OFFSET
1,3
COMMENTS
Define the Fibonacci Function F(x) and its derivative dF/dx as in A172081.
At the local maximum, dF(x)/dx = 0.
This constant x=1.0945... here satisfies this condition of vanishing first derivative.
LINKS
Gerd Lamprecht, Iterationsrechner mit Algorithmus
Gerd Lamprecht, Zahlenfolgen (sequences)
E. Weisstein, Fibonacci Number, Mathworld.
EXAMPLE
F(1.0945761052316...) = 1.0098243...
MAPLE
p := (1+sqrt(5))/2 ; F := (p^x - cos(Pi*x)/p^x )/sqrt(5);
Fpr := diff(F, x) ; Fpr2 := diff(Fpr, x) ;
Digits := 80 ; x0 := 1.0 ;
for n from 1 to 10 do
x0 := evalf(x0-subs(x=x0, Fpr)/subs(x=x0, Fpr2)) ;
end do ; # R. J. Mathar, Feb 02 2010
MATHEMATICA
digits = 105; F[x_] := (GoldenRatio^x - Cos[Pi*x]/GoldenRatio^x)/Sqrt[5]; x0 = x /. FindRoot[F'[x], {x, 1}, WorkingPrecision -> digits+1]; RealDigits[x0, 10, digits][[1]] (* Jean-François Alcover, Jan 28 2014 *)
PROG
(Other) Gerd Lamprecht online Iterationsrechner: #(@P@C1], x+x)*@C2]+cos(x*PI)*@C2]+sin(x*PI)*PI)*@P@C1], -x)/@C0]@N@C0]=@Q5); @C1]=@C0]/2+0.5; @C2]=log(@C1]); @B1]=1.09; @B2]=1.1; @B3]=Fx(@B1]); @B4]=Fx(@B2]); d=4e-16; IM=2; @N@B4]=Fx(@B2]); @B0]=(@B4]-@B3])/ (@B2]-@B1]); a=@B1]-@B3]/@B0]; b=Fx(a); if(b*@B4]%3C0){@B1]=@B2]; @B2]=a; @B3]=@B4]; }@F@B2]=a; @B3]*=@H2, @B4], b); }@N(@A@B4])%3Cd)@O(@A@B4])%3Cd)@O@A@B2]-@B1])%3Cd@N0@N1@Nif(@A@B4]) %3Cd)c=@B2]; @Eif(@A@B3])%3C1e-16)c=@B1]; @Ec=(@B1]+@B2])/2;
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 29 2010
EXTENSIONS
Edited, embedded JavaScript source code of URL removed - R. J. Mathar, Feb 02 2010
STATUS
approved