Note that most linear algebra functions operating on subspaces defined by
generating sets (such as `mathnf`

, `qflll`

, etc.) take matrices as
arguments. As usual, the generating vectors are taken to be the
*columns* of the given matrix.

Since PARI does not have a strong typing system, scalars live in
unspecified commutative base rings. It is very difficult to write
robust linear algebra routines in such a general setting. We thus
assume that the base ring is a domain and work over its field of
fractions. If the base ring is *not* a domain, one gets an error as soon
as a nonzero pivot turns out to be noninvertible. Some functions,
e.g. `mathnf`

or `mathnfmod`

, specifically assume that the base ring is
ℤ.

z being real/complex, or p-adic, finds a polynomial (in the variable
`'x`

) of degree at most
k, with integer coefficients, having z as approximate root. Note that the
polynomial which is obtained is not necessarily the "correct" one. In fact
it is not even guaranteed to be irreducible. One can check the closeness
either by a polynomial evaluation (use `subst`

), or by computing the
roots of the polynomial given by `algdep`

(use `polroots`

or
`polrootspadic`

).

Internally, `lindep`

([1,z,...,z^{k}], *flag*) is used. A nonzero value of
*flag* may improve on the default behavior if the input number is known to a
*huge* accuracy, and you suspect the last bits are incorrect: if *flag* > 0
the computation is done with an accuracy of *flag* decimal digits; to get
meaningful results, the parameter *flag* should be smaller than the number of
correct decimal digits in the input. But default values are usually
sufficient, so try without *flag* first:

? \p200 ? z = 2^(1/6)+3^(1/5); ? algdep(z, 30); \\ right in 63ms ? algdep(z, 30, 100); \\ wrong in 39ms ? algdep(z, 30, 170); \\ right in 61ms ? algdep(z, 30, 200); \\ wrong in 146ms ? \p250 ? z = 2^(1/6)+3^(1/5); \\ recompute to new, higher, accuracy ! ? algdep(z, 30); \\ right in 68ms ? algdep(z, 30, 200); \\ right in 68ms ? \p500 ? algdep(2^(1/6)+3^(1/5), 30); \\ right in 138ms ? \p1000 ? algdep(2^(1/6)+3^(1/5), 30); \\ right in 276s

The changes in `realprecision`

only affect the quality of the
initial approximation to 2^{1/6} + 3^{1/5}, `algdep`

itself uses
exact operations. The size of its operands depend on the accuracy of the
input of course: a more accurate input means slower operations.

Proceeding by increments of 5 digits of accuracy, `algdep`

with default
flag produces its first correct result at 195 digits, and from then on a
steady stream of correct results:

\\ assume T contains the correct result, for comparison forstep(d=100, 250, 5, \ localprec(d); \ print(d, " ", algdep(2^(1/6)+3^(1/5),30) == T))

This example is the test case studied in a 2000 paper by Borwein and
Lisonek: Applications of integer relation algorithms, *Discrete Math.*,
**217**, p. 65–82. The version of PARI tested there was 1.39, which
succeeded reliably from precision 265 on, in about 1000 as much time as the
current version (on slower hardware of course).

Note that this function does not work if z is a power series. The function
`seralgdep`

can be used in this case to find linear relations wich
polynomial coefficients between powers of z.

The library syntax is `GEN `

.
Also available is **algdep0**(GEN z, long k, long flag)`GEN `

(**algdep**(GEN z, long k)*flag* = 0).

T being an integral polynomial and V being a scalar, vector, or
matrix with complex coefficients, return a reasonable approximation of V
with polmods modulo T. T can also be any number field structure, in which
case the minimal polynomial attached to the structure (`T`

.pol) is
used. The *rootT* argument, if present, must be an element of
`polroots(T)`

(or `T`

.pol), i.e. a complex root of T fixing an embedding of
ℚ[x]/(T) into ℂ.

? bestapprnf(sqrt(5), polcyclo(5)) %1 = Mod(-2*x^3 - 2*x^2 - 1, x^4 + x^3 + x^2 + x + 1) ? bestapprnf(sqrt(5), polcyclo(5), exp(4*I*Pi/5)) %2 = Mod(2*x^3 + 2*x^2 + 1, x^4 + x^3 + x^2 + x + 1)

When the output has huge rational coefficients, try to
increase the working `realbitprecision`

: if the answer does not
stabilize, consider that the reconstruction failed.
Beware that if T is not Galois over ℚ, some embeddings
may not allow to reconstruct V:

? T = x^3-2; vT = polroots(T); z = 3*2^(1/3)+1; ? bestapprnf(z, T, vT[1]) %2 = Mod(3*x + 1, x^3 - 2) ? bestapprnf(z, T, vT[2]) %3 = 4213714286230872/186454048314072 \\ close to 3*2^(1/3) + 1

The library syntax is `GEN `

.**bestapprnf**(GEN V, GEN T, GEN rootT = NULL, long prec)

characteristic polynomial of A with respect to the variable v, i.e. determinant of v*I-A if A is a square matrix.

? charpoly([1,2;3,4]); %1 = x^2 - 5*x - 2 ? charpoly([1,2;3,4],, 't) %2 = t^2 - 5*t - 2

If A is not a square matrix, the function returns the characteristic polynomial of the map "multiplication by A" if A is a scalar:

? charpoly(Mod(x+2, x^3-2)) %1 = x^3 - 6*x^2 + 12*x - 10 ? charpoly(I) %2 = x^2 + 1 ? charpoly(quadgen(5)) %3 = x^2 - x - 1 ? charpoly(ffgen(ffinit(2,4))) %4 = Mod(1, 2)*x^4 + Mod(1, 2)*x^3 + Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2)

The value of *flag* is only significant for matrices, and we advise to stick
to the default value. Let n be the dimension of A.

If *flag* = 0, same method (Le Verrier's) as for computing the adjoint matrix,
i.e. using the traces of the powers of A. Assumes that n! is
invertible; uses O(n^{4}) scalar operations.

If *flag* = 1, uses Lagrange interpolation which is usually the slowest method.
Assumes that n! is invertible; uses O(n^{4}) scalar operations.

If *flag* = 2, uses the Hessenberg form. Assumes that the base ring is a field.
Uses O(n^{3}) scalar operations, but suffers from coefficient explosion
unless the base field is finite or ℝ.

If *flag* = 3, uses Berkowitz's division free algorithm, valid over any
ring (commutative, with unit). Uses O(n^{4}) scalar operations.

If *flag* = 4, x must be integral. Uses a modular algorithm: Hessenberg form
for various small primes, then Chinese remainders.

If *flag* = 5 (default), uses the "best" method given x.
This means we use Berkowitz unless the base ring is ℤ (use *flag* = 4)
or a field where coefficient explosion does not occur,
e.g. a finite field or the reals (use *flag* = 2).

The library syntax is `GEN `

where **charpoly0**(GEN A, long v = -1, long flag)`v`

is a variable number.
Also available are
`GEN `

(**charpoly**(GEN x, long v)*flag* = 5),
`GEN `

(**caract**(GEN A, long v)*flag* = 1),
`GEN `

(**carhess**(GEN A, long v)*flag* = 2),
`GEN `

(**carberkowitz**(GEN A, long v)*flag* = 3) and
`GEN `

. In this
last case, if **caradj**(GEN A, long v, GEN *pt)*pt* is not `NULL`

, `*pt`

receives the address of
the adjoint matrix of A (see `matadjoint`

), so both can be obtained at
once.

Concatenation of x and y. If x or y is
not a vector or matrix, it is considered as a one-dimensional vector. All
types are allowed for x and y, but the sizes must be compatible. Note
that matrices are concatenated horizontally, i.e. the number of rows stays
the same. Using transpositions, one can concatenate them vertically,
but it is often simpler to use `matconcat`

.

? x = matid(2); y = 2*matid(2); ? concat(x,y) %2 = [1 0 2 0] [0 1 0 2] ? concat(x~,y~)~ %3 = [1 0] [0 1] [2 0] [0 2] ? matconcat([x;y]) %4 = [1 0] [0 1] [2 0] [0 2]

To concatenate vectors sideways (i.e. to obtain a two-row or two-column
matrix), use `Mat`

instead, or `matconcat`

:

? x = [1,2]; ? y = [3,4]; ? concat(x,y) %3 = [1, 2, 3, 4] ? Mat([x,y]~) %4 = [1 2] [3 4] ? matconcat([x;y]) %5 = [1 2] [3 4]

Concatenating a row vector to a matrix having the same number of columns will add the row to the matrix (top row if the vector is x, i.e. comes first, and bottom row otherwise).

The empty matrix `[;]`

is considered to have a number of rows compatible
with any operation, in particular concatenation. (Note that this is
*not* the case for empty vectors `[ ]`

or `[ ]~`

.)

If y is omitted, x has to be a row vector or a list, in which case its elements are concatenated, from left to right, using the above rules.

? concat([1,2], [3,4]) %1 = [1, 2, 3, 4] ? a = [[1,2]~, [3,4]~]; concat(a) %2 = [1 3] [2 4] ? concat([1,2; 3,4], [5,6]~) %3 = [1 2 5] [3 4 6] ? concat([%, [7,8]~, [1,2,3,4]]) %5 = [1 2 5 7] [3 4 6 8] [1 2 3 4]

The library syntax is `GEN `

.
**gconcat**(GEN x, GEN y = NULL)`GEN `

is a shortcut for **gconcat1**(GEN x)`gconcat(x,NULL)`

.

For nonnegative n and complex number x, return the vector with n
components [1^{×},2^{×},...,n^{×}].

? dirpowers(5, 2) %1 = [1, 4, 9, 16, 25] ? dirpowers(5, 1/2) %2 = [1, 1.414..., 1.732..., 2.000..., 2.236...]

When n ≤ 0, the function returns the empty vector `[]`

.

The library syntax is `GEN `

.**dirpowers**(long n, GEN x, long prec)

q being a square and symmetric integral matrix representing a positive
definite quadratic form, evaluate `expr`

for all pairs of nonzero
vectors (-v,v) such that q(v) ≤ b. The formal variable v runs
through representatives of all such pairs in turn.

? forqfvec(v, [3,2;2,3], 3, print(v)) [0, 1]~ [1, 0]~ [-1, 1]~

The library syntax is `void `

.
The following functions are also available:
**forqfvec0**(GEN v, GEN q = NULL, GEN b)`void `

:
Evaluate **forqfvec**(void *E, long (*fun)(void *, GEN, GEN, double), GEN q, GEN b)`fun(E,U,v,m)`

on all v such that q(U v) < b, where U is a
`t_MAT`

, v is a `t_VECSMALL`

and m = q(v) is a C double. The function
`fun`

must return 0, unless `forqfvec`

should stop, in which case,
it should return 1.

`void `

:
Evaluate **forqfvec1**(void *E, long (*fun)(void *, GEN), GEN q, GEN b)`fun(E,v)`

on all v such that q(v) < b, where v is a
`t_COL`

. The function `fun`

must return 0, unless `forqfvec`

should stop, in which case, it should return 1.

finds a small nontrivial integral linear combination between components of v. If none can be found return an empty vector.

