Sparse linear algebra

Introduction

This chapter deals with sparse linear algebra over commutative rings and fields.

Sparse linear algebra, that is, linear algebra with sparse matrices, plays an important role in various algorithms in algebraic number theory. For example, it is one of the key ingredients in the computation of class groups and discrete logarithms using index calculus methods.

Sparse rows

Building blocks for sparse matrices are sparse rows, which are modelled by objects of type SRow. More precisely, the type is of parametrized form SRow{T}, where T is the element type of the base ring $R$ . For example, SRow{ZZRingElem} is the type for sparse rows over the integers.

It is important to note that sparse rows do not have a fixed number of columns, that is, they represent elements of ${(x_{i})_{i} \in R^{N} ∣ x_{i} = 0 for almost all i}$ . In particular any two sparse rows over the same base ring can be added.

Creation

# sparse_row — Method.

julia

sparse_row(R::Ring, J::Vector{Tuple{Int, T}}) -> SRow{T}

Constructs the sparse row $(a_{i})_{i}$ with $a_{i_{j}} = x_{j}$ , where $J = (i_{j}, x_{j})_{j}$ . The elements $x_{i}$ must belong to the ring $R$ .

# sparse_row — Method.

julia

sparse_row(R::Ring, J::Vector{Tuple{Int, T}}) -> SRow{T}

Constructs the sparse row $(a_{i})_{i}$ with $a_{i_{j}} = x_{j}$ , where $J = (i_{j}, x_{j})_{j}$ . The elements $x_{i}$ must belong to the ring $R$ .

julia

sparse_row(R::Ring, J::Vector{Tuple{Int, Int}}) -> SRow

Constructs the sparse row $(a_{i})_{i}$ over $R$ with $a_{i_{j}} = x_{j}$ , where $J = (i_{j}, x_{j})_{j}$ .

# sparse_row — Method.

julia

sparse_row(R::Ring, J::Vector{Int}, V::Vector{T}) -> SRow{T}

Constructs the sparse row $(a_{i})_{i}$ over $R$ with $a_{i_{j}} = x_{j}$ , where $J = (i_{j})_{j}$ and $V = (x_{j})_{j}$ .

Basic operations

Rows support the usual operations:

+, -, ==
multiplication by scalars
div, divexact

# getindex — Method.

julia

getindex(A::SRow, j::Int) -> RingElem

Given a sparse row $(a_{i})_{i}$ and an index $j$ return $a_{j}$ .

# add_scaled_row — Method.

julia

add_scaled_row(A::SRow{T}, B::SRow{T}, c::T) -> SRow{T}

Returns the row $c A + B$ .

# add_scaled_row — Method.

julia

add_scaled_row(A::SRow{T}, B::SRow{T}, c::T) -> SRow{T}

Returns the row $c A + B$ .

# transform_row — Method.

julia

transform_row(A::SRow{T}, B::SRow{T}, i::Int, j::Int, a::T, b::T, c::T, d::T)

Returns the tuple $(a A + b B, c A + d B)$ .

# length — Method.

julia

length(A::SRow)

Returns the number of nonzero entries of $A$ .

Change of base ring

# change_base_ring — Method.

julia

change_base_ring(R::Ring, A::SRow) -> SRow

Create a new sparse row by coercing all elements into the ring $R$ .

Maximum, minimum and 2-norm

# maximum — Method.

julia

maximum(A::SRow{T}) -> T

Returns the largest entry of $A$ .

# maximum — Method.

julia

maximum(A::SRow{T}) -> T

Returns the largest entry of $A$ .

# minimum — Method.

julia

minimum(A::SRow{T}) -> T

Returns the smallest entry of $A$ .

julia

  minimum(A::RelNumFieldOrderIdeal) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
  minimum(A::RelNumFieldOrderIdeal) -> RelNumFieldOrderIdeal

Returns the ideal $A \cap O$ where $O$ is the maximal order of the coefficient ideals of $A$ .

# minimum — Method.

julia

minimum(A::SRow{T}) -> T

Returns the smallest entry of $A$ .

# norm2 — Method.

julia

norm2(A::SRow{T} -> T

Returns $A \cdot A^{t}$ .

Functionality for integral sparse rows

# lift — Method.

julia

lift(A::SRow{zzModRingElem}) -> SRow{ZZRingElem}

Return the sparse row obtained by lifting all entries in $A$ .

