math::bigfloat(n)              Tcl Math Library              math::bigfloat(n)



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NAME
       math::bigfloat - Arbitrary precision floating-point numbers

SYNOPSIS
       package require Tcl  8.5

       package require math::bigfloat  ?2.0.1?

       fromstr number ?trailingZeros?

       tostr ?-nosci? number

       fromdouble double ?decimals?

       todouble number

       isInt number

       isFloat number

       int2float integer ?decimals?

       add x y

       sub x y

       mul x y

       div x y

       mod x y

       abs x

       opp x

       pow x n

       iszero x

       equal x y

       compare x y

       sqrt x

       log x

       exp x

       cos x

       sin x

       tan x

       cotan x

       acos x

       asin x

       atan x

       cosh x

       sinh x

       tanh x

       pi n

       rad2deg radians

       deg2rad degrees

       round x

       ceil x

       floor x

_________________________________________________________________

DESCRIPTION
       The bigfloat package provides arbitrary precision floating-point math
       capabilities to the Tcl language. It is designed to work with Tcl 8.5,
       but for Tcl 8.4 is provided an earlier version of this package.  See
       WHAT ABOUT TCL 8.4 ? for more explanations.  By convention, we will
       talk about the numbers treated in this library as :

       o      BigFloat for floating-point numbers of arbitrary length.

       o      integers for arbitrary length signed integers, just as basic
              integers since Tcl 8.5.

       Each BigFloat is an interval, namely [m-d, m+d], where m is the
       mantissa and d the uncertainty, representing the limitation of that
       number's precision.  This is why we call such mathematics interval
       computations.  Just take an example in physics : when you measure a
       temperature, not all digits you read are significant. Sometimes you
       just cannot trust all digits - not to mention if doubles (f.p. numbers)
       can handle all these digits.  BigFloat can handle this problem -
       trusting the digits you get - plus the ability to store numbers with an
       arbitrary precision.  BigFloats are internally represented at Tcl
       lists: this package provides a set of procedures operating against the
       internal representation in order to :

       o      perform math operations on BigFloats and (optionnaly) with
              integers.

       o      convert BigFloats from their internal representations to
              strings, and vice versa.

INTRODUCTION
       fromstr number ?trailingZeros?
              Converts number into a BigFloat. Its precision is at least the
              number of digits provided by number.  If the number contains
              only digits and eventually a minus sign, it is considered as an
              integer. Subsequently, no conversion is done at all.

              trailingZeros - the number of zeros to append at the end of the
              floating-point number to get more precision. It cannot be
              applied to an integer.


              # x and y are BigFloats : the first string contained a dot, and the second an e sign
              set x [fromstr -1.000000]
              set y [fromstr 2000e30]
              # let's see how we get integers
              set t 20000000000000
              # the old way (package 1.2) is still supported for backwards compatibility :
              set m [fromstr 10000000000]
              # but we do not need fromstr for integers anymore
              set n -39
              # t, m and n are integers


       The number's last digit is considered by the procedure to be true at
       +/-1, For example, 1.00 is the interval [0.99, 1.01], and 0.43 the
       interval [0.42, 0.44].  The Pi constant may be approximated by the
       number "3.1415".  This string could be considered as the interval
       [3.1414 , 3.1416] by fromstr.  So, when you mean 1.0 as a double, you
       may have to write 1.000000 to get enough precision.  To learn more
       about this subject, see PRECISION.

       For example :


              set x [fromstr 1.0000000000]
              # the next line does the same, but smarter
              set y [fromstr 1. 10]


       tostr ?-nosci? number
              Returns a string form of a BigFloat, in which all digits are
              exacts.  All exact digits means a rounding may occur, for
              example to zero, if the uncertainty interval does not clearly
              show the true digits.  number may be an integer, causing the
              command to return exactly the input argument.  With the -nosci
              option, the number returned is never shown in scientific
              notation, i.e. not like '3.4523e+5' but like '345230.'.


              puts [tostr [fromstr 0.99999]] ;# 1.0000
              puts [tostr [fromstr 1.00001]] ;# 1.0000
              puts [tostr [fromstr 0.002]] ;# 0.e-2


              See PRECISION for that matter.  See also iszero for how to
              detect zeros, which is useful when performing a division.

       fromdouble double ?decimals?
              Converts a double (a simple floating-point value) to a BigFloat,
              with exactly decimals digits.  Without the decimals argument, it
              behaves like fromstr.  Here, the only important feature you
              might care of is the ability to create BigFloats with a fixed
              number of decimals.


              tostr [fromstr 1.111 4]
              # returns : 1.111000 (3 zeros)
              tostr [fromdouble 1.111 4]
              # returns : 1.111


       todouble number
              Returns a double, that may be used in expr, from a BigFloat.

       isInt number
              Returns 1 if number is an integer, 0 otherwise.

       isFloat number
              Returns 1 if number is a BigFloat, 0 otherwise.