If v is a vector with real/complex entries we use a floating point
(variable precision) LLL algorithm. If *flag* = 0 the accuracy is chosen
internally using a crude heuristic. If *flag* > 0 the computation is done with
an accuracy of *flag* decimal digits. To get meaningful results in the latter
case, the parameter *flag* should be smaller than the number of correct
decimal digits in the input.

? lindep([sqrt(2), sqrt(3), sqrt(2)+sqrt(3)]) %1 = [-1, -1, 1]~

If v is p-adic, *flag* is ignored and the algorithm LLL-reduces a
suitable (dual) lattice.

? lindep([1, 2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)]) %2 = [1, -2]~

If v is a matrix (or a vector of column vectors, or a vector of row
vectors), *flag* is ignored and the function returns a non trivial kernel
vector if one exists, else an empty vector.

? lindep([1,2,3;4,5,6;7,8,9]) %3 = [1, -2, 1]~ ? lindep([[1,0], [2,0]]) %4 = [2, -1]~ ? lindep([[1,0], [0,1]]) %5 = []~

If v contains polynomials or power series over some base field, finds a linear relation with coefficients in the field.

? lindep([x*y, x^2 + y, x^2*y + x*y^2, 1]) %4 = [y, y, -1, -y^2]~

For better control, it is preferable to use `t_POL`

rather
than `t_SER`

in the input, otherwise one gets a linear combination which is
t-adically small, but not necessarily 0. Indeed, power series are first
converted to the minimal absolute accuracy occurring among the entries of v
(which can cause some coefficients to be ignored), then truncated to
polynomials:

? v = [t^2+O(t^4), 1+O(t^2)]; L=lindep(v) %1 = [1, 0]~ ? v*L %2 = t^2+O(t^4) \\ small but not 0

The library syntax is `GEN `

.**lindep0**(GEN v, long flag)

adjoint matrix of M, i.e. a matrix N
of cofactors of M, satisfying M*N = det(M)*Id. M must be a
(not necessarily invertible) square matrix of dimension n.
If *flag* is 0 or omitted, we try to use Leverrier-Faddeev's algorithm,
which assumes that n! invertible. If it fails or *flag* = 1,
computes T = `charpoly`

(M) independently first and returns
(-1)^{n-1} (T(x)-T(0))/x evaluated at M.

? a = [1,2,3;3,4,5;6,7,8] * Mod(1,4); ? matadjoint(a) %2 = [Mod(1, 4) Mod(1, 4) Mod(2, 4)] [Mod(2, 4) Mod(2, 4) Mod(0, 4)] [Mod(1, 4) Mod(1, 4) Mod(2, 4)]

Both algorithms use O(n^{4}) operations in the base ring. Over a field,
they are usually slower than computing the characteristic polynomial or
the inverse of M directly.

The library syntax is `GEN `

.
Also available are
**matadjoint0**(GEN M, long flag)`GEN `

(**adj**(GEN x)*flag* = 0) and
`GEN `

(**adjsafe**(GEN x)*flag* = 1).

The left companion matrix to the nonzero polynomial x.

The library syntax is `GEN `

.**matcompanion**(GEN x)

Returns a `t_MAT`

built from the entries of v, which may
be a `t_VEC`

(concatenate horizontally), a `t_COL`

(concatenate
vertically), or a `t_MAT`

(concatenate vertically each column, and
concatenate vertically the resulting matrices). The entries of v are always
considered as matrices: they can themselves be `t_VEC`

(seen as a row
matrix), a `t_COL`

seen as a column matrix), a `t_MAT`

, or a scalar (seen
as an 1 x 1 matrix).

? A=[1,2;3,4]; B=[5,6]~; C=[7,8]; D=9; ? matconcat([A, B]) \\ horizontal %1 = [1 2 5] [3 4 6] ? matconcat([A, C]~) \\ vertical %2 = [1 2] [3 4] [7 8] ? matconcat([A, B; C, D]) \\ block matrix %3 = [1 2 5] [3 4 6] [7 8 9]

If the dimensions of the entries to concatenate do not match up, the above rules are extended as follows:

***** each entry v_{i,j} of v has a natural length and height: 1 x
1 for a scalar, 1 x n for a `t_VEC`

of length n, n x 1
for a `t_COL`

, m x n for an m x n `t_MAT`

***** let H_{i} be the maximum over j of the lengths of the v_{i,j},
let L_{j} be the maximum over i of the heights of the v_{i,j}.
The dimensions of the (i,j)-th block in the concatenated matrix are
H_{i} x L_{j}.

***** a scalar s = v_{i,j} is considered as s times an identity matrix
of the block dimension min (H_{i},L_{j})

***** blocks are extended by 0 columns on the right and 0 rows at the
bottom, as needed.

? matconcat([1, [2,3]~, [4,5,6]~]) \\ horizontal %4 = [1 2 4] [0 3 5] [0 0 6] ? matconcat([1, [2,3], [4,5,6]]~) \\ vertical %5 = [1 0 0] [2 3 0] [4 5 6] ? matconcat([B, C; A, D]) \\ block matrix %6 = [5 0 7 8] [6 0 0 0] [1 2 9 0] [3 4 0 9] ? U=[1,2;3,4]; V=[1,2,3;4,5,6;7,8,9]; ? matconcat(matdiagonal([U, V])) \\ block diagonal %7 = [1 2 0 0 0] [3 4 0 0 0] [0 0 1 2 3] [0 0 4 5 6] [0 0 7 8 9]

The library syntax is `GEN `

.**matconcat**(GEN v)

Determinant of the square matrix x.

If *flag* = 0, uses an appropriate algorithm depending on the coefficients:

***** integer entries: modular method due to Dixon, Pernet and Stein.

***** real or p-adic entries: classical Gaussian elimination using maximal
pivot.

***** intmod entries: classical Gaussian elimination using first nonzero
pivot.

***** other cases: Gauss-Bareiss.

If *flag* = 1, uses classical Gaussian elimination with appropriate pivoting
strategy (maximal pivot for real or p-adic coefficients). This is usually
worse than the default.

The library syntax is `GEN `

.
Also available are **det0**(GEN x, long flag)`GEN `

(**det**(GEN x)*flag* = 0),
`GEN `

(**det2**(GEN x)*flag* = 1) and `GEN `

for integer
entries.**ZM_det**(GEN x)

Let B be an m x n matrix with integer coefficients. The
*determinant* D of the lattice generated by the columns of B is
the square root of det(B^{T} B) if B has maximal rank m, and 0
otherwise.

This function uses the Gauss-Bareiss algorithm to compute a positive
*multiple* of D. When B is square, the function actually returns
D = |det B|.

This function is useful in conjunction with `mathnfmod`

, which needs to
know such a multiple. If the rank is maximal but the matrix is nonsquare,
you can obtain D exactly using

matdet( mathnfmod(B, matdetint(B)) )

Note that as soon as one of the dimensions gets large (m or n is larger
than 20, say), it will often be much faster to use `mathnf(B, 1)`

or
`mathnf(B, 4)`

directly.

The library syntax is `GEN `

.**detint**(GEN B)

Given a matrix x with `t_INT`

entries and d an arbitrary positive
integer, return the determinant of x modulo d.

? A = [4,2,3; 4,5,6; 7,8,9] ? matdetmod(A,27) %2 = 9

Note that using the generic function `matdet`

on a matrix with
`t_INTMOD`

entries uses Gaussian reduction and will fail in general when
the modulus is not prime.

? matdet(A * Mod(1,27)) *** at top-level: matdet(A*Mod(1,27)) *** ^ — — — — — — *** matdet: impossible inverse in Fl_inv: Mod(3, 27).

The library syntax is `GEN `

.**matdetmod**(GEN x, GEN d)

x being a vector, creates the diagonal matrix whose diagonal entries are those of x.

? matdiagonal([1,2,3]); %1 = [1 0 0] [0 2 0] [0 0 3]

Block diagonal matrices are easily created using
`matconcat`

:

? U=[1,2;3,4]; V=[1,2,3;4,5,6;7,8,9]; ? matconcat(matdiagonal([U, V])) %1 = [1 2 0 0 0] [3 4 0 0 0] [0 0 1 2 3] [0 0 4 5 6] [0 0 7 8 9]

The library syntax is `GEN `

.**diagonal**(GEN x)

Returns the (complex) eigenvectors of x as columns of a matrix.
If *flag* = 1, return [L,H], where L contains the
eigenvalues and H the corresponding eigenvectors; multiple eigenvalues are
repeated according to the eigenspace dimension (which may be less
than the eigenvalue multiplicity in the characteristic polynomial).

This function first computes the characteristic polynomial of x and
approximates its complex roots (λ_{i}), then tries to compute the
eigenspaces as kernels of the x - λ_{i}. This algorithm is
ill-conditioned and is likely to miss kernel vectors if some roots of the
characteristic polynomial are close, in particular if it has multiple roots.

? A = [13,2; 10,14]; mateigen(A) %1 = [-1/2 2/5] [ 1 1] ? [L,H] = mateigen(A, 1); ? L %3 = [9, 18] ? H %4 = [-1/2 2/5] [ 1 1] ? A * H == H * matdiagonal(L) %5 = 1

For symmetric matrices, use `qfjacobi`

instead; for Hermitian matrices,
compute

A = real(x); B = imag(x); y = matconcat([A, -B; B, A]);

and apply `qfjacobi`

to y.

The library syntax is `GEN `

.
Also available is **mateigen**(GEN x, long flag, long prec)`GEN `

(**eigen**(GEN x, long prec)*flag* = 0)

Returns the Frobenius form of
the square matrix `M`

. If *flag* = 1, returns only the elementary divisors as
a vector of polynomials in the variable `v`

. If *flag* = 2, returns a
two-components vector [F,B] where `F`

is the Frobenius form and `B`

is
the basis change so that M = B^{-1}FB.

The library syntax is `GEN `

where **matfrobenius**(GEN M, long flag, long v = -1)`v`

is a variable number.

Returns a matrix similar to the square matrix x, which is in upper Hessenberg form (zero entries below the first subdiagonal).

The library syntax is `GEN `

.**hess**(GEN x)

Creates the Hilbert matrix of order n ≥ 0, i.e. the square matrix H whose coefficient H[i,j] is 1 / (i+j-1). This matrix is ill-conditionned but its inverse has integer entries.

The library syntax is `GEN `

.**mathilbert**(long n)

Let R be a Euclidean ring, equal to ℤ or to K[X] for some field
K. If M is a (not necessarily square) matrix with entries in R, this
routine finds the *upper triangular* Hermite normal form of M.
If the rank of M is equal to its number of rows, this is a square
matrix. In general, the columns of the result form a basis of the R-module
spanned by the columns of M.

The values of *flag* are:

***** 0 (default): only return the Hermite normal form H

***** 1 (complete output): return [H,U], where H is the Hermite
normal form of M, and U is a transformation matrix such that MU = [0|H].
The matrix U belongs to GL(R). When M has a large kernel, the
entries of U are in general huge.

For these two values, we use a naive algorithm, which behaves well
in small dimension only. Larger values correspond to different algorithms,
are restricted to *integer* matrices, and all output the unimodular
matrix U. From now on all matrices have integral entries.

