Re: [Haskell-cafe] What is a safe Haskell data type to store and manipulate Money values?

On 7/04/2017, at 10:42 AM, Bardur Arantsson wrote:

On 2017-04-07 00:35, Richard A. O'Keefe wrote:

...

For what it's worth, there are rounding algorithms that work on a whole bunch of numbers at once, ensuring that the total of the rounded numbers is equal to the total of the unrounded numbers.

REFERENCES TO LITERATURE, PLEASE!

(I'm not being sarcastic -- this would be very useful to almost anyone dealing with monetary amounts... anywhere.)

I first saw mention of such an algorithm in a journal about 40 years ago. I have a more recent memory of a paper by Knuth in a volume of his collected papers that I cannot at the moment locate, but I believe this is the article: Two-Way Rounding Author: Donald E. Knuth Published in: · Journal SIAM Journal on Discrete Mathematics archive Volume 8 Issue 2, May 1995 Pages 281 - 290 Society for Industrial and Applied Mathematics Philadelphia, PA, USA table of contents doi>10.1137/S0895480194264757 Ah, FOUND IT. Selected Papers on Design of Algorithms Donald E. Knuth CSL Lecture Notes Number 191 CSLI Publications, Stanford, California. ISBN-13: 978-1-57586-582-9 ISBN-10: 1-57686-582-3 Chapter 16, pp 219-234 "Two-Way Rounding" "Given n real numbers 0 <= x1 < 1, ..., 0 <= xn < 1, and a permutation \sigma of {1,...,n}, we can always find integers x'1 \in {0,1}, ..., x'n \in {0,1} so that the partial sums x'1+...+x'k and x'\sigma[1]+...+x'\sigma[k] differ from the unrounded values x1+...+xk and x\sigma[1...+x\sigma[k] by at most n/(n+1) for 1 <= k <= n. The latter bound is the best possible." Section 1, An Application "Sometimes it is desirable to round "spreadsheet" data to larger units while preserving row and column totals and the grand total." This application to matrices has been studied by others. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.421.2051&rep=rep1&type=pdf Rounding of Sequences and Matrices, with Applications "We show that any real matrix can be rounded to an integer matrix in such a way that the rounding errors of all row sums are less than one, and the rounding errors of all column sums as well as all sums of consecutive row entries are less than two. Such roundings can be computed in linear time." http://people.mpi-inf.mpg.de/~doerr/papers/unbimatround.pdf Unbiased Matrix Rounding "We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. ... our roundings also have a rounding error of less than one in all initial intervals of rows and columns. Consequently, arbitrary intervals have an error of at most two. ... The same result can be obtained via (dependent) randomized rounding. This has the additional advantage that the rounding is unbiased." Here's a bunch of less formal discussions of the 1D case. http://stackoverflow.com/questions/32544646/round-vector-of-numerics-to-inte... http://stackoverflow.com/questions/792460/how-to-round-floats-to-integers-wh... https://explainextended.com/2009/09/21/rounding-numbers-preserving-their-sum... https://biostatmatt.com/archives/2902