What About Unbounded End?

Yeah, what about it? The previous code have the both end bounded.

If we want a free/unbound end, we could set the condition at the with this properties (or we could choose whatever we like)

dy/dx=0

So we will have

y[1]-y[0]=0
y[0] = y[1]

if we want both free ends, we could set the other end as well

y[n] = y[n-1]

So, we just have to modify the original just a bit.

Beware though, with both ends free, we could lost the strings, :)

Waves Equation Animation in Python

I use matplotlib module to do the animation.

The main code is in def waves(y0,y1,cb) that use finite difference that solved initial value problem and boundary value problem simultaneously.    

code
from pylab import *
import matplotlib.animation as animation

fig,ax = subplots()

def waves(y0,y1,cb):
    y2 = y0
    for i in range(1,len(y0)-1):
        y2[i] = 2*y1[i]-y0[i]+cb*(y1[i+1]-2*y1[i]+y1[i-1])
    return y2

x   = linspace(0.,1.,20)
dx  = 1./(len(x))
y0  = sin(2*pi*x)
vy0 = 12.

b   = 1./32.  #dt2/dx2
dt  = sqrt(b*dx*dx)
print dt

c   = 1.

cb  = c*b

y1  = y0 + vy0*dt

print y0
print y1

line,   = ax.plot(x,y0)
def animate(i):
    global y0,y1,cb
    y2  = waves(y0,y1,cb)
    y0  = y1
    y1  = y2
    line.set_ydata(y0)
    return line,


#plot (x,y0)

ani =  animation.FuncAnimation(fig, animate, np.arange(1,200), interval =  25, blit=False)

grid(True)
ylim(-10,10)
show()

.

Gauss Jordan in Python.

Yeah, it's basically Gauss elimination (or we could call it Gauss Naif :) ) but with slight modification at the end.

So, instead using back substitutions after zeroing the lower triangle, we straight on and zeroing upper triangle as well. As addition, we could normalize the diagonal elements so we have identity matrice.

And all is well, :)

"Auto" Gauss Naif in Delphi.

After do this in Python, now it's time to bring it back to Delphi, where all of this is started, :)


The heart of code lay on this one

procedure tform1.gauss;
var i,j,k:integer; temp:real;
begin
  for i:=1 to 9 do begin
    for j:= 1 to i do begin
      if t[i,j]<>0 then begin
        temp:=t[i,j];
        for k:= 1 to 10 do begin
          if i=j then
            t[j,k]:=t[j,k]/temp
              else t[i,k]:=t[i,k]/temp - t[j,k];
        end;
      end;
    end;
  end;
  //back subtitution
  for i:=9 downto 1 do begin
    x[i]:=t[i,10];
    for j:=9  downto i do begin
      if i<>j then
        x[i]:=x[i]-x[j]*t[i,j];
    end;
  end;

You could say that it consists of zeroing lower tringle and normalizing the diagonal and then subtituting the value.

There's little failsafe code here, that is if we already have zero cell, don't proceed, or it will gave divided by zero error.

Gauss Naif in Python

Okay, we've done the manual one, how about automatize it?

It's actually just a matter of finding the pattern on that code and after we found the loop, we just have to well... loop it, :)

Manual Gauss Jordan in Python.

What if we didn't do back substitution on Gauss Naif method but eliminate the rest instead? Nah, we get the Gauss Jordan here.

The idea is after we do operation to make the  lower-triangle have zero value,  we continue the operation until all the component in the upper-triangle have zero value too, and the diagonal have value of one.

Basically, the matrix becomes identity matrix. This way, we didn't need subtitution at all since all variables already has the exact value on the right side, :)

Manual Gauss Naif Elimination using Python

How about some manual matrix using manual Gauss just like always, but in Python? Okay, here it is.

I use tuple, I think it's just the same as array for this purpose.

I created matrix a with random value.  It's like linear equation system; three unknown variables with three equation. The purpose of this code is to find x1, x2 and x3.

Oh, in this case, its x0, x1 and x2, :)