***** *flag* = 4, returns [H,U] as in "complete output" above, using a
variant of LLL reduction along the way. The matrix U is provably
small in the L_{2} sense, and often close to optimal; but the
reduction is in general slow, although provably polynomial-time.

If *flag* = 5, uses Batut's algorithm and output [H,U,P], such that H and
U are as before and P is a permutation of the rows such that P applied
to MU gives H. This is in general faster than *flag* = 4 but the matrix U
is usually worse; it is heuristically smaller than with the default algorithm.

When the matrix is dense and the dimension is large (bigger than 100, say),
*flag* = 4 will be fastest. When M has maximal rank, then

H = mathnfmod(M, matdetint(M))

will be even faster. You can then recover U as M^{-1}H.

? M = matrix(3,4,i,j,random([-5,5])) %1 = [ 0 2 3 0] [-5 3 -5 -5] [ 4 3 -5 4] ? [H,U] = mathnf(M, 1); ? U %3 = [-1 0 -1 0] [ 0 5 3 2] [ 0 3 1 1] [ 1 0 0 0] ? H %5 = [19 9 7] [ 0 9 1] [ 0 0 1] ? M*U %6 = [0 19 9 7] [0 0 9 1] [0 0 0 1]

For convenience, M is allowed to be a `t_VEC`

, which is then
automatically converted to a `t_MAT`

, as per the `Mat`

function.
For instance to solve the generalized extended gcd problem, one may use

? v = [116085838, 181081878, 314252913,10346840]; ? [H,U] = mathnf(v, 1); ? U %2 = [ 103 -603 15 -88] [-146 13 -1208 352] [ 58 220 678 -167] [-362 -144 381 -101] ? v*U %3 = [0, 0, 0, 1]

This also allows to input a matrix as a `t_VEC`

of
`t_COL`

s of the same length (which `Mat`

would concatenate to
the `t_MAT`

having those columns):

? v = [[1,0,4]~, [3,3,4]~, [0,-4,-5]~]; mathnf(v) %1 = [47 32 12] [ 0 1 0] [ 0 0 1]

The library syntax is `GEN `

.
Also available are **mathnf0**(GEN M, long flag)`GEN `

(**hnf**(GEN M)*flag* = 0) and
`GEN `

(**hnfall**(GEN M)*flag* = 1). To reduce *huge* relation matrices
(sparse with small entries, say dimension 400 or more), you can use the
pair `hnfspec`

/ `hnfadd`

. Since this is quite technical and the
calling interface may change, they are not documented yet. Look at the code
in `basemath/hnf_snf.c`

.

If x is a (not necessarily square) matrix of
maximal rank with integer entries, and d is a multiple of the (nonzero)
determinant of the lattice spanned by the columns of x, finds the
*upper triangular* Hermite normal form of x.

If the rank of x is equal to its number of rows, the result is a square
matrix. In general, the columns of the result form a basis of the lattice
spanned by the columns of x. Even when d is known, this is in general
slower than `mathnf`

but uses much less memory.

The library syntax is `GEN `

.**hnfmod**(GEN x, GEN d)

Outputs the (upper triangular) Hermite normal form of x concatenated with the diagonal matrix with diagonal d. Assumes that x has integer entries. Variant: if d is an integer instead of a vector, concatenate d times the identity matrix.

? m=[0,7;-1,0;-1,-1] %1 = [ 0 7] [-1 0] [-1 -1] ? mathnfmodid(m, [6,2,2]) %2 = [2 1 1] [0 1 0] [0 0 1] ? mathnfmodid(m, 10) %3 = [10 7 3] [ 0 1 0] [ 0 0 1]

The library syntax is `GEN `

.**hnfmodid**(GEN x, GEN d)

applies a sequence Q of Householder
transforms, as returned by `matqr`

(M,1) to the vector or matrix v.

? m = [2,1; 3,2]; \\ some random matrix ? [Q,R] = matqr(m); ? Q %3 = [-0.554... -0.832...] [-0.832... 0.554...] ? R %4 = [-3.605... -2.218...] [0 0.277...] ? v = [1, 2]~; \\ some random vector ? Q * v %6 = [-2.218..., 0.277...]~ ? [q,r] = matqr(m, 1); ? exponent(r - R) \\ r is the same as R %8 = -128 ? q \\ but q has a different structure %9 = [[0.0494..., [5.605..., 3]]]] ? mathouseholder(q, v) \\ applied to v %10 = [-2.218..., 0.277...]~

The point of the Householder structure is that it efficiently
represents the linear operator v ` ⟼ `

Q v in a more stable way
than expanding the matrix Q:

? m = mathilbert(20); v = vectorv(20,i,i^2+1); ? [Q,R] = matqr(m); ? [q,r] = matqr(m, 1); ? \p100 ? [q2,r2] = matqr(m, 1); \\ recompute at higher accuracy ? exponent(R - r) %5 = -127 ? exponent(R - r2) %6 = -127 ? exponent(mathouseholder(q,v) - mathouseholder(q2,v)) %7 = -119 ? exponent(Q*v - mathouseholder(q2,v)) %8 = 9

We see that R is OK with or without a flag to `matqr`

but that multiplying by Q is considerably less precise than applying the
sequence of Householder transforms encoded by q.

The library syntax is `GEN `

.**mathouseholder**(GEN Q, GEN v)

Creates the n x n identity matrix.

The library syntax is `GEN `

.**matid**(long n)

Gives a basis for the image of the
matrix x as columns of a matrix. A priori the matrix can have entries of
any type. If *flag* = 0, use standard Gauss pivot. If *flag* = 1, use
`matsupplement`

(much slower: keep the default flag!).

The library syntax is `GEN `

.
Also available is **matimage0**(GEN x, long flag)`GEN `

(**image**(GEN x)*flag* = 0).

Gives the vector of the column indices which
are not extracted by the function `matimage`

, as a permutation
(`t_VECSMALL`

). Hence the number of
components of `matimagecompl(x)`

plus the number of columns of
`matimage(x)`

is equal to the number of columns of the matrix x.

The library syntax is `GEN `

.**imagecompl**(GEN x)

Gives a Howell basis (unique representation for submodules
of (ℤ/dℤ)^{n})
for the image of the matrix x modulo d as columns of a matrix H. The
matrix x must have `t_INT`

entries, and d can be an arbitrary positive
integer. If U is present, set it to a matrix such that AU = H.

? A = [2,1;0,2]; ? matimagemod(A,6,&U) %2 = [1 0] [0 2] ? U %3 = [5 1] [3 4] ? (A*U)%6 %4 = [1 0] [0 2]

**Caveat.** In general the number of columns of the Howell form is not
the minimal number of generators of the submodule. Example:

? matimagemod([1;2],4) %5 = [2 1] [0 2]

**Caveat 2.** In general the matrix U is not invertible, even if A
and H have the same size. Example:

? matimagemod([4,1;0,4],8,&U) %6 = [2 1] [0 4] ? U %7 = [0 0] [2 1]

The library syntax is `GEN `

.**matimagemod**(GEN x, GEN d, GEN *U = NULL)

M being a matrix of rank r, returns a vector with two
`t_VECSMALL`

components y and z of length r giving a list of rows
and columns respectively (starting from 1) such that the extracted matrix
obtained from these two vectors using `vecextract`

(M,y,z) is
invertible. The vectors y and z are sorted in increasing order.

The library syntax is `GEN `

.**indexrank**(GEN M)

x and y being two matrices with the same number of rows, finds a basis of the vector space equal to the intersection of the spaces spanned by the columns of x and y respectively. For efficiency, the columns of x (resp. y) should be independent.

The faster function `idealintersect`

can be used to intersect
fractional ideals (projective ℤ_{K} modules of rank 1); the slower but
more general function `nfhnf`

can be used to intersect general
ℤ_{K}-modules.

The library syntax is `GEN `

.**intersect**(GEN x, GEN y)

Given a matrix x and
a column vector or matrix y, returns a preimage z of y by x if one
exists (i.e such that x z = y), an empty vector or matrix otherwise. The
complete inverse image is z + Ker x, where a basis of the kernel of
x may be obtained by `matker`

.

? M = [1,2;2,4]; ? matinverseimage(M, [1,2]~) %2 = [1, 0]~ ? matinverseimage(M, [3,4]~) %3 = []~ \\ no solution ? matinverseimage(M, [1,3,6;2,6,12]) %4 = [1 3 6] [0 0 0] ? matinverseimage(M, [1,2;3,4]) %5 = [;] \\ no solution ? K = matker(M) %6 = [-2] [1]

The library syntax is `GEN `

.**inverseimage**(GEN x, GEN y)

Computes a left inverse of the matrix x modulo d. The matrix x must
have `t_INT`

entries, and d can be an arbitrary positive integer.

? A = [3,1,2;1,2,1;3,1,1]; ? U = matinvmod(A,6) %2 = [1 1 3] [2 3 5] [1 0 5] ? (U*A)%6 %3 = [1 0 0] [0 1 0] [0 0 1] ? matinvmod(A,5) *** at top-level: matinvmod(A,5) *** ^ — — — — -- *** matinvmod: impossible inverse in gen_inv: 0.

The library syntax is `GEN `

.**matinvmod**(GEN x, GEN d)

Returns true (1) if x is a diagonal matrix, false (0) if not.

The library syntax is `GEN `

.**isdiagonal**(GEN x)

Gives a basis for the kernel of the matrix x as columns of a matrix. The matrix can have entries of any type, provided they are compatible with the generic arithmetic operations (+, x and /).

If x is known to have integral entries, set *flag* = 1.

The library syntax is `GEN `

.
Also available are **matker0**(GEN x, long flag)`GEN `

(**ker**(GEN x)*flag* = 0),
`GEN `

(**ZM_ker**(GEN x)*flag* = 1).

Gives an LLL-reduced ℤ-basis
for the lattice equal to the kernel of the matrix x with rational entries.
*flag* is deprecated, kept for backward compatibility. The function
`matsolvemod`

allows to solve more general linear systems over ℤ.

The library syntax is `GEN `

.
Use directly **matkerint0**(GEN x, long flag)`GEN `

if x is known to have
integer entries, and **kerint**(GEN x)`Q_primpart`

first otherwise.

Gives a Howell basis (unique representation for submodules
of (ℤ/dℤ)^{n},
cf. `matimagemod`

) for the kernel of the matrix x modulo d as columns
of a matrix. The matrix x must have `t_INT`

entries, and d can be an
arbitrary positive integer. If im is present, set it to a basis of the image
of x (which is computed on the way).

? A = [1,2,3;5,1,4] %1 = [1 2 3] [5 1 4] ? K = matkermod(A,6) %2 = [2 1] [2 1] [0 3] ? (A*K)%6 %3 = [0 0] [0 0]

The library syntax is `GEN `

.**matkermod**(GEN x, GEN d, GEN *im = NULL)

Product of the matrix x by the diagonal
matrix whose diagonal entries are those of the vector d. Equivalent to,
but much faster than x*`matdiagonal`

(d).

The library syntax is `GEN `

.**matmuldiagonal**(GEN x, GEN d)

Product of the matrices x and y assuming that the result is a diagonal matrix. Much faster than x*y in that case. The result is undefined if x*y is not diagonal.

