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Newin1_36
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1991-12-09
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Description of the main changes between versions 1.35 and 1.36.
(Since some intermediate releases like 1.35.01 have widely circulated, some
changes described below do not apply.)
In addition to the new programs and improvements described below, many bugs
have been corrected and improvements have been made. See the file Changes for
details.
1) Linear algebra
a) New programs
suppl (supplement a subspace), image (image of a linear map), inverseimage
(find a preimage of a vector by a linear map), indexrank (find row and column
indices for extraction of a maximal rank square submatrix), matrixqz (primitive
matrix having the same $\Bbb Q$-image), matrixqz2 (intersection of $\Bbb Z^n$
with a lattice), matrixqz3 (intersection of $\Bbb Z^n$ with a
$\Bbb Q$-subspace), kerint, kerint1 and kerint2 ($\Bbb Z$-kernel of a linear
map), intersect (intersection of subspaces), lllgramint and lllint (LLL when
the gram matrix is integral, completely accurate and fast if applicable),
lllgramkerim and lllkerim ($\Bbb Z$-kernel and image of a linear map,
generalizing lllgramint and lllint to the case where the vectors may be
dependent).
b) Improvements
Modified many programs to handle empty matrices in a reasonable way.
smith modified so as to allow singular matrices, and hermite modified so as to
accept matrices of any size, in particular with more rows than columns.
norml2 accepts now any type, in particular matrices.
The LLL algorithms have been improved and speeded up. In particular, the
linear and algebraic dependence algorithms lindep2 and algdep2 now give
considerably better results than before, since they call lllint instead of lll.
hess is now (hopefully) correct.
In a GP statement, rows and columns of a matrix can now be accessed or modified
directly using m[j,] for the j-th row and m[,k] for the k-th column.
2) Polynomials, polymods and rational funtions
a) New programs
factfq (factorization of polynomials over any finite field), cyclo (cyclotomic
polynomials), modreverse (reverse of a polymod), polred2 (polred with the
integral basis elements), polyrev (same as poly in reverse order),
smallpolred2 (smallpolred+polred2), factoredpolred2 (factoredpolred+polred2),
rootslong (a much slower but safer polynomial root finder), galois (galois
group of a polynomial of degree up to 7), galoisconj (list of conjugates of
an algebraic number belonging to the same number field), tchirnhausen (apply
a random tchirnhausen transformation), initalg (give basic information about
a number field), resultant2 (resultant for non-exact types),
factornf (factoring polynomials over number fields).
b) Improvements
gcd of two polymods or between a polymod and a polynomial is now more
reasonable, and vector/matrix arguments are allowed in gcd and lcm.
Major modifications have been introduced in the treatment of polymods which
now behave in a much more reasonable way.
The simplification of rational functions has been enhanced, resulting in
more completely simplified expressions, but also in slower speed.
3) Arithmetic functions.
a) New programs
factr (n! as a real number), centerlift (lift of mod(a,b) to a with |a|
minimum), shift (left or right shift), shiftmul (multiplication or division by
a power of 2), rhorealnod and redrealnod (same as rhoreal and redreal without
distance computation), comprealraw, sqrealraw and powrealraw (composition,
squaring and powering of binary quadratic forms with positive discriminant
without performing any reduction), isisom (test isomorphism of number fields),
isincl (test inclusion of number fields), rootsof1 (number of roots of unity
in a number field), nucomp, nudupl and nupow (composition of primitive
positive definite binary quadratic forms a la Shanks).
b) Improvements
sumdivk(n,k) now accepts k<0.
When an impossible inverse modulo occurs, the error message prints the culprit
so that one can factor the modulus if necessary.
smallfact and boundfact can now have rational arguments.
All operations (except nucomp and co.) on quadratic forms now accept
nonprimitive forms (whatever the result means however).
4) Transcendental functions
a) New programs
izeta (zeta function for integer arguments, not accessible from GP but called
automatically by zeta), incgam4 (incomplete gamma with given gamma value),
cxpolylog (polylog of complex argumnet, not accessible from GP but called
automatically by polylog).
b) Improvements
incgam(a,x) now accepts a<=0.
The polylog functions have been deeply modified. They now work in the whole
complex plane, accept negative indices, and all useful modified versions
are implemented. Also power series arguments are now accepted.
gamma and lngamma now accept power series arguments.
5) Elliptic curves
a) New programs
initell2 (fast initell, but for reasonable size coefficients), lseriesell
(fast computation of the L series of an elliptic curve), pointell (Weierstrass
P-function and its derivative).
b) Improvements
Elliptic curves with huge coefficients can now be treated.
zell is now in priciple correct and more accurate.
The coefficients of matell have been doubled, so the determinant of the
height matrix is the usual regulator.
isoncurve gives a reasonable answer in case of imprecise coefficients.
apell now accepts prime numbers larger than 65536.
6) Miscellaneous
a) New programs
vecsort (sort by some component), read (read an expression from a GP program),
random (generate a random number), permutation (list all permutations),
matsize (dimensions of a matrix), size (maximal number of decimal digits),
rounderror (maximum error made in rounding), lex (lexicographic comparison),
lexsort (lexicographic sort), simplify (remove a zero imaginary part in a
complex or quadratic, and convert a constant polynomial into its constant
term).
b) Improvements
The time to input a large file into GP by \r is much shorter.
suminf, prodinf, suminf1, prodinf1 can now have zero coefficients without
stopping the computation.
A vector can now be raised to an integral power componentwise (of course
not a matrix).
A new output format has been added (the prettymatrix format), which allows to
print matrices as boxes, the components being still in raw format. This is now
the default instead of the raw format (type \p to switch between the three
output formats, \a, \m and \b to print in raw, prettymatrix and prettyprint
format respectively).
7) Documentation and examples.
a) Two new chapters have been added to the manual (which now exceeds 103 pages,
not counting the table of contents and the index). Chapter 5 gives a complete
list of the low level functions of the PARI library, with a description of
their use. The last chapter is a tutorial to the use of the GP calculator. It
is not yet finished in this release. If you do not want it in the manual,
put a % sign in front of the line \input tutorial in the file users.tex.
b) An example of a complete and non-trivial GP program is given in the file
squfof.gp in the directory example. Just read in this file under GP by the
command \r squfof.gp, and use the function squfof on positive integers which
you want to factor. SQUFOF is a very nice factoring method invented in the
70's by D. Shanks for factoring integers, and is reasonably fast for numbers
having up to 15 or 16 digits. The squfof program which is given is a very
crude implementation. It also prints out some intermediate information as it
goes along. The final result is some factor of the number to be factored.