# mod! — Method.

julia

mod!(A::SRow{ZZRingElem}, n::ZZRingElem) -> SRow{ZZRingElem}

Inplace reduction of all entries of $A$ modulo $n$ to the positive residue system.

# mod_sym! — Method.

julia

mod_sym!(A::SRow{ZZRingElem}, n::ZZRingElem) -> SRow{ZZRingElem}

Inplace reduction of all entries of $A$ modulo $n$ to the symmetric residue system.

# mod_sym! — Method.

julia

mod_sym!(A::SRow{ZZRingElem}, n::Integer) -> SRow{ZZRingElem}

Inplace reduction of all entries of $A$ modulo $n$ to the symmetric residue system.

# maximum — Method.

julia

maximum(abs, A::SRow{ZZRingElem}) -> ZZRingElem

Returns the largest, in absolute value, entry of $A$ .

Conversion to/from julia and AbstractAlgebra types

# Vector — Method.

julia

Vector(a::SMat{T}, n::Int) -> Vector{T}

The first n entries of a, as a julia vector.

# sparse_row — Method.

julia

sparse_row(A::MatElem)

Convert A to a sparse row. nrows(A) == 1 must hold.

# dense_row — Method.

julia

dense_row(r::SRow, n::Int)

Convert r[1:n] to a dense row, that is an AbstractAlgebra matrix.

Sparse matrices

Let $R$ be a commutative ring. Sparse matrices with base ring $R$ are modelled by objects of type SMat. More precisely, the type is of parametrized form SRow{T}, where T is the element type of the base ring. For example, SMat{ZZRingElem} is the type for sparse matrices over the integers.

In contrast to sparse rows, sparse matrices have a fixed number of rows and columns, that is, they represent elements of the matrices space ${Mat}_{n \times m} (R)$ . Internally, sparse matrices are implemented as an array of sparse rows. As a consequence, unlike their dense counterparts, sparse matrices have a mutable number of rows and it is very performant to add additional rows.

Construction

# sparse_matrix — Method.

julia

sparse_matrix(R::Ring) -> SMat

Return an empty sparse matrix with base ring $R$ .

# sparse_matrix — Method.

julia

sparse_matrix(R::Ring, n::Int, m::Int) -> SMat

Return a sparse $n$ times $m$ zero matrix over $R$ .

Sparse matrices can also be created from dense matrices as well as from julia arrays:

# sparse_matrix — Method.

julia

sparse_matrix(A::MatElem; keepzrows::Bool = true)

Constructs the sparse matrix corresponding to the dense matrix $A$ . If keepzrows is false, then the constructor will drop any zero row of $A$ .

# sparse_matrix — Method.

julia

sparse_matrix(R::Ring, A::Matrix{T}) -> SMat

Constructs the sparse matrix over $R$ corresponding to $A$ .

# sparse_matrix — Method.

julia

sparse_matrix(R::Ring, A::Matrix{T}) -> SMat

Constructs the sparse matrix over $R$ corresponding to $A$ .

The normal way however, is to add rows:

# push! — Method.

julia

push!(A::SMat{T}, B::SRow{T}) where T

Appends the sparse row B to A.

Sparse matrices can also be concatenated to form larger ones:

# vcat! — Method.

julia

vcat!(A::SMat, B::SMat) -> SMat

Vertically joins $A$ and $B$ inplace, that is, the rows of $B$ are appended to $A$ .

# vcat — Method.

julia

vcat(A::SMat, B::SMat) -> SMat

Vertically joins $A$ and $B$ .

# hcat! — Method.

julia

hcat!(A::SMat, B::SMat) -> SMat

Horizontally concatenates $A$ and $B$ , inplace, changing $A$ .

# hcat — Method.

julia

hcat(A::SMat, B::SMat) -> SMat

Horizontally concatenates $A$ and $B$ .

(Normal julia $c a t$ is also supported)

There are special constructors:

# identity_matrix — Method.

julia

identity_matrix(::Type{SMat}, R::Ring, n::Int)

Return a sparse $n$ times $n$ identity matrix over $R$ .

# zero_matrix — Method.

julia

zero_matrix(::Type{SMat}, R::Ring, n::Int)

Return a sparse $n$ times $n$ zero matrix over $R$ .