The library syntax is `GEN `

.**matmultodiagonal**(GEN x, GEN y)

Creates as a matrix the lower triangular Pascal triangle of order x+1 (i.e. with binomial coefficients up to x). If q is given, compute the q-Pascal triangle (i.e. using q-binomial coefficients).

The library syntax is `GEN `

.
Also available is **matqpascal**(long n, GEN q = NULL)`GEN `

.**matpascal**(GEN x)

Permanent of the square matrix x using Ryser's formula in Gray code order.

? n = 20; m = matrix(n,n,i,j, i!=j); ? matpermanent(m) %2 = 895014631192902121 ? n! * sum(i=0,n, (-1)^i/i!) %3 = 895014631192902121

This function runs in time O(2^{n} n) for a matrix of size
n and is not implemented for n large.

The library syntax is `GEN `

.**matpermanent**(GEN x)

Returns [Q,R], the QR-decomposition of the square invertible
matrix M with real entries: Q is orthogonal and R upper triangular. If
*flag* = 1, the orthogonal matrix is returned as a sequence of Householder
transforms: applying such a sequence is stabler and faster than
multiplication by the corresponding Q matrix.
More precisely, if

[Q,R] = matqr(M); [q,r] = matqr(M, 1);

then r = R and `mathouseholder`

(q, M) is
(close to) R; furthermore

mathouseholder(q, matid(#M)) == Q~

the inverse of Q. This function raises an error if the precision is too low or x is singular.

The library syntax is `GEN `

.**matqr**(GEN M, long flag, long prec)

Rank of the matrix x.

The library syntax is `long `

.**rank**(GEN x)

Let m be a factorization matrix, i.e., a 2-column matrix whose
columns contains arbitrary "generators" and integer "exponents"
respectively. Returns the canonical form of m: the
first column is sorted with unique elements and the second one contains the
merged "exponents" (exponents of identical entries in the first column of
m are added, rows attached to 0 exponents are deleted). The generators are
sorted with respect to the universal `cmp`

routine; in particular, this
function is the identity on true integer factorization matrices, but not on
other factorizations (in products of polynomials or maximal ideals, say). It
is idempotent.

For convenience, this function also allows a vector m, which is handled as a
factorization with all exponents equal to 1, as in `factorback`

.

? A=[x,2;y,4]; B=[x,-2; y,3; 3, 4]; C=matconcat([A,B]~) %1 = [x 2] [y 4] [x -2] [y 3] [3 4] ? matreduce(C) %2 = [3 4] [y 7] ? matreduce([x,x,y,x,z,x,y]) \\ vector argument %3 = [x 4] [y 2] [z 1]

The following one-line functions will list elements occurring exactly once (resp. more than once) in the vector or list v:

unique(v) = [ x[1] | x <- matreduce(v)~, x[2] == 1 ]; duplicates(v) = [ x[1] | x <- matreduce(v)~, x[2] > 1 ]; ? v = [0,1,2,3,1,2]; ? unique(v) %2 = [0, 3] ? duplicates(v) %3 = [1, 2]

The library syntax is `GEN `

.**matreduce**(GEN m)

Creation of the
m x n matrix whose coefficients are given by the expression
*expr*. There are two formal parameters in *expr*, the first one
(X) corresponding to the rows, the second (Y) to the columns, and X
goes from 1 to m, Y goes from 1 to n. If one of the last 3 parameters
is omitted, fill the matrix with zeroes. If n is omitted, return a
square m x m matrix.

A being an m x n matrix in M_{m,n}(ℚ), let
Im_{ℚ} A (resp. Im_{ℤ} A) the ℚ-vector space
(resp. the ℤ-module) spanned by the columns of A. This function has
varying behavior depending on the sign of p:

If p ≥ 0, A is assumed to have maximal rank n ≤ m. The function
returns a matrix B ∈ M_{m,n}(ℤ), with Im_{ℚ} B =
Im_{ℚ} A,
such that the GCD of all its n x n minors is coprime to
p; in particular, if p = 0 (default), this GCD is 1.

If p = -1, returns a basis of the lattice ℤ^{m} ∩ Im_{ℤ} A.

If p = -2, returns a basis of the lattice ℤ^{m} ∩ Im_{ℚ} A.

**Caveat.** (p = -1 or -2) For efficiency reason, we do not compute
the HNF of the resulting basis.

? minors(x) = vector(#x[,1], i, matdet(x[^i,])); ? A = [3,1/7; 5,3/7; 7,5/7]; minors(A) %1 = [4/7, 8/7, 4/7] \\ determinants of all 2x2 minors ? B = matrixqz(A) %2 = [3 1] [5 2] [7 3] ? minors(%) %3 = [1, 2, 1] \\ B integral with coprime minors ? matrixqz(A,-1) %4 = [3 1] [5 3] [7 5] ? matrixqz(A,-2) %5 = [3 1] [5 2] [7 3]

The library syntax is `GEN `

.**matrixqz0**(GEN A, GEN p = NULL)

x being a vector or matrix, returns a row vector with two components, the first being the number of rows (1 for a row vector), the second the number of columns (1 for a column vector).

The library syntax is `GEN `

.**matsize**(GEN x)

If X is a (singular or nonsingular) matrix outputs the vector of
elementary divisors of X, i.e. the diagonal of the
Smith normal form of X, normalized so that d_{n} | d_{n-1} |
... | d_{1}. X must have integer or polynomial entries; in the latter
case, X must be a square matrix.

The binary digits of *flag* mean:

1 (complete output): if set, outputs [U,V,D], where U and V are two unimodular matrices such that UXV is the diagonal matrix D. Otherwise output only the diagonal of D. If X is not a square matrix, then D will be a square diagonal matrix padded with zeros on the left or the top.

4 (cleanup): if set, cleans up the output. This means that elementary divisors equal to 1 will be deleted, i.e. outputs a shortened vector D' instead of D. If complete output was required, returns [U',V',D'] so that U'XV' = D' holds. If this flag is set, X is allowed to be of the form `vector of elementary divisors' or [U,V,D] as would normally be output with the cleanup flag unset.

If v is an output from `matsnf`

and p is a power of an irreducible
element, then `snfrank(v, p)`

returns the p-rank of the attached
module.

? X = [27,0; 0,3; 1,1; 0,0]; matsnf(X) %1 = [0, 0, 3, 1] ? [U,V,D] = v = matsnf(X, 1); U*X*V == D %2 ? U %3 = [0 0 0 1] [1 9 -27 0] [0 1 0 0] [0 0 1 0] ? V %4 = [-1 1] [ 1 0] ? snfrank(v, 3) %5 = 3

Continuing the same example after cleanup:

? [U,V,D] = v = matsnf(X, 1+4); U*X*V == D %6 = 1 ? D %7 = [0] [0] [3] ? snfrank(v, 3) %8 = 3 ? snfrank(v, 2) %9 = 2

The library syntax is `GEN `

.**matsnf0**(GEN X, long flag)

Let M be a left-invertible matrix and B a column vector
such that there exists a solution X to the system of linear equations
MX = B; return the (unique) solution X. This has the same effect as, but
is faster, than M^{-1}*B. Uses Dixon p-adic lifting method if M and
B are integral and Gaussian elimination otherwise. When there is no
solution, the function returns an X such that MX - B is nonzero
although it has at least #M zero entries:

? M = [1,2;3,4;5,6]; ? B = [4,6,8]~; X = matsolve(M, B) %2 = [-2, 3]~ ? M*X == B %3 = 1 ? B = [1,2,4]~; X = matsolve(M, [1,2,4]~) %4 = [0, 1/2]~ ? M*X - B %5 = [0, 0, -1]~

Raises an exception if M is not left-invertible, even if there is a solution:

? M = [1,1;1,1]; matsolve(M, [1,1]~) *** at top-level: matsolve(M,[1,1]~) *** ^ — — — — — — *** matsolve: impossible inverse in gauss: [1, 1; 1, 1].

The function also works when B is a matrix and we return the unique matrix solution X provided it exists. Again, if there is no solution, the function returns an X such that MX - B is nonzero although it has at least #M zero rows.

The library syntax is `GEN `

.**gauss**(GEN M, GEN B)

M being any integral matrix,
D a column vector of nonnegative integer moduli, and B an integral
column vector, gives an integer solution to the system of congruences
∑_{i} m_{i,j}x_{j} = b_{i} (mod d_{i}) if one exists, otherwise
returns the integer zero. Note that we explicitly allow d_{i} = 0
corresponding to an equality in ℤ. Shorthand notation: B (resp. D)
can be given as a single integer, in which case all the b_{i}
(resp. d_{i}) above are taken to be equal to B
(resp. D). Again, D = 0 solves the linear system of equations over ℤ.

? M = [1,2;3,4]; ? matsolvemod(M, [3,4]~, [1,2]~) %2 = [10, 0]~ ? matsolvemod(M, 3, 1) \\ M X = [1,1]~ over F_{3}%3 = [2, 1]~ ? matsolvemod(M, [3,0]~, [1,2]~) \\ x + 2y = 1 (mod 3), 3x + 4y = 2 (in Z) %4 = [6, -4]~ ? matsolvemod(M, 0, [1,2]~) \\ no solution in Z for x + 2y = 1, 3x + 4y = 2

If *flag* = 1, all solutions are returned in the form of a two-component row
vector [x,u], where x is an integer solution to the system of
congruences and u is a matrix whose columns give a basis of the homogeneous
system (so that all solutions can be obtained by adding x to any linear
combination of columns of u). If no solution exists, returns zero.

The library syntax is `GEN `

.
Also available are **matsolvemod**(GEN M, GEN D, GEN B, long flag)`GEN `

(**gaussmodulo**(GEN M, GEN D, GEN B)*flag* = 0) and `GEN `

(**gaussmodulo2**(GEN M, GEN D, GEN B)*flag* = 1).

Assuming that the columns of the matrix x are linearly independent (if they are not, an error message is issued), finds a square invertible matrix whose first columns are the columns of x, i.e. supplement the columns of x to a basis of the whole space.

? matsupplement([1;2]) %1 = [1 0] [2 1]

Raises an error if x has 0 columns, since (due to a long standing design bug), the dimension of the ambient space (the number of rows) is unknown in this case:

? matsupplement(matrix(2,0)) *** at top-level: matsupplement(matrix *** ^ — — — — — — -- *** matsupplement: sorry, suppl [empty matrix] is not yet implemented.

The library syntax is `GEN `

.**suppl**(GEN x)

Transpose of x (also x~). This has an effect only on vectors and matrices.

The library syntax is `GEN `

.**gtrans**(GEN x)

minimal polynomial of A with respect to the variable v., i.e. the monic polynomial P of minimal degree (in the variable v) such that P(A) = 0.

The library syntax is `GEN `

where **minpoly**(GEN A, long v = -1)`v`

is a variable number.

Square of the L^{2}-norm of x. More precisely,
if x is a scalar, `norml2`

(x) is defined to be the square
of the complex modulus of x (real `t_QUAD`

s are not supported).
If x is a polynomial, a (row or column) vector or a matrix, `norml2(x)`

is
defined recursively as ∑_{i} `norml2`

(x_{i}), where (x_{i})
run through
the components of x. In particular, this yields the usual
∑_{i} |x_{i}|^{2} (resp. ∑_{i,j} |x_{i,j}|^{2}) if x is a
polynomial or vector (resp. matrix) with complex components.

