drygdryg/calc
1
0
Fork 0
mirror of https://github.com/lcn2/calc.git synced 2025-08-19 01:13:27 +03:00
Code Issues Packages Projects Releases Wiki Activity

convert ASCII TABs to ASCII SPACEs

Browse Source 
Converted all ASCII tabs to ASCII spaces using a 8 character
tab stop, for all files, except for all Makefiles (plus rpm.mk).
The `git diff -w` reports no changes.
This commit is contained in:
Landon Curt Noll
2024-07-11 22:03:52 -07:00
parent fe9cefe6ef
commit db77e29a23
631 changed files with 90607 additions and 90600 deletions
Download Patch File Download Diff File Expand all files Collapse all files

170
help/jacobi
View File

@@ -5,10 +5,10 @@ SYNOPSIS
jacobi(x, y)
TYPES
x integer
y integer
x integer
y integer
return 1, -1, or 0
return 1, -1, or 0
DESCRIPTION
The jacobi builtin function of calc returns the value of the
@@ -16,19 +16,19 @@ DESCRIPTION
If y is an odd prime, then the Legendre symbol (x/y) returns:
(x/y) == 0 x is divisible by y (0 == x % y)
(x/y) == 1 if x is a quadratic residue modulo y
(x/y) == -1 if x is not a quadratic residue modulo y
(x/y) == 0 x is divisible by y (0 == x % y)
(x/y) == 1 if x is a quadratic residue modulo y
(x/y) == -1 if x is not a quadratic residue modulo y
Legendre symbol often denoted as (x/y) as if x/y were a fraction.
If there exists an integer u such that:
x == u^2 modulo y (i.e., x == u^2 % y)
x == u^2 modulo y (i.e., x == u^2 % y)
then x is a quadratic residue modulo y. However, if for all integers u:
x != u^2 modulo y (i.e., x != u^2 % y)
x != u^2 modulo y (i.e., x != u^2 % y)
x is said to be a quadratic non-residue modulo y.
@@ -36,21 +36,21 @@ DESCRIPTION
That is, Jacobi symbol satisfies the same rules as the Legendre symbol.
Therefore, when y is an odd prime:
jacobi(x,y) == 0 x is divisible by y (x == 0 % y)
jacobi(x,y) == 1 if x is a quadratic residue modulo y
jacobi(x,y) == -1 if x is a quadratic non-residue modulo y
jacobi(x,y) == 0 x is divisible by y (x == 0 % y)
jacobi(x,y) == 1 if x is a quadratic residue modulo y
jacobi(x,y) == -1 if x is a quadratic non-residue modulo y
When y is even or y <= 0:
jacobi(x,y) == 0
jacobi(x,y) == 0
When y is not prime, we may express y as the product of primes:
y == p0^a0 * p1^a1 * .. * pn^an
y == p0^a0 * p1^a1 * .. * pn^an
When y is odd and NOT prime, then:
jacobi(x,y) == (x/p0)^a0 * (x/p1)^a1 * .. * (x/pn)^an
jacobi(x,y) == (x/p0)^a0 * (x/p1)^a1 * .. * (x/pn)^an
where (x/y) is Legendre symbol.
