# ExtendableSparse.jl

Sparse matrix class with efficient successive insertion of entries and entry update, supporting general number types.

## Summary

The package allows for efficient assembly of a sparse matrix without the need to handle intermediate arrays:

using ExtendableSparse
A=ExtendableSparseMatrix(10,10)
A[1,1]=1
for i = 1:9
A[i + 1, i] += -1
A[i, i + 1] += -1
A[i, i] += 1
A[i + 1, i + 1] += 1
end
b=ones(10)
x=A\b


While one could replace here ExtendableSparseMatrix(10,10) by spzeros(10,10), the later is inefficient for large problems. Without this package, the general advise is to construct a sparse matrix via the COO format.

Instead of \, the methods from LinearSolve.jl can be used:

using LinearSolve
p=LinearProblem(A,b)
LinearSolve.solve(p)


With the help of Sparspak.jl, these examples work for general number types.

sparse(A) creates a standard SparseMatrixCSC from a filled ExtendableSparseMatrix which can be used e.g. to create preconditioners. So one can instead perform e.g.

LinearSolve.solve(p, KrylovJL_CG(); Pl = ILUZero.ilu0(sparse(A)))


## Rationale

Without an intermediate data structure, efficient successive insertion/update of possibly duplicate entries in random order into a standard compressed column storage structure appears to be not possible. The package introduces ExtendableSparseMatrix, a delegating wrapper containing a Julia standard SparseMatrixCSC struct for performing linear algebra operations and a SparseMatrixLNK struct realising a linked list based (but realised in vectors) format collecting new entries.

The later is modeled after the linked list sparse matrix format described in the whitepaper by Y. Saad. See also exercise P.3-16 in his book.

Any linear algebra method on ExtendableSparseMatrix starts with a flush! method which adds the LNK entries and the existing CSC entries into a new CSC struct and resets the LNK struct.

ExtendableSparseMatrix is aimed to work as a drop-in replacement to SparseMatrixCSC in finite element and finite volume codes especially in those cases where the sparsity structure is hard to detect a priori and where working with an intermediadte COO representation appears to be not convenient.

The package provides a \ method for ExtendableSparseMatrix which dispatches to Julia's standard \ method for SparseMatrixCSC where possible. It relies on Sparspak.jl, P.Krysl's Julia MIT licensed re-implementation of Sparspak by George & Liu for number types not supported by Julia's standard implementation. Notably, this includes ForwardDiff.Dual numbers, thus supporting for automatic differentiation. When used with a non-GPL version of the system image, \ is dispatched to Sparsepak.jl in all cases.

## Caveat

This package assumes that a $m \times n$ matrix is sparse if each row and each column have less than $C$ entries with $C << n$ and $C << m$ . Adding a full matrix row will be a performance hit.

## Working with ForwardDiff

In particular, it cooperates with ForwardDiff.jl when it comes to the assembly of a sparse jacobian. For a function 'f!(y,x)' returning it's result in a vector y, one can use e.g.

x=...
y=zeros(n)
dresult=DiffResults.DiffResult(zeros(n),ExtendableSparseMatrix(n,n))
x=ForwardDiff.jacobian!(dresult,f!,y,x)
jac=DiffResults.jacobian(dresult)
h=jac\x


However, without a priori information on sparsity, ForwardDiff calls element insertion for the full range of n^2 indices, leading to a O(n^2) scaling behavior due to the nevertheless necessary search operations, see this discourse thread.

## updateindex!

In addition, the package provides a method updateindex!(A,op,v,i,j) for both SparseMatrixCSC and for ExtendableSparse which allows to update a matrix element with one index search instead of two. It allows to replace e.g. A[i,j]+=v by updateindex!(A,+,v,i,j). The former operation is lowered to

%1 = Base.getindex(A, 1, 2)
%2 = %1 + 3
Base.setindex!(A, %2, 1, 2)


triggering two index searches, one for getindex! and another one for setindex!.

See Julia issue #15630 for a discussion on this.

## Factorizations and Preconditioners

The package provides a common API for factorizations and preconditioners supporting series of solutions of similar problem as they occur during nonlinear and transient solves. For details, see the corresponding documentation.

With the advent of LinearSolve.jl, this functionality probably will be reduced to some core cases.

### Interfaces to other packages

The package directly provides interfaces to other sparse matrix solvers and preconditioners. Dependencies on these packages are handeled via Requires.jl. Currently, support includes:

For a similar approach, see Preconditioners.jl

## Alternatives

You may also evaluate alternatives to this package: