Arbitrary Precision Floating-Point Real Numbers with Error Bounds
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 ), and a consequence is that methods for generic functions that necessitate a total order tend to signal an error.
class
## 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
...: arguments passed to methods for initialize.The class generator function returns new("arb", ...).
.xData, names: inherited from virtual class flint.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 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)`.
Virtual class flint. Generic functions Mid and Rad and their replacement forms for getting and setting midpoints and radii.
The FLINT documentation of the underlying
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")
showClass("arb") showMethods(classes = "arb")