Implement missing arc-tangent for NormedAlgebra.
This can be done using e.g. the Taylor series at 0:
sum_n (-1)^n/(2n+1)x^(2n+1)
coupled with range reductions:
atan (-x) = -atan(x)
atan(1/x) = pi/2 - atan(x) for x > 0
atan(x) = atan(a) + atan((x-a)/(1+ax)) mod pi
Alternatively, we can try to solve
tan(y)-x=0 or sin(y)-x cos(y)=0
using Newton's method, which respectively yield
y'=y-cos(x)(sin(x)-xcos(x)) and y'=y-(sin(y)-xcos(y))/(cos(y)+xsin(y))=y-(tan(y)-x)/(1+x*tan(y)) .
Unfortunately, it may not be possible to compute atan of a function with a wude range (or an operator with a wide spectral radius) directly using these techniques.
Implement missing arc-tangent for NormedAlgebra.
This can be done using e.g. the Taylor series at 0:
sum_n (-1)^n/(2n+1)x^(2n+1)
coupled with range reductions:
atan (-x) = -atan(x)
atan(1/x) = pi/2 - atan(x) for x > 0
atan(x) = atan(a) + atan((x-a)/(1+ax)) mod pi
Alternatively, we can try to solve
tan(y)-x=0 or sin(y)-x cos(y)=0
using Newton's method, which respectively yield
y'=y-cos(x)(sin(x)-xcos(x)) and y'=y-(sin(y)-xcos(y))/(cos(y)+xsin(y))=y-(tan(y)-x)/(1+x*tan(y)) .
Unfortunately, it may not be possible to compute atan of a function with a wude range (or an operator with a wide spectral radius) directly using these techniques.