Merge pull request #40 from amontoison/master

Improve LinearAlgebra docstrings

docs/src/manual/linear_systems.md

@@ -97,9 +97,9 @@ Here is a list of currently supported linear solvers:
 | Linear solver type | `Tv` | System | `LSBackend` | Method |
 |:-------------------|:----:|:------:|:-------:|:-------|
 | [`DenseLinearSolver`](@ref) | `Real` | `NormalEquations` | `Lapack` | Cholesky
-| [`SparseIndefLinearSolver`](@ref) | `Float64` | `AugmentedSystem` | `Cholmod` | LDL
+| [`SparseIndefLinearSolver`](@ref) | `Float64` | `AugmentedSystem` | `Cholmod` | LDLᵀ
 | [`SparsePosDefLinearSolver`](@ref) | `Float64` | `NormalEquations` | `Cholmod` | Cholesky
-| [`LDLFLinearSolver`](@ref) | `Real` | `AugmentedSystem` | `LDLFact` | LDL
+| [`LDLFLinearSolver`](@ref) | `Real` | `AugmentedSystem` | `LDLFact` | LDLᵀ
 
 ### Default options
 If no option is specified, then the linear solver is chosen as follows:

src/LinearAlgebra/LinearSolvers/LinearSolvers.jl

@@ -3,8 +3,8 @@
 
 Abstract container for solving an augmented system
 ```
-    [-(Θ^{-1} + Rp)   A'] [dx] = [ξd]
-    [   A             Rd] [dy] = [ξp]
+    [-(Θ⁻¹ + Rp)   Aᵀ] [dx] = [ξd]
+    [   A          Rd] [dy]   [ξp]
 ```
 where `ξd` and `ξp` are given right-hand side.
 """
@@ -68,8 +68,8 @@ Update internal data, and re-compute factorization/pre-conditioner.
 
 After this call, `ls` can be used to solve the augmented system
 ```
-    [-(Θ^{-1} + Rp)   A'] [dx] = [ξd]
-    [   A             Rd] [dy] = [ξp]
+    [-(Θ⁻¹ + Rp)   Aᵀ] [dx] = [ξd]
+    [   A          Rd] [dy]   [ξp]
 ```
 for given right-hand sides `ξd` and `ξp`.
 """
@@ -81,8 +81,8 @@ function update_linear_solver! end
 
 Solve the symmetric quasi-definite augmented system
 ```
-    [-(Θ^{-1} + Rp)   A'] [dx] = [ξd]
-    [   A             Rd] [dy] = [ξp]
+    [-(Θ⁻¹ + Rp)   Aᵀ] [dx] = [ξd]
+    [   A          Rd] [dy]   [ξp]
 ```
 and over-write `dx`, `dy` with the result.
 
@@ -118,4 +118,4 @@ AbstractLinearSolver(
     ::DefaultBackend,
     ::DefaultSystem,
     A::AbstractMatrix{Tv}
-) where{Tv<:Real} = AbstractLinearSolver(LDLFact(), AugmentedSystem(), A)
+) where{Tv<:Real} = AbstractLinearSolver(LDLFact(), AugmentedSystem(), A)

src/LinearAlgebra/LinearSolvers/cholmod.jl

@@ -8,7 +8,7 @@ Use CHOLMOD backend.
 
 Options available:
 * `Float64` only
-* Augmented system with LDL factorization
+* Augmented system with LDLᵀ factorization
 * Normal equations with Cholesky factorization
 """
 struct Cholmod <: LSBackend end
@@ -22,7 +22,7 @@ struct Cholmod <: LSBackend end
 
 Linear solver for the 2x2 augmented system with ``A`` sparse.
 
-Uses an LDLt factorization of the quasi-definite augmented system.
+Uses an LDLᵀ factorization of the quasi-definite augmented system.
 """
 mutable struct SparseIndefLinearSolver <: AbstractLinearSolver{Float64}
     m::Int  # Number of rows
@@ -84,7 +84,7 @@ linear_system(::SparseIndefLinearSolver) = "Augmented system"
 """
     update_linear_solver!(ls, θ, regP, regD)
 
-Update LDLt factorization of the augmented system.
+Update LDLᵀ factorization of the augmented system.
 
