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The num library: arbitrary-precision rational arithmetic
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<H1>Chapter 19: The num library: arbitrary-precision rational arithmetic</H1>
The <TT>num</TT> library implements
exact-precision rational arithmetic. It is built upon the
state-of-the-art BigNum arbitrary-precision integer arithmetic
package, and therefore achieves very high performance.<BR>
<BR>
The functions provided in this library are fully documented in
<EM>The CAML Numbers Reference Manual</EM> by
Valrie Mnissier-Morain, technical report 141, INRIA, july 1992
(available electronically,
<TT>ftp://ftp.inria.fr/INRIA/publication/RT/RT-0141.ps.gz</TT>).
A summary of the functions is given below.<BR>
<BR>
Programs that use the <TT>num</TT> library must be linked in ``custom
runtime'' mode, as follows:
<PRE>
ocamlc -custom <I>other options</I> nums.cma <I>other files</I> -cclib -lnums
ocamlopt <I>other options</I> nums.cmxa <I>other files</I> -cclib -lnums
</PRE>
For interactive use of the <TT>nums</TT> library, do:
<PRE>
ocamlmktop -custom -o mytop nums.cma -cclib -lnums
./mytop
</PRE>
<H2>19.1 Module <TT>Num</TT>: operation on arbitrary-precision numbers</H2><A NAME="s:Num"></A>
<A NAME="@manual669"></A><PRE>
open Nat
open Big_int
open Ratio
</PRE>
<BLOCKQUOTE>
Numbers (type <CODE>num</CODE>) are arbitrary-precision rational numbers,
plus the special elements <CODE>1/0</CODE> (infinity) and <CODE>0/0</CODE> (undefined).
</BLOCKQUOTE>
<PRE>
type num = Int of int | Big_int of big_int | Ratio of ratio
</PRE>
<BLOCKQUOTE>
The type of numbers.
</BLOCKQUOTE>
<BLOCKQUOTE>
Arithmetic operations
</BLOCKQUOTE>
<PRE>
val (+/) : num -> num -> num
val add_num : num -> num -> num
</PRE>
<A NAME="@manual670"></A><A NAME="@manual671"></A><BLOCKQUOTE>
Addition
</BLOCKQUOTE>
<PRE>
val minus_num : num -> num
</PRE>
<A NAME="@manual672"></A><BLOCKQUOTE>
Unary negation.
</BLOCKQUOTE>
<PRE>
val (-/) : num -> num -> num
val sub_num : num -> num -> num
</PRE>
<A NAME="@manual673"></A><A NAME="@manual674"></A><BLOCKQUOTE>
Subtraction
</BLOCKQUOTE>
<PRE>
val (*/) : num -> num -> num
val mult_num : num -> num -> num
</PRE>
<A NAME="@manual675"></A><A NAME="@manual676"></A><BLOCKQUOTE>
Multiplication
</BLOCKQUOTE>
<PRE>
val square_num : num -> num
</PRE>
<A NAME="@manual677"></A><BLOCKQUOTE>
Squaring
</BLOCKQUOTE>
<PRE>
val (//) : num -> num -> num
val div_num : num -> num -> num
</PRE>
<A NAME="@manual678"></A><A NAME="@manual679"></A><BLOCKQUOTE>
Division
</BLOCKQUOTE>
<PRE>
val quo_num : num -> num -> num
val mod_num : num -> num -> num
</PRE>
<A NAME="@manual680"></A><A NAME="@manual681"></A><BLOCKQUOTE>
Euclidean division: quotient and remainder
</BLOCKQUOTE>
<PRE>
val (**/) : num -> num -> num
val power_num : num -> num -> num
</PRE>
<A NAME="@manual682"></A><A NAME="@manual683"></A><BLOCKQUOTE>
Exponentiation
</BLOCKQUOTE>
<PRE>
val is_integer_num : num -> bool
</PRE>
<A NAME="@manual684"></A><BLOCKQUOTE>
Test if a number is an integer
</BLOCKQUOTE>
<PRE>
val integer_num : num -> num
val floor_num : num -> num
val round_num : num -> num
val ceiling_num : num -> num
</PRE>
<A NAME="@manual685"></A><A NAME="@manual686"></A><A NAME="@manual687"></A><A NAME="@manual688"></A><BLOCKQUOTE>
Approximate a number by an integer.