? norml2( [ 1, 2, 3 ] ) \\ vector %1 = 14 ? norml2( [ 1, 2; 3, 4] ) \\ matrix %2 = 30 ? norml2( 2*I + x ) %3 = 5 ? norml2( [ [1,2], [3,4], 5, 6 ] ) \\ recursively defined %4 = 91

The library syntax is `GEN `

.**gnorml2**(GEN x)

L^{p}-norm of x; sup norm if p is omitted or `+oo`

. More precisely,
if x is a scalar, `normlp`

(x, p) is defined to be `abs`

(x).
If x is a polynomial, a (row or column) vector or a matrix:

***** if p is omitted or `+oo`

, then `normlp(x)`

is defined
recursively as max_{i} `normlp`

(x_{i})), where x_{i} runs through the
components of x. In particular, this yields the usual sup norm if x is a
polynomial or vector with complex components.

***** otherwise, `normlp(x, p)`

is defined recursively as
(∑_{i} `normlp`

^{p}(x_{i},p))^{1/p}. In particular, this yields the
usual (∑_{i} |x_{i}|^{p})^{1/p} if x is a polynomial or vector with
complex components.

? v = [1,-2,3]; normlp(v) \\ vector %1 = 3 ? normlp(v, +oo) \\ same, more explicit %2 = 3 ? M = [1,-2;-3,4]; normlp(M) \\ matrix %3 = 4 ? T = (1+I) + I*x^2; normlp(T) %4 = 1.4142135623730950488016887242096980786 ? normlp([[1,2], [3,4], 5, 6]) \\ recursively defined %5 = 6 ? normlp(v, 1) %6 = 6 ? normlp(M, 1) %7 = 10 ? normlp(T, 1) %8 = 2.4142135623730950488016887242096980786

The library syntax is `GEN `

.**gnormlp**(GEN x, GEN p = NULL, long prec)

For nonnegative n, return the vector with n+1 components
[1,x,...,x^{n}] if `x0`

is omitted, and
[x_{0}, x_{0}*x, ..., x_{0}*x^{n}] otherwise.

? powers(Mod(3,17), 4) %1 = [Mod(1, 17), Mod(3, 17), Mod(9, 17), Mod(10, 17), Mod(13, 17)] ? powers(Mat([1,2;3,4]), 3) %2 = [[1, 0; 0, 1], [1, 2; 3, 4], [7, 10; 15, 22], [37, 54; 81, 118]] ? powers(3, 5, 2) %3 = [2, 6, 18, 54, 162, 486]

When n < 0, the function returns the empty vector `[]`

.

The library syntax is `GEN `

.
Also available is
**gpowers0**(GEN x, long n, GEN x0 = NULL)`GEN `

when **gpowers**(GEN x, long n)`x0`

is `NULL`

.

G being a square and symmetric matrix with integer entries representing a
positive definite quadratic form, outputs the automorphism group of the
associate lattice.
Since this requires computing the minimal vectors, the computations can
become very lengthy as the dimension grows. G can also be given by an
`qfisominit`

structure.
See `qfisominit`

for the meaning of *fl*.

The output is a two-components vector [o,g] where o is the group order
and g is the list of generators (as a vector). For each generator H,
the equality G = {^{t}}H G H holds.

The interface of this function is experimental and will likely change in the future.

This function implements an algorithm of Plesken and Souvignier, following Souvignier's implementation.

? K = matkerint(Mat(concat([vector(23,i,2*i+1), 51, 145]))); ? M = matdiagonal(vector(25,i,if(i==25,-1,1))); ? L24 = K~ * M * K; \\ the Leech lattice ? [o,g] = qfauto(L24); o %4 = 8315553613086720000 ? #g %5 = 2

The library syntax is `GEN `

.
The function **qfauto0**(GEN G, GEN fl = NULL)`GEN `

is also available
where G is a vector of **qfauto**(GEN G, GEN fl)`zm`

matrices.

*qfa* being an automorphism group as output by
`qfauto`

, export the underlying matrix group as a string suitable
for (no flags or *flag* = 0) GAP or (*flag* = 1) Magma. The following example
computes the size of the matrix group using GAP:

? G = qfauto([2,1;1,2]) %1 = [12, [[-1, 0; 0, -1], [0, -1; 1, 1], [1, 1; 0, -1]]] ? s = qfautoexport(G) %2 = "Group([[-1, 0], [0, -1]], [[0, -1], [1, 1]], [[1, 1], [0, -1]])" ? extern("echo \"Order("s");\" | gap -q") %3 = 12

The library syntax is `GEN `

.**qfautoexport**(GEN qfa, long flag)

This function is obsolete, use `qfeval`

.

The library syntax is `GEN `

.**qfbil**(GEN x, GEN y, GEN q = NULL)

Given a square symmetric `t_MAT`

M, return R such that
^tR R = M, or [] if there is no solution.

The library syntax is `GEN `

.**qfcholesky**(GEN q, long prec)

x being a square and symmetric matrix of dimension d representing a positive definite quadratic form, and t a vector of the same dimension d. This function deals with the vectors whose squared distance to t is less than B, enumerated using the Fincke-Pohst algorithm, storing at most m vectors. There is no limit if m is omitted: beware that this may be a huge vector! The vectors are returned in no particular order.

The function searches for the closest vectors to t if B is omitted
or ≤ 0.
The behavior is undefined if x is not positive definite (a "precision too
low" error is most likely, although more precise error messages are
possible). The precise behavior depends on *flag*.

***** If *flag* = 0 (default), return [N, M, V], where N is the number of
vectors enumerated (possibly larger than m), M ≤ B is the maximum
squared distance found, and V is a matrix whose columns are found vectors.

***** If *flag* = 1, ignore m and return [M,v], where v is a nonzero
vector at squared distance M ≤ B. If no nonzero vector has distance
≤ B, return [].

In these two cases, x must have integral *small* entries: more
precisely, we definitely must have d.|x|_ oo ^2 < 2^{53} but
even that may not be enough. The implementation uses low precision floating
point computations for maximal speed and gives incorrect results when x
has large entries. That condition is checked in the code and the routine
raises an error if large rounding errors occur.

? M = [2,1;1,2]; t = [1/2, -1/2]; ? qfcvp(M, t, 0) %2 = [2, 0.5000000000000000000, [0, 1; 0, -1]] ? qfcvp(M, t, 1.5) %3 = [4, 1.5000000000000000000, [1, 0, 1, 0; 0, 0, -1, -1]]

The library syntax is `GEN `

.**qfcvp0**(GEN x, GEN t, GEN B = NULL, GEN m = NULL, long flag, long prec)

Evaluate the quadratic form q (given by a symmetric matrix) at the vector x; if y is present, evaluate the polar form at (x,y); if q omitted, use the standard Euclidean scalar product, corresponding to the identity matrix.

Roughly equivalent to `x~ * q * y`

, but a little faster and
more convenient (does not distinguish between column and row vectors):

? x = [1,2,3]~; y = [-1,3,1]~; q = [1,2,3;2,2,-1;3,-1,9]; ? qfeval(q,x,y) %2 = 23 ? for(i=1,10^6, qfeval(q,x,y)) time = 661ms ? for(i=1,10^6, x~*q*y) time = 697ms

The speedup is noticeable for the quadratic form,
compared to `x~ * q * x`

, since we save almost half the
operations:

? for(i=1,10^6, qfeval(q,x)) time = 487ms

The special case q = Id is handled faster if we omit q altogether:

? qfeval(,x,y) %6 = 8 ? q = matid(#x); ? for(i=1,10^6, qfeval(q,x,y)) time = 529 ms. ? for(i=1,10^6, qfeval(,x,y)) time = 228 ms. ? for(i=1,10^6, x~*y) time = 274 ms.

We also allow `t_MAT`

s of compatible dimensions for x,
and return `x~ * q * x`

in this case as well:

? M = [1,2,3;4,5,6;7,8,9]; qfeval(,M) \\ Gram matrix %5 = [66 78 90] [78 93 108] [90 108 126] ? q = [1,2,3;2,2,-1;3,-1,9]; ? for(i=1,10^6, qfeval(q,M)) time = 2,008 ms. ? for(i=1,10^6, M~*q*M) time = 2,368 ms. ? for(i=1,10^6, qfeval(,M)) time = 1,053 ms. ? for(i=1,10^6, M~*M) time = 1,171 ms.

If q is a `t_QFB`

, it is implicitly converted to the
attached symmetric `t_MAT`

. This is done more
efficiently than by direct conversion, since we avoid introducing a
denominator 2 and rational arithmetic:

? q = Qfb(2,3,4); x = [2,3]; ? qfeval(q, x) %2 = 62 ? Q = Mat(q) %3 = [ 2 3/2] [3/2 4] ? qfeval(Q, x) %4 = 62 ? for (i=1, 10^6, qfeval(q,x)) time = 758 ms. ? for (i=1, 10^6, qfeval(Q,x)) time = 1,110 ms.

Finally, when x is a `t_MAT`

with *integral* coefficients, we allow
a `t_QFB`

for q and return the binary
quadratic form q o M. Again, the conversion to `t_MAT`

is less
efficient in this case:

? q = Qfb(2,3,4); Q = Mat(q); x = [1,2;3,4]; ? qfeval(q, x) %2 = Qfb(47, 134, 96) ? qfeval(Q,x) %3 = [47 67] [67 96] ? for (i=1, 10^6, qfeval(q,x)) time = 701 ms. ? for (i=1, 10^6, qfeval(Q,x)) time = 1,639 ms.

The library syntax is `GEN `

.**qfeval0**(GEN q = NULL, GEN x, GEN y = NULL)

decomposition into squares of the
quadratic form represented by the symmetric matrix q. If *flag* = 0 (default),
the result is a matrix M whose diagonal entries are the coefficients of the
squares, and the off-diagonal entries on each line represent the bilinear
forms. More precisely, if (a_{ij}) denotes the output, one has
q(x) = ∑_{i} a_{i,i} (x_{i} + ∑_{j ! = i} a_{i,j} x_{j})^{2}

? qfgaussred([0,1;1,0]) %1 = [1/2 1] [-1 -1/2]

This means that 2xy = (1/2)(x+y)^{2} - (1/2)(x-y)^{2}.
Singular matrices are supported, in which case some diagonal coefficients
vanish:

? qfgaussred([1,1;1,1]) %2 = [1 1] [1 0]

This means that x^{2} + 2xy + y^{2} = (x+y)^{2}.

If *flag* = 1, return `[U,V]`

where U is a square matrix and V a vector,
such that if `D = matdiagonal(V)`

, q = {^{t}} U D U. More
precisely
q(x) = ∑_{i} D_{i} (∑_{j} U_{i,j} x_{j})^{2}
and the matrix M is recovered as M = U + D - 1.

