HP Prime and TI-84 Plus: Runge Kutta 4th Order

HP Prime and TI-84 Plus:  Runge Kutta 4^th Order

The Runge Kutta 4^th Order is a method for solving differential equations involving the form:  dy/dx = f(x,y), where:

x_n+1 = x_n + h
y_n+1 = y_n + (k1 + 2*k2 + 2*k3 + k4)/6 

Where:

k1 = h * f(x_n, y_n)
k2 = h * f(x_n + h/2, y_n + k1/2)
k3 = h * f(x_n + h/2, y_n + k2/2)
k4 = h * f(x_n + h, y_n + k3)

Variables used:

A = x_n
B = y_n
C = x_n+1
D = y_n+1
H = step
K = k1
L = k2
M = k3
N = k4

Results are stored in lists L1 and L2.  L1 represents the x coordinates, L2 represents the y coordinates.

This adopts the RK4 method that I programmed for the Casio fx-5800p, which you can see here:  http://edspi31415.blogspot.com/2015/01/fx-5800p-runge-kutta-4th-order-method.html

The screen shots show the example dy/dx = x^2 + y^2/3, with initial conditions y(0) = 1.  10 steps are plotted with step size 0.1.  

HP Prime:  DIFFTBL and RK4

Both DIFFTBL and RK4 return the same output.  The difference is that DIFFTBL uses an Input box and no-pass through arguments and RK4 uses five pass-through arguments.  This way, RK4 could be used as a subroutine.

Both return the results in a matrix, M1.  Rev. 7820 firmware is used.

DIFFTBL:

EXPORT DIFFTBL()

BEGIN

// EWS 2015-05-29

// Radian

HAngle:=0;

// Input

LOCAL f;

  

INPUT({{f,[8]},A,B,H,S},"Data Input",

{"dy/dx=","x0=","y0=","Step=",

"# Steps="});

L1:={A}; L2:={B};

// Main Loop

FOR I FROM 1 TO S DO

// k1

X:=A;

Y:=B;

K:=H*EVAL(f);

// k2

X:=A+H/2;

Y:=B+K/2;

L:=H*EVAL(f);

// k3

Y:=B+L/2;

M:=H*EVAL(f);

// k4

X:=A+H;

Y:=B+M;

N:=H*EVAL(f);

// point

A:=A+H;

B:=B+(K+2*L+2*M+N)/6;

L1:=CONCAT(L1,{A});

L2:=CONCAT(L2,{B});

END;

M1:=list2mat(CONCAT(L1,L2),

SIZE(L1));

M1:=TRN(M1);

RETURN M1;

END;

RK4:

EXPORT RK4(f,A,B,H,S)

BEGIN

// EWS 2015-05-29

// Radian

HAngle:=0;

// Input

L1:={A}; L2:={B};

// Main Loop

FOR I FROM 1 TO S DO

// k1

X:=A;

Y:=B;

K:=H*EVAL(f);

// k2

X:=A+H/2;

Y:=B+K/2;

L:=H*EVAL(f);

// k3

Y:=B+L/2;

M:=H*EVAL(f);

// k4

X:=A+H;

Y:=B+M;

N:=H*EVAL(f);

// point

A:=A+H;

B:=B+(K+2*L+2*M+N)/6;

L1:=CONCAT(L1,{A});

L2:=CONCAT(L2,{B});

END;

M1:=list2mat(CONCAT(L1,L2),

SIZE(L1));

M1:=TRN(M1);

RETURN M1;

END;

TI-84 Plus:  DIFFPLOT

DIFFPLOT plots the results in a statistics xy-plot.  The equation variable Y0 is used, but subsequently deleted.  I left out any color commands to make this easily adoptable for both the monochrome and color TI-84 Plus calculators. 

Do not forget to enclose dy/dx in quotes, unless an input error occurs

Radian:FnOff :PlotsOff

Disp "DIFF. EQUATION"

Disp "Y₀=DY/DX"

Disp "USE QUOTES"

Input Y₀

Input "X0=",A

Input "Y0=",B

Input "STEP=",H

Input "NO. STEPS=",S

{A}→L₁

{B}→L₂

For(I,1,S)

A→X:B→Y

H*Y₀→K

A+H/2→X:B+K/2→Y

H*Y₀→L

B+L/2→Y

H*Y₀→M

A+H→X:B+M→Y

H*Y₀→N

A+H→A

B+(K+2*L+2*M+N)/6→B

augment(L₁,{A})→L₁

augment(L₂,{B})→L₂

End

FnOff 0

PlotsOn 1

Plot1(xyLine,L₁,L₂)

ZoomStat

This blog is property of Edward Shore.  2015.