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Bit::Vector(3pm) - phpMan

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Vector(3pm)                    User Contributed Perl Documentation                    Vector(3pm)



NAME
       Bit::Vector - Efficient bit vector, set of integers and "big int" math library

SYNOPSIS
   OVERLOADED OPERATORS
       See Bit::Vector::Overload(3).

   MORE STRING IMPORT/EXPORT
       See Bit::Vector::String(3).

   CLASS METHODS
         Version
             $version = Bit::Vector->Version();

         Word_Bits
             $bits = Bit::Vector->Word_Bits();  #  bits in a machine word

         Long_Bits
             $bits = Bit::Vector->Long_Bits();  #  bits in an unsigned long

         new
             $vector = Bit::Vector->new($bits);  #  bit vector constructor

             @veclist = Bit::Vector->new($bits,$count);

         new_Hex
             $vector = Bit::Vector->new_Hex($bits,$string);

         new_Bin
             $vector = Bit::Vector->new_Bin($bits,$string);

         new_Dec
             $vector = Bit::Vector->new_Dec($bits,$string);

         new_Enum
             $vector = Bit::Vector->new_Enum($bits,$string);

         Concat_List
             $vector = Bit::Vector->Concat_List(@vectors);

   OBJECT METHODS
         new
             $vec2 = $vec1->new($bits);  #  alternative call of constructor

             @veclist = $vec->new($bits,$count);

         Shadow
             $vec2 = $vec1->Shadow();  #  new vector, same size but empty

         Clone
             $vec2 = $vec1->Clone();  #  new vector, exact duplicate

         Concat
             $vector = $vec1->Concat($vec2);

         Concat_List
             $vector = $vec1->Concat_List($vec2,$vec3,...);

         Size
             $bits = $vector->Size();

         Resize
             $vector->Resize($bits);
             $vector->Resize($vector->Size()+5);
             $vector->Resize($vector->Size()-5);

         Copy
             $vec2->Copy($vec1);

         Empty
             $vector->Empty();

         Fill
             $vector->Fill();

         Flip
             $vector->Flip();

         Primes
             $vector->Primes();  #  Sieve of Erathostenes

         Reverse
             $vec2->Reverse($vec1);

         Interval_Empty
             $vector->Interval_Empty($min,$max);

         Interval_Fill
             $vector->Interval_Fill($min,$max);

         Interval_Flip
             $vector->Interval_Flip($min,$max);

         Interval_Reverse
             $vector->Interval_Reverse($min,$max);

         Interval_Scan_inc
             if (($min,$max) = $vector->Interval_Scan_inc($start))

         Interval_Scan_dec
             if (($min,$max) = $vector->Interval_Scan_dec($start))

         Interval_Copy
             $vec2->Interval_Copy($vec1,$offset2,$offset1,$length);

         Interval_Substitute
             $vec2->Interval_Substitute($vec1,$off2,$len2,$off1,$len1);

         is_empty
             if ($vector->is_empty())

         is_full
             if ($vector->is_full())

         equal
             if ($vec1->equal($vec2))

         Lexicompare (unsigned)
             if ($vec1->Lexicompare($vec2) == 0)
             if ($vec1->Lexicompare($vec2) != 0)
             if ($vec1->Lexicompare($vec2) <  0)
             if ($vec1->Lexicompare($vec2) <= 0)
             if ($vec1->Lexicompare($vec2) >  0)
             if ($vec1->Lexicompare($vec2) >= 0)

         Compare (signed)
             if ($vec1->Compare($vec2) == 0)
             if ($vec1->Compare($vec2) != 0)
             if ($vec1->Compare($vec2) <  0)
             if ($vec1->Compare($vec2) <= 0)
             if ($vec1->Compare($vec2) >  0)
             if ($vec1->Compare($vec2) >= 0)

         to_Hex
             $string = $vector->to_Hex();

         from_Hex
             $vector->from_Hex($string);

         to_Bin
             $string = $vector->to_Bin();

         from_Bin
             $vector->from_Bin($string);

         to_Dec
             $string = $vector->to_Dec();

         from_Dec
             $vector->from_Dec($string);

         to_Enum
             $string = $vector->to_Enum();  #  e.g. "2,3,5-7,11,13-19"

         from_Enum
             $vector->from_Enum($string);

         Bit_Off
             $vector->Bit_Off($index);

         Bit_On
             $vector->Bit_On($index);

         bit_flip
             $bit = $vector->bit_flip($index);

         bit_test
         contains
             $bit = $vector->bit_test($index);
             $bit = $vector->contains($index);
             if ($vector->bit_test($index))
             if ($vector->contains($index))

         Bit_Copy
             $vector->Bit_Copy($index,$bit);

         LSB (least significant bit)
             $vector->LSB($bit);

         MSB (most significant bit)
             $vector->MSB($bit);

         lsb (least significant bit)
             $bit = $vector->lsb();

         msb (most significant bit)
             $bit = $vector->msb();

         rotate_left
             $carry = $vector->rotate_left();

         rotate_right
             $carry = $vector->rotate_right();

         shift_left
             $carry = $vector->shift_left($carry);

         shift_right
             $carry = $vector->shift_right($carry);

         Move_Left
             $vector->Move_Left($bits);  #  shift left "$bits" positions

         Move_Right
             $vector->Move_Right($bits);  #  shift right "$bits" positions

         Insert
             $vector->Insert($offset,$bits);

         Delete
             $vector->Delete($offset,$bits);

         increment
             $carry = $vector->increment();

         decrement
             $carry = $vector->decrement();

         inc
             $overflow = $vec2->inc($vec1);

         dec
             $overflow = $vec2->dec($vec1);

         add
             $carry = $vec3->add($vec1,$vec2,$carry);
             ($carry,$overflow) = $vec3->add($vec1,$vec2,$carry);

         subtract
             $carry = $vec3->subtract($vec1,$vec2,$carry);
             ($carry,$overflow) = $vec3->subtract($vec1,$vec2,$carry);

         Neg
         Negate
             $vec2->Neg($vec1);
             $vec2->Negate($vec1);

         Abs
         Absolute
             $vec2->Abs($vec1);
             $vec2->Absolute($vec1);

         Sign
             if ($vector->Sign() == 0)
             if ($vector->Sign() != 0)
             if ($vector->Sign() <  0)
             if ($vector->Sign() <= 0)
             if ($vector->Sign() >  0)
             if ($vector->Sign() >= 0)

         Multiply
             $vec3->Multiply($vec1,$vec2);

         Divide
             $quot->Divide($vec1,$vec2,$rest);

         GCD (Greatest Common Divisor)
             $vecgcd->GCD($veca,$vecb);
             $vecgcd->GCD($vecx,$vecy,$veca,$vecb);

         Power
             $vec3->Power($vec1,$vec2);

         Block_Store
             $vector->Block_Store($buffer);

         Block_Read
             $buffer = $vector->Block_Read();

         Word_Size
             $size = $vector->Word_Size();  #  number of words in "$vector"

         Word_Store
             $vector->Word_Store($offset,$word);

         Word_Read
             $word = $vector->Word_Read($offset);

         Word_List_Store
             $vector->Word_List_Store(@words);

         Word_List_Read
             @words = $vector->Word_List_Read();

         Word_Insert
             $vector->Word_Insert($offset,$count);

         Word_Delete
             $vector->Word_Delete($offset,$count);

         Chunk_Store
             $vector->Chunk_Store($chunksize,$offset,$chunk);

         Chunk_Read
             $chunk = $vector->Chunk_Read($chunksize,$offset);

         Chunk_List_Store
             $vector->Chunk_List_Store($chunksize,@chunks);

         Chunk_List_Read
             @chunks = $vector->Chunk_List_Read($chunksize);

         Index_List_Remove
             $vector->Index_List_Remove(@indices);

         Index_List_Store
             $vector->Index_List_Store(@indices);

         Index_List_Read
             @indices = $vector->Index_List_Read();

         Or
         Union
             $vec3->Or($vec1,$vec2);
             $set3->Union($set1,$set2);

         And
         Intersection
             $vec3->And($vec1,$vec2);
             $set3->Intersection($set1,$set2);

         AndNot
         Difference
             $vec3->AndNot($vec1,$vec2);
             $set3->Difference($set1,$set2);

         Xor
         ExclusiveOr
             $vec3->Xor($vec1,$vec2);
             $set3->ExclusiveOr($set1,$set2);

         Not
         Complement
             $vec2->Not($vec1);
             $set2->Complement($set1);

         subset
             if ($set1->subset($set2))  #  true if $set1 is subset of $set2

         Norm
             $norm = $set->Norm();
             $norm = $set->Norm2();
             $norm = $set->Norm3();

         Min
             $min = $set->Min();

         Max
             $max = $set->Max();

         Multiplication
             $matrix3->Multiplication($rows3,$cols3,
                             $matrix1,$rows1,$cols1,
                             $matrix2,$rows2,$cols2);

         Product
             $matrix3->Product($rows3,$cols3,
                      $matrix1,$rows1,$cols1,
                      $matrix2,$rows2,$cols2);

         Closure
             $matrix->Closure($rows,$cols);

         Transpose
             $matrix2->Transpose($rows2,$cols2,$matrix1,$rows1,$cols1);

IMPORTANT NOTES
       · Method naming conventions

         Method names completely in lower case indicate a boolean return value.

         (Except for the bit vector constructor method ""new()"", of course.)

       · Boolean values

         Boolean values in this module are always a numeric zero ("0") for "false" and a numeric
         one ("1") for "true".

       · Negative numbers

         All numeric input parameters passed to any of the methods in this module are regarded as
         being UNSIGNED (as opposed to the contents of the bit vectors themselves, which are
         usually considered to be SIGNED).

         As a consequence, whenever you pass a negative number as an argument to some method of
         this module, it will be treated as a (usually very large) positive number due to its
         internal two's complement binary representation, usually resulting in an "index out of
         range" error message and program abortion.

       · Bit order

         Note that bit vectors are stored least order bit and least order word first internally.

         I.e., bit #0 of any given bit vector corresponds to bit #0 of word #0 in the array of
         machine words representing the bit vector.

         (Where word #0 comes first in memory, i.e., it is stored at the least memory address in
         the allocated block of memory holding the given bit vector.)

         Note however that machine words can be stored least order byte first or last, depending
         on your system's implementation.

         When you are exporting or importing a whole bit vector at once using the methods
         ""Block_Read()"" and ""Block_Store()"" (the only time in this module where this could
         make any difference), however, a conversion to and from "least order byte first" order
         is automatically supplied.

         In other words, what ""Block_Read()"" provides and what ""Block_Store()"" expects is
         always in "least order byte first" order, regardless of the order in which words are
         stored internally on your machine.

         This is to make sure that what you export on one machine using ""Block_Read()"" can
         always be read in correctly with ""Block_Store()"" on a different machine.

         Note further that whenever bit vectors are converted to and from (binary or hexadecimal)
         strings, the RIGHTMOST bit is always the LEAST SIGNIFICANT one, and the LEFTMOST bit is
         always the MOST SIGNIFICANT bit.

         This is because in our western culture, numbers are always represented in this way
         (least significant to most significant digits go from right to left).

         Of course this requires an internal reversion of order, which the corresponding
         conversion methods perform automatically (without any additional overhead, it's just a
         matter of starting the internal loop at the bottom or the top end).

       · "Word" related methods

         Note that all methods whose names begin with ""Word_"" are MACHINE-DEPENDENT!

         They depend on the size (number of bits) of an "unsigned int" (C type) on your machine.

         Therefore, you should only use these methods if you are ABSOLUTELY CERTAIN that
         portability of your code is not an issue!

         Note that you can use arbitrarily large chunks (i.e., fragments of bit vectors) of up to
         32 bits IN A PORTABLE WAY using the methods whose names begin with ""Chunk_"".