<CODE>floor_num n</CODE> returns the largest integer smaller or equal to <CODE>n</CODE>.
<CODE>ceiling_num n</CODE> returns the smallest integer bigger or equal to <CODE>n</CODE>.
<CODE>integer_num n</CODE> returns the integer closest to <CODE>n</CODE>. In case of ties,
rounds towards zero.
<CODE>round_num n</CODE> returns the integer closest to <CODE>n</CODE>. In case of ties,
rounds off zero.
</BLOCKQUOTE>
<PRE>
val sign_num : num -> int
</PRE>
<A NAME="@manual689"></A><BLOCKQUOTE>
Return <CODE>-1</CODE>, <CODE>0</CODE> or <CODE>1</CODE> according to the sign of the argument.
</BLOCKQUOTE>
<PRE>
val (=/) : num -> num -> bool
val (</) : num -> num -> bool
val (>/) : num -> num -> bool
val (<=/) : num -> num -> bool
val (>=/) : num -> num -> bool
val (<>/) : num -> num -> bool
val eq_num : num -> num -> bool
val lt_num : num -> num -> bool
val le_num : num -> num -> bool
val gt_num : num -> num -> bool
val ge_num : num -> num -> bool
</PRE>
<A NAME="@manual690"></A><A NAME="@manual691"></A><A NAME="@manual692"></A><A NAME="@manual693"></A><A NAME="@manual694"></A><A NAME="@manual695"></A><A NAME="@manual696"></A><A NAME="@manual697"></A><A NAME="@manual698"></A><A NAME="@manual699"></A><A NAME="@manual700"></A><BLOCKQUOTE>
Usual comparisons between numbers
</BLOCKQUOTE>
<PRE>
val compare_num : num -> num -> int
</PRE>
<A NAME="@manual701"></A><BLOCKQUOTE>
Return <CODE>-1</CODE>, <CODE>0</CODE> or <CODE>1</CODE> if the first argument is less than,
equal to, or greater than the second argument.
</BLOCKQUOTE>
<PRE>
val max_num : num -> num -> num
val min_num : num -> num -> num
</PRE>
<A NAME="@manual702"></A><A NAME="@manual703"></A><BLOCKQUOTE>
Return the greater (resp. the smaller) of the two arguments.
</BLOCKQUOTE>
<PRE>
val abs_num : num -> num
</PRE>
<A NAME="@manual704"></A><BLOCKQUOTE>
Absolute value.
</BLOCKQUOTE>
<PRE>
val succ_num: num -> num
</PRE>
<A NAME="@manual705"></A><BLOCKQUOTE>
<CODE>succ n</CODE> is <CODE>n+1</CODE>
</BLOCKQUOTE>
<PRE>
val pred_num: num -> num
</PRE>
<A NAME="@manual706"></A><BLOCKQUOTE>
<CODE>pred n</CODE> is <CODE>n-1</CODE>
</BLOCKQUOTE>
<PRE>
val incr_num: num ref -> unit
</PRE>
<A NAME="@manual707"></A><BLOCKQUOTE>
<CODE>incr r</CODE> is <CODE>r:=!r+1</CODE>, where <CODE>r</CODE> is a reference to a number.
</BLOCKQUOTE>
<PRE>
val decr_num: num ref -> unit
</PRE>
<A NAME="@manual708"></A><BLOCKQUOTE>
<CODE>decr r</CODE> is <CODE>r:=!r-1</CODE>, where <CODE>r</CODE> is a reference to a number.
</BLOCKQUOTE>
<BLOCKQUOTE>
Coercions with strings
</BLOCKQUOTE>
<PRE>
val string_of_num : num -> string
</PRE>
<A NAME="@manual709"></A><BLOCKQUOTE>
Convert a number to a string, using fractional notation.
