COMMENT:
I am enclosing two document files relating to nb.c.
They are separated by rows of ####'s.
The first deals with routines in nb.c which are global, and the
second with those that are static.
They are incomplete in three ways, perhaps.
(i)
I have not yet included enough examples.
(ii)
Some of the remarks about return values may not be correct.
I have not checked them very carefully.
(iii) You may consider that, in some respects, there is too much
in them and, in particular, that the statics should not be
included.
############################################################
NB.DOC
######
ELEMENTS IN NUMBER FIELDS
------------------------This module contains routines for manipulating elements in number fields.
For the current implementation, two type of number field are available:
(i):
Cyclotomic fields,
(ii):
Square radical fields -- i.e. extensions of the rational number
field
by finitely many square roots of rational integers.
The objects in SYMMETRICA which implement these numbers are known as
cyclos and
sqrads and have the SYMMETRICA types CYCLOTOMIC and SQ_RADICAL,
respectively.
They are based on an object known as a monopoly and of type MONOPOLY
which
implements a polynomial in one variable.
In what follows, a monopoly in the variable x may be denoted by p(x) or
a_n * x:n + a_(n-1) * x:(n-1) + . . . + a_1 * x + a_0.
When this represents a cyclo, x will be the 'basic' primitive m-th root
of
unity e:(2 * pi * i / m) for an appropriate positive integer m -- m will
be
known as the index of the cyclo.
When the monopoly above represents a sqrad, the sqrad corresponds to the
number a_n * \sqrt(n) + a_(n-1) * \sqrt(n-1) + . . . + a_1.
The following gives further details of the three structures MONOPOLY,
CYCLOTOMIC, SQ_RADICAL
A monopoly is an object of type MONOPOLY, with two fields -- a self and a
next. Thus, it is syntactically the same as a list. Its next is a
monopoly or NULL. Its self is a monom whose coefficient is a scalar
(integer
or rational number) and whose self is a non-negative integer. Note that
no
check is made to see whether these constraints are satisfied, but certain
routines will behave strangely if they are violated.
There is a number object with two fields -- self and data. The self is a
monopoly. A cyclo is an number object of type CYCLOTOMIC whose data
field
is a pointer to a block of four items -- the index, the corresponding
cyclotomic polynomial, its degree and the 'automorphisms' of the field
(in fact, the integers between 1 and the index and coprime to it).
A sqrad is a number object whose data is a list of the prime divisors
(including -1) of the integers occuring as selfs of the terms of its
monopoly.
Data relating to cyclotomic numbers are held in a table (read from a file
named CYCLOS.DAT) and in a list. A newly created cyclo with a known
index
(i.e. known to the table or the list) has a data pointer pointing to the
known table or list item. A newly created cyclo with a new index has a
new data item created and appended to the list. At any time, the list of
data items may be appended to CYCLOS.DAT.
Although descriptions of the basic routines for manipulating the various
types of number, one should normally use the general routines given at
the
end of this document. For example, use add(x,y) rather than
add_scalar_cyclo(x,y), since the general routines have type-checking
built
in.
ELEMENTARY ROUTINES RELATING TO INTEGERS.
----------------------------------------NAME:
number_of_digits
SYNOPSIS:
INT number_of_digits(OP a)
DESCRIPTION:
Determines the number of digits of the integer a.
an INTEGER or a LONGINT.
RETURN:
The number of digits.
a may be
NAME:
integer_factors_to_integer
SYNOPSIS:
INT integer_factors_to_integer(OP l,OP a)
DESCRIPTION:
l is a MONOPOLY representing the factorization of an integer
into integers with integer exponents.
reform
the integer.
RETURN:
OK or ERROR
These factors are combined to
NAME:
make_coprimes
SYNOPSIS:
INT make_coprimes(OP number,OP result)
DESCRIPTION:
Given the number n, which should be a positive INTEGER or
LONGINT or a MONOPOLY representing a factorisation of an integer
greater
than 1, the result returns the list of positive integers coprime to
n.
RETURN:
OK or ERROR
NAME:
euler_phi
SYNOPSIS:
INT euler_phi(OP a,OP b)
DESCRIPTION:
Determines the number of numbers coprime to an integer a
and returns it in b. The object a may be INTEGER object or LONGINT
object.
RETURN:
OK or ERROR !!
NAME:
ganzsquareroot_longint
SYNOPSIS:
INT ganzsquareroot_longint(OP a,OP b)
DESCRIPTION:
a is a non-negative LONGINT object.
b is set to the integer part of its square root. In this case, the
return
value is OK or IMPROPER according as the integer is a perfect square
or not.
Otherwise, the return value is ERROR.
RETURN:
OK, IMPROPER or ERROR
NAME:
ganzsquareroot_integer
SYNOPSIS:
INT ganzsquareroot_integer(OP a,OP b)
DESCRIPTION:
a is a non-negative INTEGER object.
b is set to the integer part of its square root. In this case, the
return
value is OK or IMPROPER according as the integer is a perfect square
or not.
Otherwise, the return value is ERROR.
