</syntaxhighlight>
=={{header|Kotlin}}==
<syntaxhighlight lang="scala">// version 1.1.2
typealias Y = (Double) -> Double
typealias Yd = (Double, Double) -> Double
fun rungeKutta4(t0: Double, tz: Double, dt: Double, y: Y, yd: Yd) {
var tn = t0
var yn = y(tn)
val z = ((tz - t0) / dt).toInt()
for (i in 0..z) {
if (i % 10 == 0) {
val exact = y(tn)
val error = yn - exact
println("%4.1f %10f %10f %9f".format(tn, yn, exact, error))
}
if (i == z) break
val dy1 = dt * yd(tn, yn)
val dy2 = dt * yd(tn + 0.5 * dt, yn + 0.5 * dy1)
val dy3 = dt * yd(tn + 0.5 * dt, yn + 0.5 * dy2)
val dy4 = dt * yd(tn + dt, yn + dy3)
yn += (dy1 + 2.0 * dy2 + 2.0 * dy3 + dy4) / 6.0
tn += dt
}
}
fun main(args: Array<String>) {
println(" T RK4 Exact Error")
println("---- ---------- ---------- ---------")
val y = fun(t: Double): Double {
val x = t * t + 4.0
return x * x / 16.0
}
val yd = fun(t: Double, yt: Double) = t * Math.sqrt(yt)
rungeKutta4(0.0, 10.0, 0.1, y, yd)
}</syntaxhighlight>
{{out}}
<pre>
T RK4 Exact Error
---- ---------- ---------- ---------
0.0 1.000000 1.000000 0.000000
1.0 1.562500 1.562500 -0.000000
2.0 3.999999 4.000000 -0.000001
3.0 10.562497 10.562500 -0.000003
4.0 24.999994 25.000000 -0.000006
5.0 52.562489 52.562500 -0.000011
6.0 99.999983 100.000000 -0.000017
7.0 175.562476 175.562500 -0.000024
8.0 288.999968 289.000000 -0.000032
9.0 451.562459 451.562500 -0.000041
10.0 675.999949 676.000000 -0.000051
</pre>
=={{header|Liberty BASIC}}==
<syntaxhighlight lang="lb">
'[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
</syntaxhighlight>
{{Out}}
<pre>
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
</pre>
=={{header|Lua}}==
<syntaxhighlight lang="lua">local df = function (t, y)
-- derivative of function by value y at time t
return t*y^0.5
end
local dt = 0.1
local y = 1
print ("t", "realY"..' ', "y", ' '.."error")
print ("---", "-------"..' ', "---------------", ' '.."--------------------")
for i = 0, 100 do
local t = i*dt
if t%1 == 0 then
local realY = (t*t+4)^2/16
print (t, realY..' ', y, ' '..realY-y)
end
local dy1 = df(t, y)
local dy2 = df(t+dt/2, y+dt/2*dy1)
local dy3 = df(t+dt/2, y+dt/2*dy2)
local dy4 = df(t+dt, y+dt*dy3)
y = y + dt*(dy1+2*dy2+2*dy3+dy4)/6
end</syntaxhighlight>
{{Out}}
<pre>t realY y error
--- ------- --------------- --------------------
0.0 1.0 1 0.0
1.0 1.5625 1.5624998542781 1.457218921086e-007
2.0 4.0 3.9999990805208 9.1947919989011e-007
3.0 10.5625 10.562497090438 2.9095624469733e-006
4.0 25.0 24.999993765091 6.2349093639114e-006
5.0 52.5625 52.562489180303 1.0819697415343e-005
6.0 100.0 99.999983405404 1.6594596417008e-005
7.0 175.5625 175.56247648227 2.3517728749312e-005
8.0 289.0 288.9999684348 3.156520142511e-005
9.0 451.5625 451.56245927684 4.0723160338985e-005
10.0 676.0 675.99994901671 5.0983290293516e-005
</pre>
=={{header|Mathematica}}/{{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">(* Symbolic solution *)
DSolve[{y'[t] == t*Sqrt[y[t]], y[0] == 1}, y, t]
Table[{t, 1/16 (4 + t^2)^2}, {t, 0, 10}]
(* 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]]}]</syntaxhighlight>
=={{header|MATLAB}}==
The normally-used built-in solver is the ode45 function, which uses a non-fixed-step solver with 4th/5th order Runge-Kutta methods. The MathWorks Support Team released a [http://www.mathworks.com/matlabcentral/answers/98293-is-there-a-fixed-step-ordinary-differential-equation-ode-solver-in-matlab-8-0-r2012b#answer_107643 package of fixed-step RK method ODE solvers] on MATLABCentral. The ode4 function contained within uses a 4th-order Runge-Kutta method. Here is code that tests both ode4 and my own function, shows that they are the same, and compares them to the exact solution.