? q = [0,1;1,0]; ? [U,V] = qfgaussred(q,1); D = matdiagonal(V); ? U~*D*U %5 = [0 1] [1 0] ? U+D-1 %6 = [1/2 1] [ -1 -1/2]

The library syntax is `GEN `

.
See also the functions **qfgaussred0**(GEN q, long flag)`GEN `

(for **qfgaussred**(GEN a)`qfgaussred(a,0)`

),
`GEN `

(for **qfgaussred2**(GEN a)`qfgaussred0(a,1)`

). Finally,
the function
`GEN `

assumes that q is
positive definite and is a little faster; returns **qfgaussred_positive**(GEN q)`NULL`

if a vector
with negative norm occurs (non positive matrix or too many rounding errors).

G, H being square and symmetric matrices with integer entries representing
positive definite quadratic forms, return an invertible matrix S such that
G = {^{t}}S H S. This defines a isomorphism between the corresponding lattices.
Since this requires computing the minimal vectors, the computations can
become very lengthy as the dimension grows.
See `qfisominit`

for the meaning of *fl*.
If *grp* is given it must be the automorphism group of H. It will be used
to speed up the computation.

G can also be given by an `qfisominit`

structure which is preferable if
several forms H need to be compared to G.

This function implements an algorithm of Plesken and Souvignier, following Souvignier's implementation.

The library syntax is `GEN `

.
Also available is **qfisom0**(GEN G, GEN H, GEN fl = NULL, GEN grp = NULL)`GEN `

where G is a vector of **qfisom**(GEN G, GEN H, GEN fl, GEN grp)`zm`

, and H is a `zm`

, and grp is
either `NULL`

or a vector of `zm`

.

G being a square and symmetric matrix with integer entries representing a
positive definite quadratic form, return an `isom`

structure allowing to
compute isomorphisms between G and other quadratic forms faster.

The interface of this function is experimental and will likely change in future release.

If present, the optional parameter *fl* must be a `t_VEC`

with two
components. It allows to specify the invariants used, which can make the
computation faster or slower. The components are

***** `fl[1]`

Depth of scalar product combination to use.

***** `fl[2]`

Maximum level of Bacher polynomials to use.

If present, m must be the set of vectors of norm up to the maximal of the
diagonal entry of G, either as a matrix or as given by `qfminim`

.
Otherwise this function computes the minimal vectors so it become very
lengthy as the dimension of G grows.

The library syntax is `GEN `

.
Also available is
**qfisominit0**(GEN G, GEN fl = NULL, GEN m = NULL)`GEN `

where F is a vector of **qfisominit**(GEN F, GEN fl)`zm`

.

Apply Jacobi's eigenvalue algorithm to the real symmetric matrix A. This returns [L, V], where

***** L is the vector of (real) eigenvalues of A, sorted in increasing
order,

***** V is the corresponding orthogonal matrix of eigenvectors of A.

? \p19 ? A = [1,2;2,1]; mateigen(A) %1 = [-1 1] [ 1 1] ? [L, H] = qfjacobi(A); ? L %3 = [-1.000000000000000000, 3.000000000000000000]~ ? H %4 = [ 0.7071067811865475245 0.7071067811865475244] [-0.7071067811865475244 0.7071067811865475245] ? norml2( (A-L[1])*H[,1] ) \\ approximate eigenvector %5 = 9.403954806578300064 E-38 ? norml2(H*H~ - 1) %6 = 2.350988701644575016 E-38 \\ close to orthogonal

The library syntax is `GEN `

.**jacobi**(GEN A, long prec)

LLL algorithm applied to the
*columns* of the matrix x. The columns of x may be linearly
dependent. The result is by default a unimodular transformation matrix T
such that x.T is an LLL-reduced basis of the lattice generated by
the column vectors of x. Note that if x is not of maximal rank T will
not be square. The LLL parameters are (0.51,0.99), meaning that the
Gram-Schmidt coefficients for the final basis satisfy |μ_{i,j}| ≤
0.51, and the Lovász's constant is 0.99.

If *flag* = 0 (default), assume that x has either exact (integral or
rational) or real floating point entries. The matrix is rescaled, converted
to integers and the behavior is then as in *flag* = 1.
Computations involving Gram-Schmidt
vectors are approximate, with precision varying as needed (Lehmer's trick,
as generalized by Schnorr). Adapted from Nguyen and Stehlé's algorithm
and Stehlé's code (`fplll-1.3`

) as building blocks for the FLATTER
(block recursive) algorithm of Heninger and Ryan.

If *flag* = 1, disable use of FLATTER algorithm; use `fplll`

. This flag
is provided to experiment with the concrete speed-ups allowed by FLATTER,
as well as to genuinely disable it on the rare classes of lattices for which
it turns out it performs badly: many such classes are detected in the code,
which then restricts to stock `fplll`

, but new exemples may turn up.

If *flag* = 2, x should be an integer matrix whose columns are linearly
independent. Returns a partially reduced basis for x, using an unpublished
algorithm by Peter Montgomery: a basis is said to be *partially reduced*
if |v_{i} ± v_{j}| ≥ |v_{i}| for any two distinct basis vectors
v_{i}, v_{j}. This is faster than *flag* = 1, esp. when one row is huge
compared
to the other rows (knapsack-style), and should quickly produce relatively
short vectors. The resulting basis is *not* LLL-reduced in general.
If LLL reduction is eventually desired, avoid this partial reduction:
applying LLL to the partially reduced matrix is significantly *slower*
than starting from a knapsack-type lattice.

If *flag* = 3, as *flag* = 0, but the reduction is performed in place: the
routine returns x.T. This is usually faster for knapsack-type
lattices.

If *flag* = 4, as *flag* = 0, returning a vector [K, T] of matrices: the
columns of K represent a basis of the integer kernel of x
(not LLL-reduced in general) and T is the transformation
matrix such that x.T is an LLL-reduced ℤ-basis of the image
of the matrix x.

If *flag* = 5, case as *flag* = 4, but x may have polynomial coefficients.

If *flag* = 8, same as *flag* = 0, but x may have polynomial coefficients.

? \p500 realprecision = 500 significant digits ? a = 2*cos(2*Pi/97); ? C = 10^450; ? v = powers(a,48); b = round(matconcat([matid(48),C*v]~)); ? p = b * qflll(b)[,1]; \\ tiny linear combination of powers of 'a' time = 4,470 ms. ? exponent(v * p / C) %5 = -1418 ? p3 = qflll(b,3)[,1]; \\ compute in place, faster time = 3,790 ms. ? p3 == p \\ same result %7 = 1 ? p2 = b * qflll(b,2)[,1]; \\ partial reduction: faster, not as good time = 343 ms. ? exponent(v * p2 / C) %9 = -1190

The library syntax is `GEN `

.
Also available are **qflll0**(GEN x, long flag)`GEN `

(**lll**(GEN x)*flag* = 0),
`GEN `

(**lllint**(GEN x)*flag* = 1), and `GEN `

(**lllkerim**(GEN x)*flag* = 4).

Same as `qflll`

, except that the
matrix G = `x~ * x`

is the Gram matrix of some lattice vectors x,
and not the coordinates of the vectors themselves. In particular, G must
now be a square symmetric real matrix, corresponding to a positive
quadratic form (not necessarily definite: x needs not have maximal rank).
The result is a unimodular
transformation matrix T such that x.T is an LLL-reduced basis of
the lattice generated by the column vectors of x. See `qflll`

for
further details about the LLL implementation.

If *flag* = 0 (default), assume that G has either exact (integral or
rational) or real floating point entries. The matrix is rescaled, converted
to integers.
Computations involving Gram-Schmidt vectors are approximate, with precision
varying as needed (Lehmer's trick, as generalized by Schnorr). Adapted from
Nguyen and Stehlé's algorithm and Stehlé's code (`fplll-1.3`

) and
FLATTER algorithm for Heninger and Ryan.

If *flag* = 1, disable use of FLATTER algorithm.

*flag* = 4: G has integer entries, gives the kernel and reduced image of x.

*flag* = 5: same as 4, but G may have polynomial coefficients.

The library syntax is `GEN `

.
Also available are **qflllgram0**(GEN G, long flag)`GEN `

(**lllgram**(GEN G)*flag* = 0),
`GEN `

(**lllgramint**(GEN G)*flag* = 1), and `GEN `

(**lllgramkerim**(GEN G)*flag* = 4).

x being a square and symmetric matrix of dimension d representing a positive definite quadratic form, this function deals with the vectors of x whose norm is less than or equal to B, enumerated using the Fincke-Pohst algorithm, storing at most m pairs of vectors: only one vector is given for each pair ± v. There is no limit if m is omitted: beware that this may be a huge vector! The vectors are returned in no particular order.

The function searches for the minimal nonzero vectors if B is omitted.
The behavior is undefined if x is not positive definite (a "precision too
low" error is most likely, although more precise error messages are
possible). The precise behavior depends on *flag*.

***** If *flag* = 0 (default), return [N, M, V], where N is the number of
vectors enumerated (an even number, possibly larger than 2m), M ≤ B
is the maximum norm found, and V is a matrix whose columns are found
vectors.

***** If *flag* = 1, ignore m and return [M,v], where v is a nonzero
vector of length M ≤ B. If no nonzero vector has length ≤ B,
return []. If no explicit B is provided, return a vector of smallish
norm, namely the vector of smallest length (usually the first one but not
always) in an LLL-reduced basis for x.

In these two cases, x must have integral *small* entries: more
precisely, we definitely must have d.|x|_{ oo }^{2} < 2^{53} but
even that may not be enough. The implementation uses low precision floating
point computations for maximal speed and gives incorrect results when x
has large entries. That condition is checked in the code and the routine
raises an error if large rounding errors occur. A more robust, but much
slower, implementation is chosen if the following flag is used:

***** If *flag* = 2, x can have non integral real entries, but this is also
useful when x has large integral entries. Return [N, M, V] as in case
*flag* = 0, where M is returned as a floating point number. If x is
inexact and B is omitted, the "minimal" vectors in V only have
approximately the same norm (up to the internal working accuracy).
This version is very robust but still offers no hard and fast guarantee
about the result: it involves floating point operations performed at a high
floating point precision depending on your input, but done without rigorous
tracking of roundoff errors (as would be provided by interval arithmetic for
instance). No example is known where the input is exact but the function
returns a wrong result.

