Fix documentation for dense ldltfact and move ldltfact inline. [ci skip]

(cherry picked from commit b22d6f7)
ref #13030

Author: andreasnoack

base/docs/helpdb.jl

@@ -2555,22 +2555,15 @@ Right division operator: multiplication of `x` by the inverse of `y` on the righ
 Base.(:(/))
 
 doc"""
-    ldltfact(A) -> LDLtFactorization
+    ldltfact(::Union{SparseMatrixCSC,Symmetric{Float64,SparseMatrixCSC{Flaot64,SuiteSparse_long}},Hermitian{Complex{Float64},SparseMatrixCSC{Complex{Float64},SuiteSparse_long}}}; shift=0, perm=Int[]) -> CHOLMOD.Factor
 
-Compute a factorization of a positive definite matrix `A` such that `A=L*Diagonal(d)*L'` where `L` is a unit lower triangular matrix and `d` is a vector with non-negative elements.
-"""
-ldltfact(A)
-
-doc"""
-    ldltfact(A; shift=0, perm=Int[]) -> CHOLMOD.Factor
-
-Compute the LDLt factorization of a sparse symmetric or Hermitian matrix `A`. A fill-reducing permutation is used. `F = ldltfact(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*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'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`). The complete list of supported factors is `:L, :PtL, :D, :UP, :U, :LD, :DU, :PtLD, :DUP`.
+Compute the `LDLt` factorization of a sparse symmetric or Hermitian matrix. A fill-reducing permutation is used. `F = ldltfact(A)` is most frequently used to solve systems of equations `A*x = b` 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*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'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`). The complete list of supported factors is `:L, :PtL, :D, :UP, :U, :LD, :DU, :PtLD, :DUP`.
 
 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.
 """
-ldltfact(A::SparseMatrix.CHOLMOD.Sparse; shift=0, perm=Int[])
+ldltfact(A::SparseMatrixCSC; shift=0, perm=Int[])
 
 doc"""
     connect([host],port) -> TcpSocket
@@ -8955,7 +8948,7 @@ doc"""
 
 Matrix division using a polyalgorithm. For input matrices ``A`` and ``B``, the result ``X`` is such that ``A*X == B`` when ``A`` is square.  The solver that is used depends upon the structure of ``A``.  A direct solver is used for upper or lower triangular ``A``.  For Hermitian ``A`` (equivalent to symmetric ``A`` for non-complex ``A``) the ``BunchKaufman`` factorization is used.  Otherwise an LU factorization is used. For rectangular ``A`` the result is the minimum-norm least squares solution computed by a pivoted QR factorization of ``A`` and a rank estimate of ``A`` based on the R factor.
 
-When ``A`` is sparse, a similar polyalgorithm is used. For indefinite matrices, the LDLt factorization does not use pivoting during the numerical factorization and therefore the procedure can fail even for invertible matrices.
+When ``A`` is sparse, a similar polyalgorithm is used. For indefinite matrices, the ``LDLt`` factorization does not use pivoting during the numerical factorization and therefore the procedure can fail even for invertible matrices.
 ```
 """
 Base.(:(\))(A,B)

base/linalg/ldlt.jl

@@ -25,6 +25,13 @@ function ldltfact!{T<:Real}(S::SymTridiagonal{T})
     end
     return LDLt{T,SymTridiagonal{T}}(S)
 end
+
+doc"""
+    ldltfact(::SymTridiagonal) -> LDLt
+
+Compute an `LDLt` factorization of a real symmetric tridiagonal matrix such that `A = L*Diagonal(d)*L'` where `L` is a unit lower triangular matrix and `d` is a vector. The main use of an `LDLt` factorization `F = ldltfact(A)` is to solve the linear system of equations `Ax = b` with `F\b`.
+
+"""
 function ldltfact{T}(M::SymTridiagonal{T})
     S = typeof(zero(T)/one(T))
     return S == T ? ldltfact!(copy(M)) : ldltfact!(convert(SymTridiagonal{S}, M))

doc/stdlib/linalg.rst

@@ -25,7 +25,7 @@ Linear algebra functions in Julia are largely implemented by calling functions f
 
    Matrix division using a polyalgorithm. For input matrices ``A`` and ``B``, the result ``X`` is such that ``A*X == B`` when ``A`` is square.  The solver that is used depends upon the structure of ``A``.  A direct solver is used for upper or lower triangular ``A``.  For Hermitian ``A`` (equivalent to symmetric ``A`` for non-complex ``A``) the ``BunchKaufman`` factorization is used.  Otherwise an LU factorization is used. For rectangular ``A`` the result is the minimum-norm least squares solution computed by a pivoted QR factorization of ``A`` and a rank estimate of ``A`` based on the R factor.
 
-   When ``A`` is sparse, a similar polyalgorithm is used. For indefinite matrices, the LDLt factorization does not use pivoting during the numerical factorization and therefore the procedure can fail even for invertible matrices.
+   When ``A`` is sparse, a similar polyalgorithm is used. For indefinite matrices, the ``LDLt`` factorization does not use pivoting during the numerical factorization and therefore the procedure can fail even for invertible matrices.
 
 .. function:: dot(x, y)
               ⋅(x,y)
@@ -139,17 +139,17 @@ Linear algebra functions in Julia are largely implemented by calling functions f
 
    ``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)``.
 
-.. function:: ldltfact(A) -> LDLtFactorization
+.. function:: ldltfact(::SymTridiagonal) -> LDLt
 
    .. Docstring generated from Julia source
 
-   Compute a factorization of a positive definite matrix ``A`` such that ``A=L*Diagonal(d)*L'`` where ``L`` is a unit lower triangular matrix and ``d`` is a vector with non-negative elements.
+   Compute an ``LDLt`` factorization of a real symmetric tridiagonal matrix such that ``A = L*Diagonal(d)*L'`` where ``L`` is a unit lower triangular matrix and ``d`` is a vector. The main use of an ``LDLt`` factorization ``F = ldltfact(A)`` is to solve the linear system of equations ``Ax = b`` with ``F\b``\ .
 
-.. function:: ldltfact(A; shift=0, perm=Int[]) -> CHOLMOD.Factor
+.. function:: ldltfact(::Union{SparseMatrixCSC,Symmetric{Float64,SparseMatrixCSC{Flaot64,SuiteSparse_long}},Hermitian{Complex{Float64},SparseMatrixCSC{Complex{Float64},SuiteSparse_long}}}; shift=0, perm=Int[]) -> CHOLMOD.Factor
 
    .. Docstring generated from Julia source
 
-   Compute the LDLt factorization of a sparse symmetric or Hermitian matrix ``A``\ . A fill-reducing permutation is used. ``F = ldltfact(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*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'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``\ ). The complete list of supported factors is ``:L, :PtL, :D, :UP, :U, :LD, :DU, :PtLD, :DUP``\ .
+   Compute the ``LDLt`` factorization of a sparse symmetric or Hermitian matrix. A fill-reducing permutation is used. ``F = ldltfact(A)`` is most frequently used to solve systems of equations ``A*x = b`` 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*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'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``\ ). The complete list of supported factors is ``:L, :PtL, :D, :UP, :U, :LD, :DU, :PtLD, :DUP``\ .
 
    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).