QuasiTriangular

A Julia package for quasi upper triangular matrices strongly inspired by https://github.com/JuliaLang/julia/blob/master/stdlib/LinearAlgebra/src/triangular.jl

WORK IN PROGRESS

Type QuasiUpperTriangular stores a quasi upper triangular matrix in a square matrix. The various algorithms ignore the lower zero elements

Functions

Matrix - vector product

• A_mul_B!(a::QuasiUpperTriangular,b::AbstractVector,work::AbstractVector) uses BLAS trmv and corrects the results for lower nonzero elements. For small matrices, (n < 27), uses regular matrix - vector product
• A_mul_B!(A::QuasiUpperTriangular, B::AbstractVector) pure Julia implementation. Doesn't need extra workspace.

Vector - matrix produc

• A_mul_B!(A::AbstractVector, B::QuasiUpperTriangular) pure Julia implementation. Doesn't need extra workspace.

Transposed matrix -vector product

• At_mul_B!(A::QuasiUpperTriangular, B::AbstractVector) pure Julia implementation. Doesn't need extra workspace.

Vector - transposed matrix product

• A_mul_Bt!(A::AbstractVector, B::QuasiUpperTriangular) pure Julia implementation. Doesn't need extra workspace.

Matrix - matrix product

• A_mul_B!(C::AbstractMatrix, A::QuasiUpperTriangular, B::AbstractMatrix): C = A*B, A is quasi upper triangular
• A_mul_B!(C::AbstractMatrix, A::QuasiUpperTriangular, b::AbstractMatrix): C = αA*B, A is quasi upper triangular
• At_mul_B!(C::AbstractMatrix, A::QuasiUpperTriangular, B::AbstractMatrix): C = transpose(A)*B, A is quasi upper triangular
• At_mul_B!(C::AbstractMatrix, A::QuasiUpperTriangular, b::AbstractMatrix): C = α*transpose(A)*B, A is quasi upper triangular
• At_mul_B!(A::QuasiUpperTriangular, B::AbstractMatrix) B = transpose(A)*B in place computation, A is quasi upper triangular.
• A_mul_B!(C::AbstractMatrix, A::AbstractMatrix, B::QuasiUpperTriangular): C = A*B, B is quasi upper triangular.
• A_mul_B!(A::AbstractMatrix, B::QuasiUpperTriangular): C = A*B, in place computation, B is quasi upper triangular.
• A_mul_Bt!(C::AbstractMatrix, A::AbstractMatrix, B::QuasiUpperTriangular): C = A*transpose(B), B is quasi upper triangular.
• A_mul_Bt!(A::AbstractMatrix, B::QuasiUpperTriangular): C = A*transpose(B), in place computation, B is quasi upper triangular.
• A_mul_B!(C:QuasiUpperTriangular, A::QuasiUpperTriangular, B::QuasiUpperTriangular), A, B and C are quasi upper triangular.

Linear problem solver

• A_ldiv_B!(A::QuasiUpperTriangular, B::AbstractMatrix) solves A*X = B, where A is quasi upper triangular. Solves by back substitution. Lower off-diagonal elements make 2 * 2 problems that are solved explicitely.
• A_rdiv_B!(A::AbstractMatrix, B::QuasiUpperTriangular) solves X*B = A, where B is quasi upper triangular. Solves by back substitution. Lower off-diagonal elements make 2 * 2 problems that are solved explicitely.
• A_rdiv_Bt!(A::AbstractMatrix, B::QuasiUpperTriangular) solves X*transpose(B) = A, where B is quasi upper triangular. Solves by back substitution. Lower off-diagonal elements make 2 * 2 problems that are solved explicitely.
• I_plus_rA_ldiv_B!(r::Float64,a::QuasiUpperTriangular, b::AbstractVector) solves (I + r*A)*x = b where A is quasi upper triangular.

TODO

• replace function names for product by mul!
• introduce lazy transpose evaluation
• handle quasi lower triangular matrices
• benchmark cases that have two different implementations