Lambda Example: Generate Fibonacci series

davidleal 

I rarely use recursion these days, preferring instead to use Lambda helper functions.  Occasionally it might be a built-in SCAN or BYROW but, more often, it is a mocked-up version built from REDUCE/VSTACK since it is unusual to require an array of scalars; an array of arrays is far more useful and more frequently encountered.

For the Fibonacci series, one needs to take an array of 2 values forward so

Fibonacciλ(priorTerms, [currentData])
= LET(
    F₁, INDEX(priorTerms, 1), 
    F₂, INDEX(priorTerms, 2), 
    HSTACK(F₁ + F₂, F₁)
  )

The vertical SCAN is achieved by using

SCANV(initialValues, scannedArray, Fnλ)
= REDUCE(
     initialValues,
     SEQUENCE(1, ROWS(scannedArray)),
     LAMBDA(accumulation, parameter,
        LET(
            priorResult, CHOOSEROWS(accumulation, -1),
            currentData, CHOOSEROWS(scannedArray, parameter),
            newResult, Fnλ(priorResult, currentData),
            VSTACK(accumulation, newResult)
        )
     )
)

In this instance, the scanned array does not figure in the calculation so the function could be simplified, but at the cost of generality.  For the Fibonacci series only one column of the results is needed, so

= LET(
    k, SEQUENCE(n-1),
    f, SCANV({1,1}, k, Fibonacciλ),
    TAKE(f,,-1)
  )

returns the first 1,000 Fibonacci numbers in 100 ms.  I couldn't extend the test that much without hitting #NUM! overflow errors.

Out of interest, I had lifted the SCANV function directly from a workbook that calculated the motion of a container slung from an overhead crane, by tenths of a second for 20 secs.  That required 4 variables at each step, namely the cart position, the pendulum angle subtended by the container and the corresponding velocities.  The name of the game was to provide force inputs that led to a final state in which the container was hanging vertically with zero velocity since placing a swinging cargo on a truck is tricky.

The parameters were optimised by using Solver to reduce the terminal energy of the system, though lori_m has written optimisation routines as Lambda functions!

= SCANV(X₀, t, RK4Stepλ(Dλ))