<syntaxhighlight lang="matlab">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</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|Maxima}}==
<syntaxhighlight lang="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]);</syntaxhighlight>
=={{header|МК-61/52}}==
<pre>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</pre>
''Input:'' 1/2 (h/2) - Р5, 1 (y<sub>0</sub>) - Р8 and Р7, 0 (t<sub>0</sub>) - Р6.
=={{header|Nim}}==
<syntaxhighlight lang="nim">import math
proc fn(t, y: float): float =
result = t * math.sqrt(y)
proc solution(t: float): float =
result = (t^2 + 4)^2 / 16
proc rk(start, stop, step: float) =
let nsteps = int(round((stop - start) / step)) + 1
let delta = (stop - start) / float(nsteps - 1)
var cur_y = 1.0
for i in 0..(nsteps - 1):
let cur_t = start + delta * float(i)
if abs(cur_t - math.round(cur_t)) < 1e-5:
echo "y(", cur_t, ") = ", cur_y, ", error = ", solution(cur_t) - cur_y
let dy1 = step * fn(cur_t, cur_y)
let dy2 = step * fn(cur_t + 0.5 * step, cur_y + 0.5 * dy1)
let dy3 = step * fn(cur_t + 0.5 * step, cur_y + 0.5 * dy2)
let dy4 = step * fn(cur_t + step, cur_y + dy3)
import math, strformat
proc fn(t, y: float): float =
result = t * math.sqrt(y)
proc solution(t: float): float =
result = (t^2 + 4)^2 / 16
proc rk(start, stop, step: float) =
let nsteps = int(round((stop - start) / step)) + 1
let delta = (stop - start) / float(nsteps - 1)
var cur_y = 1.0
for i in 0..<nsteps:
let cur_t = start + delta * float(i)
if abs(cur_t - math.round(cur_t)) < 1e-5:
echo &"y({cur_t}) = {cur_y}, error = {solution(cur_t) - cur_y}"
let dy1 = step * fn(cur_t, cur_y)
let dy2 = step * fn(cur_t + 0.5 * step, cur_y + 0.5 * dy1)
let dy3 = step * fn(cur_t + 0.5 * step, cur_y + 0.5 * dy2)
let dy4 = step * fn(cur_t + step, cur_y + dy3)
cur_y += (dy1 + 2 * (dy2 + dy3) + dy4) / 6
rk(start = 0, stop = 10, step = 0.1)
cur_y += (dy1 + 2.0 * (dy2 + dy3) + dy4) </syntaxhighlight>
{{out}}
<pre>y(0.0) = 1.0, error = 0.0
y(1.0) = 1.562499854278108, error = 1.457218921085968e-07
y(2.0) = 3.9999990805208, error = 9.194792003341945e-07
y(3.0) = 10.56249709043755, error = 2.909562448749625e-06
y(4.0) = 24.99999376509064, error = 6.234909363911356e-06
y(5.0) = 52.56248918030258, error = 1.081969741534294e-05
y(6.0) = 99.99998340540358, error = 1.659459641700778e-05
y(7.0) = 175.5624764822713, error = 2.351772874931157e-05
y(8.0) = 288.9999684347986, error = 3.156520142510999e-05
y(9.0) = 451.5624592768397, error = 4.07231603389846e-05
y(10.0) = 675.9999490167097, error = 5.098329029351589e-05</pre>
=={{header|Objeck}}==
<syntaxhighlight lang="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();
}
}</syntaxhighlight>
Output:
<pre>
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
</pre>
=={{header|OCaml}}==
<syntaxhighlight lang="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)</syntaxhighlight>
{{out}}
<pre>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</pre>
=={{header|Octave}}==
<syntaxhighlight lang="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
</syntaxhighlight>
{{out}}
<pre>
Time Exact Value ODE4 Value Num. Error
0 1.00000 1.00000 0
1 1.56250 1.56250 1.457e-007
2 4.00000 4.00000 9.195e-007
3 10.56250 10.56250 2.91e-006
4 25.00000 24.99999 6.235e-006
5 52.56250 52.56249 1.082e-005
6 100.00000 99.99998 1.659e-005
7 175.56250 175.56248 2.352e-005
8 289.00000 288.99997 3.157e-005
9 451.56250 451.56246 4.072e-005
10 676.00000 675.99995 5.098e-005</pre>
=={{header|PARI/GP}}==
{{trans|C}}
<syntaxhighlight lang="parigp">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()</syntaxhighlight>
{{out}}
<pre>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</pre>
=={{header|Pascal}}==
{{trans|Ada}}
This code has been compiled using Free Pascal 2.6.2.
<syntaxhighlight lang="pascal">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.</syntaxhighlight>
