EXPLANATION FILE OF PROGRAM TEQUDIF1
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  The Runge-Kutta Method
  ----------------------


  We present here the Runge-Kutta method of order 4 to integrate an ODE

  of order 1:       Y' = F(X, Y)

  The development of Y around x  coincidates with its Taylor development
                               n
  of order 4:

       y    = y  + h y' + (h^2/2) y" + (h^3/6) y"' + (h^4/24) y""
        n+1    n      n            n            n              n

  Other orders may also be used for the Runge-Kutta method, for instance

  the Euler's method is a Runge-Kutta of order one.

  Order 4 is often used because it is a good compromise between speed and
  
  accuracy.

  From a current point (x , y ), the next point is determined by:
                         n   n

  	K  = h f(x , y )
         0        n   n

	k  = h f(x + h/2, y + K /2)
         1        n        n   0

	k  = h f(x + h/2, y + K /2)
         2        n        n   1

	k  = h f(x + h, y + K )
         3        n      n   2

  and   y   = y  + (1/6)(K + 2 K + K )
         n+1   n          0     1   3

  It can be proved that the error at each step is about h^5.

  The only drawback is the time of calculation: at each step, four function

  evaluations are necessary.

  From [BIBLI 03].

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End of file rkutta.txt.