RETURN:
OK, IMPROPER or ERROR
NAME:
primep
SYNOPSIS:
INT primep(OP a)
DESCRIPTION:
returns TRUE if a is prime number FALSE else
COMMENT:
INTEGER FACTORISATION
--------------------There is a simple routine for prime factorization of integers. A table of
prime factors is either read from a file PRIMES.DAT or constructed
directly.
An integer is factored by first using the primes in the table, and then
continuing through all odd numbers following the greatest prime in the
table.
NAME:
setup_prime_table
SYNOPSIS:
INT setup_prime_table()
DESCRIPTION:
Creates a table of rational prime numbers.
If the source is compiled with PRIME_FILE #defined, it searches for a
file
in the current directory named PRIMES.DAT, and reads the table of
primes
from that file. The first entry should be the number of primes in the
file,
then the list of primes (assumes that INTs are longs).
If PRIME_FILE is not #defined, a table of the first 15 primes is set
up.
Returns OK if the table is set; otherwise, returns ERROR.
RETURN:
OK or ERROR
NAME:
first_prime_factor
SYNOPSIS:
INT first_prime_factor(OP a,OP first_prime)
DESCRIPTION:
This routine finds the smallest prime factor of an integer
a. The prime found is returned as first_prime.
RETURN:
OK or ERROR
NAME:
square_free_part
SYNOPSIS:
INT square_free_part(OP a,OP b,OP c,OP la,OP lb,OP lc)
DESCRIPTION:
This routine find the square-free part of the integer a,
i.e. the product of the prime factors which occur to an odd exponent
and -1,
if the integer is negative. a may be either an INTEGER, LONGINT or a
MONOPOLY containing the prime factorization of an integer.
b and c return the the square-free part the square-root of the square
part,
respectively. Thus, a = b * c ** 2, and b has no repeated prime
factors.
If a is not a MONOPOLY and la is not NULL, la returns a MONOPOLY
containing
the prime factorization of a. If they are not NULL, lb and lc return
MONOPOLYs containing the prime factorizations of b and c.
The parameters a,b,c must be distinct. If la,lb,lc are not NULL they
must
be distinct also.
This routine makes use of the ancillary routine square_free_part_0.
RETURN:
OK or ERROR
NAME:
square_free_part_0
SYNOPSIS:
INT square_free_part_0( OP la,lb,lc)
DESCRIPTION:
This routine find the square-free part of the integer, which
is given as a prime factors list la -- a MONOPOLY containing the
prime
factorization of the integer. lb and lc return MONOPOLYs containing
the
prime factorization of the square-free part and square-root of the
square
part, respectively.
That is, lb has no prime (including -1), occurring with an exponent >
1.
The parameters la,lb,lc must be distinct.
RETURN:
OK or ERROR
NAME:
jacobi
SYNOPSIS:
INT jacobi(a,b,c) OP a,b,c;
DESCRIPTION:
The Jacobi Symbol: (a/b) b odd.
a and b are integers.
c must
point to a location different from a and b. If a and b have a common
factor,
c is set to 0 and ERROR is returned. Otherwise, c is set to the the
jacobi
symbol (a/b). Note that b must be odd.
RETURN:
OK or ERROR
NAME:
kronecker
SYNOPSIS:
INT kronecker(a,b,c) OP a,b,c;
DESCRIPTION:
The Kronecker Symbol: (a/b). a square-free and congruent to
0 or 1 mod 4. a and b are integers. c must point to a location
different
from a and b. If a and b have a common factor, c is set to 0 and
ERROR is
returned. Otherwise, c is set to the the kronecker symbol (a/b).
Note
that b must be odd.
RETURN:
OK or ERROR
NAME:
b_skn_mp
SYNOPSIS:
INT b_skn_mp( OP s,k,n,e)
DESCRIPTION:
Build a monopoly whose self is s, coefficient is
k and next is n. b_skn_mp uses the objects supplied as arguments,
while
m_skn_mp uses copies of them.
RETURN:
OK or ERROR (!)
NAME:
m_skn_mp
SYNOPSIS:
INT m_skn_mp( OP s,k,n,e)
DESCRIPTION:
Make a monopoly whose self is s, coefficient is
k and next is n. b_skn_mp uses the objects supplied as arguments,
while
m_skn_mp uses copies of them.
RETURN:
OK or ERROR (!)
EXAMPLE:
Making the monopoly corresponding to -1 * x : 10 + 4 * x : 17.
It is assumed that all OP identifiers point to objects created by
callocobject.