       int2float integer ?decimals?
              Converts an integer to a BigFloat with decimals trailing zeros.
              The default, and minimal, number of decimals is 1.  When
              converting back to string, one decimal is lost:


              set n 10
              set x [int2float $n]; # like fromstr 10.0
              puts [tostr $x]; # prints "10."
              set x [int2float $n 3]; # like fromstr 10.000
              puts [tostr $x]; # prints "10.00"


ARITHMETICS
       add x y

       sub x y

       mul x y
              Return the sum, difference and product of x by y.  x - may be
              either a BigFloat or an integer y - may be either a BigFloat or
              an integer When both are integers, these commands behave like
              expr.

       div x y

       mod x y
              Return the quotient and the rest of x divided by y.  Each
              argument (x and y) can be either a BigFloat or an integer, but
              you cannot divide an integer by a BigFloat Divide by zero throws
              an error.

       abs x  Returns the absolute value of x

       opp x  Returns the opposite of x

       pow x n
              Returns x taken to the nth power.  It only works if n is an
              integer.  x might be a BigFloat or an integer.

COMPARISONS
       iszero x
              Returns 1 if x is :

              o      a BigFloat close enough to zero to raise "divide by
                     zero".

              o      the integer 0.

              See here how numbers that are close to zero are converted to
              strings:


              tostr [fromstr 0.001] ; # -> 0.e-2
              tostr [fromstr 0.000000] ; # -> 0.e-5
              tostr [fromstr -0.000001] ; # -> 0.e-5
              tostr [fromstr 0.0] ; # -> 0.
              tostr [fromstr 0.002] ; # -> 0.e-2

              set a [fromstr 0.002] ; # uncertainty interval : 0.001, 0.003
              tostr  $a ; # 0.e-2
              iszero $a ; # false

              set a [fromstr 0.001] ; # uncertainty interval : 0.000, 0.002
              tostr  $a ; # 0.e-2
              iszero $a ; # true


       equal x y
              Returns 1 if x and y are equal, 0 elsewhere.

       compare x y
              Returns 0 if both BigFloat arguments are equal, 1 if x is
              greater than y, and -1 if x is lower than y.  You would not be
              able to compare an integer to a BigFloat : the operands should
              be both BigFloats, or both integers.

ANALYSIS
       sqrt x

       log x

       exp x

       cos x

       sin x

       tan x

       cotan x

       acos x

       asin x

       atan x

       cosh x

       sinh x

       tanh x The above functions return, respectively, the following : square
              root, logarithm, exponential, cosine, sine, tangent, cotangent,
              arc cosine, arc sine, arc tangent, hyperbolic cosine, hyperbolic
              sine, hyperbolic tangent, of a BigFloat named x.

       pi n   Returns a BigFloat representing the Pi constant with n digits
              after the dot.  n is a positive integer.

       rad2deg radians

       deg2rad degrees
              radians - angle expressed in radians (BigFloat)

              degrees - angle expressed in degrees (BigFloat)

              Convert an angle from radians to degrees, and vice versa.

ROUNDING
       round x

       ceil x

       floor x
              The above functions return the x BigFloat, rounded like with the
              same mathematical function in expr, and returns it as an
              integer.

PRECISION
       How do conversions work with precision ?

       o      When a BigFloat is converted from string, the internal
              representation holds its uncertainty as 1 at the level of the
              last digit.

       o      During computations, the uncertainty of each result is
              internally computed the closest to the reality, thus saving the
              memory used.

       o      When converting back to string, the digits that are printed are
              not subject to uncertainty. However, some rounding is done, as
              not doing so causes severe problems.

       Uncertainties are kept in the internal representation of the number ;
       it is recommended to use tostr only for outputting data (on the screen
       or in a file), and NEVER call fromstr with the result of tostr.  It is
       better to always keep operands in their internal representation.  Due
       to the internals of this library, the uncertainty interval may be
       slightly wider than expected, but this should not cause false digits.

       Now you may ask this question : What precision am I going to get after
       calling add, sub, mul or div?  First you set a number from the string
       representation and, by the way, its uncertainty is set:


              set a [fromstr 1.230]
              # $a belongs to [1.229, 1.231]
              set a [fromstr 1.000]
              # $a belongs to [0.999, 1.001]
              # $a has a relative uncertainty of 0.1% : 0.001(the uncertainty)/1.000(the medium value)

       The uncertainty of the sum, or the difference, of two numbers, is the
       sum of their respective uncertainties.


              set a [fromstr 1.230]
              set b [fromstr 2.340]
              set sum [add $a $b]]
              # the result is : [3.568, 3.572] (the last digit is known with an uncertainty of 2)
              tostr $sum ; # 3.57

       But when, for example, we add or substract an integer to a BigFloat,
       the relative uncertainty of the result is unchanged. So it is desirable
       not to convert integers to BigFloats:


              set a [fromstr 0.999999999]
              # now something dangerous
              set b [fromstr 2.000]
              # the result has only 3 digits
              tostr [add $a $b]

              # how to keep precision at its maximum
              puts [tostr [add $a 2]]


       For multiplication and division, the relative uncertainties of the
       product or the quotient, is the sum of the relative uncertainties of
       the operands.  Take care of division by zero : check each divider with
       iszero.


              set num [fromstr 4.00]
              set denom [fromstr 0.01]

              puts [iszero $denom];# true
              set quotient [div $num $denom];# error : divide by zero

              # opposites of our operands
              puts [compare $num [opp $num]]; # 1
              puts [compare $denom [opp $denom]]; # 0 !!!
              # No suprise ! 0 and its opposite are the same...