# zero_matrix — Method.

julia

zero_matrix(::Type{SMat}, R::Ring, n::Int, m::Int)

Return a sparse $n$ times $m$ zero matrix over $R$ .

# block_diagonal_matrix — Method.

julia

block_diagonal_matrix(xs::Vector{SMat})

Return the block diagonal matrix with the matrices in xs on the diagonal. Requires all blocks to have the same base ring.

Slices:

# sub — Method.

julia

sub(A::SMat, r::AbstractUnitRange, c::AbstractUnitRange) -> SMat

Return the submatrix of $A$ , where the rows correspond to $r$ and the columns correspond to $c$ .

Transpose:

# transpose — Method.

julia

transpose(A::SMat) -> SMat

Returns the transpose of $A$ .

Elementary Properties

# sparsity — Method.

julia

sparsity(A::SMat) -> Float64

Return the sparsity of A, that is, the number of zero-valued elements divided by the number of all elements.

# density — Method.

julia

density(A::SMat) -> Float64

Return the density of A, that is, the number of nonzero-valued elements divided by the number of all elements.

# nnz — Method.

julia

nnz(A::SMat) -> Int

Return the number of non-zero entries of $A$ .

# number_of_rows — Method.

julia

number_of_rows(A::SMat) -> Int

Return the number of rows of $A$ .

# number_of_columns — Method.

julia

number_of_columns(A::SMat) -> Int

Return the number of columns of $A$ .

# isone — Method.

julia

isone(A::SMat)

Tests if $A$ is an identity matrix.

# iszero — Method.

julia

iszero(A::SMat)

Tests if $A$ is a zero matrix.

# is_upper_triangular — Method.

julia

is_upper_triangular(A::SMat)

Returns true if and only if $A$ is upper (right) triangular.

# maximum — Method.

julia

maximum(A::SMat{T}) -> T

Finds the largest entry of $A$ .

# minimum — Method.

julia

minimum(A::SMat{T}) -> T

Finds the smallest entry of $A$ .

# maximum — Method.

julia

maximum(abs, A::SMat{ZZRingElem}) -> ZZRingElem

Finds the largest, in absolute value, entry of $A$ .

# elementary_divisors — Method.

julia

elementary_divisors(A::SMat{ZZRingElem}) -> Vector{ZZRingElem}

The elementary divisors of $A$ , i.e. the diagonal elements of the Smith normal form of $A$ .

# solve_dixon_sf — Method.

julia

solve_dixon_sf(A::SMat{ZZRingElem}, b::SRow{ZZRingElem}, is_int::Bool = false) -> SRow{ZZRingElem}, ZZRingElem
solve_dixon_sf(A::SMat{ZZRingElem}, B::SMat{ZZRingElem}, is_int::Bool = false) -> SMat{ZZRingElem}, ZZRingElem

For a sparse square matrix $A$ of full rank and a sparse matrix (row), find a sparse matrix (row) $x$ and an integer $d$ s.th. $x A = b d$ holds. The algorithm is a Dixon-based linear p-adic lifting method. If \code{is_int} is given, then $d$ is assumed to be $1$ . In this case rational reconstruction is avoided.

# hadamard_bound2 — Method.

julia

hadamard_bound2(A::SMat{T}) -> T

The square of the product of the norms of the rows of $A$ .

# echelon_with_transform — Method.

julia

echelon_with_transform(A::SMat{zzModRingElem}) -> SMat, SMat

Find a unimodular matrix $T$ and an upper-triangular $E$ s.th. $T A = E$ holds.

# reduce_full — Method.

julia

reduce_full(A::SMat{ZZRingElem}, g::SRow{ZZRingElem},
                      trafo = Val{false}) -> SRow{ZZRingElem}, Vector{Int}

Reduces $g$ modulo $A$ and assumes that $A$ is upper triangular.

The second return value is the array of pivot elements of $A$ that changed.

If trafo is set to Val{true}, then additionally an array of transformations is returned.

# hnf! — Method.

julia

hnf!(A::SMat{ZZRingElem})

Inplace transform of $A$ into upper right Hermite normal form.

# hnf — Method.

julia

hnf(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}

Return the upper right Hermite normal form of $A$ .

# snf — Method.

julia

snf(A::SMat{ZZRingElem})

The Smith normal form (snf) of $A$ .