@@ -73,113 +73,113 @@ DESCRIPTION
We have these identities:
jacobi(x,y) == 0 if x <= 0
jacobi(x,y) == 0 if y <= 0
jacobi(x,y) == 0 if y is even
jacobi(x,y) == 0 if x <= 0
jacobi(x,y) == 0 if y <= 0
jacobi(x,y) == 0 if y is even
jacobi(x,b) == 0 if gcd(x,b) > 1
jacobi(x,b) == 0 if gcd(x,b) > 1
jacobi(0,b) == 0
jacobi(1,b) == 1
jacobi(0,b) == 0
jacobi(1,b) == 1
jacobi(x,b) == jacobi(y,b) if x == y % b
jacobi(x,b) == jacobi(y,b) if x == y % b
jacobi(a,b) == jacobi(2*a,b) if b == 1 % 8 OR b == 7 % 8
jacobi(a,b) == -jacobi(2*a,b) if b != 1 & 8 AND b != 7 % 8
jacobi(a,b) == jacobi(2*a,b) if b == 1 % 8 OR b == 7 % 8
jacobi(a,b) == -jacobi(2*a,b) if b != 1 & 8 AND b != 7 % 8
jacobi(a,b) == jacobi(b,a) if a == 3 % 4 AND b == 3 % 4
jacobi(a,b) == -jacobi(b,a) if a != 3 % 4 OR b != 3 % 4
jacobi(a,b) == jacobi(b,a) if a == 3 % 4 AND b == 3 % 4
jacobi(a,b) == -jacobi(b,a) if a != 3 % 4 OR b != 3 % 4
jacobi(-1,b) == 1 if b == 1 % 4
jacobi(-1,b) == -1 if b == 3 % 4
jacobi(-1,b) == 1 if b == 1 % 4
jacobi(-1,b) == -1 if b == 3 % 4
jacobi(x,b) * jacobi(y,b) == jacobi(x*y,b)
jacobi(x,a) * jacobi(x,b) == jacobi(x,a*b)
jacobi(x,b) * jacobi(y,b) == jacobi(x*y,b)
jacobi(x,a) * jacobi(x,b) == jacobi(x,a*b)
Technically the Jacobi symbol jacobi(x,y) is not defined when
x == 0, x < 0, y is even, y == 0, and when y < 0. So by convention:
jacobi(x,y) == 0 if x <= 0
jacobi(x,y) == 0 if y <= 0
jacobi(x,y) == 0 if y == 0 % 2
jacobi(x,y) == 0 if x <= 0
jacobi(x,y) == 0 if y <= 0
jacobi(x,y) == 0 if y == 0 % 2
It is also worth noting that:
jacobi(x,y) == 0 if gcd(y,x) != 1
jacobi(y,y) == 0 if y > 1
jacobi(x,y) == 0 if gcd(y,x) != 1
jacobi(y,y) == 0 if y > 1
Based on the generalization of the quadratic reciprocity theorem,
when a and b are odd, b >= 3, and gcd(a,b) == 1:
jacobi(a,b) == jacobi(b,a) * ((-1) ^ ((a-1) * (b-1) / 4))
jacobi(a,b) == jacobi(b,a) * ((-1) ^ ((a-1) * (b-1) / 4))
Therefore, when a and b are odd, b >= 3, and gcd(a,b) == 1:
jacobi(a,b) == jacobi(b,a) if a == 1 % 4 OR b == 1 % 4
jacobi(a,b) == -jacobi(b,a) if a == 3 % 4 AND b == 3 % 4
jacobi(a,b) == jacobi(b,a) if a == 1 % 4 OR b == 1 % 4
jacobi(a,b) == -jacobi(b,a) if a == 3 % 4 AND b == 3 % 4
Jacobi symbol satisfies the same rules as the Legendre symbol such as:
jacobi(x,y0*y1) == jacobi(x,y0) * jacobi(x,y1)
jacobi(x0*x1,y) == jacobi(x0,y) * jacobi(x1,y)
jacobi(x,y0*y1) == jacobi(x,y0) * jacobi(x,y1)
jacobi(x0*x1,y) == jacobi(x0,y) * jacobi(x1,y)
When b is odd:
jacobi(x^2,b) == 1 if gcd(x,b) == 1
jacobi(x,b^2) == 1 if gcd(x,b) == 1
jacobi(x^2,b) == 1 if gcd(x,b) == 1
jacobi(x,b^2) == 1 if gcd(x,b) == 1
jacobi(x0,b) == jacobi(x1,b) if x0 == x1 % b
jacobi(x0,b) == jacobi(x1,b) if x0 == x1 % b
jacobi(-1,b) == (-1) ^ ((b-1) / 2) == 1 if b == 1 % 4
jacobi(-1,b) == (-1) ^ ((b-1) / 2) == -1 if b == 3 % 4
jacobi(-1,b) == (-1) ^ ((b-1) / 2) == 1 if b == 1 % 4
jacobi(-1,b) == (-1) ^ ((b-1) / 2) == -1 if b == 3 % 4
jacobi(2,b) == (-1) ^ ((b^2 - 1) / 8) == 1 if b == 1 % 8 OR