(1)
m_i_i(17L,b);
m_i_i(4L,c);
b_skn_mp(b,c,d,a);
b = callocobject();
c = callocobject();
m_i_i(10L,b);
m_i_i(-1L,c);
b_skn_mp(b,c,NULL,d);
/* need new objects for these pointers
*/
/* at this stage the monopoly is complete
*/
The statement 'println(a);' causes the output 4 17
-1 10
The statement 'objectwrite(stdout,a);' causes the output where, for
convenience,
some lines have been joined and the components have been annotated:
126
1
21
1
4
1
17
1
MONOPOLY
nonNULL MONOM
INTEGER value
INTEGER value
more terms
126
1
21
1
-1
1
10
0
MONOPOLY
nonNULL MONOM
INTEGER value
INTEGER value
no more terms
(2) The following creates the same monopoly, the insert routine carries
out some sorting.
init(MONOPOLY,a);
m_i_i(10L,b);
m_i_i(-1L,c);
h = callocobject();
m_sk_mo(b,c,h);
insert(h,a,add_koeff,NULL);
m_i_i(17L,b);
m_i_i(4L,c);
h = callocobject();
m_sk_mo(b,c,h);
insert(h,a,add_koeff,NULL);
The statement 'println(a);' causes the output -1 10
4 17
The statement 'objectwrite(stdout,a);' causes the output:ce,
126 1 21 1 -1 1 10 1 126 1 21 1 4 1 17 0
NAME:
scan_monopoly
SYNOPSIS:
INT scan_monopoly(a) OP a;
DESCRIPTION:
Routines for inputting a monopoly from stdin. scan_monopoly
requests the type of self and coefficient. It then transfers control
to
SCMPCO which requests the number of terms in the monopoly and inputs
the
terms one by one. This is a subroutine of scan. It is better to
use the general routine scan.
RETURN:
OK or ERROR
NAME:
remove_zero_terms
SYNOPSIS:
INT remove_zero_terms(a) OP a;
DESCRIPTION:
Removes those terms from a MONOPOLY with zero coefficients
unless this makes the list empty. In this case, one term with self
and
coefficient both 0 is left.
RETURN:
OK or ERROR
NAME:
add_scalar_monopoly
SYNOPSIS:
INT add_scalar_monopoly( OP a,b,c)
DESCRIPTION:
subroutine of add
NAME:
mult_scalar_monopoly
SYNOPSIS:
INT mult_scalar_monopoly( OP a,b,c)
DESCRIPTION:
subroutine of mult
NAME:
add_monopoly_monopoly
SYNOPSIS:
INT add_monopoly_monopoly( OP a, b, c)
DESCRIPTION:
subroutine of add
NAME:
mult_monopoly_monopoly
SYNOPSIS:
INT mult_monopoly_monopoly(OP a, b, c)
DESCRIPTION:
subroutine of mult
NAME:
add_monopoly
SYNOPSIS:
INT add_monopoly( OP a,b,c)
DESCRIPTION:
subroutine of add
NAME:
add_apply_monopol
SYNOPSIS:
INT add_apply_monopoly( OP a,b)
DESCRIPTION:
Addition of objects of type INTEGER,
LONGINT, BRUCH and MONOPOLY. Subroutine of add_apply
NAME:
mult_monopoly
SYNOPSIS: INT mult_monopoly( OP a,b,c)
DESCRIPTION:
for the multiplication of a object a of type MONOPOLY
with an arbitray object b, the result is the object c.
Subroutine of mult
RETURN:
OK if no error occured.
NAME:
mult_apply_monopoly
SYNOPSIS:
INT mult_apply_monopoly(OP a,b)
DESCRIPTION:
Multiplication of objects of type INTEGER, LONGINT,
BRUCH, MATRIX, MONOM, POLYNOM, SCHUBERT, VECTOR and MONOPOLY,
Better to use the general routine mult_apply
NAME:
addinvers_monopoly
SYNOPSIS:
INT addinvers_monopoly( OP a,b)
DESCRIPTION:
subroutine of the general routine addinvers
NAME:
addinvers_apply_monopoly
SYNOPSIS:
INT addinvers_apply_monopoly(OP a)
DESCRITPION:
subroutine of the general routine addinvers_apply
NAME:
nullp
SYNOPSIS:
INT nullp_monopoly(OP a)
DESCRIPTION:
subroutine of the general routine nullp.
NAME:
comp_monopoly
SYNOPSIS:
INT comp_monopoly(OP a,b)
DESCRIPTION:
Compares monopolies as lists, using comp_list()
RETURN:
0 (= equal) <0 if a<b >0 if a>b
NAME:
quores_monopoly
SYNOPSIS:
INT quores_monopoly( OP poly,dpoly,qpoly,rpoly)
DESCRIPTION:
Carries out the division algorithm on polynomials of one
variable to find the quotient (qpoly) and remainder (rpoly). The
result
is poly = dpoly * qpoly + rpoly, where rpoly has degree less than
dpoly,
or represents 0. The parameters poly, dpoly, qpoly and rpoly must
all be
different.
RETURN:
OK or ERROR
NAME:
raise_power_monopoly
SYNOPSIS:
INT raise_power_monopoly( OP a, b)
DESCRIPTION:
Multiplies all the self components of the terms of the
monopoly b by the scalar a. Viewing the monopoly as a polynomial
p(x),
this has the effect of replacing it by p(x**a).
RETURN:
OK or ERROR
NAME:
scale_monopoly
SYNOPSIS:
INT scale_monopoly(a,b) OP a, b;
DESCRIPTION:
Viewing the monopoly b as a polynomial p(x), the effect of
this routine is to replace it by p(a*x).