       · Chunk sizes

         Note that legal chunk sizes for all methods whose names begin with ""Chunk_"" range from
         "1" to ""Bit::Vector->Long_Bits();"" bits ("0" is NOT allowed!).

         In order to make your programs portable, however, you shouldn't use chunk sizes larger
         than 32 bits, since this is the minimum size of an "unsigned long" (C type) on all
         systems, as prescribed by ANSI C.

       · Matching sizes

         In general, for methods involving several bit vectors at the same time, all bit vector
         arguments must have identical sizes (number of bits), or a fatal "size mismatch" error
         will occur.

         Exceptions from this rule are the methods ""Concat()"", ""Concat_List()"", ""Copy()"",
         ""Interval_Copy()"" and ""Interval_Substitute()"", where no conditions at all are
         imposed on the size of their bit vector arguments.

         In method ""Multiply()"", all three bit vector arguments must in principle obey the rule
         of matching sizes, but the bit vector in which the result of the multiplication is to be
         stored may be larger than the two bit vector arguments containing the factors for the
         multiplication.

         In method ""Power()"", the bit vector for the result must be the same size or greater
         than the base of the exponentiation term. The exponent can be any size.

       · Index ranges

         All indices for any given bits must lie between "0" and ""$vector->Size()-1"", or a
         fatal "index out of range" error will occur.

       · Object persistence

         Since version 6.5, "Bit::Vector" objects can be serialized and de-serialized
         automatically with "Storable", out-of-the-box, without requiring any further user action
         for this to work.

         This is also true for nested data structures (since version 6.8).

         See the Storable(3) documentation for more details.

DESCRIPTION
   OVERLOADED OPERATORS
       See Bit::Vector::Overload(3).

   MORE STRING IMPORT/EXPORT
       See Bit::Vector::String(3).

   CLASS METHODS
       · "$version = Bit::Vector->Version();"

         Returns the current version number of this module.

       · "$bits = Bit::Vector->Word_Bits();"

         Returns the number of bits of an "unsigned int" (C type) on your machine.

         (An "unsigned int" is also called a "machine word", hence the name of this method.)

       · "$bits = Bit::Vector->Long_Bits();"

         Returns the number of bits of an "unsigned long" (C type) on your machine.

       · "$vector = Bit::Vector->new($bits);"

         This is the bit vector constructor method.

         Call this method to create a new bit vector containing "$bits" bits (with indices
         ranging from "0" to ""$bits-1"").

         Note that - in contrast to previous versions - bit vectors of length zero (i.e., with
         "$bits = 0") are permitted now.

         The method returns a reference to the newly created bit vector.

         A new bit vector is always initialized so that all bits are cleared (turned off).

         An exception will be raised if the method is unable to allocate the necessary memory.

         Note that if you specify a negative number for "$bits" it will be interpreted as a large
         positive number due to its internal two's complement binary representation.

         In such a case, the bit vector constructor method will obediently attempt to create a
         bit vector of that size, probably resulting in an exception, as explained above.

       · "@veclist = Bit::Vector->new($bits,$count);"

         You can also create more than one bit vector at a time if you specify the optional
         second parameter "$count".

         The method returns a list of "$count" bit vectors which all have the same number of bits
         "$bits" (and which are all initialized, i.e., all bits are cleared).

         If "$count" is zero, an empty list is returned.

         If "$bits" is zero, a list of null-sized bit vectors is returned.

         Note again that if you specify a negative number for "$count" it will be interpreted as
         a large positive number due to its internal two's complement binary representation.

         In such a case, the bit vector constructor method will obediently attempt to create that
         many bit vectors, probably resulting in an exception ("out of memory").

       · "$vector = Bit::Vector->new_Hex($bits,$string);"

         This method is an alternative constructor which allows you to create a new bit vector
         object (with "$bits" bits) and to initialize it all in one go.

         The method internally first calls the bit vector constructor method ""new()"" and then
         passes the given string to the method ""from_Hex()"".

         However, this method is more efficient than performing these two steps separately: First
         because in this method, the memory area occupied by the new bit vector is not
         initialized to zeros (which is pointless in this case), and second because it saves you
         from the associated overhead of one additional method invocation.

         An exception will be raised if the necessary memory cannot be allocated (see the
         description of the method ""new()"" immediately above for possible causes) or if the
         given string cannot be converted successfully (see the description of the method
         ""from_Hex()"" further below for details).

         In the latter case, the memory occupied by the new bit vector is released first (i.e.,
         "free"d) before the exception is actually raised.

       · "$vector = Bit::Vector->new_Bin($bits,$string);"

         This method is an alternative constructor which allows you to create a new bit vector
         object (with "$bits" bits) and to initialize it all in one go.

         The method internally first calls the bit vector constructor method ""new()"" and then
         passes the given string to the method ""from_Bin()"".

         However, this method is more efficient than performing these two steps separately: First
         because in this method, the memory area occupied by the new bit vector is not
         initialized to zeros (which is pointless in this case), and second because it saves you
         from the associated overhead of one additional method invocation.

         An exception will be raised if the necessary memory cannot be allocated (see the
         description of the method ""new()"" above for possible causes) or if the given string
         cannot be converted successfully (see the description of the method ""from_Bin()""
         further below for details).

         In the latter case, the memory occupied by the new bit vector is released first (i.e.,
         "free"d) before the exception is actually raised.

       · "$vector = Bit::Vector->new_Dec($bits,$string);"

         This method is an alternative constructor which allows you to create a new bit vector
         object (with "$bits" bits) and to initialize it all in one go.

         The method internally first calls the bit vector constructor method ""new()"" and then
         passes the given string to the method ""from_Dec()"".

         However, this method is more efficient than performing these two steps separately: First
         because in this method, ""new()"" does not initialize the memory area occupied by the
         new bit vector with zeros (which is pointless in this case, because ""from_Dec()"" will
         do it anyway), and second because it saves you from the associated overhead of one
         additional method invocation.

         An exception will be raised if the necessary memory cannot be allocated (see the
         description of the method ""new()"" above for possible causes) or if the given string
         cannot be converted successfully (see the description of the method ""from_Dec()""
         further below for details).

         In the latter case, the memory occupied by the new bit vector is released first (i.e.,
         "free"d) before the exception is actually raised.

       · "$vector = Bit::Vector->new_Enum($bits,$string);"

         This method is an alternative constructor which allows you to create a new bit vector
         object (with "$bits" bits) and to initialize it all in one go.

         The method internally first calls the bit vector constructor method ""new()"" and then
         passes the given string to the method ""from_Enum()"".

         However, this method is more efficient than performing these two steps separately: First
         because in this method, ""new()"" does not initialize the memory area occupied by the
         new bit vector with zeros (which is pointless in this case, because ""from_Enum()"" will
         do it anyway), and second because it saves you from the associated overhead of one
         additional method invocation.

         An exception will be raised if the necessary memory cannot be allocated (see the
         description of the method ""new()"" above for possible causes) or if the given string
         cannot be converted successfully (see the description of the method ""from_Enum()""
         further below for details).

         In the latter case, the memory occupied by the new bit vector is released first (i.e.,
         "free"d) before the exception is actually raised.

       · "$vector = Bit::Vector->Concat_List(@vectors);"

         This method creates a new vector containing all bit vectors from the argument list in
         concatenated form.

         The argument list may contain any number of arguments (including zero); the only
         condition is that all arguments must be bit vectors.

         There is no condition concerning the length (in number of bits) of these arguments.

         The vectors from the argument list are not changed in any way.

         If the argument list is empty or if all arguments have length zero, the resulting bit
         vector will also have length zero.

         Note that the RIGHTMOST bit vector from the argument list will become the LEAST
         significant part of the resulting bit vector, and the LEFTMOST bit vector from the
         argument list will become the MOST significant part of the resulting bit vector.

   OBJECT METHODS
       · "$vec2 = $vec1->new($bits);"

         "@veclist = $vec->new($bits);"

         This is an alternative way of calling the bit vector constructor method.

         Vector "$vec1" (or "$vec") is not affected by this, it just serves as an anchor for the
         method invocation mechanism.

         In fact ALL class methods in this module can be called this way, even though this is
         probably considered to be "politically incorrect" by OO ("object-orientation")
         aficionados. ;-)

         So even if you are too lazy to type ""Bit::Vector->"" instead of ""$vec1->"" (and even
         though laziness is - allegedly - a programmer's virtue ":-)"), maybe it is better not to
         use this feature if you don't want to get booed at. ;-)

       · "$vec2 = $vec1->Shadow();"

         Creates a NEW bit vector "$vec2" of the SAME SIZE as "$vec1" but which is EMPTY.

         Just like a shadow that has the same shape as the object it originates from, but is flat
         and has no volume, i.e., contains nothing.

       · "$vec2 = $vec1->Clone();"

         Creates a NEW bit vector "$vec2" of the SAME SIZE as "$vec1" which is an EXACT COPY of
         "$vec1".

       · "$vector = $vec1->Concat($vec2);"

         This method returns a new bit vector object which is the result of the concatenation of
         the contents of "$vec1" and "$vec2".

         Note that the contents of "$vec1" become the MOST significant part of the resulting bit
         vector, and "$vec2" the LEAST significant part.

         If both bit vector arguments have length zero, the resulting bit vector will also have
         length zero.

       · "$vector = $vec1->Concat_List($vec2,$vec3,...);"

         This is an alternative way of calling this (class) method as an object method.

         The method returns a new bit vector object which is the result of the concatenation of
         the contents of "$vec1 . $vec2 . $vec3 . ..."

         See the section "class methods" above for a detailed description of this method.

         Note that the argument list may be empty and that all arguments must be bit vectors if
         it isn't.

       · "$bits = $vector->Size();"

         Returns the size (number of bits) the given vector was created with (or ""Resize()""d
         to).

       · "$vector->Resize($bits);"

         Changes the size of the given vector to the specified number of bits.

         This method allows you to change the size of an existing bit vector, preserving as many
         bits from the old vector as will fit into the new one (i.e., all bits with indices
         smaller than the minimum of the sizes of both vectors, old and new).

         If the number of machine words needed to store the new vector is smaller than or equal
         to the number of words needed to store the old vector, the memory allocated for the old
         vector is reused for the new one, and only the relevant book-keeping information is
         adjusted accordingly.

         This means that even if the number of bits increases, new memory is not necessarily
         being allocated (i.e., if the old and the new number of bits fit into the same number of
         machine words).

         If the number of machine words needed to store the new vector is greater than the number
         of words needed to store the old vector, new memory is allocated for the new vector, the
         old vector is copied to the new one, the remaining bits in the new vector are cleared
         (turned off) and the old vector is deleted, i.e., the memory that was allocated for it
         is released.

         (An exception will be raised if the method is unable to allocate the necessary memory
         for the new vector.)

         As a consequence, if you decrease the size of a given vector so that it will use fewer
         machine words, and increase it again later so that it will use more words than
         immediately before but still less than the original vector, new memory will be allocated
         anyway because the information about the size of the original vector is lost whenever
         you resize it.

         Note also that if you specify a negative number for "$bits" it will be interpreted as a
         large positive number due to its internal two's complement binary representation.

         In such a case, "Resize()" will obediently attempt to create a bit vector of that size,
         probably resulting in an exception, as explained above.

         Finally, note that - in contrast to previous versions - resizing a bit vector to a size
         of zero bits (length zero) is now permitted.

       · "$vec2->Copy($vec1);"

         Copies the contents of bit vector "$vec1" to bit vector "$vec2".

         The previous contents of bit vector "$vec2" get overwritten, i.e., are lost.

         Both vectors must exist beforehand, i.e., this method does not CREATE any new bit vector
         object.

         The two vectors may be of any size.

         If the source bit vector is larger than the target, this method will copy as much of the
         least significant bits of the source vector as will fit into the target vector, thereby
         discarding any extraneous most significant bits.