       Effects of the precision of a number considered equal to zero to the
       cos function:


              puts [tostr [cos [fromstr 0. 10]]]; # -> 1.000000000
              puts [tostr [cos [fromstr 0. 5]]]; # -> 1.0000
              puts [tostr [cos [fromstr 0e-10]]]; # -> 1.000000000
              puts [tostr [cos [fromstr 1e-10]]]; # -> 1.000000000

       BigFloats with different internal representations may be converted to
       the same string.

       For most analysis functions (cosine, square root, logarithm, etc.),
       determining the precision of the result is difficult.  It seems however
       that in many cases, the loss of precision in the result is of one or
       two digits.  There are some exceptions : for example,


              tostr [exp [fromstr 100.0 10]]
              # returns : 2.688117142e+43 which has only 10 digits of precision, although the entry
              # has 14 digits of precision.


WHAT ABOUT TCL 8.4 ?
       If your setup do not provide Tcl 8.5 but supports 8.4, the package can
       still be loaded, switching back to math::bigfloat 1.2. Indeed, an
       important function introduced in Tcl 8.5 is required - the ability to
       handle bignums, that we can do with expr.  Before 8.5, this ability was
       provided by several packages, including the pure-Tcl math::bignum
       package provided by tcllib.  In this case, all you need to know, is
       that arguments to the commands explained here, are expected to be in
       their internal representation.  So even with integers, you will need to
       call fromstr and tostr in order to convert them between string and
       internal representations.


              #
              # with Tcl 8.5
              # ============
              set a [pi 20]
              # round returns an integer and 'everything is a string' applies to integers
              # whatever big they are
              puts [round [mul $a 10000000000]]
              #
              # the same with Tcl 8.4
              # =====================
              set a [pi 20]
              # bignums (arbitrary length integers) need a conversion hook
              set b [fromstr 10000000000]
              # round returns a bignum:
              # before printing it, we need to convert it with 'tostr'
              puts [tostr [round [mul $a $b]]]


NAMESPACES AND OTHER PACKAGES
       We have not yet discussed about namespaces because we assumed that you
       had imported public commands into the global namespace, like this:


              namespace import ::math::bigfloat::*

       If you matter much about avoiding names conflicts, I considere it
       should be resolved by the following :


              package require math::bigfloat
              # beware: namespace ensembles are not available in Tcl 8.4
              namespace eval ::math::bigfloat {namespace ensemble create -command ::bigfloat}
              # from now, the bigfloat command takes as subcommands all original math::bigfloat::* commands
              set a [bigfloat sub [bigfloat fromstr 2.000] [bigfloat fromstr 0.530]]
              puts [bigfloat tostr $a]


EXAMPLES
       Guess what happens when you are doing some astronomy. Here is an
       example :


              # convert acurrate angles with a millisecond-rated accuracy
              proc degree-angle {degrees minutes seconds milliseconds} {
                  set result 0
                  set div 1
                  foreach factor {1 1000 60 60} var [list $milliseconds $seconds $minutes $degrees] {
                      # we convert each entry var into milliseconds
                      set div [expr {$div*$factor}]
                      incr result [expr {$var*$div}]
                  }
                  return [div [int2float $result] $div]
              }
              # load the package
              package require math::bigfloat
              namespace import ::math::bigfloat::*
              # work with angles : a standard formula for navigation (taking bearings)
              set angle1 [deg2rad [degree-angle 20 30 40   0]]
              set angle2 [deg2rad [degree-angle 21  0 50 500]]
              set opposite3 [deg2rad [degree-angle 51  0 50 500]]
              set sinProduct [mul [sin $angle1] [sin $angle2]]
              set cosProduct [mul [cos $angle1] [cos $angle2]]
              set angle3 [asin [add [mul $sinProduct [cos $opposite3]] $cosProduct]]
              puts "angle3 : [tostr [rad2deg $angle3]]"


BUGS, IDEAS, FEEDBACK
       This document, and the package it describes, will undoubtedly contain
       bugs and other problems.  Please report such in the category math ::
       bignum :: float of the Tcllib SF Trackers
       [http://sourceforge.net/tracker/?group_id=12883].  Please also report
       any ideas for enhancements you may have for either package and/or
       documentation.

KEYWORDS
       computations, floating-point, interval, math, multiprecision, tcl

CATEGORY
       Mathematics

COPYRIGHT
       Copyright (c) 2004-2008, by Stephane Arnold <stephanearnold at yahoo dot fr>




math                                 2.0.1                   math::bigfloat(n)