RETURN:
OK or ERROR
NAME:
objectread_monopoly
SYNOPSIS:
INT objectread_monopoly(f,a) FILE *f; OP a;
DESCRIPTION:
Reads a monopoly a from the stream f.
RETURN:
OK or ERROR
NAME:
tex_monopoly
SYNOPSIS:
INT tex_monopoly(a) OP a;
DESCRIPTION:
Outputs a monopoly in a form suitable for TeX processing.
It is treated as a polynomial in x. Subroutine of the general routine
tex.
RETURN:
OK or ERROR
NAME:
make_unitary0_monopoly
SYNOPSIS:
INT make_unitary0_monopoly(number,result) OP number,
result;
DESCRIPTION:
Given the number n, which should be an positive INTEGER or
LONGINT, the result returns the monopoly corresponding to x**n-1.
RETURN:
OK or ERROR
NAME:
make_unitary1_monopoly
SYNOPSIS:
INT make_unitary1_monopoly(number,result) OP number,
result;
DESCRIPTION:
Given the number n, which should be an positive INTEGER or
LONGINT, the result returns the MONOPOLY x**(n-1) + x**(n-2) + ... +
x + 1.
RETURN:
OK or ERROR
NAME:
make_cyclotomic_monopoly
SYNOPSIS:
INT make_cyclotomic_monopoly(number,result) OP number,
result;
DESCRIPTION:
Given the number n, which should be an positive INTEGER or
LONGINT or a MONOPOLY representing a factorisation of an integer
greater
than 1, the result returns the cyclotomic polynomial of index n,
phi_n(x).
RETURN:
OK or ERROR
NAME:
t_MONOPOLY_POLYNOM
SYNOPSIS:
INT t_MONOPOLY_POLYNOM(OP a,b)
DESCRIPTION:
converts a MONOPOLY object
with one variable.
a into a POLYNOM object b
NAME:
eq_fieldobject_int
SYNOPSIS:
INT eq_fieldobject_int(type,a,i) OBJECTKIND type; OP a;
INT i;
DESCRIPTION:
Determines if the 'field object' (a monopoly, sqrad or cyclo)
is equal to the integer i. Returns OK for equality. There are six
associated macros:
EINSP_MONOPOLY(a)
eq_fieldobject_int(MONOPOLY,(a),1L)
EINSP_CYCLO(a)
eq_fieldobject_int(CYCLOTOMIC,(a),1L)
EINSP_SQRAD(a)
eq_fieldobject_int(SQ_RADICAL,(a),1L)
NEGEINSP_MONOPOLY(a)
eq_fieldobject_int(MONOPOLY,(a),-1L)
NEGEINSP_CYCLO(a)
eq_fieldobject_int(CYCLOTOMIC,(a),-1L)
NEGEINSP_SQRAD(a)
eq_fieldobject_int(SQ_RADICAL,(a),-1L)
RETURN:
OK or ERROR
NAME:
b_ksd_n
SYNOPSIS:
INT b_ksd_n(kind,self,data,result) OBJECTKIND kind;
OP self,data,result;
DESCRIPTION:
build a number object (sqrad or cyclo), whose type
is 'kind', and with the given self and data. b_ksd_n uses the
objects
supplied as arguments, while m_ksd_n uses copies of them.
RETURN:
OK or ERROR (!)
NAME:
m_ksd_n
SYNOPSIS:
INT m_ksd_n(kind,self,data,result) OBJECTKIND kind;
OP self,data,result;
DESCRIPTION:
Make a number object (sqrad or cyclo), whose type
is 'kind', and with the given self and data. b_ksd_n uses the
objects
supplied as arguments, while m_ksd_n uses copies of them.
RETURN:
OK or ERROR (!)
NAME:
objectwrite_number
SYNOPSIS:
INT objectwrite_number(f,number) FILE *f; OP number;
DESCRIPTION:
writes a number (sqrad or cyclo)
to a stream. In the case of a cyclo, the only part of the data
transferred is the index.
RETURN:
OK or ERROR
NAME:
objectread_number
SYNOPSIS:
INT objectread_number( FILE *f; OP number; OBJECTKIND type)
DESCRIPTION:
Reads a number (sqrad or cyclo) from
a stream. In the case of a cyclo, the only part of the data
transferred is the index. There are two associated macros:
OBJECTREAD_CYCLO(f,a)
objectread_number((f),(a),CYCLOTOMIC)
OBJECTREAD_SQRAD(f,a)
objectread_number((f),(a),SQ_RADICAL)
RETURN:
OK or ERROR
NAME:
fprint_number
SYNOPSIS:
INT fprint_number(f,n) FILE *f; OP n;
DESCRIPTION:
Prints the number n on the stream f. The self is printed
first and is separated by a colon from the data list in the case of a
sqrad
and the index in the case of a cyclo.