? x = matid(2); ? qfminim(x) \\ 4 minimal vectors of norm 1: ±[0,1], ±[1,0] %2 = [4, 1, [0, 1; 1, 0]] ? { x = \\ The Leech lattice [4, 2, 0, 0, 0,-2, 0, 0, 0, 0, 0, 0, 1,-1, 0, 0, 0, 1, 0,-1, 0, 0, 0,-2; 2, 4,-2,-2, 0,-2, 0, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1, 0, 1,-1,-1; 0,-2, 4, 0,-2, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 1, 0, 0, 1,-1,-1, 0, 0; 0,-2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1,-1, 0, 0, 0, 1,-1, 0, 1,-1, 1, 0; 0, 0,-2, 0, 4, 0, 0, 0, 1,-1, 0, 0, 1, 0, 0, 0,-2, 0, 0,-1, 1, 1, 0, 0; -2, -2,0, 0, 0, 4,-2, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,-1, 1, 1; 0, 0, 0, 0, 0,-2, 4,-2, 0, 0, 0, 0, 0, 1, 0, 0, 0,-1, 0, 0, 0, 1,-1, 0; 0, 0, 0, 0, 0, 0,-2, 4, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1,-1,-1, 0, 1, 0; 0, 0, 0, 0, 1,-1, 0, 0, 4, 0,-2, 0, 1, 1, 0,-1, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0,-1, 0, 0, 0, 0, 4, 0, 0, 1, 1,-1, 1, 0, 0, 0, 1, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0,-2, 0, 4,-2, 0,-1, 0, 0, 0,-1, 0,-1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-2, 4,-1, 1, 0, 0,-1, 1, 0, 1, 1, 1,-1, 0; 1, 0,-1, 1, 1, 0, 0,-1, 1, 1, 0,-1, 4, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1,-1; -1,-1, 1,-1, 0, 0, 1, 0, 1, 1,-1, 1, 0, 4, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1; 0, 0, 0, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 1, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 1, 1, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0; 0, 0, 1, 0,-2, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 4, 1, 1, 1, 0, 0, 1, 1; 1, 0, 0, 1, 0, 0,-1, 0, 1, 0,-1, 1, 1, 0, 0, 0, 1, 4, 0, 1, 1, 0, 1, 0; 0, 0, 0,-1, 0, 1, 0,-1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 4, 0, 1, 1, 0, 1; -1, -1,1, 0,-1, 1, 0,-1, 0, 1,-1, 1, 0, 1, 0, 0, 1, 1, 0, 4, 0, 0, 1, 1; 0, 0,-1, 1, 1, 0, 0,-1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 4, 1, 0, 1; 0, 1,-1,-1, 1,-1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 4, 0, 1; 0,-1, 0, 1, 0, 1,-1, 1, 0, 1, 0,-1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 4, 1; -2,-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 4]; } ? qfminim(x,,0) \\ 0: don't store minimal vectors time = 121 ms. %4 = [196560, 4, [;]] \\ 196560 minimal vectors of norm 4 ? qfminim(x) \\ store all minimal vectors ! time = 821 ms. ? qfminim(x,,0,2); \\ safe algorithm. Slower and unnecessary here. time = 5,540 ms. %6 = [196560, 4.000061035156250000, [;]] ? qfminim(x,,,2); \\ safe algorithm; store all minimal vectors time = 6,602 ms.

In this example, storing 0 vectors limits memory use; storing all of them
requires a `parisize`

about 50MB. All minimal vectors are nevertheless
enumerated in both cases of course, which means the speedup is likely to be
marginal.

The library syntax is `GEN `

.
Also available are
**qfminim0**(GEN x, GEN B = NULL, GEN m = NULL, long flag, long prec)`GEN `

(**minim**(GEN x, GEN B = NULL, GEN m = NULL)*flag* = 0),
`GEN `

(**minim2**(GEN x, GEN B = NULL, GEN m = NULL)*flag* = 1).
`GEN `

(do not perform LLL
reduction on x and return **minim_raw**(GEN x, GEN B = NULL, GEN m = NULL)`NULL`

on accuracy error).
`GEN `

(**minim_zm**(GEN x, GEN B = NULL, GEN m = NULL)*flag* = 0, return vectors as
`t_VECSMALL`

to save memory)

Given a square symmetric matrix G with rational coefficients, and
non-zero determinant, of dimension n ≥ 1, return `[H,U,c]`

such that
`H = c*U~*G*U`

for some rational c, and H integral with minimal
determinant. The coefficients of U are usually nonintegral.

? G = matdiagonal([650, -104329, -104329]); ? [H,U,c]=qfminimize(G); H %2 = [-1,0,0;0,-1,0;0,0,1] ? U %3 = [0,0,1/5;5/323,-1/323,0;-1/323,-5/323,0] ? c %4 = 1/26 ? c * U~ * G * U %4 = [-1,0,0;0,-1,0;0,0,1]

The library syntax is `GEN `

.**qfminimize**(GEN G)

This function is obsolete, use `qfeval`

.

The library syntax is `GEN `

.**qfnorm**(GEN x, GEN q = NULL)

Return the orbits of V under the action of the group of linear transformation generated by the set G. It is assumed that G contains minus identity, and only one vector in {v, -v} should be given. If G does not stabilize V, the function return 0.

In the example below, we compute representatives and lengths of the orbits of
the vectors of norm ≤ 3 under the automorphisms of the lattice ℤ^{6}.

? Q=matid(6); G=qfauto(Q); V=qfminim(Q,3); ? apply(x->[x[1],#x],qforbits(G,V)) %2 = [[[0,0,0,0,0,1]~,6],[[0,0,0,0,1,-1]~,30],[[0,0,0,1,-1,-1]~,80]]

The library syntax is `GEN `

.**qforbits**(GEN G, GEN V)

Coefficients of binary quadratic forms that parametrize the
solutions of the ternary quadratic form G, using the particular
solution *sol*.
*flag* is optional and can be 1, 2, or 3, in which case the *flag*-th form is
reduced. The default is *flag* = 0 (no reduction).

? G = [1,0,0;0,1,0;0,0,-34]; ? M = qfparam(G, qfsolve(G)) %2 = [ 3 -10 -3] [-5 -6 5] [ 1 0 1]

Indeed, the solutions can be parametrized as
(3x^{2}-10xy-3y^{2})^{2} + (-5x^{2}-6xy+5y^{2})^{2} -34(x^{2}+y^{2})^{2}
= 0.

? v = y^2 * M*[1,x/y,(x/y)^2]~ %3 = [3*x^2 - 10*y*x - 3*y^2, -5*x^2 - 6*y*x + 5*y^2, -x^2 - y^2]~ ? v~*G*v %4 = 0

The library syntax is `GEN `

.**qfparam**(GEN G, GEN sol, long flag)

G being a square and symmetric matrix with integer entries
representing a positive definite quadratic form, outputs the perfection rank
of the form. That is, gives the rank of the family of the s symmetric
matrices v{^{t}}v, where v runs through the minimal vectors.

A form is perfect if and only if its perfection rank is d(d+1)/2 where d is the dimension of G.

The algorithm computes the minimal vectors and its runtime is exponential in d.

The library syntax is `GEN `

.**qfperfection**(GEN G)

q being a square and symmetric matrix with integer entries representing a positive definite quadratic form, count the vectors representing successive integers.

***** If *flag* = 0, count all vectors. Outputs the vector whose i-th
entry, 1 ≤ i ≤ B is half the number of vectors v such that q(v) = i.

***** If *flag* = 1, count vectors of even norm. Outputs the vector
whose i-th entry, 1 ≤ i ≤ B is half the number of vectors such
that q(v) = 2i.

? q = [2, 1; 1, 3]; ? qfrep(q, 5) %2 = Vecsmall([0, 1, 2, 0, 0]) \\ 1 vector of norm 2, 2 of norm 3, etc. ? qfrep(q, 5, 1) %3 = Vecsmall([1, 0, 0, 1, 0]) \\ 1 vector of norm 2, 0 of norm 4, etc.

This routine uses a naive algorithm based on `qfminim`

, and
will fail if any entry becomes larger than 2^{31} (or 2^{63}).

The library syntax is `GEN `

.**qfrep0**(GEN q, GEN B, long flag)

Returns [p,m] the signature of the quadratic form represented by the symmetric matrix x. Namely, p (resp. m) is the number of positive (resp. negative) eigenvalues of x. The result is computed using Gaussian reduction.

The library syntax is `GEN `

.**qfsign**(GEN x)

Given a square symmetric matrix G of dimension n ≥ 1, solve over
ℚ the quadratic equation {^{t}}X G X = 0. The matrix G must have rational
coefficients. When G is integral, the argument can also be a vector [G,F]
where F is the factorization matrix of the absolute value of the determinant
of G.

The solution might be a single nonzero column vector
(`t_COL`

) or a matrix (whose columns generate a totally isotropic
subspace).

If no solution exists, returns an integer, that can be a prime p such that there is no local solution at p, or -1 if there is no real solution, or -2 if n = 2 and -det G is not a square (which implies there is a real solution, but no local solution at some p dividing det G).

? G = [1,0,0;0,1,0;0,0,-34]; ? qfsolve(G) %1 = [-3, -5, 1]~ ? qfsolve([1,0; 0,2]) %2 = -1 \\ no real solution ? qfsolve([1,0,0;0,3,0; 0,0,-2]) %3 = 3 \\ no solution in Q_{3}? qfsolve([1,0; 0,-2]) %4 = -2 \\ no solution, n = 2

The library syntax is `GEN `

.**qfsolve**(GEN G)

The set whose elements are the f(x,y), where x,y run through X,Y. respectively. If Y is omitted, assume that X = Y and that f is symmetric: f(x,y) = f(y,x) for all x,y in X.

? X = [1,2,3]; Y = [2,3,4]; ? setbinop((x,y)->x+y, X,Y) \\ set X + Y %2 = [3, 4, 5, 6, 7] ? setbinop((x,y)->x-y, X,Y) \\ set X - Y %3 = [-3, -2, -1, 0, 1] ? setbinop((x,y)->x+y, X) \\ set 2X = X + X %2 = [2, 3, 4, 5, 6]

The library syntax is `GEN `

.**setbinop**(GEN f, GEN X, GEN Y = NULL)

Symmetric difference of the two sets x and y (see `setisset`

).
If x or y is not a set, the result is undefined.

? a=[1,2,2,3];b=[4,2,3,4]; ? setdelta(Set(a), Set(b)) %2 = [1, 4] \\ the symmetric difference of the two sets ? setdelta(a,b) %3 = [1, 2, 2, 3, 4, 2, 3, 4] \\ undefined result

The library syntax is `GEN `

.**setdelta**(GEN x, GEN y)

Intersection of the two sets x and y (see `setisset`

).
If x or y is not a set, the result is undefined.

The library syntax is `GEN `

.**setintersect**(GEN x, GEN y)

Returns true (1) if x is a set, false (0) if
not. In PARI, a set is a row vector whose entries are strictly
increasing with respect to a (somewhat arbitrary) universal comparison
function. To convert any object into a set (this is most useful for
vectors, of course), use the function `Set`

.

? a = [3, 1, 1, 2]; ? setisset(a) %2 = 0 ? Set(a) %3 = [1, 2, 3]

The library syntax is `long `

.**setisset**(GEN x)

Difference of the two sets x and y (see `setisset`

),
i.e. set of elements of x which do not belong to y.
If x or y is not a set, the result is undefined.

The library syntax is `GEN `

.**setminus**(GEN x, GEN y)

Determines whether x belongs to the set or sorted list S
(see `setisset`

).

We first describe the default behavior, when *flag* is zero or omitted. If x
belongs to the set S, returns the index j such that S[j] = x, otherwise
returns 0.

? T = [7,2,3,5]; S = Set(T); ? setsearch(S, 2) %2 = 1 ? setsearch(S, 4) \\ not found %3 = 0 ? setsearch(T, 7) \\ search in a randomly sorted vector %4 = 0 \\ WRONG !

If S is not a set, we also allow sorted lists with
respect to the `cmp`

sorting function, without repeated entries,
as per `listsort`

(L,1); otherwise the result is undefined.

