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kernel: simplify and sped up ModRat using InverseModInt
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Before (on a 64 bit machine):

gap> q:=1/3;;n:=7;; for i in [1..2000000] do x:= q mod n; od; time;
258
gap> q:=1/3;;n:=2^59;; for i in [1..2000000] do x:= q mod n; od; time;
295
gap> q:=1/3;;n:=2^60;; for i in [1..2000000] do x:= q mod n; od; time;
569
gap> q:=1/3^60;;n:=7;; for i in [1..2000000] do x:= q mod n; od; time;
537
gap> q:=1/3^60;;n:=2^59;; for i in [1..2000000] do x:= q mod n; od; time;
2364
gap> q:=1/3^60;;n:=2^60;; for i in [1..2000000] do x:= q mod n; od; time;
2821
gap> q:=1/2^6000;;n:=3^6000;; for i in [1..6000] do x:= q mod n; od; time;
19358

After:

gap> q:=1/3;;n:=7;; for i in [1..2000000] do x:= q mod n; od; time;
255
gap> q:=1/3;;n:=2^59;; for i in [1..2000000] do x:= q mod n; od; time;
244
gap> q:=1/3;;n:=2^60;; for i in [1..2000000] do x:= q mod n; od; time;
392
gap> q:=1/3^60;;n:=7;; for i in [1..2000000] do x:= q mod n; od; time;
245
gap> q:=1/3^60;;n:=2^59;; for i in [1..2000000] do x:= q mod n; od; time;
1219
gap> q:=1/3^60;;n:=2^60;; for i in [1..2000000] do x:= q mod n; od; time;
1292
gap> q:=1/2^6000;;n:=3^6000;; for i in [1..6000] do x:= q mod n; od; time;
692
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@fingolfin
fingolfin committed Jan 8, 2018
1 parent 14d9ca1 commit 79ec5f3
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Showing 3 changed files with 39 additions and 36 deletions.
8 changes: 8 additions & 0 deletions src/integer.h
View file
Original file line number Diff line number Diff line change
Expand Up @@ -335,6 +335,14 @@ extern Obj GcdInt( Obj opL, Obj opR );
extern Obj AInvInt( Obj op );


/****************************************************************************
**
*F InverseModInt( <op> ) . . . . mult. inverse of an integer modulo another
**
*/
extern Obj InverseModInt(Obj base, Obj mod);


/****************************************************************************
**
** Compute log2 of the absolute value of an Int, i.e. the index of the highest
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49 changes: 13 additions & 36 deletions src/rational.c
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Original file line number Diff line number Diff line change
Expand Up @@ -611,10 +611,10 @@ Obj QuoRat (

/****************************************************************************
**
*F ModRat( <opL>, <opR> ) . . . . . . . . remainder of fraction mod integer
*F ModRat( <opL>, <n> ) . . . . . . . . remainder of fraction mod integer
**
** 'ModRat' returns the remainder of the fraction <opL> modulo the integer
** <opR>. The remainder is always an integer.
** <n>. The remainder is always an integer.
**
** '<r> / <s> mod <n>' yields the remainder of the fraction '<p> / <q>'
** modulo the integer '<n>', where '<p> / <q>' is the reduced form of
Expand All @@ -634,43 +634,20 @@ Obj QuoRat (
** such that $0 \<= t/s \< n$ and $r/s - t/s$ is a multiple of $n$. This is
** rarely needed while computing modular inverses is very useful.
*/
Obj ModRat (
Obj opL,
Obj opR )
Obj ModRat(Obj opL, Obj n)
{
Obj a, aL, b, bL, c, cL, hdQ;

/* make the integer positive */
if ( IS_NEG_INT(opR) ) {
opR = AInvInt( opR );
}

/* let <p>/<q> represent <r>/<s> in reduced form */
/* invert the denominator <q> modulo <n> with Euclids algorithm */
a = opR; aL = INTOBJ_INT( 0L ); /* a = <n> */
b = DEN_RAT(opL); bL = INTOBJ_INT( 1L ); /* b = <q> */
while ( a != INTOBJ_INT( 1L ) ) {
while ( b != INTOBJ_INT( 0L ) ) {
hdQ = QuoInt( a, b );
c = b; cL = bL;
b = DiffInt( a, ProdInt( hdQ, b ) );
bL = DiffInt( aL, ProdInt( hdQ, bL ) );
a = c; aL = cL;
}

/* check whether the denominator <q> really was invertible mod <n> */
if ( a != INTOBJ_INT( 1L ) ) {
opR = ErrorReturnObj(
"ModRat: for <r>/<s> mod <n>, <s>/gcd(<r>,<s>) and <n> must be coprime",
0L, 0L,
"you can replace the integer <n> via 'return <n>;'" );
a = opR; aL = INTOBJ_INT( 0L ); /* a = <n> */
b = DEN_RAT(opL); bL = INTOBJ_INT( 1L ); /* b = <q> */
}
// invert the denominator
Obj d = InverseModInt( DEN_RAT(opL), n );

// check whether the denominator of <opL> really was invertible mod <n> */
if ( d == Fail ) {
ErrorMayQuit(
"ModRat: for <r>/<s> mod <n>, <s>/gcd(<r>,<s>) and <n> must be coprime",
0, 0 );
}

/* return the remainder */
return ModInt( ProdInt( NUM_RAT(opL), aL ), opR );
// return the remainder
return ModInt( ProdInt( NUM_RAT(opL), d ), n );
}


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18 changes: 18 additions & 0 deletions tst/testinstall/rationals.tst
View file
Original file line number Diff line number Diff line change
Expand Up @@ -3,6 +3,10 @@ gap> NumeratorRat(1/2);
1
gap> DenominatorRat(1/2);
2

#
# 'Rat' -- converting things to a rational
#
gap> Rat(1/2) = 1/2;
true
gap> Rat(1) = 1;
Expand All @@ -19,3 +23,17 @@ gap> Rat(3.7e-1);
37/100

#
# modular inverses of rationals
#
gap> 1/2 mod 3;
2
gap> 1/2 mod -3;
2
gap> -1/2 mod 3;
1
gap> -1/2 mod -3;
1
gap> 1/2 mod 2;
Error, ModRat: for <r>/<s> mod <n>, <s>/gcd(<r>,<s>) and <n> must be coprime

#

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