1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
|
\name{sumexact, cumsumexact}
\alias{sumexact}
\alias{cumsumexact}
\title{Basic Sum Operations without Round-off Errors}
\description{
Functions for performing basic sum operations without round-off errors
}
\usage{
sumexact(..., na.rm = FALSE)
cumsumexact(x)
}
\arguments{
\item{x}{numeric vector}
\item{...}{numeric vector(s), numbers or other objects to be summed}
\item{na.rm}{logical. Should missing values be removed?}
}
\details{
All three functions use full precision summation using multiple doubles for
intermediate values. The sum of numbers x & y is a=x+y with error term
b=error(a+b). That way a+b is equal exactly x+y, so sum of 2 numbers is stored
as 2 or fewer values, which when added would under-flow. By extension sum of n
numbers is calculated with intermediate results stored as array of numbers
that can not be added without introducing an error. Only final result is
converted to a single number
}
\value{
Function \code{sumexact} returns single number. Function \code{cumsumexact}
returns vector of the same length as \code{x}.
}
\references{
Round-off error correction is based on:
Shewchuk, Jonathan, \emph{Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates}
McCullough, D.B., (1998) \emph{Assessing the Reliability of Statistical
Software, Part I}, The American Statistician, Vol. 52 No.
McCullough, D.B., (1999) \emph{Assessing the Reliability of Statistical
Software, Part II}, The American Statistician, Vol. 53 No 2
NIST Statistical Reference Datasets (StRD) website
}
\author{Jarek Tuszynski (SAIC) \email{jaroslaw.w.tuszynski@saic.com} based on
code by Vadim Ogranovich, which is based on algorithms described in
references, pointed out by Gabor Grothendieck.
}
\seealso{
\itemize{
\item \code{\link[base]{sum}} is faster but not error-save version of
\code{sumexact}
\item \code{\link[base]{cumsum}} is equivalent to \code{cumsumexact}
}
}
\examples{
x = c(1, 1e20, 1e40, -1e40, -1e20, -1)
a = sum(x); print(a)
b = sumexact(x); print(b)
stopifnot(b==0)
a = cumsum(x); print(a)
b = cumsumexact(x); print(b)
stopifnot(b[6]==0)
}
\keyword{ts}
\keyword{smooth}
\keyword{array}
\keyword{utilities}
\concept{sum with no round-off errors}
\concept{cumulative sum with no round-off errors}
|