RETURN:
OK or ERROR
COMMENT:
There are the standard routines and macros
NAME
MACRO
DESCRIPTION
RETURN
TYPE
-------------------------------------------------------------------------c_n_s
C_N_S
change number self
INT
c_n_d
C_N_D
change number data
INT
s_n_s
S_N_S
select number self
OP
s_n_d
S_N_D
select number sqrad data
OP
s_n_dci
S_DCI_I
select number cyclo data:index
OP
s_n_dcd
S_DCI_D
select number cyclo data:degree OP
s_n_dcp
S_DCI_P
select number cyclo data:poly
OP
NAME:
mult_lists
SYNOPSIS:
INT mult_lists(a,b,c) OP a, b, c;
DESCRIPTION:
Multiplies the entries in two lists pairwise, putting the
resulting objects in a list. Duplicate objects are ignored.
RETURN:
OK or ERROR
NAME:
tidy
SYNOPSIS:
INT tidy(a) OP a;
DESCRIPTION:
Tidies up an object which contains cyclos in some of its
components. Such cyclos are reduced modulo the cyclotomic
polynomial.
RETURN:
OK or ERROR
NAME:
make_monopoly_sqrad
SYNOPSIS:
INT make_monopoly_sqrad(a,b) OP a,b;
DESCRIPTION:
Makes b a sqrad whose self is a copy of the monopoly a.
Also determines the data of the sqrad.
RETURN:
OK or ERROR
NAME:
make_scalar_sqrad
SYNOPSIS:
INT make_scalar_sqrad(a,b) OP a,b;
DESCRIPTION:
Makes b a sqrad whose self is 1 and whose coefficient is a.
RETURN:
OK or ERROR
NAME:
scan_sqrad
SYNOPSIS:
INT scan_sqrad(a) OP a;
DESCRIPTION:
Input a sqrad directly from standard input.
RETURN:
OK or ERROR
NAME:
add_scalar_sqrad
SYNOPSIS:
INT add_scalar_sqrad(a,b,c) OP a,b,c;
DESCRIPTION:
this is a subroutine of the general routine add
NAME:
mult_scalar_sqrad
SYNOPSIS:
INT mult_scalar_sqrad(a,b,c) OP a, b, c;
DESCRIPTION:
this is a subroutine
of the general routine mult
NAME:
add_sqrad_sqrad
SYNOPSIS:
INT add_sqrad_sqrad(a,b,c) OP a, b, c;
DESCRIPTION:
this is a subroutine of the general routine add
NAME:
mult_sqrad_sqrad
SYNOPSIS:
INT mult_sqrad_sqrad(a,b,c) OP a, b, c;
DESCRIPTION:
this is a subroutine of the general routine mult
NAME:
add_sqrad
SYNOPSIS:
INT add_sqrad(a,b,c) OP a,b,c;
DESCRIPTION:
this is a subroutine of the general routine add
NAME:
add_apply_sqrad
SYNOPSIS:
INT add_apply_sqrad(a,b) OP a,b;
DESCRIPTION:
Addition of objects of type INTEGER, LONGINT, BRUCH, POLYNOM,
SQ_RADICA or CYCLOTOMIC.
This is a subroutine of the general routine add_apply
NAME:
mult_sqrad
SYNOPSIS:
INT mult_sqrad(a,b,c) OP a,b,c;
DESCRIPTION:
this is a subroutine of the general routine mult
NAME:
mult_apply_sqrad
SYNOPSIS:
INT mult_apply_sqrad(a,b) OP a,b;
DESCRIPTION:
Multiplication of objects the first of type SQ_RADICAL and the second
of
type INTEGER, LONGINT, CYCLOTOMIC, BRUCH, MATRIX, MONOM, VECTOR,
SQ_RADICAL, POLYNOM or SCHUBERT.
this is a subroutine of the general routine mult_apply
NAME:
addinvers_sqrad
SYNOPSIS:
INT addinvers_sqrad(a,b) OP a,b;
DESCRIPTION:
this is a subroutine of the general routine addinvers
NAME:
addinvers_apply_sqrad
SYNOPSIS:
INT addinvers_apply_sqrad(a) OP a;
DESCRIPTION:
this is a subroutine of the general routine addinvers_apply
NAME:
invers_sqrad
SYNOPSIS:
INT invers_sqrad(a,b) OP a,b;
DESCRIPTION:
this is a subroutine of the general routine invers
NAME:
nullp_sqrad
SYNOPSIS:
INT nullp_sqrad(a) OP a;
DESCRIPTION:
this is a subroutine of the general routine nullp
SYNOPSIS:
INT comp_sqrad(a,b) OP a,b;
DESCRIPTION:
Uses comp_list on the self fields.
NAME:
tex_sqrad
SYNOPSIS:
INT tex_sqrad(a) OP a;
DESCRIPTION:
Outputs a sqrad in a form suitable for TeX processing.
Each term of the self is expressed in the form: coefficient * \sqrt
(self).