         BEWARE that this causes a brutal cutoff in the middle of your data, and it will also
         leave you with an almost unpredictable sign if subsequently the number in the target
         vector is going to be interpreted as a number! (You have been warned!)

         If the target bit vector is larger than the source, this method fills up the remaining
         most significant bits in the target bit vector with either 0's or 1's, depending on the
         sign (= the most significant bit) of the source bit vector. This is also known as "sign
         extension".

         This makes it possible to copy numbers from a smaller bit vector into a larger one while
         preserving the number's absolute value as well as its sign (due to the two's complement
         binary representation of numbers).

       · "$vector->Empty();"

         Clears all bits in the given vector.

       · "$vector->Fill();"

         Sets all bits in the given vector.

       · "$vector->Flip();"

         Flips (i.e., complements) all bits in the given vector.

       · "$vector->Primes();"

         Clears the given bit vector and sets all bits whose indices are prime numbers.

         This method uses the algorithm known as the "Sieve of Erathostenes" internally.

       · "$vec2->Reverse($vec1);"

         This method copies the given vector "$vec1" to the vector "$vec2", thereby reversing the
         order of all bits.

         I.e., the least significant bit of "$vec1" becomes the most significant bit of "$vec2",
         whereas the most significant bit of "$vec1" becomes the least significant bit of
         "$vec2", and so forth for all bits in between.

         Note that in-place processing is also possible, i.e., "$vec1" and "$vec2" may be
         identical.

         (Internally, this is the same as "$vec1->Interval_Reverse(0,$vec1->Size()-1);".)

       · "$vector->Interval_Empty($min,$max);"

         Clears all bits in the interval "[$min..$max]" (including both limits) in the given
         vector.

         "$min" and "$max" may have the same value; this is the same as clearing a single bit
         with ""Bit_Off()"" (but less efficient).

         Note that "$vector->Interval_Empty(0,$vector->Size()-1);" is the same as
         "$vector->Empty();" (but less efficient).

       · "$vector->Interval_Fill($min,$max);"

         Sets all bits in the interval "[$min..$max]" (including both limits) in the given
         vector.

         "$min" and "$max" may have the same value; this is the same as setting a single bit with
         ""Bit_On()"" (but less efficient).

         Note that "$vector->Interval_Fill(0,$vector->Size()-1);" is the same as
         "$vector->Fill();" (but less efficient).

       · "$vector->Interval_Flip($min,$max);"

         Flips (i.e., complements) all bits in the interval "[$min..$max]" (including both
         limits) in the given vector.

         "$min" and "$max" may have the same value; this is the same as flipping a single bit
         with ""bit_flip()"" (but less efficient).

         Note that "$vector->Interval_Flip(0,$vector->Size()-1);" is the same as
         "$vector->Flip();" and "$vector->Complement($vector);" (but less efficient).

       · "$vector->Interval_Reverse($min,$max);"

         Reverses the order of all bits in the interval "[$min..$max]" (including both limits) in
         the given vector.

         I.e., bits "$min" and "$max" swap places, and so forth for all bits in between.

         "$min" and "$max" may have the same value; this has no effect whatsoever, though.

       · "if (($min,$max) = $vector->Interval_Scan_inc($start))"

         Returns the minimum and maximum indices of the next contiguous block of set bits (i.e.,
         bits in the "on" state).

         The search starts at index "$start" (i.e., "$min" >= "$start") and proceeds upwards
         (i.e., "$max" >= "$min"), thus repeatedly increments the search pointer "$start"
         (internally).

         Note though that the contents of the variable (or scalar literal value) "$start" is NOT
         altered. I.e., you have to set it to the desired value yourself prior to each call to
         ""Interval_Scan_inc()"" (see also the example given below).

         Actually, the bit vector is not searched bit by bit, but one machine word at a time, in
         order to speed up execution (which means that this method is quite efficient).

         An empty list is returned if no such block can be found.

         Note that a single set bit (surrounded by cleared bits) is a valid block by this
         definition. In that case the return values for "$min" and "$max" are the same.

         Typical use:

             $start = 0;
             while (($start < $vector->Size()) &&
                 (($min,$max) = $vector->Interval_Scan_inc($start)))
             {
                 $start = $max + 2;

                 # do something with $min and $max
             }

       · "if (($min,$max) = $vector->Interval_Scan_dec($start))"

         Returns the minimum and maximum indices of the next contiguous block of set bits (i.e.,
         bits in the "on" state).

         The search starts at index "$start" (i.e., "$max" <= "$start") and proceeds downwards
         (i.e., "$min" <= "$max"), thus repeatedly decrements the search pointer "$start"
         (internally).

         Note though that the contents of the variable (or scalar literal value) "$start" is NOT
         altered. I.e., you have to set it to the desired value yourself prior to each call to
         ""Interval_Scan_dec()"" (see also the example given below).

         Actually, the bit vector is not searched bit by bit, but one machine word at a time, in
         order to speed up execution (which means that this method is quite efficient).

         An empty list is returned if no such block can be found.

         Note that a single set bit (surrounded by cleared bits) is a valid block by this
         definition. In that case the return values for "$min" and "$max" are the same.

         Typical use:

             $start = $vector->Size() - 1;
             while (($start >= 0) &&
                 (($min,$max) = $vector->Interval_Scan_dec($start)))
             {
                 $start = $min - 2;

                 # do something with $min and $max
             }

       · "$vec2->Interval_Copy($vec1,$offset2,$offset1,$length);"

         This method allows you to copy a stretch of contiguous bits (starting at any position
         "$offset1" you choose, with a length of "$length" bits) from a given "source" bit vector
         "$vec1" to another position "$offset2" in a "target" bit vector "$vec2".

         Note that the two bit vectors "$vec1" and "$vec2" do NOT need to have the same
         (matching) size!

         Consequently, any of the two terms ""$offset1 + $length"" and ""$offset2 + $length"" (or
         both) may exceed the actual length of its corresponding bit vector (""$vec1->Size()""
         and ""$vec2->Size()"", respectively).

         In such a case, the "$length" parameter is automatically reduced internally so that both
         terms above are bounded by the number of bits of their corresponding bit vector.

         This may even result in a length of zero, in which case nothing is copied at all.

         (Of course the value of the "$length" parameter, supplied by you in the initial method
         call, may also be zero right from the start!)

         Note also that "$offset1" and "$offset2" must lie within the range "0" and,
         respectively, ""$vec1->Size()-1"" or ""$vec2->Size()-1"", or a fatal "offset out of
         range" error will occur.

         Note further that "$vec1" and "$vec2" may be identical, i.e., you may copy a stretch of
         contiguous bits from one part of a given bit vector to another part.

         The source and the target interval may even overlap, in which case the copying is
         automatically performed in ascending or descending order (depending on the direction of
         the copy - downwards or upwards in the bit vector, respectively) to handle this
         situation correctly, i.e., so that no bits are being overwritten before they have been
         copied themselves.

       · "$vec2->Interval_Substitute($vec1,$off2,$len2,$off1,$len1);"

         This method is (roughly) the same for bit vectors (i.e., arrays of booleans) as what the
         "splice" function in Perl is for lists (i.e., arrays of scalars).

         (See "splice" in perlfunc for more details about this function.)

         The method allows you to substitute a stretch of contiguous bits (defined by a position
         (offset) "$off1" and a length of "$len1" bits) from a given "source" bit vector "$vec1"
         for a different stretch of contiguous bits (defined by a position (offset) "$off2" and a
         length of "$len2" bits) in another, "target" bit vector "$vec2".

         Note that the two bit vectors "$vec1" and "$vec2" do NOT need to have the same
         (matching) size!

         Note further that "$off1" and "$off2" must lie within the range "0" and, respectively,
         ""$vec1->Size()"" or ""$vec2->Size()"", or a fatal "offset out of range" error will
         occur.

         Alert readers will have noticed that these upper limits are NOT ""$vec1->Size()-1"" and
         ""$vec2->Size()-1"", as they would be for any other method in this module, but that
         these offsets may actually point to one position PAST THE END of the corresponding bit
         vector.

         This is necessary in order to make it possible to APPEND a given stretch of bits to the
         target bit vector instead of REPLACING something in it.

         For reasons of symmetry and generality, the same applies to the offset in the source bit
         vector, even though such an offset (one position past the end of the bit vector) does
         not serve any practical purpose there (but does not cause any harm either).

         (Actually this saves you from the need of testing for this special case, in certain
         circumstances.)

         Note that whenever the term ""$off1 + $len1"" exceeds the size ""$vec1->Size()"" of bit
         vector "$vec1" (or if ""$off2 + $len2"" exceeds ""$vec2->Size()""), the corresponding
         length ("$len1" or "$len2", respectively) is automatically reduced internally so that
         ""$off1 + $len1 <= $vec1->Size()"" (and ""$off2 + $len2 <= $vec2->Size()"") holds.

         (Note that this does NOT alter the intended result, even though this may seem counter-
         intuitive at first!)

         This may even result in a length ("$len1" or "$len2") of zero.

         A length of zero for the interval in the SOURCE bit vector (""$len1 == 0"") means that
         the indicated stretch of bits in the target bit vector (starting at position "$off2") is
         to be replaced by NOTHING, i.e., is to be DELETED.

         A length of zero for the interval in the TARGET bit vector ("$len2 == 0") means that
         NOTHING is replaced, and that the stretch of bits from the source bit vector is simply
         INSERTED into the target bit vector at the indicated position ("$off2").

         If both length parameters are zero, nothing is done at all.

         Note that in contrast to any other method in this module (especially
         ""Interval_Copy()"", ""Insert()"" and ""Delete()""), this method IMPLICITLY and
         AUTOMATICALLY adapts the length of the resulting bit vector as needed, as given by

                 $size = $vec2->Size();   #  before
                 $size += $len1 - $len2;  #  after

         (The only other method in this module that changes the size of a bit vector is the
         method ""Resize()"".)

         In other words, replacing a given interval of bits in the target bit vector with a
         longer or shorter stretch of bits from the source bit vector, or simply inserting
         (""$len2 == 0"") a stretch of bits into or deleting (""$len1 == 0"") an interval of bits
         from the target bit vector will automatically increase or decrease, respectively, the
         size of the target bit vector accordingly.

         For the sake of generality, this may even result in a bit vector with a size of zero
         (containing no bits at all).

         This is also the reason why bit vectors of length zero are permitted in this module in
         the first place, starting with version 5.0.

         Finally, note that "$vec1" and "$vec2" may be identical, i.e., in-place processing is
         possible.

         (If you think about that for a while or if you look at the code, you will see that this
         is far from trivial!)

       · "if ($vector->is_empty())"

         Tests whether the given bit vector is empty, i.e., whether ALL of its bits are cleared
         (in the "off" state).

         In "big integer" arithmetic, this is equivalent to testing whether the number stored in
         the bit vector is zero ("0").

         Returns "true" ("1") if the bit vector is empty and "false" ("0") otherwise.

         Note that this method also returns "true" ("1") if the given bit vector has a length of
         zero, i.e., if it contains no bits at all.

       · "if ($vector->is_full())"

         Tests whether the given bit vector is full, i.e., whether ALL of its bits are set (in
         the "on" state).

         In "big integer" arithmetic, this is equivalent to testing whether the number stored in
         the bit vector is minus one ("-1").

         Returns "true" ("1") if the bit vector is full and "false" ("0") otherwise.

         If the given bit vector has a length of zero (i.e., if it contains no bits at all), this
         method returns "false" ("0").

       · "if ($vec1->equal($vec2))"

         Tests the two given bit vectors for equality.

         Returns "true" ("1") if the two bit vectors are exact copies of one another and "false"
         ("0") otherwise.

       · "$cmp = $vec1->Lexicompare($vec2);"

         Compares the two given bit vectors, which are regarded as UNSIGNED numbers in binary
         representation.

