Seeing this tweet sparked a thought that iteration could be cast as a sort of dual of recursion. The idea in @CentrlPotential’s tweet is called factoring out the characteristic equation of a recurrence relation.
The characteristic equation extracted out can be used to arrive at Binet’s formula via Umbral Calculus. This formula is used to calculate the nth Fibonacci number.
Conversely, golden ratio is approximated through iteration of ratios of successive Fibonacci numbers.
Making this post, transported me 8 years back to 2012, when I posted this question on Quora: quora.com/Can-all-iterat…
It would be cool if math equations are represented in a computation language which can be evaluated partially. Intermediate results would be stored as continuations which as per the user’s wish can be resumed at a different time or machine.
A compiler here would act as a bookkeeper of facts that will track fixed points for recurrence relation like fib(1) = 1. This means when fib(1) gets performed it will use the stored state. This idea eliminates the need for a certain class of conditionals in programs/equations.
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