cholfact(A)

cholfact(A; shift=0, perm=Int[]) -> CHOLMOD.Factor

Compute the Cholesky factorization of a sparse positive definite matrix A. A fill-reducing 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=1e-5);

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 matrix A 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 matrix A 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 matrix A 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.

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cholfact(A)

Examples

See Also

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