RETURN:
OK or ERROR
NAME:
squareroot_integer
SYNOPSIS:
INT squareroot_integer(a,b) OP a,b;
DESCRIPTION:
b is a sqrad whose square is the scalar a, which is a
INTEGER object. This is a
subroutine of the generalroutine squareroot
RETURN:
OK or ERROR
NAME:
squareroot_longint
SYNOPSIS:
INT squareroot_longint(a,b) OP a,b;
DESCRIPTION:
b is a sqrad whose square is the scalar a, which is a
LONGINT object. This is a
subroutine of the generalroutine squareroot
RETURN:
OK or ERROR
NAME:
squareroot_bruch
SYNOPSIS:
INT squareroot_bruch(a,b) OP a,b;
DESCRIPTION:
b is a sqrad whose square is the scalar a, which is a
BRUCH object. This is a
subroutine of the generalroutine squareroot
RETURN:
OK or ERROR
NAME:
convert_radical_cyclo
SYNOPSIS:
INT convert_radical_cyclo(a,b) OP a,b;
DESCRIPTION:
Converts the square root of an integer a to a cyclo b.
RETURN:
OK
NAME:
trans_index_monopoly_cyclo
SYNOPSIS:
INT trans_index_monopoly_cyclo(a,b,c) OP a,b,c;
DESCRIPTION:
Given a positive integer a and a monopoly b corresponding to
the polynomial p(x), a cyclo c is constructed whose index is a and
which
repersents the cyclotomic number p(x) where x is the basic primitive
a-th
root of unity. p(x) is reduced modulo x:a - 1 but not modulo
phi_a(x).
RETURN:
OK or ERROR
NAME:
field_check_cyclo
SYNOPSIS:
INT field_check_cyclo(a) OP a;
DESCRIPTION:
Check if element of field element , the CYCLOTOMIC
object a, is essentially an INTEGER,
and if so, transform to an object of type INTEGER.
RETURN:
OK or ERROR
NAME:
field_check_sqrad
SYNOPSIS:
INT field_check_sqrad(a) OP a;
DESCRIPTION:
Check if element of field element, the SQ_RADICAL
object a, is essentially an INTEGER,
and if so, transform to an object of type INTEGER.
RETURN:
OK or ERROR
NAME:
make_scalar_cyclo
SYNOPSIS:
INT make_scalar_cyclo(a,b) OP a,b;
DESCRIPTION:
transfer a scalar object l into an CYCLOTOMIC object b.
NAME:
make_index_coeff_power_cyclo
SYNOPSIS:
INT make_index_coeff_power_cyclo(a,b,c,d) OP a,b,c,d;
DESCRIPTION:
The monomial b * x:c is treated as a cyclotomic number,
where x is the basic primitive a-th root of unity. A cyclo d is
constructed corresponding to this number.
RETURN:
OK or ERROR
NAME:
scan_cyclo
SYNOPSIS:
INT scan_cyclo(a) OP a;
DESCRIPTION:
Input a cyclo directly from standard input.
Subroutine of the general routine scan.
RETURN:
OK or ERROR
NAME:
add_scalar_cyclo
SYNOPSIS:
INT add_scalar_cyclo(a,b,c) OP a,b,c;
DESCRIPTION:
this is a subroutine of the general routine add
NAME:
mult_scalar_cyclo
SYNOPSIS:
INT mult_scalar_cyclo(a,b,c) OP a, b, c;
DESCRIPTION:
this is a subroutine of the general routine mult
NAME:
add_cyclo_cyclo
SYNOPSIS:
INT add_cyclo_cyclo(a,b,c) OP a,b,c;
DESCRIPTION:
c is completely tidied.
this is a subroutine of the general routine add
NAME:
mult_cyclo_cyclo
SYNOPSIS:
INT mult_cyclo_cyclo(a,b,c) OP a,b,c;
DESCRIPTION:
c is completely tidied.
this is a subroutine of the general routine mult
NAME:
add_cyclo
SYNOPSIS:
INT add_cyclo(a,b,c) OP a,b,c;
DESCRIPTION:
this is a subroutine of the general routine add
NAME:
add_apply_cyclo
SYNOPSIS:
INT add_apply_cyclo(a,b) OP a,b;
DESCRIPTION:
Adds a cyclo to an object of type INTEGER, LONGINT, BRUCH,
SQ_RADICAL,
CYCLOTOMIC or POLYNOM.
this is a subroutine of the general routine add_apply
NAME:
mult_cyclo
SYNOPSIS:
INT mult_cyclo(a,b,c) OP a,b,c;
DESCRIPTION:
this is a subroutine of the general routine mult
NAME:
mult_apply_cyclo
SYNOPSIS:
INT mult_apply_cyclo(a,b) OP a,b;
DESCRIPTION:
Multiplies a cyclo with an object of type INTEGER, LONGINT, BRUCH,
SQ_RADICAL, CYCLOTOMIC, POLYNOM, SCHUBERT, VECTOR or MATRIX.
this is a subroutine of the general routine mult_apply
NAME:
addinvers_cyclo
SYNOPSIS:
INT addinvers_cyclo(a,b) OP a,b;
DESCRIPTION:
this is a subroutine of the general routine addinvers
NAME:
addinvers_apply_cyclo
SYNOPSIS:
INT addinvers_apply_cyclo(a) OP a;
DESCRIPTION:
this is a subroutine of the general routine addinvers_apply
NAME:
invers_cyclo
SYNOPSIS:
INT invers_cyclo(a,b) OP a,b;
DESCRIPTION:
this is a subroutine of the general routine invers
NAME:
nullp_cyclo
SYNOPSIS:
INT nullp_cyclo(a) OP a;
DESCRIPTION:
this is a subroutine of the general routine nullp
NAME:
comp_cyclo
SYNOPSIS:
INT comp_cyclo(a) OP a;
DESCRIPTION:
Uses comp_list on the self fields.