         The method returns ""-1"" if the first bit vector is smaller than the second bit vector,
         "0" if the two bit vectors are exact copies of one another and "1" if the first bit
         vector is greater than the second bit vector.

       · "$cmp = $vec1->Compare($vec2);"

         Compares the two given bit vectors, which are regarded as SIGNED numbers in binary
         representation.

         The method returns ""-1"" if the first bit vector is smaller than the second bit vector,
         "0" if the two bit vectors are exact copies of one another and "1" if the first bit
         vector is greater than the second bit vector.

       · "$string = $vector->to_Hex();"

         Returns a hexadecimal string representing the given bit vector.

         Note that this representation is quite compact, in that it only needs at most twice the
         number of bytes needed to store the bit vector itself, internally.

         Note also that since a hexadecimal digit is always worth four bits, the length of the
         resulting string is always a multiple of four bits, regardless of the true length (in
         bits) of the given bit vector.

         Finally, note that the LEAST significant hexadecimal digit is located at the RIGHT end
         of the resulting string, and the MOST significant digit at the LEFT end.

       · "$vector->from_Hex($string);"

         Allows one to read in the contents of a bit vector from a hexadecimal string, such as
         returned by the method ""to_Hex()"" (see above).

         Remember that the least significant bits are always to the right of a hexadecimal
         string, and the most significant bits to the left. Therefore, the string is actually
         read in from right to left while the bit vector is filled accordingly, 4 bits at a time,
         starting with the least significant bits and going upward to the most significant bits.

         If the given string contains less hexadecimal digits than are needed to completely fill
         the given bit vector, the remaining (most significant) bits are all cleared.

         This also means that, even if the given string does not contain enough digits to
         completely fill the given bit vector, the previous contents of the bit vector are erased
         completely.

         If the given string is longer than it needs to fill the given bit vector, the
         superfluous characters are simply ignored.

         (In fact they are ignored completely - they are not even checked for proper syntax. See
         also below for more about that.)

         This behaviour is intentional so that you may read in the string representing one bit
         vector into another bit vector of different size, i.e., as much of it as will fit.

         If during the process of reading the given string any character is encountered which is
         not a hexadecimal digit, a fatal syntax error ensues ("input string syntax error").

       · "$string = $vector->to_Bin();"

         Returns a binary string representing the given bit vector.

         Example:

           $vector = Bit::Vector->new(8);
           $vector->Primes();
           $string = $vector->to_Bin();
           print "'$string'\n";

         This prints:

           '10101100'

         (Bits #7, #5, #3 and #2 are set.)

         Note that the LEAST significant bit is located at the RIGHT end of the resulting string,
         and the MOST significant bit at the LEFT end.

       · "$vector->from_Bin($string);"

         This method allows you to read in the contents of a bit vector from a binary string,
         such as returned by the method ""to_Bin()"" (see above).

         Note that this method assumes that the LEAST significant bit is located at the RIGHT end
         of the binary string, and the MOST significant bit at the LEFT end. Therefore, the
         string is actually read in from right to left while the bit vector is filled
         accordingly, one bit at a time, starting with the least significant bit and going upward
         to the most significant bit.

         If the given string contains less binary digits ("0" and "1") than are needed to
         completely fill the given bit vector, the remaining (most significant) bits are all
         cleared.

         This also means that, even if the given string does not contain enough digits to
         completely fill the given bit vector, the previous contents of the bit vector are erased
         completely.

         If the given string is longer than it needs to fill the given bit vector, the
         superfluous characters are simply ignored.

         (In fact they are ignored completely - they are not even checked for proper syntax. See
         also below for more about that.)

         This behaviour is intentional so that you may read in the string representing one bit
         vector into another bit vector of different size, i.e., as much of it as will fit.

         If during the process of reading the given string any character is encountered which is
         not either "0" or "1", a fatal syntax error ensues ("input string syntax error").

       · "$string = $vector->to_Dec();"

         This method returns a string representing the contents of the given bit vector converted
         to decimal (base 10).

         Note that this method assumes the given bit vector to be SIGNED (and to contain a number
         in two's complement binary representation).

         Consequently, whenever the most significant bit of the given bit vector is set, the
         number stored in it is regarded as being NEGATIVE.

         The resulting string can be fed into ""from_Dec()"" (see below) in order to copy the
         contents of this bit vector to another one (or to restore the contents of this one).
         This is not advisable, though, since this would be very inefficient (there are much more
         efficient methods for storing and copying bit vectors in this module).

         Note that such conversion from binary to decimal is inherently slow since the bit vector
         has to be repeatedly divided by 10 with remainder until the quotient becomes 0 (each
         remainder in turn represents a single decimal digit of the resulting string).

         This is also true for the implementation of this method in this module, even though a
         considerable effort has been made to speed it up: instead of repeatedly dividing by 10,
         the bit vector is repeatedly divided by the largest power of 10 that will fit into a
         machine word. The remainder is then repeatedly divided by 10 using only machine word
         arithmetics, which is much faster than dividing the whole bit vector ("divide and rule"
         principle).

         According to my own measurements, this resulted in an 8-fold speed increase over the
         straightforward approach.

         Still, conversion to decimal should be used only where absolutely necessary.

         Keep the resulting string stored in some variable if you need it again, instead of
         converting the bit vector all over again.

         Beware that if you set the configuration for overloaded operators to "output=decimal",
         this method will be called for every bit vector enclosed in double quotes!

       · "$vector->from_Dec($string);"

         This method allows you to convert a given decimal number, which may be positive or
         negative, into two's complement binary representation, which is then stored in the given
         bit vector.

         The decimal number should always be provided as a string, to avoid possible truncation
         (due to the limited precision of integers in Perl) or formatting (due to Perl's use of
         scientific notation for large numbers), which would lead to errors.

         If the binary representation of the given decimal number is too big to fit into the
         given bit vector (if the given bit vector does not contain enough bits to hold it), a
         fatal "numeric overflow error" occurs.

         If the input string contains other characters than decimal digits ("0-9") and an
         optional leading sign (""+"" or ""-""), a fatal "input string syntax error" occurs.

         Beware that large positive numbers which cause the most significant bit to be set (e.g.
         "255" in a bit vector with 8 bits) will be printed as negative numbers when converted
         back to decimal using the method "to_Dec()" (e.g.  "-1", in our example), because
         numbers with the most significant bit set are considered to be negative in two's
         complement binary representation.

         Note also that while it is possible to thusly enter negative numbers as large positive
         numbers (e.g. "255" for "-1" in a bit vector with 8 bits), the contrary isn't, i.e., you
         cannot enter "-255" for "+1", in our example.  A fatal "numeric overflow error" will
         occur if you try to do so.

         If possible program abortion is unwanted or intolerable, use ""eval"", like this:

           eval { $vector->from_Dec("1152921504606846976"); };
           if ($@)
           {
               # an error occurred
           }

         There are four possible error messages:

           if ($@ =~ /item is not a string/)

           if ($@ =~ /input string syntax error/)

           if ($@ =~ /numeric overflow error/)

           if ($@ =~ /unable to allocate memory/)

         Note that the conversion from decimal to binary is costly in terms of processing time,
         since a lot of multiplications have to be carried out (in principle, each decimal digit
         must be multiplied with the binary representation of the power of 10 corresponding to
         its position in the decimal number, i.e., 1, 10, 100, 1000, 10000 and so on).

         This is not as time consuming as the opposite conversion, from binary to decimal (where
         successive divisions have to be carried out, which are even more expensive than
         multiplications), but still noticeable.

         Again (as in the case of ""to_Dec()""), the implementation of this method in this module
         uses the principle of "divide and rule" in order to speed up the conversion, i.e., as
         many decimal digits as possible are first accumulated (converted) in a machine word and
         only then stored in the given bit vector.

         Even so, use this method only where absolutely necessary if speed is an important
         consideration in your application.

         Beware that if you set the configuration for overloaded operators to "input=decimal",
         this method will be called for every scalar operand you use!

       · "$string = $vector->to_Enum();"

         Converts the given bit vector or set into an enumeration of single indices and ranges of
         indices (".newsrc" style), representing the bits that are set ("1") in the bit vector.

         Example:

           $vector = Bit::Vector->new(20);
           $vector->Bit_On(2);
           $vector->Bit_On(3);
           $vector->Bit_On(11);
           $vector->Interval_Fill(5,7);
           $vector->Interval_Fill(13,19);
           print "'", $vector->to_Enum(), "'\n";

         which prints

           '2,3,5-7,11,13-19'

         If the given bit vector is empty, the resulting string will also be empty.

         Note, by the way, that the above example can also be written a little handier, perhaps,
         as follows:

           Bit::Vector->Configuration("out=enum");
           $vector = Bit::Vector->new(20);
           $vector->Index_List_Store(2,3,5,6,7,11,13,14,15,16,17,18,19);
           print "'$vector'\n";

       · "$vector->from_Enum($string);"

         This method first empties the given bit vector and then tries to set the bits and ranges
         of bits specified in the given string.

         The string "$string" must only contain unsigned integers or ranges of integers (two
         unsigned integers separated by a dash "-"), separated by commas (",").

         All other characters are disallowed (including white space!)  and will lead to a fatal
         "input string syntax error".

         In each range, the first integer (the lower limit of the range) must always be less than
         or equal to the second integer (the upper limit), or a fatal "minimum > maximum index"
         error occurs.

         All integers must lie in the permitted range for the given bit vector, i.e., they must
         lie between "0" and ""$vector->Size()-1"".

         If this condition is not met, a fatal "index out of range" error occurs.

         If possible program abortion is unwanted or intolerable, use ""eval"", like this:

           eval { $vector->from_Enum("2,3,5-7,11,13-19"); };
           if ($@)
           {
               # an error occurred
           }

         There are four possible error messages:

           if ($@ =~ /item is not a string/)

           if ($@ =~ /input string syntax error/)

           if ($@ =~ /index out of range/)

           if ($@ =~ /minimum > maximum index/)

         Note that the order of the indices and ranges is irrelevant, i.e.,

           eval { $vector->from_Enum("11,5-7,3,13-19,2"); };

         results in the same vector as in the example above.

         Ranges and indices may also overlap.

         This is because each (single) index in the string is passed to the method ""Bit_On()"",
         internally, and each range to the method ""Interval_Fill()"".

         This means that the resulting bit vector is just the union of all the indices and ranges
         specified in the given string.

       · "$vector->Bit_Off($index);"

         Clears the bit with index "$index" in the given vector.

       · "$vector->Bit_On($index);"

         Sets the bit with index "$index" in the given vector.

       · "$vector->bit_flip($index)"

         Flips (i.e., complements) the bit with index "$index" in the given vector.

         Moreover, this method returns the NEW state of the bit in question, i.e., it returns "0"
         if the bit is cleared or "1" if the bit is set (AFTER flipping it).

       · "if ($vector->bit_test($index))"

         "if ($vector->contains($index))"

         Returns the current state of the bit with index "$index" in the given vector, i.e.,
         returns "0" if it is cleared (in the "off" state) or "1" if it is set (in the "on"
         state).

       · "$vector->Bit_Copy($index,$bit);"

         Sets the bit with index "$index" in the given vector either to "0" or "1" depending on
         the boolean value "$bit".

       · "$vector->LSB($bit);"

         Allows you to set the least significant bit in the given bit vector to the value given
         by the boolean parameter "$bit".

         This is a (faster) shortcut for ""$vector->Bit_Copy(0,$bit);"".

       · "$vector->MSB($bit);"

         Allows you to set the most significant bit in the given bit vector to the value given by
         the boolean parameter "$bit".

         This is a (faster) shortcut for ""$vector->Bit_Copy($vector->Size()-1,$bit);"".

       · "$bit = $vector->lsb();"

         Returns the least significant bit of the given bit vector.