? L = List([1,4,2,3,2]); setsearch(L, 4) %1 = 0 \\ WRONG ! ? listsort(L, 1); L \\ sort L first %2 = List([1, 2, 3, 4]) ? setsearch(L, 4) %3 = 4 \\ now correct

If *flag* is nonzero, this function returns the index j where x should be
inserted, and 0 if it already belongs to S. This is meant to be used for
dynamically growing (sorted) lists, in conjunction with `listinsert`

.

? L = List([1,5,2,3,2]); listsort(L,1); L %1 = List([1,2,3,5]) ? j = setsearch(L, 4, 1) \\ 4 should have been inserted at index j %2 = 4 ? listinsert(L, 4, j); L %3 = List([1, 2, 3, 4, 5])

The library syntax is `long `

.**setsearch**(GEN S, GEN x, long flag)

Union of the two sets x and y (see `setisset`

).
If x or y is not a set, the result is undefined.

The library syntax is `GEN `

.**setunion**(GEN x, GEN y)

Assuming that D is a Smith normal form
(i.e. vector of elementary divisors) for some module and q a power of an
irreducible element or 0, returns the minimal number of generators for
D/qD. For instance, if q = p^{n} where p is a prime number, this is the
dimension of (p^{n-1}D)/p^{n}D as an 𝔽_{p}-vector space. An argument q = 0
may be omitted.

? snfrank([4,4,2], 2) %1 = 3 ? snfrank([4,4,2], 4) %2 = 2 ? snfrank([4,4,2], 8) %3 = 0 ? snfrank([4,4,2]) \\ or snfrank([4,4,2], 0) %4 = 3

The function also works for K[x]-modules:

? D=matsnf([-x-5,-1,-1,0; 0,x^2+10*x+26,-1,-x-5; 1,-x-5,-x-5,1; -1,0,0,1]); ? snfrank(D, x^2 + 10*x + 27) %6 = 2 ? A=matdiagonal([x-1,x^2+1,x-1,(x^2+1)^2,x,(x-1)^2]); D=matsnf(A); ? snfrank(D,x-1) %8 = 3 ? snfrank(D,(x-1)^2) %9 = 1 ? snfrank(D,(x-1)^3) %9 = 0 ? snfrank(D,x^2+1) %10 = 2

Finally this function supports any output from `matsnf`

(e.g., with transformation matrices included, with or without cleanup).

The library syntax is `long `

.**snfrank**(GEN D, GEN q = NULL)

This applies to quite general x. If x is not a matrix, it is equal to the sum of x and its conjugate, except for polmods where it is the trace as an algebraic number.

For x a square matrix, it is the ordinary trace. If x is a nonsquare matrix (but not a vector), an error occurs.

The library syntax is `GEN `

.**gtrace**(GEN x)

Extraction of components of the vector or matrix x according to y.
In case x is a matrix, its components are the *columns* of x. The
parameter y is a component specifier, which is either an integer, a string
describing a range, or a vector.

If y is an integer, it is considered as a mask: the binary bits of y are
read from right to left, but correspond to taking the components from left to
right. For example, if y = 13 = (1101)_{2} then the components 1,3 and 4 are
extracted.

If y is a vector (`t_VEC`

, `t_COL`

or `t_VECSMALL`

), which must have
integer entries, these entries correspond to the component numbers to be
extracted, in the order specified.

If y is a string, it can be

***** a single (nonzero) index giving a component number (a negative
index means we start counting from the end).

***** a range of the form `"a..b"`

, where a and b are
indexes as above. Any of a and b can be omitted; in this case, we take
as default values a = 1 and b = -1, i.e. the first and last components
respectively. We then extract all components in the interval [a,b], in
reverse order if b < a.

In addition, if the first character in the string is `^`

, the
complement of the given set of indices is taken.

If z is not omitted, x must be a matrix. y is then the *row*
specifier, and z the *column* specifier, where the component specifier
is as explained above.

? v = [a, b, c, d, e]; ? vecextract(v, 5) \\ mask %1 = [a, c] ? vecextract(v, [4, 2, 1]) \\ component list %2 = [d, b, a] ? vecextract(v, "2..4") \\ interval %3 = [b, c, d] ? vecextract(v, "-1..-3") \\ interval + reverse order %4 = [e, d, c] ? vecextract(v, "^2") \\ complement %5 = [a, c, d, e] ? vecextract(matid(3), "2..", "..") %6 = [0 1 0] [0 0 1]

The range notations `v[i..j]`

and `v[^i]`

(for `t_VEC`

or
`t_COL`

) and `M[i..j, k..l]`

and friends (for `t_MAT`

) implement a
subset of the above, in a simpler and *faster* way, hence should be
preferred in most common situations. The following features are not
implemented in the range notation:

***** reverse order,

***** omitting either a or b in `a..b`

.

The library syntax is `GEN `

.**extract0**(GEN x, GEN y, GEN z = NULL)

Return the product of the components of the vector v. Return 1 on an empty vector.

? vecprod([1,2,3]) %1 = 6 ? vecprod([]) %2 = 1

The library syntax is `GEN `

.**vecprod**(GEN v)

Determines whether x belongs to the sorted vector or list v: return the (positive) index where x was found, or 0 if it does not belong to v.

If the comparison function cmpf is omitted, we assume that v is sorted in
increasing order, according to the standard comparison function `lex`

,
thereby restricting the possible types for x and the elements of v
(integers, fractions, reals, and vectors of such). We also transparently
allow a `t_VECSMALL`

x in this case, for the natural ordering of the
integers.

If `cmpf`

is present, it is understood as a comparison function and we
assume that v is sorted according to it, see `vecsort`

for how to
encode comparison functions.

? v = [1,3,4,5,7]; ? vecsearch(v, 3) %2 = 2 ? vecsearch(v, 6) %3 = 0 \\ not in the list ? vecsearch([7,6,5], 5) \\ unsorted vector: result undefined %4 = 0

Note that if we are sorting with respect to a key which is expensive to compute (e.g. a discriminant), one should rather precompute all keys, sort that vector and search in the vector of keys, rather than searching in the original vector with respect to a comparison function.

By abuse of notation, x is also allowed to be a matrix, seen as a vector
of its columns; again by abuse of notation, a `t_VEC`

is considered
as part of the matrix, if its transpose is one of the matrix columns.

? v = vecsort([3,0,2; 1,0,2]) \\ sort matrix columns according to lex order %1 = [0 2 3] [0 2 1] ? vecsearch(v, [3,1]~) %2 = 3 ? vecsearch(v, [3,1]) \\ can search for x or x~ %3 = 3 ? vecsearch(v, [1,2]) %4 = 0 \\ not in the list

The library syntax is `long `

.**vecsearch**(GEN v, GEN x, GEN cmpf = NULL)

Sorts the vector x in ascending order, using a mergesort method. x must be a list, vector or matrix (seen as a vector of its columns). Note that mergesort is stable, hence the initial ordering of "equal" entries (with respect to the sorting criterion) is not changed.

If `cmpf`

is omitted, we use the standard comparison function
`lex`

, thereby restricting the possible types for the elements of x
(integers, fractions or reals and vectors of those). We also transparently
allow a `t_VECSMALL`

x in this case, for the standard ordering on the
integers.

If `cmpf`

is present, it is understood as a comparison function and we
sort according to it. The following possibilities exist:

***** an integer k: sort according to the value of the k-th
subcomponents of the components of x.

***** a vector: sort lexicographically according to the components listed in
the vector. For example, if `cmpf`

= `[2,1,3]`

, sort with respect to
the second component, and when these are equal, with respect to the first,
and when these are equal, with respect to the third.

***** a comparison function: `t_CLOSURE`

with two arguments x and y,
and returning a real number which is < 0, > 0 or = 0 if x < y, x > y or
x = y respectively.

***** a key: `t_CLOSURE`

with one argument x and returning
the value f(x) with respect to which we sort.

? vecsort([3,0,2; 1,0,2]) \\ sort columns according to lex order %1 = [0 2 3] [0 2 1] ? vecsort(v, (x,y)->y-x) \\ reverse sort ? vecsort(v, (x,y)->abs(x)-abs(y)) \\ sort by increasing absolute value ? vecsort(v, abs) \\ sort by increasing absolute value, using key ? cmpf(x,y) = my(dx = poldisc(x), dy = poldisc(y)); abs(dx) - abs(dy); ? v = [x^2+1, x^3-2, x^4+5*x+1] vecsort(v, cmpf) \\ comparison function ? vecsort(v, x->abs(poldisc(x))) \\ key

The `abs`

and `cmpf`

examples show how to use a named function
instead of an anonymous function. It is preferable to use a *key*
whenever possible rather than include it in the comparison function as above
since the key is evaluated O(n) times instead of O(nlog n),
where n is the number of entries.

A direct approach is also possible and equivalent to using a sorting key:

? T = [abs(poldisc(x)) | x<-v]; ? perm = vecsort(T,,1); \\ indirect sort ? vecextract(v, perm)

This also provides the vector T of all keys, which is
interesting for instance in later `vecsearch`

calls: it is more
efficient to sort T (`T = vecextract(T, perm)`

) then search for a key
in T rather than to search in v using a comparison function or a key.
Note also that `mapisdefined`

is often easier to use and faster than
`vecsearch`

.

The binary digits of *flag* mean:

***** 1: indirect sorting of the vector x, i.e. if x is an
n-component vector, returns a permutation of [1,2,...,n] which
applied to the components of x sorts x in increasing order.
For example, `vecextract(x, vecsort(x,,1))`

is equivalent to
`vecsort(x)`

.

***** 4: use descending instead of ascending order.

***** 8: remove "duplicate" entries with respect to the sorting function
(keep the first occurring entry). For example:

? vecsort([Pi,Mod(1,2),z], (x,y)->0, 8) \\ make everything compare equal %1 = [3.141592653589793238462643383] ? vecsort([[2,3],[0,1],[0,3]], 2, 8) %2 = [[0, 1], [2, 3]]

The library syntax is `GEN `

.**vecsort0**(GEN x, GEN cmpf = NULL, long flag)

Return the sum of the components of the vector v. Return 0 on an empty vector.

? vecsum([1,2,3]) %1 = 6 ? vecsum([]) %2 = 0

The library syntax is `GEN `

.**vecsum**(GEN v)

Creates a row vector (type
`t_VEC`

) with n components whose components are the expression
*expr* evaluated at the integer points between 1 and n. If the last
two arguments are omitted, fills the vector with zeroes.

? vector(3,i, 5*i) %1 = [5, 10, 15] ? vector(3) %2 = [0, 0, 0]

The variable X is lexically scoped to each evaluation of *expr*. Any
change to X within *expr* does not affect subsequent evaluations, it
still runs 1 to n. A local change allows for example different indexing:

vector(10, i, i=i-1; f(i)) \\ i = 0, ..., 9 vector(10, i, i=2*i; f(i)) \\ i = 2, 4, ..., 20

This per-element scope for X differs from `for`

loop evaluations,
as the following example shows:

n = 3 v = vector(n); vector(n, i, i++) — -> [2, 3, 4] v = vector(n); for (i = 1, n, v[i] = i++) — -> [2, 0, 4]

Creates a row vector of small integers (type `t_VECSMALL`

) with n
components whose components are the expression *expr* evaluated at the
integer points between 1 and n.

As `vector`

, but returns a column vector (type `t_COL`

).