Package 'PreciseSums'

Title: Accurate Floating Point Sums and Products
Description: Most of the time floating point arithmetic does approximately the right thing. When adding sums or having products of numbers that greatly differ in magnitude, the floating point arithmetic may be incorrect. This package implements the Kahan (1965) sum <doi:10.1145/363707.363723>, Neumaier (1974) sum <doi:10.1002/zamm.19740540106>, pairwise-sum (adapted from 'NumPy', See Castaldo (2008) <doi:10.1137/070679946> for a discussion of accuracy), and arbitrary precision sum (adapted from the fsum in 'Python' ; Shewchuk (1997) <https://people.eecs.berkeley.edu/~jrs/papers/robustr.pdf>). In addition, products are changed to long double precision for accuracy, or changed into a log-sum for accuracy.
Authors: Matthew Fidler [aut, cre, cph], Raymond Hettinger [cph, aut], Jonathan Shewchuk [cph, aut], Julian Taylor [cph, aut], Nathaniel Smith [cph, aut], NumPy Team [cph], Python Team [cph]
Maintainer: Matthew Fidler <[email protected]>
License: GPL (>= 2)
Version: 0.7
Built: 2024-12-16 03:43:03 UTC
Source: https://github.com/nlmixr2/PreciseSums

Help Index


Return an accurate floating point sum of values

Description

This method avoids loss of precision by tracking multiple intermediate partial sums. Based on python's math.fsum

Usage

fsum(numbers)

Arguments

numbers

A vector of numbers to sum.

Value

Sum of numbers without loss of precision

The algorithm's accuracy depends on IEEE-754 arithmetic guarantees and the typical case where the rounding mode is half-even. On some non-Windows builds, the underlying C library uses extended precision addition and may occasionally double-round an intermediate sum causing it to be off in its least significant bit.

Author(s)

Matthew Fidler (R implementation), Raymond Hettinger, Jonathan Shewchuk, Python Team

References

https://docs.python.org/2/library/math.html https://github.com/python/cpython/blob/a0ce375e10b50f7606cb86b072fed7d8cd574fe7/Modules/mathmodule.c

Shewchuk, JR. (1996) Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates. http://www-2.cs.cmu.edu/afs/cs/project/quake/public/papers/robust-arithmetic.ps

Examples

sum(c(1,1e100,1,-1e100)) ## Should be 2, gives 0
fsum(c(1,1e100,1,-1e100)) ## Gives 2.

Using the Kahan method, take a more accurate sum

Description

Using the Kahan method, take a more accurate sum

Usage

kahanSum(numbers)

Arguments

numbers

A vector of numbers to sum.

Value

Sum of numbers

References

https://en.wikipedia.org/wiki/Kahan_summation_algorithm

Examples

sum(c(1,1e100,1,-1e100)) ## Should be 2, gives 0
kahanSum(c(1,1e100,1,-1e100)) ## Not accurate enough for the correct result. (still = 0)

Using the Neumaier method, take a more accurate sum

Description

Using the Neumaier method, take a more accurate sum

Usage

neumaierSum(numbers)

Arguments

numbers

A vector of numbers to sum.

Value

Sum of numbers, a bit more accurate than kahanSum

References

https://en.wikipedia.org/wiki/Kahan_summation_algorithm

Examples

sum(c(1,1e100,1,-1e100)) ## Should be 2, gives 0
neumaierSum(c(1,1e100,1,-1e100)) ## Gives 2

Return an accurate floating point sum of values

Description

This was taken by NumPy and adapted for use here. It is more accurate than a standard sum, but still has numerical issues. Its main benefit is that it is about the same amount of time as a standard time with the added accuracy.

Usage

pairwiseSum(numbers)

Arguments

numbers

A vector of numbers to sum.

Value

A sum of numbers with a rounding error of O(lg n) instead of O(n).

Author(s)

Matthew Fidler (R implementation), Julian Taylor, Nathaniel J Smith, and NumPy team. #' @examples sum(c(1,1e100,1,-1e100)) ## Should be 2, gives 0 pairwiseSum(c(1,1e100,1,-1e100)) ## Should be 2, still 0

References

https://github.com/juliantaylor/numpy/blob/b0bc01275cac04483e6df021211c1fa2ba65eaa3/numpy/core/src/umath/loops.c.src

https://github.com/numpy/numpy/pull/3685


Using PreciceSums's default method, take a product

Description

Using PreciceSums's default method, take a product

Usage

psProd(numbers)

Arguments

numbers

A vector of numbers to sum.

Value

Product of numbers


Choose the type of product to use in PreciceSums. These are used in the PreciceSums prod blocks

Description

Choose the type of product to use in PreciceSums. These are used in the PreciceSums prod blocks

Usage

psSetProd(type = c("long double", "double", "logify"))

Arguments

type

Product to use for prod() in PreciceSums blocks

long double converts to long double, performs the multiplication and then converts back.

double uses the standard double scale for multiplication.

Value

nothing

Author(s)

Matthew L. Fidler


Choose the type of sums to use for PreciceSums.

Description

Choose the types of sums to use in PreciceSums. These are used in the PreciceSums sum blocks and the psSum function

Usage

psSetSum(type = c("pairwise", "fsum", "kahan", "neumaier", "klein", "c"))

Arguments

type

Sum type to use for psSum and sum() in PreciceSums code blocks.

pairwise uses the pairwise sum (fast, default)

fsum uses Python's fsum function (most accurate)

kahan uses kahan correction

neumaier uses Neumaier correction

klein uses Klien correction

c uses no correction, bud default/native summing

Value

nothing

Author(s)

Matthew L. Fidler


Using PreciceSums's default method, take a sum

Description

Using PreciceSums's default method, take a sum

Usage

psSum(numbers)

Arguments

numbers

A vector of numbers to sum.

Value

Sum of numbers