Improve the docstrings of block-Krylov processes

src/block_krylov_processes.jl

@@ -10,7 +10,7 @@
 #### Output arguments
 
 * `V`: a dense n × p(k+1) matrix;
-* `Ψ`: a dense p × p upper triangular matrix;
+* `Ψ`: a dense p × p upper triangular matrix such that V₁Ψ = B;
 * `T`: a sparse p(k+1) × pk block tridiagonal matrix with a bandwidth p.
 """
 function hermitian_lanczos(A, B::AbstractMatrix{FC}, k::Int; algo::String="mgs") where FC <: FloatOrComplex
@@ -66,10 +66,10 @@ end
 #### Output arguments
 
 * `V`: a dense n × p(k+1) matrix;
-* `Ψ`: a dense p × p upper triangular matrix;
+* `Ψ`: a dense p × p upper triangular matrix such that V₁Ψ = B;
 * `T`: a sparse p(k+1) × pk block tridiagonal matrix with a bandwidth p;
 * `U`: a dense n × p(k+1) matrix;
-* `Φᴴ`: a dense p × p upper triangular matrix;
+* `Φᴴ`: a dense p × p upper triangular matrix such that U₁Φᴴ = C;
 * `Tᴴ`: a sparse p(k+1) × pk block tridiagonal matrix with a bandwidth p.
 """
 function nonhermitian_lanczos(A, B::AbstractMatrix{FC}, C::AbstractMatrix{FC}, k::Int) where FC <: FloatOrComplex
@@ -158,7 +158,7 @@ end
 #### Output arguments
 
 * `V`: a dense n × p(k+1) matrix;
-* `Γ`: a dense p × p upper triangular matrix;
+* `Γ`: a dense p × p upper triangular matrix such that V₁Γ = B;
 * `H`: a dense p(k+1) × pk block upper Hessenberg matrix with a lower bandwidth p.
 """
 function arnoldi(A, B::AbstractMatrix{FC}, k::Int; algo::String="mgs", reorthogonalization::Bool=false) where FC <: FloatOrComplex
@@ -223,7 +223,7 @@ end
 
 * `V`: a dense n × p(k+1) matrix;
 * `U`: a dense m × p(k+1) matrix;
-* `Ψ`: a dense p × p upper triangular matrix;
+* `Ψ`: a dense p × p upper triangular matrix such that U₁Ψ = B;
 * `L`: a sparse p(k+1) × p(k+1) block lower bidiagonal matrix with a lower bandwidth p.
 """
 function golub_kahan(A, B::AbstractMatrix{FC}, k::Int; algo::String="mgs") where FC <: FloatOrComplex
@@ -287,10 +287,10 @@ end
 #### Output arguments
 
 * `V`: a dense m × p(k+1) matrix;
-* `Ψ`: a dense p × p upper triangular matrix;
+* `Ψ`: a dense p × p upper triangular matrix such that V₁Ψ = B;
 * `T`: a sparse p(k+1) × pk block tridiagonal matrix with a bandwidth p;
 * `U`: a dense n × p(k+1) matrix;
-* `Φᴴ`: a dense p × p upper triangular matrix;
+* `Φᴴ`: a dense p × p upper triangular matrix such that U₁Φᴴ = C;
 * `Tᴴ`: a sparse p(k+1) × pk block tridiagonal matrix with a bandwidth p.
 """
 function saunders_simon_yip(A, B::AbstractMatrix{FC}, C::AbstractMatrix{FC}, k::Int; algo::String="mgs") where FC <: FloatOrComplex
@@ -375,10 +375,10 @@ end
 #### Output arguments
 
 * `V`: a dense m × p(k+1) matrix;
-* `Γ`: a dense p × p upper triangular matrix;
+* `Γ`: a dense p × p upper triangular matrix such that V₁Γ = D;
 * `H`: a dense p(k+1) × pk block upper Hessenberg matrix with a lower bandwidth p;
 * `U`: a dense n × p(k+1) matrix;
-* `Λ`: a dense p × p upper triangular matrix;
+* `Λ`: a dense p × p upper triangular matrix such that U₁Λ = C;
 * `F`: a dense p(k+1) × pk block upper Hessenberg matrix with a lower bandwidth p.
 """
 function montoison_orban(A, B, D::AbstractMatrix{FC}, C::AbstractMatrix{FC}, k::Int; algo::String="mgs", reorthogonalization::Bool=false) where FC <: FloatOrComplex