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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 $\{ (x_i)_i \in R^{\mathbb{N}} \mid x_i = 0 \text{ for almost all }i\}$. In particular any two sparse rows over the same base ring can be added.

Creation

sparse_row — Method
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 $R$.

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sparse_row — Method
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 $J = (i_j, x_j)_j$.

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sparse_row — Method
sparse_row(R::NCRing, 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 $V = (x_j)_j$.

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Basic operations

Rows support the usual operations:

  • +, -, ==
  • multiplication by scalars
  • div, divexact
getindex — Method
getindex(A::SRow, j::Int) -> RingElem

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

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

Returns the row $c A + B$.

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

Returns the row $c A + B$.

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

Returns the tuple $(aA + bB, cA + dB)$.

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length — Method
length(A::SRow)

Returns the number of nonzero entries of $A$.

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Change of base ring

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

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

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Maximum, minimum and 2-norm

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

Returns the largest entry of $A$.

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

Returns the largest entry of $A$.

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

Returns the smallest entry of $A$.

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  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 $A$.

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

Returns the smallest entry of $A$.

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norm2 — Method
norm2(A::SRow{T} -> T

Returns $A \cdot A^t$.

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Functionality for integral sparse rows

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

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

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

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

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

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

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

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

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

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

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Conversion to/from julia and AbstractAlgebra types

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

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

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sparse_row — Method
sparse_row(A::MatElem)

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

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

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

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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 $\mathrm{Mat}_{n\times m}(R)$. 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
sparse_matrix(R::Ring) -> SMat

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

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

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

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

sparse_matrix — Method
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 $A$.

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

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

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

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

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The normal way however, is to add rows:

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

Appends the sparse row B to A.

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Sparse matrices can also be concatenated to form larger ones:

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

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

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

Vertically joins $A$ and $B$.

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

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

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

Horizontally concatenates $A$ and $B$.

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(Normal julia $cat$ is also supported)

There are special constructors:

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

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

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

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

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

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

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block_diagonal_matrix — Method
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.

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Slices:

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

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

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Transpose:

transpose — Method
transpose(A::SMat) -> SMat

Returns the transpose of $A$.

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Elementary Properties

sparsity — Method
sparsity(A::SMat) -> Float64

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

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density — Method
density(A::SMat) -> Float64

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

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nnz — Method
nnz(A::SMat) -> Int

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

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isone — Method
isone(A::SMat)

Tests if $A$ is an identity matrix.

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iszero — Method
iszero(A::SMat)

Tests if $A$ is a zero matrix.

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

Finds the largest entry of $A$.

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

Finds the smallest entry of $A$.

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

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

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

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

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solve_dixon_sf — Method
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. $x A = bd$ 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.

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

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

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

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

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reduce_full — Method
reduce_full(A::SMat{ZZRingElem}, g::SRow{ZZRingElem},
                      with_transform = 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 with_transform is set to Val(true), then additionally an array of transformations is returned.

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hnf! — Method
hnf!(A::SMat{ZZRingElem})

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

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

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

\[truncate\]

indicates if zero rows should be returned or disgarded, when $full_hnf$ is set to $false$ only an upper triangular matrix is computed, ie. the upwards reduction is not performed.

\[auto\]

if set to true selects a different elemination strategy - here the upwards reduction is temporarily switched off.

\[limit\]

marks the last column where size reduction happens, so calling $hnf$ on $hcat(A, identity_matrix(SMat, ZZ, nrows(A)))$ with $limit$ set to $ncols(A)$ should produce a non-reduced transformation matrix in the right. If $limit$ is not set, the transformation matrix will also be partly triangular.

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snf — Method
snf(A::SMat{ZZRingElem})

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

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hnf_extend! — Method
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 $b$.

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is_diagonal — Method
is_diagonal(A::SMat) -> Bool

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

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det — Method
det(A::SMat{ZZRingElem})

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

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det_mc — Method
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 $p$.

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valence_mc — Method
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 $transpose(A)*A$, 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.

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

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

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

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

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

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diagonal_form — Method
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.

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Manipulation/ Access

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

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

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

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

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setindex! — Method
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 $b$.

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

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

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

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

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

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

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add_scaled_col! — Method
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, $A_j \rightarrow A_j + c \cdot A_i$, where $(A_i)_i$ denote the columns of $A$.

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add_scaled_row! — Method
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, $A_j \rightarrow A_j + c \cdot A_i$, where $(A_i)_i$ denote the rows of $A$.

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

Applies the transformation $(A_i, A_j) \rightarrow (aA_i + bA_j, cA_i + dA_j)$ to $A$, where $(A_i)_i$ are the rows of $A$.

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

The diagonal elements of $A$ in an array.

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

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

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find_row_starting_with — Method
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.

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

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.

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reduce — Method
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 $A$.

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reduce — Method
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 $A$.

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rand_row — Method
rand_row(A::SMat) -> SRow

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

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Changing of the ring:

map_entries — Method
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 $A$.

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

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

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Arithmetic

Matrices support the usual operations as well

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

Various products:

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

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

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

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

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

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

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

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

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mul_sparse — Method
mul_sparse(A::SMat{T}, B::SMat{T}) -> SMat{T}

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

The product of two sparse matrices is, in general, not sparse, so depending on the context, it might be more efficient to use mul_dense(::SMat{T}, ::SMat{T}) where {T} instead.

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mul_dense — Method
mul_dense(A::SMat{T}, B::SMat{T}) -> MatElem{T}

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

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

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

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

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

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

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

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Other:

sparse — Method
sparse(A::SMat) -> SparseMatrixCSC

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

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ZZMatrix — Method
ZZMatrix(A::SMat{ZZRingElem})

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

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ZZMatrix — Method
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 $\mathbb ZZ$.

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

The same matrix, but as a julia matrix.

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

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

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