# hnf_extend! — Method.

julia

hnf_extend!(A::SMat{ZZRingElem}, b::SMat{ZZRingElem}, offset::Int = 0) -> SMat{ZZRingElem}

Given a matrix $A$ in HNF, extend this to get the HNF of the concatenation with $b$ .

# is_diagonal — Method.

julia

is_diagonal(A::SMat) -> Bool

True iff only the i-th entry in the i-th row is non-zero.

# det — Method.

julia

det(A::SMat{ZZRingElem})

The determinant of $A$ using a modular algorithm. Uses the dense (zzModMatrix) determinant on $A$ for various primes $p$ .

# det_mc — Method.

julia

det_mc(A::SMat{ZZRingElem})

Computes the determinant of $A$ using a LasVegas style algorithm, i.e. the result is not proven to be correct. Uses the dense (zzModMatrix) determinant on $A$ for various primes $p$ .

# valence_mc — Method.

julia

valence_mc{T}(A::SMat{T}; extra_prime = 2, trans = Vector{SMatSLP_add_row{T}}()) -> T

Uses a Monte-Carlo algorithm to compute the valence of $A$ . The valence is the valence of the minimal polynomial $f$ of $t r a n s p o s e (A) * A$ , thus the last non-zero coefficient, typically $f (0)$ .

The valence is computed modulo various primes until the computation stabilises for extra_prime many.

trans, if given, is a SLP (straight-line-program) in GL(n, Z). Then the valence of trans * $A$ is computed instead.

# saturate — Method.

julia

saturate(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}

Computes the saturation of $A$ , that is, a basis for $Q \otimes M \meet Z^{n}$ , where $M$ is the row span of $A$ and $n$ the number of rows of $A$ .

Equivalently, return $T A$ for an invertible rational matrix $T$ , such that $T A$ is integral and the elementary divisors of $T A$ are all trivial.

# hnf_kannan_bachem — Method.

julia

hnf_kannan_bachem(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}

Compute the Hermite normal form of $A$ using the Kannan-Bachem algorithm.

# diagonal_form — Method.

julia

diagonal_form(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}

A matrix $D$ that is diagonal and obtained via unimodular row and column operations. Like a snf without the divisibility condition.

Manipulation/ Access

# getindex — Method.

julia

getindex(A::SMat, i::Int, j::Int)

Given a sparse matrix $A = (a_{i j})_{i, j}$ , return the entry $a_{i j}$ .

# getindex — Method.

julia

getindex(A::SMat, i::Int) -> SRow

Given a sparse matrix $A$ and an index $i$ , return the $i$ -th row of $A$ .

# setindex! — Method.

julia

setindex!(A::SMat, b::SRow, i::Int)

Given a sparse matrix $A$ , a sparse row $b$ and an index $i$ , set the $i$ -th row of $A$ equal to $b$ .

# swap_rows! — Method.

julia

swap_rows!(A::SMat{T}, i::Int, j::Int)

Swap the $i$ -th and $j$ -th row of $A$ inplace.

# swap_cols! — Method.

julia

swap_cols!(A::SMat, i::Int, j::Int)

Swap the $i$ -th and $j$ -th column of $A$ inplace.

# scale_row! — Method.

julia

scale_row!(A::SMat{T}, i::Int, c::T)

Multiply the $i$ -th row of $A$ by $c$ inplace.

# add_scaled_col! — Method.

julia

add_scaled_col!(A::SMat{T}, i::Int, j::Int, c::T)

Add $c$ times the $i$ -th column to the $j$ -th column of $A$ inplace, that is, $A_{j} \to A_{j} + c \cdot A_{i}$ , where $(A_{i})_{i}$ denote the columns of $A$ .

# add_scaled_row! — Method.

julia

add_scaled_row!(A::SMat{T}, i::Int, j::Int, c::T)

Add $c$ times the $i$ -th row to the $j$ -th row of $A$ inplace, that is, $A_{j} \to A_{j} + c \cdot A_{i}$ , where $(A_{i})_{i}$ denote the rows of $A$ .

# transform_row! — Method.

julia

transform_row!(A::SMat{T}, i::Int, j::Int, a::T, b::T, c::T, d::T)

Applies the transformation $(A_{i}, A_{j}) \to (a A_{i} + b A_{j}, c A_{i} + d A_{j})$ to $A$ , where $(A_{i})_{i}$ are the rows of $A$ .