         This is a (faster) shortcut for ""$bit = $vector->bit_test(0);"".

       · "$bit = $vector->msb();"

         Returns the most significant bit of the given bit vector.

         This is a (faster) shortcut for ""$bit = $vector->bit_test($vector->Size()-1);"".

       · "$carry_out = $vector->rotate_left();"

           carry             MSB           vector:           LSB
            out:
           +---+            +---+---+---+---     ---+---+---+---+
           |   |  <---+---  |   |   |   |    ...    |   |   |   |  <---+
           +---+      |     +---+---+---+---     ---+---+---+---+      |
                      |                                                |
                      +------------------------------------------------+

         The least significant bit (LSB) is the bit with index "0", the most significant bit
         (MSB) is the bit with index ""$vector->Size()-1"".

       · "$carry_out = $vector->rotate_right();"

                   MSB           vector:           LSB            carry
                                                                   out:
                  +---+---+---+---     ---+---+---+---+           +---+
           +--->  |   |   |   |    ...    |   |   |   |  ---+---> |   |
           |      +---+---+---+---     ---+---+---+---+     |     +---+
           |                                                |
           +------------------------------------------------+

         The least significant bit (LSB) is the bit with index "0", the most significant bit
         (MSB) is the bit with index ""$vector->Size()-1"".

       · "$carry_out = $vector->shift_left($carry_in);"

           carry         MSB           vector:           LSB         carry
            out:                                                      in:
           +---+        +---+---+---+---     ---+---+---+---+        +---+
           |   |  <---  |   |   |   |    ...    |   |   |   |  <---  |   |
           +---+        +---+---+---+---     ---+---+---+---+        +---+

         The least significant bit (LSB) is the bit with index "0", the most significant bit
         (MSB) is the bit with index ""$vector->Size()-1"".

       · "$carry_out = $vector->shift_right($carry_in);"

           carry         MSB           vector:           LSB         carry
            in:                                                       out:
           +---+        +---+---+---+---     ---+---+---+---+        +---+
           |   |  --->  |   |   |   |    ...    |   |   |   |  --->  |   |
           +---+        +---+---+---+---     ---+---+---+---+        +---+

         The least significant bit (LSB) is the bit with index "0", the most significant bit
         (MSB) is the bit with index ""$vector->Size()-1"".

       · "$vector->Move_Left($bits);"

         Shifts the given bit vector left by "$bits" bits, i.e., inserts "$bits" new bits at the
         lower end (least significant bit) of the bit vector, moving all other bits up by "$bits"
         places, thereby losing the "$bits" most significant bits.

         The inserted new bits are all cleared (set to the "off" state).

         This method does nothing if "$bits" is equal to zero.

         Beware that the whole bit vector is cleared WITHOUT WARNING if "$bits" is greater than
         or equal to the size of the given bit vector!

         In fact this method is equivalent to

           for ( $i = 0; $i < $bits; $i++ ) { $vector->shift_left(0); }

         except that it is much more efficient (for "$bits" greater than or equal to the number
         of bits in a machine word on your system) than this straightforward approach.

       · "$vector->Move_Right($bits);"

         Shifts the given bit vector right by "$bits" bits, i.e., deletes the "$bits" least
         significant bits of the bit vector, moving all other bits down by "$bits" places,
         thereby creating "$bits" new bits at the upper end (most significant bit) of the bit
         vector.

         These new bits are all cleared (set to the "off" state).

         This method does nothing if "$bits" is equal to zero.

         Beware that the whole bit vector is cleared WITHOUT WARNING if "$bits" is greater than
         or equal to the size of the given bit vector!

         In fact this method is equivalent to

           for ( $i = 0; $i < $bits; $i++ ) { $vector->shift_right(0); }

         except that it is much more efficient (for "$bits" greater than or equal to the number
         of bits in a machine word on your system) than this straightforward approach.

       · "$vector->Insert($offset,$bits);"

         This method inserts "$bits" fresh new bits at position "$offset" in the given bit
         vector.

         The "$bits" most significant bits are lost, and all bits starting with bit number
         "$offset" up to and including bit number ""$vector->Size()-$bits-1"" are moved up by
         "$bits" places.

         The now vacant "$bits" bits starting at bit number "$offset" (up to and including bit
         number ""$offset+$bits-1"") are then set to zero (cleared).

         Note that this method does NOT increase the size of the given bit vector, i.e., the bit
         vector is NOT extended at its upper end to "rescue" the "$bits" uppermost (most
         significant) bits - instead, these bits are lost forever.

         If you don't want this to happen, you have to increase the size of the given bit vector
         EXPLICITLY and BEFORE you perform the "Insert" operation, with a statement such as the
         following:

           $vector->Resize($vector->Size() + $bits);

         Or use the method ""Interval_Substitute()"" instead of ""Insert()"", which performs
         automatic growing and shrinking of its target bit vector.

         Note also that "$offset" must lie in the permitted range between "0" and
         ""$vector->Size()-1"", or a fatal "offset out of range" error will occur.

         If the term ""$offset + $bits"" exceeds ""$vector->Size()-1"", all the bits starting
         with bit number "$offset" up to bit number ""$vector->Size()-1"" are simply cleared.

       · "$vector->Delete($offset,$bits);"

         This method deletes, i.e., removes the bits starting at position "$offset" up to and
         including bit number ""$offset+$bits-1"" from the given bit vector.

         The remaining uppermost bits (starting at position ""$offset+$bits"" up to and including
         bit number ""$vector->Size()-1"") are moved down by "$bits" places.

         The now vacant uppermost (most significant) "$bits" bits are then set to zero (cleared).

         Note that this method does NOT decrease the size of the given bit vector, i.e., the bit
         vector is NOT clipped at its upper end to "get rid of" the vacant "$bits" uppermost
         bits.

         If you don't want this, i.e., if you want the bit vector to shrink accordingly, you have
         to do so EXPLICITLY and AFTER the "Delete" operation, with a couple of statements such
         as these:

           $size = $vector->Size();
           if ($bits > $size) { $bits = $size; }
           $vector->Resize($size - $bits);

         Or use the method ""Interval_Substitute()"" instead of ""Delete()"", which performs
         automatic growing and shrinking of its target bit vector.

         Note also that "$offset" must lie in the permitted range between "0" and
         ""$vector->Size()-1"", or a fatal "offset out of range" error will occur.

         If the term ""$offset + $bits"" exceeds ""$vector->Size()-1"", all the bits starting
         with bit number "$offset" up to bit number ""$vector->Size()-1"" are simply cleared.

       · "$carry = $vector->increment();"

         This method increments the given bit vector.

         Note that this method regards bit vectors as being unsigned, i.e., the largest possible
         positive number is directly followed by the smallest possible (or greatest possible,
         speaking in absolute terms) negative number:

           before:  2 ^ (b-1) - 1    (= "0111...1111")
           after:   2 ^ (b-1)        (= "1000...0000")

         where ""b"" is the number of bits of the given bit vector.

         The method returns "false" ("0") in all cases except when a carry over occurs (in which
         case it returns "true", i.e., "1"), which happens when the number "1111...1111" is
         incremented, which gives "0000...0000" plus a carry over to the next higher (binary)
         digit.

         This can be used for the terminating condition of a "while" loop, for instance, in order
         to cycle through all possible values the bit vector can assume.

       · "$carry = $vector->decrement();"

         This method decrements the given bit vector.

         Note that this method regards bit vectors as being unsigned, i.e., the smallest possible
         (or greatest possible, speaking in absolute terms) negative number is directly followed
         by the largest possible positive number:

           before:  2 ^ (b-1)        (= "1000...0000")
           after:   2 ^ (b-1) - 1    (= "0111...1111")

         where ""b"" is the number of bits of the given bit vector.

         The method returns "false" ("0") in all cases except when a carry over occurs (in which
         case it returns "true", i.e., "1"), which happens when the number "0000...0000" is
         decremented, which gives "1111...1111" minus a carry over to the next higher (binary)
         digit.

         This can be used for the terminating condition of a "while" loop, for instance, in order
         to cycle through all possible values the bit vector can assume.

       · "$overflow = $vec2->inc($vec1);"

         This method copies the contents of bit vector "$vec1" to bit vector "$vec2" and
         increments the copy (not the original).

         If by incrementing the number its sign becomes invalid, the return value ("overflow"
         flag) will be true ("1"), or false ("0") if not. (See the description of the method
         "add()" below for a more in-depth explanation of what "overflow" means).

         Note that in-place operation is also possible, i.e., "$vec1" and "$vec2" may be
         identical.

       · "$overflow = $vec2->dec($vec1);"

         This method copies the contents of bit vector "$vec1" to bit vector "$vec2" and
         decrements the copy (not the original).

         If by decrementing the number its sign becomes invalid, the return value ("overflow"
         flag) will be true ("1"), or false ("0") if not. (See the description of the method
         "subtract()" below for a more in-depth explanation of what "overflow" means).

         Note that in-place operation is also possible, i.e., "$vec1" and "$vec2" may be
         identical.

       · "$carry = $vec3->add($vec1,$vec2,$carry);"

         "($carry,$overflow) = $vec3->add($vec1,$vec2,$carry);"

         This method adds the two numbers contained in bit vector "$vec1" and "$vec2" with carry
         "$carry" and stores the result in bit vector "$vec3".

         I.e.,
                     $vec3 = $vec1 + $vec2 + $carry

         Note that the "$carry" parameter is a boolean value, i.e., only its least significant
         bit is taken into account. (Think of it as though ""$carry &= 1;"" was always executed
         internally.)

         In scalar context, the method returns a boolean value which indicates if a carry over
         (to the next higher bit position) has occured. In list context, the method returns the
         carry and the overflow flag (in this order).

         The overflow flag is true ("1") if the sign (i.e., the most significant bit) of the
         result is wrong. This can happen when adding two very large positive numbers or when
         adding two (by their absolute value) very large negative numbers. See also further
         below.

         The carry in- and output is needed mainly for cascading, i.e., to add numbers that are
         fragmented into several pieces.

         Example:

           # initialize

           for ( $i = 0; $i < $n; $i++ )
           {
               $a[$i] = Bit::Vector->new($bits);
               $b[$i] = Bit::Vector->new($bits);
               $c[$i] = Bit::Vector->new($bits);
           }

           # fill @a and @b

           # $a[  0 ] is low order part,
           # $a[$n-1] is high order part,
           # and same for @b

           # add

           $carry = 0;
           for ( $i = 0; $i < $n; $i++ )
           {
               $carry = $c[$i]->add($a[$i],$b[$i],$carry);
           }

         Note that it makes no difference to this method whether the numbers in "$vec1" and
         "$vec2" are unsigned or signed (i.e., in two's complement binary representation).

         Note however that the return value (carry flag) is not meaningful when the numbers are
         SIGNED.

         Moreover, when the numbers are signed, a special type of error can occur which is
         commonly called an "overflow error".

         An overflow error occurs when the sign of the result (its most significant bit) is
         flipped (i.e., falsified) by a carry over from the next-lower bit position ("MSB-1").

         In fact matters are a bit more complicated than that: the overflow flag is set to "true"
         whenever there is a carry over from bit position MSB-1 to the most significant bit (MSB)
         but no carry over from the MSB to the output carry flag, or vice-versa, i.e., when there
         is no carry over from bit position MSB-1 to the most significant bit (MSB) but a carry
         over to the output carry flag.

         Thus the overflow flag is the result of an exclusive-or operation between incoming and
         outgoing carry over at the most significant bit position.

       · "$carry = $vec3->subtract($vec1,$vec2,$carry);"

         "($carry,$overflow) = $vec3->subtract($vec1,$vec2,$carry);"

         This method subtracts the two numbers contained in bit vector "$vec1" and "$vec2" with
         carry "$carry" and stores the result in bit vector "$vec3".