{{out}}
<pre>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
</pre>
=={{header|Perl}}==
There are many ways of doing this. Here we define the runge_kutta function
as a function of <math>y'</math> and <math>\delta t</math>, returning a closure
which itself takes <math>(t, y)</math> as argument and returns the next <math>(t, y)</math>.
Notice how we have to use sprintf to deal with floating point rounding. See perlfaq4.
<syntaxhighlight lang="perl">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", $t) =~ /0000$/;
}</syntaxhighlight>
{{out}}
<pre>y( 0) = 1.000000 ± 0.000000e+00
y( 1) = 1.562500 ± 1.457219e-07
y( 2) = 3.999999 ± 9.194792e-07
y( 3) = 10.562497 ± 2.909562e-06
y( 4) = 24.999994 ± 6.234909e-06
y( 5) = 52.562489 ± 1.081970e-05
y( 6) = 99.999983 ± 1.659460e-05
y( 7) = 175.562476 ± 2.351773e-05
y( 8) = 288.999968 ± 3.156520e-05
y( 9) = 451.562459 ± 4.072316e-05
y(10) = 675.999949 ± 5.098329e-05</pre>
=={{header|Phix}}==
{{trans|ERRE}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">dt</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0.1</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1.0</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" x true/actual y calculated y relative error\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" --- ------------- ------------- --------------\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">100</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">*</span><span style="color: #000000;">dt</span>
<span style="color: #008080;">if</span> <span style="color: #004080;">integer</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">act</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">16</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%4.1f %14.9f %14.9f %.9e\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">act</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">-</span><span style="color: #000000;">act</span><span style="color: #0000FF;">)})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">k1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">k2</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">k1</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">k3</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">k2</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">k4</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">dt</span><span style="color: #0000FF;">*</span><span style="color: #000000;">k3</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">y</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">dt</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">k1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">k2</span><span style="color: #0000FF;">+</span><span style="color: #000000;">k3</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">k4</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">6</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
x true/actual y calculated y relative error
--- ------------- ------------- --------------
0.0 1.000000000 1.000000000 0.000000000e+0
1.0 1.562500000 1.562499854 1.457218921e-7
2.0 4.000000000 3.999999081 9.194791999e-7
3.0 10.562500000 10.562497090 2.909562447e-6
4.0 25.000000000 24.999993765 6.234909363e-6
5.0 52.562500000 52.562489180 1.081969741e-5
6.0 100.000000000 99.999983405 1.659459641e-5
7.0 175.562500000 175.562476482 2.351772874e-5
8.0 289.000000000 288.999968435 3.156520142e-5
9.0 451.562500000 451.562459277 4.072316033e-5
10.0 676.000000000 675.999949017 5.098329030e-5
</pre>
=={{header|PL/I}}==
<syntaxhighlight lang="pl/i">
Runge_Kutta: procedure options (main); /* 10 March 2014 */
declare (y, dy1, dy2, dy3, dy4) float (18);
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;
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|PowerShell}}==
