Sparse Linear Algebra (API)

cholesky(A::SparseMatrixCSC; shift = 0.0, check = true, perm = nothing) -> CHOLMOD.Factor

Compute the Cholesky factorization of a sparse positive definite matrix A. A must be a SparseMatrixCSC or a Symmetric/Hermitian view of a SparseMatrixCSC. Note that even if A doesn't have the type tag, it must still be symmetric or Hermitian. If perm is not given, a fill-reducing permutation is used. F = cholesky(A) is most frequently used to solve systems of equations with F\b, but also the methods diag, det, and logdet are defined for F. You can also extract individual factors from F, using F.L. However, since pivoting is on by default, the factorization is internally represented as A == P'*L*L'*P with a permutation matrix P; using just L without accounting for P will give incorrect answers. To include the effects of permutation, it's typically preferable to extract "combined" factors like PtL = F.PtL (the equivalent of P'*L) and LtP = F.UP (the equivalent of L'*P).

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

Setting the optional shift keyword argument computes the factorization of A+shift*I instead of A. If the perm argument is provided, it should be a permutation of 1:size(A,1) giving the ordering to use (instead of CHOLMOD's default AMD ordering).

Examples

In the following example, the fill-reducing permutation used is [3, 2, 1]. If perm is set to 1:3 to enforce no permutation, the number of nonzero elements in the factor is 6.

julia> A = [2 1 1; 1 2 0; 1 0 2]
3×3 Matrix{Int64}:
 2  1  1
 1  2  0
 1  0  2

julia> C = cholesky(sparse(A))
SparseArrays.CHOLMOD.Factor{Float64, Int64}
type:    LLt
method:  simplicial
maxnnz:  5
nnz:     5
success: true

julia> C.p
3-element Vector{Int64}:
 3
 2
 1

julia> L = sparse(C.L);

julia> Matrix(L)
3×3 Matrix{Float64}:
 1.41421   0.0       0.0
 0.0       1.41421   0.0
 0.707107  0.707107  1.0

julia> L * L' ≈ A[C.p, C.p]
true

julia> P = sparse(1:3, C.p, ones(3))
3×3 SparseMatrixCSC{Float64, Int64} with 3 stored entries:
  ⋅    ⋅   1.0
  ⋅   1.0   ⋅
 1.0   ⋅    ⋅

julia> P' * L * L' * P ≈ A
true

julia> C = cholesky(sparse(A), perm=1:3)
SparseArrays.CHOLMOD.Factor{Float64, Int64}
type:    LLt
method:  simplicial
maxnnz:  6
nnz:     6
success: true

julia> L = sparse(C.L);

julia> Matrix(L)
3×3 Matrix{Float64}:
 1.41421    0.0       0.0
 0.707107   1.22474   0.0
 0.707107  -0.408248  1.1547

julia> L * L' ≈ A
true

cholesky!(F::CHOLMOD.Factor, A::SparseMatrixCSC; shift = 0.0, check = true) -> CHOLMOD.Factor

Compute the Cholesky ($LL'$) factorization of A, reusing the symbolic factorization F. A must be a SparseMatrixCSC or a Symmetric/ Hermitian view of a SparseMatrixCSC. Note that even if A doesn't have the type tag, it must still be symmetric or Hermitian.

See also cholesky.

lowrankdowndate(F::CHOLMOD.Factor, C::AbstractArray) -> FF::CHOLMOD.Factor

Get an LDLt Factorization of A + C*C' given an LDLt or LLt factorization F of A.

The returned factor is always an LDLt factorization.

See also lowrankdowndate!, lowrankupdate, lowrankupdate!.

lowrankdowndate!(F::CHOLMOD.Factor, C::AbstractArray)

Update an LDLt or LLt Factorization F of A to a factorization of A - C*C'.

LLt factorizations are converted to LDLt.

See also lowrankdowndate, lowrankupdate, lowrankupdate!.

lowrankupdowndate!(F::CHOLMOD.Factor, C::Sparse, update::Cint)

Update an LDLt or LLt Factorization F of A to a factorization of A ± C*C'.