         I.e.,
                     $vec3 = $vec1 - $vec2 - $carry

         Note that the "$carry" parameter is a boolean value, i.e., only its least significant
         bit is taken into account. (Think of it as though ""$carry &= 1;"" was always executed
         internally.)

         In scalar context, the method returns a boolean value which indicates if a carry over
         (to the next higher bit position) has occured. In list context, the method returns the
         carry and the overflow flag (in this order).

         The overflow flag is true ("1") if the sign (i.e., the most significant bit) of the
         result is wrong. This can happen when subtracting a very large negative number from a
         very large positive number or vice-versa. See also further below.

         The carry in- and output is needed mainly for cascading, i.e., to subtract numbers that
         are fragmented into several pieces.

         Example:

           # initialize

           for ( $i = 0; $i < $n; $i++ )
           {
               $a[$i] = Bit::Vector->new($bits);
               $b[$i] = Bit::Vector->new($bits);
               $c[$i] = Bit::Vector->new($bits);
           }

           # fill @a and @b

           # $a[  0 ] is low order part,
           # $a[$n-1] is high order part,
           # and same for @b

           # subtract

           $carry = 0;
           for ( $i = 0; $i < $n; $i++ )
           {
               $carry = $c[$i]->subtract($a[$i],$b[$i],$carry);
           }

         Note that it makes no difference to this method whether the numbers in "$vec1" and
         "$vec2" are unsigned or signed (i.e., in two's complement binary representation).

         Note however that the return value (carry flag) is not meaningful when the numbers are
         SIGNED.

         Moreover, when the numbers are signed, a special type of error can occur which is
         commonly called an "overflow error".

         An overflow error occurs when the sign of the result (its most significant bit) is
         flipped (i.e., falsified) by a carry over from the next-lower bit position ("MSB-1").

         In fact matters are a bit more complicated than that: the overflow flag is set to "true"
         whenever there is a carry over from bit position MSB-1 to the most significant bit (MSB)
         but no carry over from the MSB to the output carry flag, or vice-versa, i.e., when there
         is no carry over from bit position MSB-1 to the most significant bit (MSB) but a carry
         over to the output carry flag.

         Thus the overflow flag is the result of an exclusive-or operation between incoming and
         outgoing carry over at the most significant bit position.

       · "$vec2->Neg($vec1);"

         "$vec2->Negate($vec1);"

         This method calculates the two's complement of the number in bit vector "$vec1" and
         stores the result in bit vector "$vec2".

         Calculating the two's complement of a given number in binary representation consists of
         inverting all bits and incrementing the result by one.

         This is the same as changing the sign of the given number from ""+"" to ""-"" or vice-
         versa. In other words, applying this method twice on a given number yields the original
         number again.

         Note that in-place processing is also possible, i.e., "$vec1" and "$vec2" may be
         identical.

         Most importantly, beware that this method produces a counter-intuitive result if the
         number contained in bit vector "$vec1" is "2 ^ (n-1)" (i.e., "1000...0000"), where ""n""
         is the number of bits the given bit vector contains: The negated value of this number is
         the number itself!

       · "$vec2->Abs($vec1);"

         "$vec2->Absolute($vec1);"

         Depending on the sign (i.e., the most significant bit) of the number in bit vector
         "$vec1", the contents of bit vector "$vec1" are copied to bit vector "$vec2" either with
         the method ""Copy()"" (if the number in bit vector "$vec1" is positive), or with
         ""Negate()"" (if the number in bit vector "$vec1" is negative).

         In other words, this method calculates the absolute value of the number in bit vector
         "$vec1" and stores the result in bit vector "$vec2".

         Note that in-place processing is also possible, i.e., "$vec1" and "$vec2" may be
         identical.

         Most importantly, beware that this method produces a counter-intuitive result if the
         number contained in bit vector "$vec1" is "2 ^ (n-1)" (i.e., "1000...0000"), where ""n""
         is the number of bits the given bit vector contains: The absolute value of this number
         is the number itself, even though this number is still negative by definition (the most
         significant bit is still set)!

       · "$sign = $vector->Sign();"

         This method returns "0" if all bits in the given bit vector are cleared, i.e., if the
         given bit vector contains the number "0", or if the given bit vector has a length of
         zero (contains no bits at all).

         If not all bits are cleared, this method returns ""-1"" if the most significant bit is
         set (i.e., if the bit vector contains a negative number), or "1" otherwise (i.e., if the
         bit vector contains a positive number).

       · "$vec3->Multiply($vec1,$vec2);"

         This method multiplies the two numbers contained in bit vector "$vec1" and "$vec2" and
         stores the result in bit vector "$vec3".

         Note that this method regards its arguments as SIGNED.

         If you want to make sure that a large number can never be treated as being negative by
         mistake, make your bit vectors at least one bit longer than the largest number you wish
         to represent, right from the start, or proceed as follows:

             $msb1 = $vec1->msb();
             $msb2 = $vec2->msb();
             $vec1->Resize($vec1->Size()+1);
             $vec2->Resize($vec2->Size()+1);
             $vec3->Resize($vec3->Size()+1);
             $vec1->MSB($msb1);
             $vec2->MSB($msb2);
             $vec3->Multiply($vec1,$vec2);

         Note also that all three bit vector arguments must in principle obey the rule of
         matching sizes, but that the bit vector "$vec3" may be larger than the two factors
         "$vec1" and "$vec2".

         In fact multiplying two binary numbers with ""n"" bits may yield a result which is at
         most ""2n"" bits long.

         Therefore, it is usually a good idea to let bit vector "$vec3" have twice the size of
         bit vector "$vec1" and "$vec2", unless you are absolutely sure that the result will fit
         into a bit vector of the same size as the two factors.

         If you are wrong, a fatal "numeric overflow error" will occur.

         Finally, note that in-place processing is possible, i.e., "$vec3" may be identical with
         "$vec1" or "$vec2", or both.

       · "$quot->Divide($vec1,$vec2,$rest);"

         This method divides the two numbers contained in bit vector "$vec1" and "$vec2" and
         stores the quotient in bit vector "$quot" and the remainder in bit vector "$rest".

         I.e.,
                     $quot = $vec1 / $vec2;  #  div
                     $rest = $vec1 % $vec2;  #  mod

         Therefore, "$quot" and "$rest" must be two DISTINCT bit vectors, or a fatal "result
         vector(s) must be distinct" error will occur.

         Note also that a fatal "division by zero error" will occur if "$vec2" is equal to zero.

         Note further that this method regards its arguments as SIGNED.

         If you want to make sure that a large number can never be treated as being negative by
         mistake, make your bit vectors at least one bit longer than the largest number you wish
         to represent, right from the start, or proceed as follows:

             $msb1 = $vec1->msb();
             $msb2 = $vec2->msb();
             $vec1->Resize($vec1->Size()+1);
             $vec2->Resize($vec2->Size()+1);
             $quot->Resize($quot->Size()+1);
             $rest->Resize($rest->Size()+1);
             $vec1->MSB($msb1);
             $vec2->MSB($msb2);
             $quot->Divide($vec1,$vec2,$rest);

         Finally, note that in-place processing is possible, i.e., "$quot" may be identical with
         "$vec1" or "$vec2" or both, and "$rest" may also be identical with "$vec1" or "$vec2" or
         both, as long as "$quot" and "$rest" are distinct. (!)

       · "$vecgcd->GCD($veca,$vecb);"

         This method calculates the "Greatest Common Divisor" of the two numbers contained in bit
         vector "$veca" and "$vecb" and stores the result in bit vector "$vecgcd".

         The method uses Euklid's algorithm internally:

             int GCD(int a, int b)
             {
                 int t;

                 while (b != 0)
                 {
                     t = a % b; /* = remainder of (a div b) */
                     a = b;
                     b = t;
                 }
                 return(a);
             }

         Note that "GCD(z,0) == GCD(0,z) == z".

       · "$vecgcd->GCD($vecx,$vecy,$veca,$vecb);"

         This variant of the "GCD" method calculates the "Greatest Common Divisor" of the two
         numbers contained in bit vector "$veca" and "$vecb" and stores the result in bit vector
         "$vecgcd".

         Moreover, it determines the two factors which are necessary in order to represent the
         greatest common divisor as a linear combination of its two arguments, i.e., the two
         factors "x" and "y" so that "GCD(a,b) == x * a + y * b", and stores them in bit vector
         "$vecx" and "$vecy", respectively.

         For example:

           a = 2322
           b =  654

           GCD( 2322, 654 ) == 6

           x =  20
           y = -71

           20 * 2322 - 71 * 654 == 6

         Please see http://www.cut-the-knot.org/blue/extension.shtml for an explanation of how
         this extension of Euklid's algorithm works.

       · "$vec3->Power($vec1,$vec2);"

         This method calculates the exponentiation of base "$vec1" elevated to the "$vec2" power,
         i.e., ""$vec1 ** $vec2"", and stores the result in bit vector "$vec3".

         The method uses an efficient divide-and-conquer algorithm:

         Suppose the exponent is (decimal) 13, for example. The binary representation of this
         exponent is "1101".

         This means we want to calculate

           $vec1 * $vec1 * $vec1 * $vec1 * $vec1 * $vec1 * $vec1 * $vec1 *
           $vec1 * $vec1 * $vec1 * $vec1 *
           $vec1

         That is, "$vec1" multiplied with itself 13 times. The grouping into lines above is no
         coincidence. The first line comprises 8 factors, the second contains 4, and the last
         line just one. This just happens to be the binary representation of 13. ";-)"

         We then calculate a series of squares (of squares of squares...) of the base, i.e.,

           $power[0] = $vec1;
           $power[1] = $vec1 * $vec1;
           $power[2] = $power[1] * $power[1];
           $power[3] = $power[2] * $power[2];
           etc.

         To calculate the power of our example, we simply initialize our result with 1 and
         consecutively multiply it with the items of the series of powers we just calculated, if
         the corresponding bit of the binary representation of the exponent is set:

           $result = 1;
           $result *= $power[0] if ($vec2 & 1);
           $result *= $power[1] if ($vec2 & 2);
           $result *= $power[2] if ($vec2 & 4);
           $result *= $power[3] if ($vec2 & 8);
           etc.

         The bit vector "$vec3" must be of the same size as the base "$vec1" or greater. "$vec3"
         and "$vec1" may be the same vector (i.e., in-place calculation as in ""$vec1 **=
         $vec2;"" is possible), but "$vec3" and "$vec2" must be distinct. Finally, the exponent
         "$vec2" must be positive. A fatal error occurs if any of these conditions is not met.

       · "$vector->Block_Store($buffer);"

         This method allows you to load the contents of a given bit vector in one go.

         This is useful when you store the contents of a bit vector in a file, for instance
         (using method ""Block_Read()""), and when you want to restore the previously saved bit
         vector.

         For this, "$buffer" MUST be a string (NO automatic conversion from numeric to string is
         provided here as would normally in Perl!)  containing the bit vector in "low order byte
         first" order.

         If the given string is shorter than what is needed to completely fill the given bit
         vector, the remaining (most significant) bytes of the bit vector are filled with zeros,
         i.e., the previous contents of the bit vector are always erased completely.

         If the given string is longer than what is needed to completely fill the given bit
         vector, the superfluous bytes are simply ignored.

         See "sysread" in perlfunc for how to read in the contents of "$buffer" from a file prior
         to passing it to this method.

       · "$buffer = $vector->Block_Read();"

         This method allows you to export the contents of a given bit vector in one block.

         This is useful when you want to save the contents of a bit vector for later, for
         instance in a file.

         The advantage of this method is that it allows you to do so in the compactest possible
         format, in binary.

         The method returns a Perl string which contains an exact copy of the contents of the
         given bit vector in "low order byte first" order.

         See "syswrite" in perlfunc for how to write the data from this string to a file.