{{works with|PowerShell|4.0}}
<syntaxhighlight lang="powershell">
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 $tEnd
</syntaxhighlight>
<b>Output:</b>
<pre>
t y error
- - -----
0 1 0
1 1.56249985427811 1.45721892108597E-07
2 3.9999990805208 9.19479200778284E-07
3 10.5624970904376 2.90956244874963E-06
4 24.9999937650906 6.23490936391136E-06
5 52.5624891803026 1.08196974153429E-05
6 99.9999834054036 1.65945964170078E-05
7 175.562476482271 2.35177287493116E-05
8 288.999968434799 3.156520142511E-05
9 451.56245927684 4.07231603389846E-05
10 675.99994901671 5.09832902935159E-05
</pre>
=={{header|PureBasic}}==
{{trans|BBC Basic}}
<syntaxhighlight lang="purebasic">EnableExplicit
Define.i i
Define.d y=1.0, k1=0.0, k2=0.0, k3=0.0, k4=0.0, t=0.0
If OpenConsole()
For i=0 To 100
t=i/10
If Not i%10
PrintN("y("+RSet(StrF(t,0),2," ")+") ="+RSet(StrF(y,4),9," ")+#TAB$+"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</syntaxhighlight>
{{out}}
<pre>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...</pre>
=={{header|Python}}==
<syntaxhighlight lang="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</syntaxhighlight>
=={{header|R}}==
<syntaxhighlight lang="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</syntaxhighlight>
=={{header|Racket}}==
See [[Euler method#Racket]] for implementation of simple general ODE-solver.
The Runge-Kutta method
<syntaxhighlight lang="racket">
(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))))))
</syntaxhighlight>
The method modifier which divides each time-step into ''n'' sub-steps:
<syntaxhighlight lang="racket">
(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))))
</syntaxhighlight>
Usage:
<syntaxhighlight lang="racket">
(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))))
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
Graphical representation:
<syntaxhighlight lang="racket">
> (require plot)
> (plot (list (function exact-solution 0 10 #:label "Exact solution")
(points numeric-solution #:label "Runge-Kutta method"))
#:x-label "t" #:y-label "y(t)")
</syntaxhighlight>
[[File:runge-kutta.png]]
=={{header|Raku}}==
(formerly Perl 6)
{{Works with|rakudo|2016.03}}
<syntaxhighlight lang="raku" line>sub runge-kutta(&yp) {
return -> \t, \y, \δt {
my $a = δt * yp( t, y );
my $b = δt * yp( t + δt/2, y + $a/2 );
my $c = δt * yp( t + δt/2, y + $b/2 );
my $d = δt * yp( t + δt, y + $c );
($a + 2*($b + $c) + $d) / 6;
}
}
constant δt = .1;
my &δy = runge-kutta { $^t * sqrt($^y) };
loop (
my ($t, $y) = (0, 1);
$t <= 10;
($t, $y) »+=« (δt, δy($t, $y, δt))
) {
printf "y(%2d) = %12f ± %e\n", $t, $y, abs($y - ($t**2 + 4)**2 / 16)
if $t %% 1;
}</syntaxhighlight>
{{out}}
<pre>y( 0) = 1.000000 ± 0.000000e+00
y( 1) = 1.562500 ± 1.457219e-07
y( 2) = 3.999999 ± 9.194792e-07
y( 3) = 10.562497 ± 2.909562e-06
y( 4) = 24.999994 ± 6.234909e-06
y( 5) = 52.562489 ± 1.081970e-05
y( 6) = 99.999983 ± 1.659460e-05
y( 7) = 175.562476 ± 2.351773e-05
y( 8) = 288.999968 ± 3.156520e-05
y( 9) = 451.562459 ± 4.072316e-05
y(10) = 675.999949 ± 5.098329e-05</pre>
=={{header|REXX}}==
The Runge─Kutta method is used to solve the following differential equation:
<big><big> y'(t) = t<sup>2</sup> √<span style="text-decoration: overline"> y(t) </span></big></big>
The exact solution: <big><big> y(t) = (t<sup>2</sup>+4)<sup>2</sup> ÷ 16 </big></big>
<syntaxhighlight lang="rexx">/*REXX program uses the Runge─Kutta method to solve the equation: y'(t) = t² √[y(t)] */
numeric digits 40; f= digits() % 4 /*use 40 decimal digs, but only show 10*/
x0= 0; x1= 10; dx= .1 /*define variables: X0 X1 DX */
n=1 + (x1-x0) / dx
y.=1; 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</syntaxhighlight>
Programming note: the '''fmt''' function is used to
align the output with attention paid to the different ways some
<br>REXXes format numbers that are in floating point representation.