If sparsity preserving factorization is used, i.e. L*L' == P*A*P' then the new factor will be L*L' == P*A*P' + C'*C

update: Cint(1) for A + CC', Cint(0) for A - CC'

ldlt(A::SparseMatrixCSC; shift = 0.0, check = true, perm=nothing) -> CHOLMOD.Factor

Compute the $LDL'$ factorization of a sparse matrix A. A must be a SparseMatrixCSC or a Symmetric/Hermitian view of a SparseMatrixCSC. Note that even if A doesn't have the type tag, it must still be symmetric or Hermitian. A fill-reducing permutation is used. F = ldlt(A) is most frequently used to solve systems of equations A*x = b with F\b. The returned factorization object F also supports the methods diag, det, logdet, and inv. You can extract individual factors from F using F.L. However, since pivoting is on by default, the factorization is internally represented as A == P'*L*D*L'*P with a permutation matrix P; using just L without accounting for P will give incorrect answers. To include the effects of permutation, it is typically preferable to extract "combined" factors like PtL = F.PtL (the equivalent of P'*L) and LtP = F.UP (the equivalent of L'*P). The complete list of supported factors is :L, :PtL, :D, :UP, :U, :LD, :DU, :PtLD, :DUP.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

Setting the optional shift keyword argument computes the factorization of A+shift*I instead of A. If the perm argument is provided, it should be a permutation of 1:size(A,1) giving the ordering to use (instead of CHOLMOD's default AMD ordering).

qr(A::SparseMatrixCSC; tol=_default_tol(A), ordering=ORDERING_DEFAULT) -> QRSparse

Compute the QR factorization of a sparse matrix A. Fill-reducing row and column permutations are used such that F.R = F.Q'*A[F.prow,F.pcol]. The main application of this type is to solve least squares or underdetermined problems with \. The function calls the C library SPQR^[ACM933].

Examples

julia> A = sparse([1,2,3,4], [1,1,2,2], [1.0,1.0,1.0,1.0])
4×2 SparseMatrixCSC{Float64, Int64} with 4 stored entries:
 1.0   ⋅
 1.0   ⋅
  ⋅   1.0
  ⋅   1.0

julia> qr(A)
SparseArrays.SPQR.QRSparse{Float64, Int64}
Q factor:
4×4 SparseArrays.SPQR.QRSparseQ{Float64, Int64}
R factor:
2×2 SparseMatrixCSC{Float64, Int64} with 2 stored entries:
 -1.41421    ⋅
   ⋅       -1.41421
Row permutation:
4-element Vector{Int64}:
 1
 3
 4
 2
Column permutation:
2-element Vector{Int64}:
 1
 2

lu(A::AbstractSparseMatrixCSC; check = true, q = nothing, control = get_umfpack_control()) -> F::UmfpackLU

Compute the LU factorization of a sparse matrix A.

For sparse A with real or complex element type, the return type of F is UmfpackLU{Tv, Ti}, with Tv = Float64 or ComplexF64 respectively and Ti is an integer type (Int32 or Int64).

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

The permutation q can either be a permutation vector or nothing. If no permutation vector is provided or q is nothing, UMFPACK's default is used. If the permutation is not zero-based, a zero-based copy is made.

The control vector defaults to the Julia SparseArrays package's default configuration for UMFPACK (NB: this is modified from the UMFPACK defaults to disable iterative refinement), but can be changed by passing a vector of length UMFPACK_CONTROL, see the UMFPACK manual for possible configurations. For example to reenable iterative refinement:

umfpack_control = SparseArrays.UMFPACK.get_umfpack_control(Float64, Int64) # read Julia default configuration for a Float64 sparse matrix
SparseArrays.UMFPACK.show_umf_ctrl(umfpack_control) # optional - display values
umfpack_control[SparseArrays.UMFPACK.JL_UMFPACK_IRSTEP] = 2.0 # reenable iterative refinement (2 is UMFPACK default max iterative refinement steps)

Alu = lu(A; control = umfpack_control)
x = Alu \ b   # solve Ax = b, including UMFPACK iterative refinement

The individual components of the factorization F can be accessed by indexing:

The relation between F and A is

F.L*F.U == (F.Rs .* A)[F.p, F.q]

F further supports the following functions:

• \
• det

See also lu!