       · "$size = $vector->Word_Size();"

         Each bit vector is internally organized as an array of machine words.

         The methods whose names begin with "Word_" allow you to access this internal array of
         machine words.

         Note that because the size of a machine word may vary from system to system, these
         methods are inherently MACHINE-DEPENDENT!

         Therefore, DO NOT USE these methods unless you are absolutely certain that portability
         of your code is not an issue!

         You have been warned!

         To be machine-independent, use the methods whose names begin with ""Chunk_"" instead,
         with chunk sizes no greater than 32 bits.

         The method ""Word_Size()"" returns the number of machine words that the internal array
         of words of the given bit vector contains.

         This is similar in function to the term ""scalar(@array)"" for a Perl array.

       · "$vector->Word_Store($offset,$word);"

         This method allows you to store a given value "$word" at a given position "$offset" in
         the internal array of words of the given bit vector.

         Note that "$offset" must lie in the permitted range between "0" and
         ""$vector->Word_Size()-1"", or a fatal "offset out of range" error will occur.

         This method is similar in function to the expression ""$array[$offset] = $word;"" for a
         Perl array.

       · "$word = $vector->Word_Read($offset);"

         This method allows you to access the value of a given machine word at position "$offset"
         in the internal array of words of the given bit vector.

         Note that "$offset" must lie in the permitted range between "0" and
         ""$vector->Word_Size()-1"", or a fatal "offset out of range" error will occur.

         This method is similar in function to the expression ""$word = $array[$offset];"" for a
         Perl array.

       · "$vector->Word_List_Store(@words);"

         This method allows you to store a list of values "@words" in the internal array of
         machine words of the given bit vector.

         Thereby the LEFTMOST value in the list ("$words[0]") is stored in the LEAST significant
         word of the internal array of words (the one with offset "0"), the next value from the
         list ("$words[1]") is stored in the word with offset "1", and so on, as intuitively
         expected.

         If the list "@words" contains fewer elements than the internal array of words of the
         given bit vector contains machine words, the remaining (most significant) words are
         filled with zeros.

         If the list "@words" contains more elements than the internal array of words of the
         given bit vector contains machine words, the superfluous values are simply ignored.

         This method is comparable in function to the expression ""@array = @words;"" for a Perl
         array.

       · "@words = $vector->Word_List_Read();"

         This method allows you to retrieve the internal array of machine words of the given bit
         vector all at once.

         Thereby the LEFTMOST value in the returned list ("$words[0]") is the LEAST significant
         word from the given bit vector, and the RIGHTMOST value in the returned list
         ("$words[$#words]") is the MOST significant word of the given bit vector.

         This method is similar in function to the expression ""@words = @array;"" for a Perl
         array.

       · "$vector->Word_Insert($offset,$count);"

         This method inserts "$count" empty new machine words at position "$offset" in the
         internal array of words of the given bit vector.

         The "$count" most significant words are lost, and all words starting with word number
         "$offset" up to and including word number ""$vector->Word_Size()-$count-1"" are moved up
         by "$count" places.

         The now vacant "$count" words starting at word number "$offset" (up to and including
         word number ""$offset+$count-1"") are then set to zero (cleared).

         Note that this method does NOT increase the size of the given bit vector, i.e., the bit
         vector is NOT extended at its upper end to "rescue" the "$count" uppermost (most
         significant) words - instead, these words are lost forever.

         If you don't want this to happen, you have to increase the size of the given bit vector
         EXPLICITLY and BEFORE you perform the "Insert" operation, with a statement such as the
         following:

           $vector->Resize($vector->Size() + $count * Bit::Vector->Word_Bits());

         Note also that "$offset" must lie in the permitted range between "0" and
         ""$vector->Word_Size()-1"", or a fatal "offset out of range" error will occur.

         If the term ""$offset + $count"" exceeds ""$vector->Word_Size()-1"", all the words
         starting with word number "$offset" up to word number ""$vector->Word_Size()-1"" are
         simply cleared.

       · "$vector->Word_Delete($offset,$count);"

         This method deletes, i.e., removes the words starting at position "$offset" up to and
         including word number ""$offset+$count-1"" from the internal array of machine words of
         the given bit vector.

         The remaining uppermost words (starting at position ""$offset+$count"" up to and
         including word number ""$vector->Word_Size()-1"") are moved down by "$count" places.

         The now vacant uppermost (most significant) "$count" words are then set to zero
         (cleared).

         Note that this method does NOT decrease the size of the given bit vector, i.e., the bit
         vector is NOT clipped at its upper end to "get rid of" the vacant "$count" uppermost
         words.

         If you don't want this, i.e., if you want the bit vector to shrink accordingly, you have
         to do so EXPLICITLY and AFTER the "Delete" operation, with a couple of statements such
         as these:

           $bits = $vector->Size();
           $count *= Bit::Vector->Word_Bits();
           if ($count > $bits) { $count = $bits; }
           $vector->Resize($bits - $count);

         Note also that "$offset" must lie in the permitted range between "0" and
         ""$vector->Word_Size()-1"", or a fatal "offset out of range" error will occur.

         If the term ""$offset + $count"" exceeds ""$vector->Word_Size()-1"", all the words
         starting with word number "$offset" up to word number ""$vector->Word_Size()-1"" are
         simply cleared.

       · "$vector->Chunk_Store($chunksize,$offset,$chunk);"

         This method allows you to set more than one bit at a time with different values.

         You can access chunks (i.e., ranges of contiguous bits) between one and at most
         ""Bit::Vector->Long_Bits()"" bits wide.

         In order to be portable, though, you should never use chunk sizes larger than 32 bits.

         If the given "$chunksize" does not lie between "1" and ""Bit::Vector->Long_Bits()"", a
         fatal "chunk size out of range" error will occur.

         The method copies the "$chunksize" least significant bits from the value "$chunk" to the
         given bit vector, starting at bit position "$offset" and proceeding upwards until bit
         number ""$offset+$chunksize-1"".

         (I.e., bit number "0" of "$chunk" becomes bit number "$offset" in the given bit vector,
         and bit number ""$chunksize-1"" becomes bit number ""$offset+$chunksize-1"".)

         If the term ""$offset+$chunksize-1"" exceeds ""$vector->Size()-1"", the corresponding
         superfluous (most significant) bits from "$chunk" are simply ignored.

         Note that "$offset" itself must lie in the permitted range between "0" and
         ""$vector->Size()-1"", or a fatal "offset out of range" error will occur.

         This method (as well as the other ""Chunk_"" methods) is useful, for example, when you
         are reading in data in chunks of, say, 8 bits, which you need to access later, say,
         using 16 bits at a time (like audio CD wave files, for instance).

       · "$chunk = $vector->Chunk_Read($chunksize,$offset);"

         This method allows you to read the values of more than one bit at a time.

         You can read chunks (i.e., ranges of contiguous bits) between one and at most
         ""Bit::Vector->Long_Bits()"" bits wide.

         In order to be portable, though, you should never use chunk sizes larger than 32 bits.

         If the given "$chunksize" does not lie between "1" and ""Bit::Vector->Long_Bits()"", a
         fatal "chunk size out of range" error will occur.

         The method returns the "$chunksize" bits from the given bit vector starting at bit
         position "$offset" and proceeding upwards until bit number ""$offset+$chunksize-1"".

         (I.e., bit number "$offset" of the given bit vector becomes bit number "0" of the
         returned value, and bit number ""$offset+$chunksize-1"" becomes bit number
         ""$chunksize-1"".)

         If the term ""$offset+$chunksize-1"" exceeds ""$vector->Size()-1"", the non-existent
         bits are simply not returned.

         Note that "$offset" itself must lie in the permitted range between "0" and
         ""$vector->Size()-1"", or a fatal "offset out of range" error will occur.

       · "$vector->Chunk_List_Store($chunksize,@chunks);"

         This method allows you to fill the given bit vector with a list of data packets
         ("chunks") of any size ("$chunksize") you like (within certain limits).

         In fact the given "$chunksize" must lie in the range between "1" and
         ""Bit::Vector->Long_Bits()"", or a fatal "chunk size out of range" error will occur.

         In order to be portable, though, you should never use chunk sizes larger than 32 bits.

         The given bit vector is thereby filled in ascending order: The first chunk from the list
         (i.e., "$chunks[0]") fills the "$chunksize" least significant bits, the next chunk from
         the list ("$chunks[1]") fills the bits number "$chunksize" to number ""2*$chunksize-1"",
         the third chunk ("$chunks[2]") fills the bits number ""2*$chunksize"", to number
         ""3*$chunksize-1"", and so on.

         If there a less chunks in the list than are needed to fill the entire bit vector, the
         remaining (most significant) bits are cleared, i.e., the previous contents of the given
         bit vector are always erased completely.

         If there are more chunks in the list than are needed to fill the entire bit vector,
         and/or if a chunk extends beyond ""$vector->Size()-1"" (which happens whenever
         ""$vector->Size()"" is not a multiple of "$chunksize"), the superfluous chunks and/or
         bits are simply ignored.

         The method is useful, for example (and among many other applications), for the
         conversion of packet sizes in a data stream.

         This method can also be used to store an octal string in a given bit vector:

           $vector->Chunk_List_Store(3, split(//, reverse $string));

         Note however that unlike the conversion methods ""from_Hex()"", ""from_Bin()"",
         ""from_Dec()"" and ""from_Enum()"", this statement does not include any syntax checking,
         i.e., it may fail silently, without warning.

         To perform syntax checking, add the following statements:

           if ($string =~ /^[0-7]+$/)
           {
               # okay, go ahead with conversion as shown above
           }
           else
           {
               # error, string contains other than octal characters
           }

         Another application is to store a repetitive pattern in a given bit vector:

           $pattern = 0xDEADBEEF;
           $length = 32;            # = length of $pattern in bits
           $size = $vector->Size();
           $factor = int($size / $length);
           if ($size % $length) { $factor++; }
           $vector->Chunk_List_Store($length, ($pattern) x $factor);

       · "@chunks = $vector->Chunk_List_Read($chunksize);"

         This method allows you to access the contents of the given bit vector in form of a list
         of data packets ("chunks") of a size ("$chunksize") of your choosing (within certain
         limits).

         In fact the given "$chunksize" must lie in the range between "1" and
         ""Bit::Vector->Long_Bits()"", or a fatal "chunk size out of range" error will occur.

         In order to be portable, though, you should never use chunk sizes larger than 32 bits.

         The given bit vector is thereby read in ascending order: The "$chunksize" least
         significant bits (bits number "0" to ""$chunksize-1"") become the first chunk in the
         returned list (i.e., "$chunks[0]"). The bits number "$chunksize" to ""2*$chunksize-1""
         become the next chunk in the list ("$chunks[1]"), and so on.

         If ""$vector->Size()"" is not a multiple of "$chunksize", the last chunk in the list
         will contain fewer bits than "$chunksize".

         BEWARE that for large bit vectors and/or small values of "$chunksize", the number of
         returned list elements can be extremely large! BE CAREFUL!

         You could blow up your application with lack of memory (each list element is a full-
         grown Perl scalar, internally, with an associated memory overhead for its
         administration!) or at least cause a noticeable, more or less long-lasting "freeze" of
         your application!

         Possible applications:

         The method is especially useful in the conversion of packet sizes in a data stream.

         This method can also be used to convert a given bit vector to a string of octal numbers:

           $string = reverse join('', $vector->Chunk_List_Read(3));

       · "$vector->Index_List_Remove(@indices);"

         This method allows you to specify a list of indices of bits which should be turned off
         in the given bit vector.

         In fact this method is a shortcut for

             foreach $index (@indices)
             {
                 $vector->Bit_Off($index);
             }

         In contrast to all other import methods in this module, this method does NOT clear the
         given bit vector before processing its list of arguments.

         Instead, this method allows you to accumulate the results of various consecutive calls.

