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Linear Solver Helpers

Gentleman's Solver for Least Squares

Abstract Types

RLinearAlgebra.GentData — Type
GentData{S, M<:Matrix{S}, V<:Vector{S}}

This is a preallocated data structure for solving least squares problems with Gentleman's algorithm. This data structure is not exported and thus is not designed to be manipulated by the user.

Fields

  • A::M, the matrix to be solved.
  • B::M, the block of rows of A that are used to update the QR decomposition.
  • R::UpperTriangular, the upper triangular part of B.
  • tab::SubArray, the last column of B where the constant vector entries corresponding

to the rows of A that were brought to B are stored.

  • v::V, a buffer vector that stores the scaling constants from the Householder reflections,

see LAPACK.geqrf! in LinearAlgebra for more information.

  • bsize::Int64, the number of rows transported to B at each iteration.

Calling Gent(A, bsize) will allocate a B matrix with the n + 1 columns and n + bsize + 1 rows where n is the number of columns of A. It also allocates a buffer vector v with n + 1 entries. Finally, it allocates a view of the upper triangular part of B[1:n, 1:n] and a view of the last column of B.

Miller, Alan J. “Algorithm AS 274: Least Squares Routines to Supplement Those of Gentleman.” Journal of the Royal Statistical Society. Series C (Applied Statistics), vol. 41, no. 2, 1992, pp. 458–78. JSTOR, https://doi.org/10.2307/2347583.

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Solver

RLinearAlgebra.gentleman! — Function
gentleman!(G::GentData)

A function that updates the GenData structure with the Gentleman's algorithm applied to a single row block of A.

Inputs

  • G::GentData, a data structure with the information to be updated

by an iteration of Gentlemans.

Output

The function will result in updates to the upper triangular part R and the tab vector of the GentData structure.

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LinearAlgebra.ldiv! — Function
ldiv!(x::AbstractVector, G::GentData, b::AbstractVector)

Function that takes the allocated GentData structure to solve the least squares problem $\min_x \|A x-b\|_2^2$. The function overwrites x with the solution.

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RLinearAlgebra.copy_block_from_mat! — Function
copyBlockFromMat!(B::AbstractMatrix, A::AbstractMatrix, b::AbstractVector, index::Union{UnitRange{Int64}, Vector{Int64}})

Function that updates matrix B with the entries at indicies index corresponding to the rows of the A matrix and the entries of the b constant vector of the linear system.

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