{{out|output|text= when using Regina REXX:}}
<pre>
════X═════ ══════════Y═══════════ ═══════relative error═══════
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
</pre>
{{out|output|text= when using PC/REXX, Personal REXX, ROO, or R4 REXX:}}
<pre>
════X═════ ══════════Y═══════════ ═══════relative error═══════
0 1 0
1 1.5624998543 -0.0000000933
2 3.9999990805 -0.0000002299
3 10.5624970904 -0.0000002755
4 24.9999937651 -0.0000002494
5 52.5624891803 -0.0000002058
6 99.9999834054 -0.0000001659
7 175.5624764823 -0.000000134
8 288.9999684348 -0.0000001092
9 451.5624592768 -0.0000000902
10 675.9999490167 -0.0000000754
</pre>
=={{header|Ring}}==
<syntaxhighlight lang="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
</syntaxhighlight>
Output:
<pre>
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
</pre>
=={{header|Ruby}}==
<syntaxhighlight lang="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</syntaxhighlight>
{{Out}}
<pre>
y( 0.0) = 1.000000 error: 0.000000e+00
y( 1.0) = 1.562500 error: 1.457219e-07
y( 2.0) = 3.999999 error: 9.194792e-07
y( 3.0) = 10.562497 error: 2.909562e-06
y( 4.0) = 24.999994 error: 6.234909e-06
y( 5.0) = 52.562489 error: 1.081970e-05
y( 6.0) = 99.999983 error: 1.659460e-05
y( 7.0) = 175.562476 error: 2.351773e-05
y( 8.0) = 288.999968 error: 3.156520e-05
y( 9.0) = 451.562459 error: 4.072316e-05
y(10.0) = 675.999949 error: 5.098329e-05
</pre>
=={{header|Run BASIC}}==
<syntaxhighlight lang="runbasic">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</syntaxhighlight>
{{out}}
<pre>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
</pre>
=={{header|Rust}}==
This is a translation of the javascript solution with some minor differences.