NAME:
conj_cyclo
SYNOPSIS:
INT conj_cyclo(a,b,c) OP a,b,c;
DESCRIPTION:
If a represents the cyclotomic number p(x), where x is the
basic primitive n-th root of unity, c is a cyclo representing the
number
p(x:b). If b is coprime to n, this is an algebraic conjugate of
p(x).
RETURN:
OK or ERROR
NAME:
tex_cyclo
SYNOPSIS:
INT tex_cyclo(a) OP a;
DESCRIPTION:
Outputs a cyclo in a form suitable for TeX processing.
Each term of the self is expressed in the form:
coefficient * \omega_{index} : (self).
RETURN:
OK or ERROR
NAME:
setup_cyclotomic_table
SYNOPSIS:
INT setup_cyclotomic_table()
DESCRIPTION:
Reads the table of cyclos from the file CYCLOS.DAT. The first
entry should be no_cyclos, then the list of cyclo_data. Returns OK
if the
table is set; otherwise, returns ERROR.
RETURN:
OK or ERROR
NAME:
print_cyclo_data
SYNOPSIS:
INT print_cyclo_data(ptr) CYCLO_DATA *ptr;
DESCRIPTION:
Prints at stdout the cyclotomic data pointed to by ptr,
prefacing the entries by Index, Degree, Polynomial and Automorphism
exponents respectively.
RETURN:
NAME:
print_cyclo_table
SYNOPSIS:
INT print_cyclo_table()
DESCRIPTION:
Prints the data corresponding to each item on the cyclo
table.
RETURN:
OK or ERROR
NAME:
print_cyclo_list
SYNOPSIS:
INT print_cyclo_list()
DESCRIPTION:
Prints the data corresponding to each item on the cyclo list.
RETURN:
OK or ERROR
NAME:
save_cyclo_list
SYNOPSIS:
INT save_cyclo_list()
DESCRIPTION:
Appends the data corresponding to each item on the cyclo list
to the file CYCLOS.DAT -- the file is created if it does not exist.
There
is no check for duplication of data.
RETURN:
OK or ERROR
COMMENT:
--------------------------------------------------------------------GENERAL ROUTINES
---------------NAME
DESCRIPTION
--------------------------------------------------------------------add()
mult()
addinvers()
invers()
nullp()
comp()
copy()
fprint()
fprintln()
freeall()
freeself()
objectread()
objectwrite()
scan()
tex()
COMMENT:
############################################################
STATIC.DOC
##########
THE STATIC ROUTINES
------------------NAME:
SYNOPSIS:
integer_factor_0
static INT integer_factor_0(a,l,g,m,first_prime)
OP a,l,g,m, first_prime;
This routine factorizes an integer using the
DESCRIPTION:
table of primes.
a is the integer to be factored; l is a monopoly in which the
prime factors
of a, which are contained in the table, and their exponents are
inserted as
monomials with the primes as the selfs and the exponents as the
koeffs;
g is the remaining factor; m is the last number tried as a
factor.
If it is non-NULL, first_prime is set to the first prime factor
and the
routine returns as soon as it is found. For a full
factorization, it
must be set to NULL.
The parameters a,l,g,m and first_prime
must be
different.
RETURN:
OK or ERROR
NAME:
integer_factor_1
SYNOPSIS:
static INT
integer_factor_1(a,f1,f2,b,l,first_prime)
OP a,f1,f2,b,l,first_prime;
DESCRIPTION:
This routine finds all prime factors of the
integer a
between two bounds f1 (the lower) and f2. If f1 is even, it is
replaced
by f1 + 1. l is a monopoly in which the prime factors of a,
between the
given bounds, and their exponents are inserted as monomials
with the
primes as the selfs and the exponents as the koeffs. b is set
to the
remaining factor. If it is non-NULL, first_prime is set to the
first
prime factor and the routine returns as soon as it is found.
For a full factorization, it must be set to NULL. The
parameters a,l,b,
f1,f2 and first_prime must be different.
RETURN:
OK or ERROR
NAME:
integer_factor_2 /*
deleted
*/
SYNOPSIS:
DESCRIPTION:
This routine finds a partial prime factorisation
of the
integer a with all the primes on a list l0. l is a monopoly to
hold the
factorisation and b is set to the remaining factor. a,l,b,l0
must be
different.
