arb-class: Arbitrary Precision Floating-Point Real Numbers with Error Bounds

Description

Class arb extends virtual class flint. It represents vectors of arbitrary precision floating-point real numbers with error bounds. Elements are specified by a pair of mixed format floating-point numbers: an arf midpoint and a mag radius.

Arithmetic on arb vectors is midpoint-radius interval arithmetic, also known as ball arithmetic, enabling computation with rigorous propagation of errors. Logic and comparison involving arb vectors are defined as follows: unary op(x) is true if and only if op is true for all elements of the interval x, and binary op(x, y) is true if and only if op is true for all elements of the Cartesian product of the intervals x and y. A corollary is that the operator <= does not define a total order on the range of arb (that is, the set of intervals \([m-r,m+r]\)), and a consequence is that methods for generic functions that necessitate a total order tend to signal an error.

Usage

## The class generator function:
.arb(...)
## Mode 1: initialize with zeros
## .arb(length = 0L)
##
## Mode 2: initialize with vector
## .arb(length = length(x), x)
##
## Mode 3: initialize by parts
## .arb(length = max0(length(mid), length(rad)), mid, rad)
##
## where max0 <- function(m, n) if (min(m, n)) max(m, n) else 0L

Value

The class generator function returns new("arb", ...).

Arguments

...: arguments passed to methods for initialize.

Slots

.xData, names: inherited from virtual class flint.

Methods

Due to constraints imposed by generic functions, methods typically do not provide a formal argument prec allowing for a precision to be indicated in the function call. Such methods use the current default precision set by flintPrec.

!: signature(x = "arb"):
equivalent to (but faster than) x == 0.
+: signature(e1 = "arb", e2 = "missing"):
returns a copy of the argument.
-: signature(e1 = "arb", e2 = "missing"):
returns the negation of the argument.
Complex: signature(z = "arb"):
mathematical functions of one argument; see S4groupGeneric.
Math: signature(x = "arb"):
mathematical functions of one argument; see S4groupGeneric.
Math2: signature(x = "arb"):
decimal rounding according to a second argument digits; see S4groupGeneric. There are just two member member functions: round, signif.
Ops: signature(e1 = "ANY", e2 = "arb"):
signature(e1 = "arb", e2 = "ANY"):
binary arithmetic, comparison, and logical operators; see S4groupGeneric. The “other” operand must be atomic or inherit from virtual class flint. The operands are recycled and promoted as necessary.
Summary: signature(x = "arb"):
univariate summary statistics; see S4groupGeneric. The return value is a logical vector of length 1 (any, all) or an arb vector of length 1 or 2 (sum, prod, min, max, range).
anyNA: signature(x = "arb"):
returns TRUE if any element of x has midpoint NaN, FALSE otherwise.
as.vector: signature(x = "arb"):
returns as.vector(y, mode), where y is a double vector containing the result of converting the midpoints of x to the range of double, rounding if the value is not exactly representable in double precision. The rounding mode is to the nearest representable number (with precedence to even significands in case of ties), unless a midpoint exceeds .Machine[["double.xmax"]] in absolute value, in which case -Inf or Inf is introduced with a warning. Coercion to types "character", "symbol" (synonym "name"), "pairlist", "list", and "expression", which are not “number-like”, is handled specially.
coerce: signature(from = "ANY", to = "arb"):
returns .arb(x = from). An error is signaled if the class or type of from is not supported by the method for initialize.
format: signature(x = "arb"):
returns a character vector suitable for printing, using string format "(m +/- r)" and scientific format for m and r. Optional arguments control the output; see format-methods.
initialize: signature(.Object = "arb", length = "numeric"):
returns .Object after setting its .xData slot. The new slot value is an external pointer to an array of the corresponding C type, which is newly allocated and initialized entirely with zeros (zero midpoint, zero radius). .Object is not copied, so all references are affected.
initialize: signature(.Object = "arb", length = "numeric", x = "atomic|flint"):
as above, except that the new slot value points to an array initialized by the elements of x.
initialize: signature(.Object = "arb", length = "numeric", mid = "atomic|flint", rad = "atomic|flint"):
as above, except that the new slot value points to an array whose midpoints and radii are initialized separately with values from mid and rad, which are coerced to classes arf and mag and recycled as necessary.
is.finite: signature(x = "arb"):
returns a logical vector indicating which elements of x do not have midpoint NaN, -Inf, or Inf or radius Inf.
is.infinite: signature(x = "arb"):
returns a logical vector indicating which elements of x have midpoint -Inf or Inf or radius Inf.
is.na, is.nan: signature(x = "arb"):
returns a logical vector indicating which elements of x have midpoint NaN.
is.unsorted: signature(x = "arb"):
signals an error indicating that <= is not a total order on the range of arb; see xtfrm below.
log: signature(x = "arb"):
returns the logarithm of the argument. The natural logarithm is computed by default (when optional argument base is unset).
mean: signature(x = "arb"):
returns the arithmetic mean.
xtfrm: signature(x = "arb"):
signals an error indicating that <= is not a total order on the range of arb: a <= b || b <= a is is not TRUE for all finite a and b of class arb. Thus, direct sorting of arb is not supported. Users wanting to order the midpoints should operate on Mid(x).

References

The FLINT documentation of the underlying C type: https://flintlib.org/doc/arb.html

Johansson, F. (2017). Arb: efficient arbitrary-precision midpoint-radius interval arithmetic. IEEE Transactions on Computers, 66(8), 1281-1292. tools:::Rd_expr_doi("10.1109/TC.2017.2690633")

Examples

Run this code

showClass("arb")
showMethods(classes = "arb")

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