GenericLinearAlgebra.jl

svd!(A[, tol, full])::SVD

A generic singular value decomposition (SVD). The implementation only uses Julia functions so the SVD can be computed for any element type provided that the necessary arithmetic operations are supported by the element type.

• tol: The relative tolerance for determining convergence. The default value is eltype(T) where T is the element type of the input matrix bidiagonal (i.e. after converting the matrix to bidiagonal form).

• full: If set to true then all the left and right singular vectors are returned. If set to false then only the vectors corresponding to the number of rows and columns of the input matrix A are returned (the default).

Algorithm

...tomorrow

Example

julia> svd(big.([1 2; 3 4]))
SVD{BigFloat, BigFloat, Matrix{BigFloat}}
U factor:
2×2 Matrix{BigFloat}:
 -0.404554   0.914514
 -0.914514  -0.404554
singular values:
2-element Vector{BigFloat}:
 5.464985704219042650451188493284182533042584640492784181017488774646871847029449
 0.3659661906262578204229643842614005434788136943931877734325179702209382149672422
Vt factor:
2×2 Matrix{BigFloat}:
 -0.576048  -0.817416
 -0.817416   0.576048

svdvals!(A [, tol])

Generic computation of singular values.

julia> using LinearAlgebra, GenericLinearAlgebra, Quaternions

julia> n = 20;

julia> H = [big(1)/(i + j - 1) for i in 1:n, j in 1:n]; # The Hilbert matrix

julia> Float64(svdvals(H)[end]/svdvals(Float64.(H))[end] - 1) # The relative error of the LAPACK based solution in 64 bit floating point.
-0.9999999999447275

julia> A = qr([Quaternion(randn(4)...) for i in 1:3, j in 1:3]).Q *
           Diagonal([3, 2, 1]) *
           qr([Quaternion(randn(4)...) for i in 1:3, j in 1:3]).Q'; # A quaternion matrix with the singular value 1, 2, and 3.

julia> svdvals(A) ≈ [3, 2, 1]
true