<syntaxhighlight lang="rust">fn runge_kutta4(fx: &dyn Fn(f64, f64) -> f64, x: f64, y: f64, dx: f64) -> f64 {
let k1 = dx * fx(x, y);
let k2 = dx * fx(x + dx / 2.0, y + k1 / 2.0);
let k3 = dx * fx(x + dx / 2.0, y + k2 / 2.0);
let k4 = dx * fx(x + dx, y + k3);
y + (k1 + 2.0 * k2 + 2.0 * k3 + k4) / 6.0
}
fn f(x: f64, y: f64) -> f64 {
x * y.sqrt()
}
fn actual(x: f64) -> f64 {
(1.0 / 16.0) * (x * x + 4.0).powi(2)
}
fn main() {
let mut y = 1.0;
let mut x = 0.0;
let step = 0.1;
let max_steps = 101;
let sample_every_n = 10;
for steps in 0..max_steps {
if steps % sample_every_n == 0 {
println!("y({}):\t{:.10}\t\t {:E}", x, y, actual(x) - y)
}
y = runge_kutta4(&f, x, y, step);
x = ((x * 10.0) + (step * 10.0)) / 10.0;
}
}</syntaxhighlight>
<pre>
y(0): 1.0000000000 0E0
y(1): 1.5624998543 1.4572189210859676E-7
y(2): 3.9999990805 9.194792007782837E-7
y(3): 10.5624970904 2.9095624487496252E-6
y(4): 24.9999937651 6.234909363911356E-6
y(5): 52.5624891803 1.0819697415342944E-5
y(6): 99.9999834054 1.659459641700778E-5
y(7): 175.5624764823 2.3517728749311573E-5
y(8): 288.9999684348 3.156520142510999E-5
y(9): 451.5624592768 4.07231603389846E-5
y(10): 675.9999490167 5.098329029351589E-5
</pre>
=={{header|Scala}}==
<syntaxhighlight lang="scala">object Main extends App {
val f = (t: Double, y: Double) => t * Math.sqrt(y) // Runge-Kutta solution
val g = (t: Double) => Math.pow(t * t + 4, 2) / 16 // Exact solution
new Calculator(f, Some(g)).compute(100, 0, .1, 1)
}
class Calculator(f: (Double, Double) => Double, g: Option[Double => Double] = None) {
def compute(counter: Int, tn: Double, dt: Double, yn: Double): Unit = {
if (counter % 10 == 0) {
val c = (x: Double => Double) => (t: Double) => {
val err = Math.abs(x(t) - yn)
f" Error: $err%7.5e"
}
val s = g.map(c(_)).getOrElse((x: Double) => "") // If we don't have exact solution, just print nothing
println(f"y($tn%4.1f) = $yn%12.8f${s(tn)}") // Else, print Error estimation here
}
if (counter > 0) {
val dy1 = dt * f(tn, yn)
val dy2 = dt * f(tn + dt / 2, yn + dy1 / 2)
val dy3 = dt * f(tn + dt / 2, yn + dy2 / 2)
val dy4 = dt * f(tn + dt, yn + dy3)
val y = yn + (dy1 + 2 * dy2 + 2 * dy3 + dy4) / 6
val t = tn + dt
compute(counter - 1, t, dt, y)
}
}
}</syntaxhighlight>
<pre>
y( 0.0) = 1.00000000 Error: 0.00000e+00
y( 1.0) = 1.56249985 Error: 1.45722e-07
y( 2.0) = 3.99999908 Error: 9.19479e-07
y( 3.0) = 10.56249709 Error: 2.90956e-06
y( 4.0) = 24.99999377 Error: 6.23491e-06
y( 5.0) = 52.56248918 Error: 1.08197e-05
y( 6.0) = 99.99998341 Error: 1.65946e-05
y( 7.0) = 175.56247648 Error: 2.35177e-05
y( 8.0) = 288.99996843 Error: 3.15652e-05
y( 9.0) = 451.56245928 Error: 4.07232e-05
y(10.0) = 675.99994902 Error: 5.09833e-05
</pre>
=={{header|Sidef}}==
{{trans|Raku}}
<syntaxhighlight lang="ruby">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;
}</syntaxhighlight>
{{out}}
<pre>
y( 0) = 1.000000 ± 0.000000e+00
y( 1) = 1.562500 ± 1.457219e-07
y( 2) = 3.999999 ± 9.194792e-07
y( 3) = 10.562497 ± 2.909562e-06
y( 4) = 24.999994 ± 6.234909e-06
y( 5) = 52.562489 ± 1.081970e-05
y( 6) = 99.999983 ± 1.659460e-05
y( 7) = 175.562476 ± 2.351773e-05
y( 8) = 288.999968 ± 3.156520e-05
y( 9) = 451.562459 ± 4.072316e-05
y(10) = 675.999949 ± 5.098329e-05
</pre>
=={{header|Standard ML}}==
<syntaxhighlight lang="sml">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'</syntaxhighlight>
{{out}}
<pre>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</pre>
=={{header|Stata}}==
<syntaxhighlight lang="stata">function rk4(f, t0, y0, t1, n) {
h = (t1-t0)/(n-1)
a = J(n, 2, 0)
a[1, 1] = t = t0
a[1, 2] = y = y0
for (i=2; i<=n; i++) {
k1 = h*(*f)(t, y)
k2 = h*(*f)(t+0.5*h, y+0.5*k1)
k3 = h*(*f)(t+0.5*h, y+0.5*k2)
k4 = h*(*f)(t+h, y+k3)
t = t+h
y = y+(k1+2*k2+2*k3+k4)/6
a[i, 1] = t
a[i, 2] = y
}
return(a)
}
function f(t, y) {
return(t*sqrt(y))
}
a = rk4(&f(), 0, 1, 10, 101)
t = a[., 1]
a = a, a[., 2]:-(t:^2:+4):^2:/16
a[range(1,101,10), .]