# diagonal — Method.

julia

diagonal(A::SMat) -> ZZRingElem[]

The diagonal elements of $A$ in an array.

# reverse_rows! — Method.

julia

reverse_rows!(A::SMat)

Inplace inversion of the rows of $A$ .

# mod_sym! — Method.

julia

mod_sym!(A::SMat{ZZRingElem}, n::ZZRingElem)

Inplace reduction of all entries of $A$ modulo $n$ to the symmetric residue system.

# find_row_starting_with — Method.

julia

find_row_starting_with(A::SMat, p::Int) -> Int

Tries to find the index $i$ such that $A_{i, p} \neq 0$ and $A_{i, p - j} = 0$ for all $j > 1$ . It is assumed that $A$ is upper triangular. If such an index does not exist, find the smallest index larger.

# reduce — Method.

julia

reduce(A::SMat{ZZRingElem}, g::SRow{ZZRingElem}, m::ZZRingElem) -> SRow{ZZRingElem}

Given an upper triangular matrix $A$ over the integers, a sparse row $g$ and an integer $m$ , this function reduces $g$ modulo $A$ and returns $g$ modulo $m$ with respect to the symmetric residue system.

# reduce — Method.

julia

reduce(A::SMat{ZZRingElem}, g::SRow{ZZRingElem}) -> SRow{ZZRingElem}

Given an upper triangular matrix $A$ over a field and a sparse row $g$ , this function reduces $g$ modulo $A$ .

# reduce — Method.

julia

reduce(A::SMat{T}, g::SRow{T}) -> SRow{T}

Given an upper triangular matrix $A$ over a field and a sparse row $g$ , this function reduces $g$ modulo $A$ .

# rand_row — Method.

julia

rand_row(A::SMat) -> SRow

Return a random row of the sparse matrix $A$ .

Changing of the ring:

# map_entries — Method.

julia

map_entries(f, A::SMat) -> SMat

Given a sparse matrix $A$ and a callable object $f$ , this function will construct a new sparse matrix by applying $f$ to all elements of $A$ .

# change_base_ring — Method.

julia

change_base_ring(R::Ring, A::SMat)

Create a new sparse matrix by coercing all elements into the ring $R$ .

Arithmetic

Matrices support the usual operations as well

+, -, ==
div, divexact by scalars
multiplication by scalars

Various products:

# * — Method.

julia

*(A::SMat{T}, b::AbstractVector{T}) -> Vector{T}

Return the product $A \cdot b$ as a dense vector.

# * — Method.

julia

*(A::SMat{T}, b::AbstractMatrix{T}) -> Matrix{T}

Return the product $A \cdot b$ as a dense array.

# * — Method.

julia

*(A::SMat{T}, b::MatElem{T}) -> MatElem

Return the product $A \cdot b$ as a dense matrix.

# * — Method.

julia

*(A::SRow, B::SMat) -> SRow

Return the product $A \cdot B$ as a sparse row.

# dot — Method.

julia

dot(x::SRow{T}, A::SMat{T}, y::SRow{T}) where T -> T

Return the generalized dot product dot(x, A*y).

# dot — Method.

julia

dot(x::MatrixElem{T}, A::SMat{T}, y::MatrixElem{T}) where T -> T

Return the generalized dot product dot(x, A*y).

# dot — Method.

julia

dot(x::AbstractVector{T}, A::SMat{T}, y::AbstractVector{T}) where T -> T

Return the generalized dot product dot(x, A*y).

Other:

# sparse — Method.

julia

sparse(A::SMat) -> SparseMatrixCSC

The same matrix, but as a sparse matrix of julia type SparseMatrixCSC.

# ZZMatrix — Method.

julia

ZZMatrix(A::SMat{ZZRingElem})

The same matrix $A$ , but as an ZZMatrix.

# ZZMatrix — Method.

julia

ZZMatrix(A::SMat{T}) where {T <: Integer}

The same matrix $A$ , but as an ZZMatrix. Requires a conversion from the base ring of $A$ to $Z Z$ .

# Matrix — Method.

julia

Matrix(A::SMat{T}) -> Matrix{T}

The same matrix, but as a julia matrix.

# Array — Method.

julia

Array(A::SMat{T}) -> Matrix{T}

The same matrix, but as a two-dimensional julia array.