[Haskell-cafe] Horner's Rule, foldl, and direct recursion

R J rj248842 at hotmail.com
Tue Mar 10 19:58:45 EDT 2009

Given a list of decimal digits represented by Integers between 0 and 9--for example, the list [1,2,3, 4]--with the high-order digit at the left, the list can be converted to a decimal integer n using the following formula, an instance of Horner's rule:

n = 10 *  10 * 10 * 1 + 10 * 10 * 2 + 10 * 3 + 4
= 10 * (10 * 10 * 1 + 10 * 2 + 3) + 4
= 10 * (10 *(10 * 1 + 2) + 3) + 4

In Haskell, the foldl function neatly captures this pattern:

horner          :: [Integer] -> Integer
horner          =  myFoldl timesPlus 0
where timesPlus x y = 10 * x + y

What is the direct recursive calculation of this function without using the call to foldl?  In other words, what's the second equation of:

horner2          :: [Integer] -> Integer
horner2 []       =  0
horner2 (x : xs) =  ?

Given that we've already got the definition using foldl, it ought to be easy to express the second equation, but it's eluding me.

Thanks.

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