 Update diagonal scaling ``\\theta``, primal-dual regularizations, and re-compute
     the factorization.
@@ -179,16 +179,16 @@ end
     SparsePosDefLinearSolver
 
 Linear solver for the 2x2 augmented system
-```math
-    [-(Θ^{-1} + Rp)   A'] [dx] = [xi_d]
-    [   A             Rd] [dy] = [xi_p]
+```
+    [-(Θ⁻¹ + Rp)   Aᵀ] [dx] = [xi_d]
+    [   A          Rd] [dy]   [xi_p]
 ```
 with ``A`` sparse.
 
 Uses a Cholesky factorization of the positive definite normal equations system
 ```
-(A*(Θ^{-1} + Rp)^{-1}*A' + Rd)  dy = xi_p + A*(Θ^{-1} + Rp)^{-1}*xi_d
-                                dx = (Θ^{-1} + Rp)^{-1} * (A' dy - xi_d)
+(A * ((Θ⁻¹ + Rp)⁻¹ * Aᵀ + Rd) dy = xi_p + A * (θ⁻¹ + Rp)⁻¹ * xi_d
+                              dx = (Θ⁻¹ + Rp)⁻¹ * (Aᵀ * dy - xi_d)
 ```
 """
 mutable struct SparsePosDefLinearSolver <: AbstractLinearSolver{Float64}
@@ -307,4 +307,4 @@ function solve_augmented_system!(
 
 
     return nothing
-end
+end

src/LinearAlgebra/LinearSolvers/lapack.jl

@@ -18,8 +18,8 @@ struct Lapack <: LSBackend end
 
 Linear solver for the 2x2 augmented system
 ```
-    [-(Θ^{-1} + Rp)   A'] [dx] = [xi_d]
-    [   A             Rd] [dy] = [xi_p]
+    [-(Θ⁻¹ + Rp)   Aᵀ] [dx] = [xi_d]
+    [   A          Rd] [dy]   [xi_p]
 ```
 with ``A`` dense.
 
@@ -214,4 +214,4 @@ function solve_augmented_system!(
 
     # TODO: Iterative refinement
     return nothing
-end
+end

src/LinearAlgebra/LinearSolvers/ldlfact.jl

@@ -9,7 +9,7 @@ Use LDLFactorizations backend.
 
 Options available:
 * Any numerical type `T`
-* Augmented system with LDL factorization
+* Augmented system with LDLᵀ factorization
 """
 struct LDLFact <: LSBackend end
 
@@ -22,7 +22,7 @@ struct LDLFact <: LSBackend end
 
 Linear solver for the 2x2 augmented system with ``A`` sparse.
 
-Uses LDLFactorizations.jl to compute an LDLt factorization of the quasi-definite augmented system.
+Uses LDLFactorizations.jl to compute an LDLᵀ factorization of the quasi-definite augmented system.
 """
 mutable struct LDLFLinearSolver{Tv<:Real} <: AbstractLinearSolver{Tv}
     m::Int  # Number of rows
@@ -69,7 +69,7 @@ linear_system(::LDLFLinearSolver) = "Augmented system"
 """
     update_linear_solver!(ls, θ, regP, regD)
 
-Update LDLt factorization of the augmented system.
+Update LDLᵀ factorization of the augmented system.
 
 Update diagonal scaling ``\\theta``, primal-dual regularizations, and re-compute
     the factorization.

src/LinearAlgebra/unitBlockAngular.jl

@@ -367,7 +367,7 @@ end
 """
     mul!(y, A, x)
 
-In-place computation of `y = A*x`
+In-place computation of `y = A * x`
 """
 function mul!(
     y::Vector,
@@ -386,9 +386,9 @@ function mul!(
 end
 
 """
-    mul!(x, At, y)
+    mul!(x, Aᵀ, y)
 
-Compute Matrix-vector product `A'*y` and overwrites the result in `x`.
+Compute Matrix-vector product `Aᵀ * y` and overwrites the result in `x`.
 """
 function mul!(
     x::Vector,
@@ -520,7 +520,7 @@ end
 """
     factor_normaleq(A, θ)
 
-Compute Cholesky factorization of `A*Θ*A'`, where `Θ = Diag(θ)`.
+Compute Cholesky factorization of `A*Θ*Aᵀ`, where `Θ = Diag(θ)`.
 """
 function factor_normaleq(A::UnitBlockAngular{Tv}, θ::Vector{Tv}) where Tv<:Real
 
@@ -556,7 +556,7 @@ end
 """
     factor_normaleq!(F, A, θ)
 
-In-place computation of Cholesky factorization of `A*Θ*A'`, where `Θ = Diag(θ)`.
+In-place computation of Cholesky factorization of `A*Θ*Aᵀ`, where `Θ = Diag(θ)`.
 """
 function factor_normaleq!(
     A::UnitBlockAngular{Tv},