         (The same holds for the method ""Index_List_Store()"". As a consequence, you can "wipe
         out" what you did using the method ""Index_List_Remove()"" by passing the identical
         argument list to the method ""Index_List_Store()"".)

       · "$vector->Index_List_Store(@indices);"

         This method allows you to specify a list of indices of bits which should be turned on in
         the given bit vector.

         In fact this method is a shortcut for

             foreach $index (@indices)
             {
                 $vector->Bit_On($index);
             }

         In contrast to all other import methods in this module, this method does NOT clear the
         given bit vector before processing its list of arguments.

         Instead, this method allows you to accumulate the results of various consecutive calls.

         (The same holds for the method ""Index_List_Remove()"". As a consequence, you can "wipe
         out" what you did using the method ""Index_List_Store()"" by passing the identical
         argument list to the method ""Index_List_Remove()"".)

       · "@indices = $vector->Index_List_Read();"

         This method returns a list of Perl scalars.

         The list contains one scalar for each set bit in the given bit vector.

         BEWARE that for large bit vectors, this can result in a literally overwhelming number of
         list elements! BE CAREFUL! You could run out of memory or slow down your application
         considerably!

         Each scalar contains the number of the index corresponding to the bit in question.

         These indices are always returned in ascending order.

         If the given bit vector is empty (contains only cleared bits) or if it has a length of
         zero (if it contains no bits at all), the method returns an empty list.

         This method can be useful, for instance, to obtain a list of prime numbers:

             $limit = 1000; # or whatever
             $vector = Bit::Vector->new($limit+1);
             $vector->Primes();
             @primes = $vector->Index_List_Read();

       · "$vec3->Or($vec1,$vec2);"

         "$set3->Union($set1,$set2);"

         This method calculates the union of "$set1" and "$set2" and stores the result in
         "$set3".

         This is usually written as ""$set3 = $set1 u $set2"" in set theory (where "u" is the
         "cup" operator).

         (On systems where the "cup" character is unavailable this operator is often denoted by a
         plus sign "+".)

         In-place calculation is also possible, i.e., "$set3" may be identical with "$set1" or
         "$set2" or both.

       · "$vec3->And($vec1,$vec2);"

         "$set3->Intersection($set1,$set2);"

         This method calculates the intersection of "$set1" and "$set2" and stores the result in
         "$set3".

         This is usually written as ""$set3 = $set1 n $set2"" in set theory (where "n" is the
         "cap" operator).

         (On systems where the "cap" character is unavailable this operator is often denoted by
         an asterisk "*".)

         In-place calculation is also possible, i.e., "$set3" may be identical with "$set1" or
         "$set2" or both.

       · "$vec3->AndNot($vec1,$vec2);"

         "$set3->Difference($set1,$set2);"

         This method calculates the difference of "$set1" less "$set2" and stores the result in
         "$set3".

         This is usually written as ""$set3 = $set1 \ $set2"" in set theory (where "\" is the
         "less" operator).

         In-place calculation is also possible, i.e., "$set3" may be identical with "$set1" or
         "$set2" or both.

       · "$vec3->Xor($vec1,$vec2);"

         "$set3->ExclusiveOr($set1,$set2);"

         This method calculates the symmetric difference of "$set1" and "$set2" and stores the
         result in "$set3".

         This can be written as ""$set3 = ($set1 u $set2) \ ($set1 n $set2)"" in set theory (the
         union of the two sets less their intersection).

         When sets are implemented as bit vectors then the above formula is equivalent to the
         exclusive-or between corresponding bits of the two bit vectors (hence the name of this
         method).

         Note that this method is also much more efficient than evaluating the above formula
         explicitly since it uses a built-in machine language instruction internally.

         In-place calculation is also possible, i.e., "$set3" may be identical with "$set1" or
         "$set2" or both.

       · "$vec2->Not($vec1);"

         "$set2->Complement($set1);"

         This method calculates the complement of "$set1" and stores the result in "$set2".

         In "big integer" arithmetic, this is equivalent to calculating the one's complement of
         the number stored in the bit vector "$set1" in binary representation.

         In-place calculation is also possible, i.e., "$set2" may be identical with "$set1".

       · "if ($set1->subset($set2))"

         Returns "true" ("1") if "$set1" is a subset of "$set2" (i.e., completely contained in
         "$set2") and "false" ("0") otherwise.

         This means that any bit which is set ("1") in "$set1" must also be set in "$set2", but
         "$set2" may contain set bits which are not set in "$set1", in order for the condition of
         subset relationship to be true between these two sets.

         Note that by definition, if two sets are identical, they are also subsets (and also
         supersets) of each other.

       · "$norm = $set->Norm();"

         Returns the norm (number of bits which are set) of the given vector.

         This is equivalent to the number of elements contained in the given set.

         Uses a byte lookup table for calculating the number of set bits per byte, and thus needs
         a time for evaluation (and a number of loops) linearly proportional to the length of the
         given bit vector (in bytes).

         This should be the fastest algorithm on average.

       · "$norm = $set->Norm2();"

         Returns the norm (number of bits which are set) of the given vector.

         This is equivalent to the number of elements contained in the given set.

         This does the same as the method ""Norm()"" above, only with a different algorithm:

         This method counts the number of set and cleared bits at the same time and will stop
         when either of them has been exhausted, thus needing at most half as many loops per
         machine word as the total number of bits in a machine word - in fact it will need a
         number of loops equal to the minimum of the number of set bits and the number of cleared
         bits.

         This might be a faster algorithm than of the method ""Norm()"" above on some systems,
         depending on the system's architecture and the compiler and optimisation used, for bit
         vectors with sparse set bits and for bit vectors with sparse cleared bits (i.e.,
         predominantly set bits).

       · "$norm = $set->Norm3();"

         Returns the norm (number of bits which are set) of the given vector.

         This is equivalent to the number of elements contained in the given set.

         This does the same as the two methods ""Norm()"" and ""Norm2()"" above, however with a
         different algorithm.

         In fact this is the implementation of the method ""Norm()"" used in previous versions of
         this module.

         The method needs a number of loops per machine word equal to the number of set bits in
         that machine word.

         Only for bit vectors with sparse set bits will this method be fast; it will depend on a
         system's architecture and compiler whether the method will be faster than any of the two
         methods above in such cases.

         On average however, this is probably the slowest method of the three.

       · "$min = $set->Min();"

         Returns the minimum of the given set, i.e., the minimum of all indices of all set bits
         in the given bit vector "$set".

         If the set is empty (no set bits), plus infinity (represented by the constant "MAX_LONG"
         on your system) is returned.

         (This constant is usually 2 ^ (n-1) - 1, where ""n"" is the number of bits of an
         unsigned long on your machine.)

       · "$max = $set->Max();"

         Returns the maximum of the given set, i.e., the maximum of all indices of all set bits
         in the given bit vector "$set".

         If the set is empty (no set bits), minus infinity (represented by the constant
         "MIN_LONG" on your system) is returned.

         (This constant is usually -(2 ^ (n-1) - 1) or -(2 ^ (n-1)), where ""n"" is the number of
         bits of an unsigned long on your machine.)

       · "$m3->Multiplication($r3,$c3,$m1,$r1,$c1,$m2,$r2,$c2);"

         This method multiplies two boolean matrices (stored as bit vectors) "$m1" and "$m2" and
         stores the result in matrix "$m3".

         The method uses the binary "xor" operation (""^"") as the boolean addition operator
         (""+"").

         An exception is raised if the product of the number of rows and columns of any of the
         three matrices differs from the actual size of their underlying bit vector.

         An exception is also raised if the numbers of rows and columns of the three matrices do
         not harmonize in the required manner:

           rows3 == rows1
           cols3 == cols2
           cols1 == rows2

         This method is used by the module "Math::MatrixBool".

         See Math::MatrixBool(3) for details.

       · "$m3->Product($r3,$c3,$m1,$r1,$c1,$m2,$r2,$c2);"

         This method multiplies two boolean matrices (stored as bit vectors) "$m1" and "$m2" and
         stores the result in matrix "$m3".

         This special method uses the binary "or" operation (""|"") as the boolean addition
         operator (""+"").

         An exception is raised if the product of the number of rows and columns of any of the
         three matrices differs from the actual size of their underlying bit vector.

         An exception is also raised if the numbers of rows and columns of the three matrices do
         not harmonize in the required manner:

           rows3 == rows1
           cols3 == cols2
           cols1 == rows2

         This method is used by the module "Math::MatrixBool".

         See Math::MatrixBool(3) for details.

       · "$matrix->Closure($rows,$cols);"

         This method calculates the reflexive transitive closure of the given boolean matrix
         (stored as a bit vector) using Kleene's algorithm.

         (See Math::Kleene(3) for a brief introduction into the theory behind Kleene's
         algorithm.)

         The reflexive transitive closure answers the question whether a path exists between any
         two vertices of a graph whose edges are given as a matrix:

         If a (directed) edge exists going from vertex "i" to vertex "j", then the element in the
         matrix with coordinates (i,j) is set to "1" (otherwise it remains set to "0").

         If the edges are undirected, the resulting matrix is symmetric, i.e., elements (i,j) and
         (j,i) always contain the same value.

         The matrix representing the edges of the graph only answers the question whether an EDGE
         exists between any two vertices of the graph or not, whereas the reflexive transitive
         closure answers the question whether a PATH (a series of adjacent edges) exists between
         any two vertices of the graph!

         Note that the contents of the given matrix are modified by this method, so make a copy
         of the initial matrix in time if you are going to need it again later.

         An exception is raised if the given matrix is not quadratic, i.e., if the number of rows
         and columns of the given matrix is not identical.

         An exception is also raised if the product of the number of rows and columns of the
         given matrix differs from the actual size of its underlying bit vector.

         This method is used by the module "Math::MatrixBool".

         See Math::MatrixBool(3) for details.

       · "$matrix2->Transpose($rows2,$cols2,$matrix1,$rows1,$cols1);"

         This method calculates the transpose of a boolean matrix "$matrix1" (stored as a bit
         vector) and stores the result in matrix "$matrix2".

         The transpose of a boolean matrix, representing the edges of a graph, can be used for
         finding the strongly connected components of that graph.

         An exception is raised if the product of the number of rows and columns of any of the
         two matrices differs from the actual size of its underlying bit vector.

         An exception is also raised if the following conditions are not met:

           rows2 == cols1
           cols2 == rows1

         Note that in-place processing ("$matrix1" and "$matrix2" are identical) is only possible
         if the matrix is quadratic. Otherwise, a fatal "matrix is not quadratic" error will
         occur.

         This method is used by the module "Math::MatrixBool".

         See Math::MatrixBool(3) for details.

SEE ALSO
       Bit::Vector::Overload(3), Bit::Vector::String(3), Storable(3).

       Set::IntRange(3), Math::MatrixBool(3), Math::MatrixReal(3), DFA::Kleene(3),
       Math::Kleene(3), Graph::Kruskal(3).

VERSION
       This man page documents "Bit::Vector" version 7.3.

AUTHOR
         Steffen Beyer
         mailto:STBEY AT cpan.org
         http://www.engelschall.com/u/sb/download/

COPYRIGHT
       Copyright (c) 1995 - 2013 by Steffen Beyer. All rights reserved.

LICENSE
       This package is free software; you can redistribute it and/or modify it under the same
       terms as Perl itself, i.e., under the terms of the "Artistic License" or the "GNU General
       Public License".

       The C library at the core of this Perl module can additionally be redistributed and/or
       modified under the terms of the "GNU Library General Public License".

       Please refer to the files "Artistic.txt", "GNU_GPL.txt" and "GNU_LGPL.txt" in this
       distribution for details!

DISCLAIMER
       This package is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY;
       without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

       See the "GNU General Public License" for more details.



perl v5.20.0                                2014-08-15                                Vector(3pm)


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