1 2 3
+----------------------------------------------+
1 | 0 1 0 |
2 | 1 1.562499854 -1.45722e-07 |
3 | 2 3.999999081 -9.19479e-07 |
4 | 3 10.56249709 -2.90956e-06 |
5 | 4 24.99999377 -6.23491e-06 |
6 | 5 52.56248918 -.0000108197 |
7 | 6 99.99998341 -.0000165946 |
8 | 7 175.5624765 -.0000235177 |
9 | 8 288.9999684 -.0000315652 |
10 | 9 451.5624593 -.0000407232 |
11 | 10 675.999949 -.0000509833 |
+----------------------------------------------+</syntaxhighlight>
=={{header|Swift}}==
{{trans|C}}
<syntaxhighlight lang="swift">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))
}</syntaxhighlight>
{{out}}
<pre> 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</pre>
=={{header|Tcl}}==
<
# Hack to bring argument function into expression
Line 349 ⟶ 3,732:
printvals $t $y
}
}</
{{out}}
<pre>y(0.0) = 1.00000000 Error: 0.00000000e+00
Line 362 ⟶ 3,745:
y(9.0) = 451.56245928 Error: 4.07231581e-05
y(10.0) = 675.99994902 Error: 5.09832864e-05</pre>
=={{header|V (Vlang)}}==
{{trans|Ring}}
<syntaxhighlight lang="Zig">
import math
fn main() {
mut t, mut k1, mut k2, mut k3, mut k4, mut y := 0.0, 0.0, 0.0, 0.0, 0.0, 1.0
for i in 0..101 {
t = i / 10.0
if t == math.floor(t) {
actual := math.pow((math.pow(t, 2) + 4), 2)/16
println("y(${t:.0}) = ${y:.8f} error = ${(actual - y):.8f}")
}
k1 = t * math.sqrt(y)
k2 = (t + 0.05) * math.sqrt(y + 0.05 * k1)
k3 = (t + 0.05) * math.sqrt(y + 0.05 * k2)
k4 = (t + 0.10) * math.sqrt(y + 0.10 * k3)
y += 0.1 * (k1 + 2 * (k2 + k3) + k4) / 6
}
}
</syntaxhighlight>
{{out}}
<pre>
y(0) = 1.00000000 error = 0.00000000
y(1) = 1.56249985 error = 0.00000015
y(2) = 3.99999908 error = 0.00000092
y(3) = 10.56249709 error = 0.00000291
y(4) = 24.99999377 error = 0.00000623
y(5) = 52.56248918 error = 0.00001082
y(6) = 99.99998341 error = 0.00001659
y(7) = 175.56247648 error = 0.00002352
y(8) = 288.99996843 error = 0.00003157
y(9) = 451.56245928 error = 0.00004072
y(10) = 675.99994902 error = 0.00005098
</pre>
=={{header|Wren}}==
{{trans|Kotlin}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="wren">import "./fmt" for Fmt
var rungeKutta4 = Fn.new { |t0, tz, dt, y, yd|
var tn = t0
var yn = y.call(tn)
var z = ((tz - t0)/dt).truncate
for (i in 0..z) {
if (i % 10 == 0) {
var exact = y.call(tn)
var error = yn - exact
Fmt.print("$4.1f $10f $10f $9f", tn, yn, exact, error)
