秩亏矩阵的 QR 分解

Question

如果我们计算像 A=[1 2 3;4 5 6;7 8 9] 这样的秩亏矩阵的 QR 分解，R 矩阵的一些对角项将“非常小”。然而，当在 Julia 中进行数值计算时，这些对角线条目必须有多小才能被视为零（因此排名不足）？我正在尝试找到一个公式，而不是一个绝对数字。

Answer 1

A

可以被认为是

完全

秩不足，但有一些加性噪声，则应忽略的R对角线的小错误非零元素将与以下大小大致相同

A

中的噪音。例如，

julia> d = Diagonal(-5*log.(rand(5)))
5×5 Diagonal{Float64, Vector{Float64}}:
 5.56654    ⋅       ⋅        ⋅         ⋅ 
  ⋅       17.6294   ⋅        ⋅         ⋅ 
  ⋅         ⋅      2.20874   ⋅         ⋅ 
  ⋅         ⋅       ⋅       2.91583    ⋅ 
  ⋅         ⋅       ⋅        ⋅       26.5664
julia> A = [d zeros(5,3); zeros(3,8)]
8×8 SparseArrays.SparseMatrixCSC{Float64, Int64} with 5 stored entries:
 5.56654    ⋅       ⋅        ⋅         ⋅       ⋅    ⋅    ⋅ 
  ⋅       17.6294   ⋅        ⋅         ⋅       ⋅    ⋅    ⋅ 
  ⋅         ⋅      2.20874   ⋅         ⋅       ⋅    ⋅    ⋅ 
  ⋅         ⋅       ⋅       2.91583    ⋅       ⋅    ⋅    ⋅ 
  ⋅         ⋅       ⋅        ⋅       26.5664   ⋅    ⋅    ⋅ 
  ⋅         ⋅       ⋅        ⋅         ⋅       ⋅    ⋅    ⋅ 
  ⋅         ⋅       ⋅        ⋅         ⋅       ⋅    ⋅    ⋅ 
  ⋅         ⋅       ⋅        ⋅         ⋅       ⋅    ⋅    ⋅ 

julia> diag(qr(A).R)
8-element SparseArrays.SparseVector{Float64, Int64} with 5 stored entries:
  [1]  =  5.56654
  [2]  =  17.6294
  [3]  =  2.20874
  [4]  =  2.91583
  [5]  =  26.5664

julia> diag(qr(A+rand(8,8)*1e-6).R)
8-element Vector{Float64}:
  -5.566543507821689
 -17.629411781406233
  -2.20873750628613
  -2.915828986361413
 -26.56641073082019
  -1.1241491098747558e-6
  -6.122032725632311e-7
  -2.3444092119580006e-7