MATLAB TUTORIAL for the First Course. Part III: Hamming Method

Prof. Vladimir A. Dobrushkin

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Hamming Method

In 1959, Richard Wesley Hamming from Bell Telephone Laboratories, Murray Hill, New Jersey, proposed a stable predictor-corrector method for ordinary differential equations, which now bears his name. He improved classical Milne--Simpson method by replacing unstable corrector rule by a stable one.
Journal of the ACM, Volume 6, Issue 1, Jan. 1959, pages 37-47.

The Hamming multistep method is used to approximate solution of the initial vaue problem \( y' = f(x,y), \quad y(x_0 ) = y_0 \) over interval [a,b] (where usually x0 = a) by using the predictor
\[ p_{n+1} = y_{n-3} + \frac{4h}{3} \left( 2\,f_{n-2} - f_{n-1} + 2\,f_n \right) , \qquad n=3,4,\ldots . \]
and the corrector
\[ y_{n+1} = \frac{9\, y_n - y_{n-2}}{8} + \frac{3h}{8} \left( f_{n+1} - f_{n-1} + 2\,f_n \right) , \qquad n=3,4,\ldots . \]
As usual, we implement a unifom grid \( x_n = x_0 + n\,h , \) with step length h. Here yn is an approximation to the true value \( \phi (x_n ) \) of the actual solution \( y = \phi (x) . \)

Example. Let us start with the Riccati equation \( y' = x^2 + y^2 \) subject to the initial condition \( y(0) =-1 . \) Its solution is expressed through Bessel functions:

\[ d(x)\, y(x) = J_{-3/4} \left( \frac{x^2}{2} \right) \left( \Gamma^2 \left( \frac{3}{4} \right) + \pi \right) - Y_{-3/4} \left( \frac{x^2}{2} \right) \Gamma^2 \left( \frac{3}{4} \right) . \]
where the denominator
\[ d(x) = Y_{1/4} \left( \frac{x^2}{2} \right) \Gamma^2 \left( \frac{3}{4} \right) - J_{1/4} \left( \frac{x^2}{2} \right) \left( \Gamma^2 \left( \frac{3}{4} \right) +\pi \right) \]
has the first positive null at x = 2.223378383 as the figure shows
d[x_] = BesselY[1/4, x^2 /2]*Gamma[3/4]^2 - BesselJ[1/4, x^2 /2]*(Gamma[3/4]^2 + Pi)
Plot[d[x], {x, 0, 5.5}, PlotStyle -> Thick]
The sollution blows up near the point 2.223378. 

function H=hamming(f,t,y)
% Input:    f is the function entered as a string 'f'
%           t is the vector of abscissas
%           y is the vector of ordinates
% Remark:   The first four coordinates of t and y must have 
%           starting values obtained with RK4
% Output:   H=[t',y'] where t is the vector of abscissas
%           y is the vector of ordinates
n=length(t);
if n < 5, break,end;
F=zeros(1,4); 
F=feval(f,t(1:4),y(1:4));
h=t(2)-t(1);
pold=0;
yold=0;
for k=4:n-1
    % Predictor
    pnew=y(k-3)+(4*h/3)*(F(2:4)*[2 -1 2]'); 
    % Modifier
    pmod=pnew+112*(yold-pold)/121;
    t(k+1)=t(1)+h*k;
    F=[F(2) F(3) F(4) feval(f,t(k+1),pmod)];
    % Corrector
    cnew=(9*y(k)-y(k-2)+3*h*(F(2:4)*[-1 2 1]'))/8; 
    y(k+1)=cnew+9*(pnew-cnew)/121;
    pold=pnew;
    yold=cnew;
    F(4)=feval(f,t(k+1),y(k+1));
end
H=[t',y'];

Example. Hamming's solution to \( y' = 30-5y, \quad y(0) =1 \) over interval [0,10] with N = 50 steps produces oscillations. It is stabilized when N = 70. 


Complete