diff options
Diffstat (limited to 'modular.txi')
| -rw-r--r-- | modular.txi | 56 |
1 files changed, 20 insertions, 36 deletions
diff --git a/modular.txi b/modular.txi index bf2cd52..666f52a 100644 --- a/modular.txi +++ b/modular.txi @@ -2,31 +2,6 @@ @ftindex modular -@defun mod x1 x2 -@defunx rem x1 x2 - -These procedures implement the Common-Lisp functions of the same names. -The real number @var{x2} must be non-zero. -@code{mod} returns @code{(- @var{x1} (* @var{x2} (floor (/ @var{x1} @var{x2}))))}. -@code{rem} returns @code{(- @var{x1} (* @var{x2} (truncate (/ @var{x1} @var{x2}))))}. - -If @var{x1} and @var{x2} are integers, then @code{mod} behaves like -@code{modulo} and @code{rem} behaves like @code{remainder}. - -@format -@t{(mod -90 360) @result{} 270 -(rem -90 180) @result{} -90 - -(mod 540 360) @result{} 180 -(rem 540 360) @result{} 180 - -(mod (* 5/2 pi) (* 2 pi)) @result{} 1.5707963267948965 -(rem (* -5/2 pi) (* 2 pi)) @result{} -1.5707963267948965 -} -@end format -@end defun - - @defun extended-euclid n1 n2 Returns a list of 3 integers @code{(d x y)} such that d = gcd(@var{n1}, @@ -34,13 +9,18 @@ Returns a list of 3 integers @code{(d x y)} such that d = gcd(@var{n1}, @end defun -@defun symmetric:modulus n +@defun symmetric:modulus m -Returns @code{(quotient (+ -1 n) -2)} for positive odd integer @var{n}. +For odd positive integer @var{m}, returns an object suitable for passing +as the first argument to @code{modular:} procedures, directing them +to return a symmetric modular number, ie. an @var{n} such that +@example +(<= (quotient @var{m} -2) @var{n} (quotient @var{m} 2) +@end example @end defun -@defun modulus->integer modulus +@defun modular:characteristic modulus Returns the non-negative integer characteristic of the ring formed when @var{modulus} is used with @code{modular:} procedures. @@ -49,7 +29,7 @@ Returns the non-negative integer characteristic of the ring formed when @defun modular:normalize modulus n -Returns the integer @code{(modulo @var{n} (modulus->integer +Returns the integer @code{(modulo @var{n} (modular:characteristic @var{modulus}))} in the representation specified by @var{modulus}. @end defun @@ -61,18 +41,22 @@ arguments are constrained by the following table: For all of these functions, if the first argument (@var{modulus}) is: @table @code @item positive? -Work as before. The result is between 0 and @var{modulus}. +Integers mod @var{modulus}. The result is between 0 and +@var{modulus}. @item zero? The arguments are treated as integers. An integer is returned. +@end table -@item negative? -The arguments and result are treated as members of the integers modulo -@code{(+ 1 (* -2 @var{modulus}))}, but with @dfn{symmetric} +@noindent +Otherwise, if @var{modulus} is a value returned by +@code{(symmetric:modulus @var{radix})}, then the arguments and +result are treated as members of the integers modulo @var{radix}, +but with @dfn{symmetric} representation; i.e. @cindex symmetric -representation; i.e. @code{(<= (- @var{modulus}) @var{n} -@var{modulus})}. -@end table +@example +(<= (quotient @var{radix} 2) @var{n} (quotient (- -1 @var{radix}) 2) +@end example @noindent If all the arguments are fixnums the computation will use only fixnums. |