</BLOCKQUOTE>
<PRE>
val approx_num_fix : int -> num -> string
val approx_num_exp : int -> num -> string
</PRE>
<A NAME="@manual710"></A><A NAME="@manual711"></A><BLOCKQUOTE>
Approximate a number by a decimal. The first argument is the
required precision. The second argument is the number to
approximate. <CODE>approx_fix</CODE> uses decimal notation; the first
argument is the number of digits after the decimal point.
<CODE>approx_exp</CODE> uses scientific (exponential) notation; the
first argument is the number of digits in the mantissa.
</BLOCKQUOTE>
<PRE>
val num_of_string : string -> num
</PRE>
<A NAME="@manual712"></A><BLOCKQUOTE>
Convert a string to a number.
</BLOCKQUOTE>
<BLOCKQUOTE>
Coercions between numerical types
</BLOCKQUOTE>
<PRE>
val int_of_num : num -> int
val num_of_int : int -> num
val nat_of_num : num -> nat
val num_of_nat : nat -> num
val num_of_big_int : big_int -> num
val big_int_of_num : num -> big_int
val ratio_of_num : num -> ratio
val num_of_ratio : ratio -> num
val float_of_num : num -> float
</PRE>
<A NAME="@manual713"></A><A NAME="@manual714"></A><A NAME="@manual715"></A><A NAME="@manual716"></A><A NAME="@manual717"></A><A NAME="@manual718"></A><A NAME="@manual719"></A><A NAME="@manual720"></A><A NAME="@manual721"></A>
<H2>19.2 Module <TT>Arith_status</TT>: flags that control rational arithmetic</H2><A NAME="s:Arithstatus"></A>
<A NAME="@manual722"></A><PRE>
val arith_status: unit -> unit
</PRE>
<A NAME="@manual723"></A><BLOCKQUOTE>
Print the current status of the arithmetic flags.
</BLOCKQUOTE>
<PRE>
val get_error_when_null_denominator : unit -> bool
val set_error_when_null_denominator : bool -> unit
</PRE>
<A NAME="@manual724"></A><A NAME="@manual725"></A><BLOCKQUOTE>
Get or set the flag <CODE>null_denominator</CODE>. When on, attempting to
create a rational with a null denominator raises an exception.
When off, rationals with null denominators are accepted.
Initially: on.
</BLOCKQUOTE>
<PRE>
val get_normalize_ratio : unit -> bool
val set_normalize_ratio : bool -> unit
</PRE>
<A NAME="@manual726"></A><A NAME="@manual727"></A><BLOCKQUOTE>
Get or set the flag <CODE>normalize_ratio</CODE>. When on, rational
numbers are normalized after each operation. When off,
rational numbers are not normalized until printed.
Initially: off.
</BLOCKQUOTE>
<PRE>
val get_normalize_ratio_when_printing : unit -> bool
val set_normalize_ratio_when_printing : bool -> unit
</PRE>
<A NAME="@manual728"></A><A NAME="@manual729"></A><BLOCKQUOTE>
Get or set the flag <CODE>normalize_ratio_when_printing</CODE>.
When on, rational numbers are normalized before being printed.
When off, rational numbers are printed as is, without normalization.
Initially: on.
</BLOCKQUOTE>
<PRE>
val get_approx_printing : unit -> bool
val set_approx_printing : bool -> unit
</PRE>
<A NAME="@manual730"></A><A NAME="@manual731"></A><BLOCKQUOTE>
Get or set the flag <CODE>approx_printing</CODE>.
When on, rational numbers are printed as a decimal approximation.
When off, rational numbers are printed as a fraction.
Initially: off.
</BLOCKQUOTE>
<PRE>
val get_floating_precision : unit -> int
val set_floating_precision : int -> unit
</PRE>
<A NAME="@manual732"></A><A NAME="@manual733"></A><BLOCKQUOTE>
Get or set the parameter <CODE>floating_precision</CODE>.
This parameter is the number of digits displayed when
<CODE>approx_printing</CODE> is on.
Initially: 12.
</BLOCKQUOTE>
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