Dory.jl Documentation

Dory is a package to extend the functionality of Hecke.jl.

The main features are:

Convenience functions, such as various matrix get/set index methods for matrices supporting various types.
Generic eigenvector methods.
p-adic Linear algebra.

Broadcasting

Broadcasting is enabled for the AbstractAlgebra.Generic.MatElem{T} type. the syntax is your_function.(A).

The return type is either of type AbstractAlgebra.Generic.MatElem{T} or an array if the output type of your_function is not a subtype of NCRingElem.

pAdic linear algebra.

Dory.padic_qr — Function.

padic_qr(A :: Hecke.Generic.MatElem{padic} ; col_pivot :: Union{Val{true},Val{false}}) --> F :: QRPadicPivoted

The return type of F is a QRPadicPivoted, with fields F.Q, F.R, F.p, F.q described below.

Compute the p-adic QR factorization of A. More precisely, compute matrices Q,R, and an arrays p, q such that

A[F.p,F.q] = Q*R

If col_pivot=Val(false), then F.q = [1,2,...,size(A,2)].

#–––––––––-

INPUTS: A – a matrix over Qp col_pivot – a type, either Val(true) or Val(false), indicating whether column permutations should be used to move p-adically large entries to the pivot position.

source

padic_qr(A; col_pivot) --> F

The return type of F is a QRPadicPivoted, with fields F.Q, F.R, F.p, F.q described below.

Compute the p-adic QR factorization of A. More precisely, compute matrices Q,R, and an arrays p, q such that

A[F.p,F.q] = Q*R

If col_pivot=Val(false), then F.q = [1,2,...,size(A,2)].

#–––––––––-

INPUTS: A – a matrix over Qp, A::Hecke.Generic.MatElem{padic} col_pivot – a type, either Val(true) or Val(false), indicating whether column permutations should be used to move p-adically large entries to the pivot position.

source

Dory.rectangular_solve — Function.

rectangular_solve(A::Hecke.MatElem{T}, b::Hecke.MatElem{T}; suppress_error=false) where T 
                                                                                --> x ::Hecke.MatElem{T}

Solve the possibly overdetermined linear equation Ax = b. If no solution exists returns an error. Generally not intended for use.

WARNINGS:

This function is very unsafe. It does not do basic sanity checks and will fail if the top nxn block is singular.

source

rectangular_solve(A::Hecke.MatElem{padic}, b_input::Hecke.MatElem{padic}; stable::Bool=false)
                                                                    --> (nu :: Int64,N::Hecke.MatElem{padic})

Solves the linear system A*N = b. The output nu is the dimension of the nullspace. Parameter stable determines whether padic_qr or svd method is used. Default is qr (for speed).

WARNINGS: If A,b_input have different precisions, maximal precision output is not guarenteed. Underdetermined solve not implemented.

source

Dory.eigen — Function.

eigen(A::nmod_mat)

Computes the Eigenvalue decomposition of A. Requires factorization of polynomials implemented over the base ring.

(Depreciated. eigspaces is better to use.)

source

Dory.eigvecs — Function.

eigvecs( A :: Hecke.Generic.MatElem{T}) where T -> A :: Hecke.Generic.MatElem{T}

Return a matrix M whose columns are the eigenvectors of A. (The kth eigenvector can be obtained from the slice M[:, k].)

source

Dory.eigspaces — Function.

eigspaces(A::Hecke.Generic.MatElem{T}) where T --> EigenSpaceDec{T}

Computes the eigen spaces of a generic matrix, and returns a list of matrices whose columns are generators for the eigen spaces.

source

Dory.singular_values — Function.

singular_values(A::Hecke.MatElem{padic}) -> Array{padic, 1}

Returns the list of diagonal elements in the singular value decomposition of the matrix A.

source