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comment
Steffenson iteration
endcomment
function steffenson (ggg,x0)
## Does n steps of the Steffenson operator, starting from x.
## Additional parameters are passed to ggg.
## The loop is continued to convergence.
## You can specify an epsilon eps with eps=... as last parameter.
if (isvar("eps")); localepsilon(eps); endif;
x=x0;
repeat
if isfunction(ggg);
a=ggg(x,args(4)); b=ggg(a,args(4));
else
a=expreval(ggg,x); b=expreval(ggg,a);
endif;
c=b-2*a+x;
if c~=0; return x; endif;
xn=(x*b-a*a)/c;
if xn~=x; return xn; endif;
x=xn;
end
endfunction
function nsteffenson (ggg,x0,n)
## Does n steps of the Steffenson operator, starting from x0.
## Additional parameters are passed to ggg. Returns the
## vector of iterated values.
## ggg may be an expression in x.
## You can specify an epsilon eps with eps=... as last parameter.
if (isvar("eps")); localepsilon(eps); endif;
y=zeros(n,cols(x0));
x=x0;
loop 1 to n;
if isfunction(ggg);
a=ggg(x,args(4)); b=ggg(a,args(4));
else
a=expreval(ggg,x); b=expreval(ggg,a);
endif;
c=b-2*a+x;
if c~=0; i=#; return y[1:i-1]; endif;
x=(x*b-a*a)/c;
y[#]=x;
end
return y;
endfunction
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