RETURN:
OK or ERROR
NAME:
integer_factor
SYNOPSIS:
static INT integer_factor(a,l) OP a,l;
DESCRIPTION:
This is the main integer factorization routine.
a is the integer to be factored. l is a monopoly to hold the
factorisation.
l need not be initialized to a MONOPOLY.
RETURN:
OK or ERROR
NAME:
SYNOPSIS:
DESCRIPTION:
callocnumber
static struct number * callocnumber()
Creates a number object and returns a pointer to
RETURN:
A pointer or NULL
NAME:
SYNOPSIS:
insert_zero_into_monopoly
static INT insert_zero_into_monopoly(a) OP a;
it.
DESCRIPTION:
RETURN:
Converts an empty monopoly into a non-empty one.
OK or ERROR
NAME:
SYNOPSIS:
DESCRIPTION:
find_sqrad_data
static INT find_sqrad_data(a) OP a;
Finds the list of prime factors of the radicals
of a
sqrad a and -1 if one of these radicals is negative, and
inserts this list
in the appropriate field of a.
RETURN:
OK or ERROR
NAME:
adjust_sqrad_data
SYNOPSIS:
static INT adjust_sqrad_data(a) OP a;
DESCRIPTION:
This adjusts an incomplete data list of primes
for the
sqrad a to make it complete. It may result in the list having
too many
primes.
RETURN:
OK or ERROR
NAME:
conj_sqrad
SYNOPSIS:
static INT conj_sqrad(a,b,c) OP a,b,c;
DESCRIPTION:
Obtains the conjugate of the sqrad a with respect
to the
automorphism x + y * \sqrt(b) -> x - y * \sqrt(b) of the number
field
F = E(\sqrt(b)), where [E:F] = 2, and F is generated by the
square roots
of the elements on the data list of a. The variable c returns
the
conjugate.
RETURN:
OK or ERROR
NAME:
convert_sqrad_scalar
SYNOPSIS:
static INT convert_sqrad_scalar(a) OP a;
DESCRIPTION:
If a is a sqrad not involving radicals, it is
converted to a
scalar -- the coefficient of \sqrt(1).
RETURN:
OK or ERROR
NAME:
SYNOPSIS:
make_index_monopoly_cyclo
static INT make_index_monopoly_cyclo(a,b,c,tid)
OP a,b,c; int tid;
Given a positive integer a and a monopoly b
DESCRIPTION:
corresponding to
the polynomial p(x), a cyclo c is constructed whose index is a
and which
repersents the cyclotomic number p(x) where x is the basic
primitive a-th
root of unity. The parameter tid specifies the amount of
tidying up
required -- see the standardise_cyclo routine.
RETURN:
OK or ERROR
NAME:
SYNOPSIS:
standardise_cyclo
static INT standardise_cyclo(a,tid) OP a; int
tid;
DESCRIPTION:
This routine carries out a tidying-up process on
the
monopoly p(x) corresponding to the cyclo a.
If tid = 0, the
monopoly
p(x) is not tidied up in any way. if tid = 1, p(x) is reduced
modulo
x:n - 1; and if tid = 2, it is reduced modulo phi_n(x), where n
denotes
the index.
RETURN:
NAME:
SYNOPSIS:
int tid;
OK or ERROR
add_cyclo_cyclo_0
mult_cyclo_cyclo_0
static INT add_cyclo_cyclo_0(a,b,c,tid) OP a,b,c;
static INT mult_cyclo_cyclo_0(a,b,c,tid) OP a,b,c; int
tid;
DESCRIPTION:
RETURN:
Adding and multiplying with tidying facilities.
OK or ERROR
NAME:
SYNOPSIS:
DESCRIPTION:
and returns
them as b and
its
conjugates in
of a.
RETURN:
invers_cyclo_norm
static INT invers_cyclo_norm(a,b,c) OP a,b,c;
Calculates the inverse and norm of the cyclo a
c respectively. The norm of a is the product of
the cyclotomic field corresponding to the index
OK or ERROR
NAME:
add_cyclo_data
SYNOPSIS:
static CYCLO_DATA *add_cyclo_data(index) OP
index;
DESCRIPTION:
Creates the data associated with a new cyclotomic
index
and appends it to the global list of cyclotomic data. A pointer
to this
data is returned.
RETURN:
pointer or NULL
NAME:
cyclo_ptr
SYNOPSIS:
static CYCLO_DATA *cyclo_ptr(index) OP index;
DESCRIPTION:
DESCRIPTION:
Returns a pointer to data associated with the
cyclotomic index
given -- the pointer points to a table item or a list item -or NULL if
the data is neither on the table or the list.
RETURN:
pointer or NULL
static INT convert_cyclo_scalar(a) OP a;
NAME:
convert_cyclo_scalar
SYNOPSIS:
static INT convert_cyclo_scalar(a) OP a;
DESCRIPTION:
If a is a cyclo not involving roots of unity, it
is converted
to a scalar -- the coefficient of x:0.
RETURN:
OK or ERROR
############################################################