b == 7 % 8
jacobi(2,b) == (-1) ^ ((b^2 - 1) / 8) == -1 if b == 3 % 8 OR
b == 5 % 8
jacobi(2,b) == (-1) ^ ((b^2 - 1) / 8) == 1 if b == 1 % 8 OR
b == 7 % 8
jacobi(2,b) == (-1) ^ ((b^2 - 1) / 8) == -1 if b == 3 % 8 OR
b == 5 % 8
jacobi(-3,b) == 1 if b == 1 % 6
jacobi(-3,b) == -1 if b == 5 % 6
jacobi(-3,b) == 1 if b == 1 % 6
jacobi(-3,b) == -1 if b == 5 % 6
jacobi(5,b) == 1 if b == 1 % 10 OR
b == 9 % 10
jacobi(5,b) == -1 if b == 3 % 10 OR
b == 7 % 10
jacobi(5,b) == 1 if b == 1 % 10 OR
b == 9 % 10
jacobi(5,b) == -1 if b == 3 % 10 OR
b == 7 % 10
If a and b are both odd primes, OR
if gcd(a,b) = 1 and, a > 0, and a is odd:
jacobi(a,b) == jacobi(b,a) * ((-1) ^ (((a-1)/2) * ((b-1)/2)))
jacobi(a,b) * jacobi(b,a) == ((-1) ^ (((a-1)/2) * ((b-1)/2)))
jacobi(a,b) == jacobi(b,a) * ((-1) ^ (((a-1)/2) * ((b-1)/2)))
jacobi(a,b) * jacobi(b,a) == ((-1) ^ (((a-1)/2) * ((b-1)/2)))
If b is an odd prime:
jacobi(a,b) == (a ^ ((b-1)/2)) % b
jacobi(a,b) == (a ^ ((b-1)/2)) % b
Combining all of the above, computing jacobi(x,b) may be expressed
in the following algorithm when b is odd and 0 < x < b:
j = 1;
while (x != 0) {
while (iseven(x)) {
x = x / 2;
t = b % 8;
if (t == 3 || t == 5) {
j = -j;
}
}
swap(x,b);
if (((x % 4) == 3) && ((b % 4) == 3)) {
j = -j;
}
x = x % b;
}
j = 1;
while (x != 0) {
while (iseven(x)) {
x = x / 2;
t = b % 8;
if (t == 3 || t == 5) {
j = -j;
}
}
swap(x,b);
if (((x % 4) == 3) && ((b % 4) == 3)) {
j = -j;
}
x = x % b;
}
if (b == 1) {
return j;
} else {
return 0;
}
if (b == 1) {
return j;
} else {
return 0;
}
The above calc pseudo code is provided to help illustrate the
properties of the jacobi symbol only. The use of calc's jacobi
@@ -188,9 +188,9 @@ DESCRIPTION
For more information on the Legendre symbol and the Jacobi symbol:
http://mathworld.wolfram.com/LegendreSymbol.html
http://mathworld.wolfram.com/JacobiSymbol.html
http://primes.utm.edu/glossary/xpage/JacobiSymbol.html
http://mathworld.wolfram.com/LegendreSymbol.html
http://mathworld.wolfram.com/JacobiSymbol.html
http://primes.utm.edu/glossary/xpage/JacobiSymbol.html
EXAMPLE
; print jacobi(2,3), jacobi(2,5), jacobi(2,15)
@@ -216,7 +216,7 @@ SEE ALSO
##
## Calc 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 Lesser General
## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
## Public License for more details.
##
## A copy of version 2.1 of the GNU Lesser General Public License is
@@ -224,8 +224,8 @@ SEE ALSO
## received a copy with calc; if not, write to Free Software Foundation, Inc.
## 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
##
## Under source code control: 1995/12/18 12:34:57
## File existed as early as: 1995
## Under source code control: 1995/12/18 12:34:57
## File existed as early as: 1995
##
## chongo <was here> /\oo/\ http://www.isthe.com/chongo/
## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
## chongo <was here> /\oo/\ http://www.isthe.com/chongo/
## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/