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Compute the Cholesky factorization of a sparse positive definite matrix ``A``. A fill-reducing permutation is used. The main application of this type is to solve systems of equations with ``\``, but also the methods ``diag``, ``det``, ``logdet`` are defined. The function calls the C library CHOLMOD and many other functions from the library are wrapped but not exported.
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Compute the Cholesky factorization of a sparse positive definite
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matrix ``A``. A fill-reducing permutation is used. ``F =
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cholfact(A)`` is most frequently used to solve systems of equations
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with ``F\b``, but also the methods ``diag``, ``det``, ``logdet``
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are defined for ``F``. You can also extract individual factors
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from ``F``, using ``F[:L]``. However, since pivoting is on by
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default, the factorization is internally represented as ``A ==
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P'*L*L'*P`` with a permutation matrix ``P``; using just ``L``
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without accounting for ``P`` will give incorrect answers. To
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include the effects of permutation, it's typically preferable to
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extact "combined" factors like ``PtL = F[:PtL]`` (the equivalent of
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``P'*L``) and ``LtP = F[:UP]`` (the equivalent of ``L'*P``).
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Setting optional ``shift`` keyword argument computes the factorization
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of ``A+shift*I`` instead of ``A``. If the ``perm`` argument is nonempty,
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it should be a permutation of `1:size(A,1)` giving the ordering to use
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(instead of CHOLMOD's default AMD ordering).
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The function calls the C library CHOLMOD and many other functions
``cholfact!`` is the same as :func:`cholfact`, but saves space by overwriting the input ``A``, instead of creating a copy. ``cholfact!`` can also reuse the symbolic factorization from a different matrix ``F`` with the same structure when used as: ``cholfact!(F::CholmodFactor, A)``.
@@ -121,13 +135,29 @@ Linear algebra functions in Julia are largely implemented by calling functions f
Compute the LDLt factorization of a sparse symmetric or Hermitian matrix ``A``. A fill-reducing permutation is used. The main application of this type is to solve systems of equations with ``\``, but also the methods ``diag``, ``det``, ``logdet`` are defined. The function calls the C library CHOLMOD and many other functions from the library are wrapped but not exported.
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Compute the LDLt factorization of a sparse symmetric or Hermitian
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matrix ``A``. A fill-reducing permutation is used. ``F =
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ldltfact(A)`` is most frequently used to solve systems of equations
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with ``F\b``, but also the methods ``diag``, ``det``, ``logdet``
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are defined for ``F``. You can also extract individual factors from
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``F``, using ``F[:L]``. However, since pivoting is on by default,
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the factorization is internally represented as ``A == P'*L*D*L'*P``
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with a permutation matrix ``P``; using just ``L`` without
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accounting for ``P`` will give incorrect answers. To include the
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effects of permutation, it's typically preferable to extact
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"combined" factors like ``PtL = F[:PtL]`` (the equivalent of
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``P'*L``) and ``LtP = F[:UP]`` (the equivalent of ``L'*P``). The
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complete list of supported factors is ``:L, :PtL, :D, :UP, :U, :LD,
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:DU, :PtLD, :DUP``.
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Setting optional ``shift`` keyword argument computes the factorization
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of ``A+shift*I`` instead of ``A``. If the ``perm`` argument is nonempty,
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it should be a permutation of `1:size(A,1)` giving the ordering to use
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(instead of CHOLMOD's default AMD ordering).
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The function calls the C library CHOLMOD and many other functions
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from the library are wrapped but not exported.
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.. function:: qr(A [,pivot=Val{false}][;thin=true]) -> Q, R, [p]
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Compute the (pivoted) QR factorization of ``A`` such that either ``A = Q*R`` or ``A[:,p] = Q*R``. Also see ``qrfact``. The default is to compute a thin factorization. Note that ``R`` is not extended with zeros when the full ``Q`` is requested.
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