}
if (i == z) break
var dy1 = dt * yd.call(tn, yn)
var dy2 = dt * yd.call(tn + 0.5 * dt, yn + 0.5 * dy1)
var dy3 = dt * yd.call(tn + 0.5 * dt, yn + 0.5 * dy2)
var dy4 = dt * yd.call(tn + dt, yn + dy3)
yn = yn + (dy1 + 2.0 * dy2 + 2.0 * dy3 + dy4) / 6.0
tn = tn + dt
}
}
System.print(" T RK4 Exact Error")
System.print("---- --------- ---------- ---------")
var y = Fn.new { |t|
var x = t * t + 4.0
return x * x / 16.0
}
var yd = Fn.new { |t, yt| t * yt.sqrt }
rungeKutta4.call(0, 10, 0.1, y, yd)</syntaxhighlight>
{{out}}
<pre>
T RK4 Exact Error
---- --------- ---------- ---------
0.0 1.000000 1.000000 0.000000
1.0 1.562500 1.562500 -0.000000
2.0 3.999999 4.000000 -0.000001
3.0 10.562497 10.562500 -0.000003
4.0 24.999994 25.000000 -0.000006
5.0 52.562489 52.562500 -0.000011
6.0 99.999983 100.000000 -0.000017
7.0 175.562476 175.562500 -0.000024
8.0 288.999968 289.000000 -0.000032
9.0 451.562459 451.562500 -0.000041
10.0 675.999949 676.000000 -0.000051
</pre>
=={{header|XPL0}}==
<syntaxhighlight lang "XPL0">func real Y_(T, Y);
real T, Y;
return T*sqrt(Y);
def DT = 0.1;
real T, Y, Exact, DY1, DY2, DY3, DY4;
[Text(0, " T RK Exact Error^m^j");
T:= 0.; Y:= 1.;
repeat if Mod(T+.001, 1.) < .01 then
[Format(2, 1);
RlOut(0, T);
Format(5, 7);
RlOut(0, Y);
Exact:= sq(T*T+4.)/16.;
RlOut(0, Exact);
RlOut(0, Y-Exact);
CrLf(0);
];
DY1:= DT * Y_(T, Y);
DY2:= DT * Y_(T+DT/2., Y+DY1/2.);
DY3:= DT * Y_(T+DT/2., Y+DY2/2.);
DY4:= DT * Y_(T+DT, Y+DY3);
Y:= Y + (DY1 + 2.*DY2 + 2.*DY3 + DY4) / 6.;
T:= T + DT;
until T > 10.;
]</syntaxhighlight>
{{out}}
<pre>
T RK Exact Error
0.0 1.0000000 1.0000000 0.0000000
1.0 1.5624999 1.5625000 -0.0000001
2.0 3.9999991 4.0000000 -0.0000009
3.0 10.5624971 10.5625000 -0.0000029
4.0 24.9999938 25.0000000 -0.0000062
5.0 52.5624892 52.5625000 -0.0000108
6.0 99.9999834 100.0000000 -0.0000166
7.0 175.5624765 175.5625000 -0.0000235
8.0 288.9999684 289.0000000 -0.0000316
9.0 451.5624593 451.5625000 -0.0000407
10.0 675.9999490 676.0000000 -0.0000510
</pre>
=={{header|zkl}}==
{{trans|OCaml}}
<syntaxhighlight lang="zkl">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
}</syntaxhighlight>
{{out}}
<pre>
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
</pre>
{{omit from|Brlcad}}
{{omit from|GUISS}}
{{omit from|Lilypond}}
{{omit from|Openscad}}
{{omit from|TPP}}
| |||