cholfact(A)
cholfact(A; shift=0, perm=Int[]) > CHOLMOD.Factor
Compute the Cholesky factorization of a sparse positive definite matrix A
. A fillreducing permutation is used. F = cholfact(A)
is most frequently used to solve systems of equations with F\b
, but also the methods diag
, det
, 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 extact "combined" factors like PtL = F[:PtL]
(the equivalent of P'*L
) and LtP = F[:UP]
(the equivalent of L'*P
).
Setting optional shift
keyword argument computes the factorization of A+shift*I
instead of A
. If the perm
argument is nonempty, it should be a permutation of 1:size(A,1)
giving the ordering to use (instead of CHOLMOD's default AMD ordering).
The function calls the C library CHOLMOD and many other functions from the library are wrapped but not exported.
Examples
# Cholesky factorization of a matrix A
# Return type depends on the value of pivot
function cholfact(A, LU=:U, pivot=Val{false}; tol=1.0)
Examples:

Compute Cholesky factorization of a matrix:
julia> A = [4.0 12.0 16.0; 12.0 37.0 43.0; 16.0 43.0 98.0]; julia> F = cholfact(A); julia> F[:U] 3×3 UpperTriangular{Float64,Array{Float64,2}}: 2.0 6.0 8.0 ⋅ 1.0 5.0 ⋅ ⋅ 3.0

Compute Cholesky factorization with lower triangular matrix:
julia> A = [4.0 12.0 16.0; 12.0 37.0 43.0; 16.0 43.0 98.0]; julia> F = cholfact(A, :L); julia> F[:L] 3×3 LowerTriangular{Float64,Array{Float64,2}}: 2.0 ⋅ ⋅ 6.0 1.0 ⋅ 8.0 5.0 3.0

Compute Cholesky factorization with pivoting:
julia> A = [4.0 12.0 16.0; 12.0 37.0 43.0; 16.0 43.0 98.0]; julia> F = cholfact(A, pivot=Val{true}); julia> F[:U] 3×3 UpperTriangular{Float64,Array{Float64,2}}: 9.89949 1.21268 1.61696 ⋅ 6.26343 0.225991 ⋅ ⋅ 9.88671
 Compute Cholesky factorization with custom tolerance:
julia> A = [4.0 12.0 16.0; 12.0 37.0 43.0; 16.0 43.0 98.0]; julia> F = cholfact(A, tol=1e5);
Additional Functions for Cholesky
Objects:

size
: Get the size of the Cholesky factorization.julia> F = cholfact(A); julia> size(F) (3, 3)

inv
: Compute the inverse of the original matrixA
using the Cholesky factorization.julia> F = cholfact(A); julia> inv(F) 3×3 Array{Float64,2}: 0.25 0.666667 0.666667 0.666667 0.777778 0.259259 0.666667 0.259259 0.925926
det
: Compute the determinant of the original matrixA
using the Cholesky factorization.julia> F = cholfact(A); julia> det(F) 454.9999999999999
Additional Functions for CholeskyPivoted
Objects:
rank
: Compute the rank of the original matrixA
using the Cholesky factorization with pivoting.julia> F = cholfact(A, pivot=Val{true}); julia> rank(F) 3
Common Mistake:
julia> A = [1 2; 2 1];
julia> F = cholfact(A);
ERROR: PosDefException: matrix is not positive definite
In this example, the matrix A
is not positive definite, but the cholfact
function is called with the default pivot=Val{false}
argument. If the matrix is not positive definite, a PosDefException
exception is thrown. Make sure to check the positive definiteness of the matrix before using cholfact
.
See Also
User Contributed Notes
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