Update LU factors 10/11/2013 Zack Replace one column (or one row) - PowerPoint PPT Presentation

Update LU factors 10/11/2013 Zack

Replace one column (or one row) 𝑈 𝐵 = 𝐵 + ( 𝑏 − 𝑏)𝑓 𝑠 Update LU Factors Add a rank-1 matrix 𝜏𝑤𝑥 𝑈 𝐵 = 𝐵 + 𝜏𝑤𝑥 𝑈

Bartels-Golub-Reid update (BGR) Replace one column Forrest-Tomlin update (FT) Block-LU update (BLU)

BGR update 𝑈 𝐵 = 𝐵 + 𝑏 − 𝑏 𝑓 𝑠 𝑈 = 𝑀 𝑉 + 𝑣 − 𝑣 𝑓 𝑠 = 𝑀𝑉′ the last nonzero element of 𝑣 will be in row 𝑚 , and to restore triangularity we need to nd an 𝑀𝑉 factorization of 𝑉’.

BGR update let 𝑄 be a cyclic permutation that moves the 𝑠 th column and row of 𝑉′ to position 𝑚 and shifts the intervening columns and rows forward.

BGR update • Ideally the existing diagonals in rows 𝑠 ∶ 𝑚 − 1 will be large enough to serve as pivots, but to ensure stability we must allow row interchanges .

FT update • the FT update is equivalent to the BGR update with the restriction that 𝑚 = 𝑛 and row interchanges are not allowed in the factorization 𝑉’. • numerical stability of the FT update is not completely reliable. Nevertheless, the ease of implementation means that the FT update has been adopted in virtually all sparse implementations of the simplex method, except LA05, LA15, and LUSOL.

FT update • The reason why FT update works well in practice                          1 1 1 1                              1 1 1 1                    1 1 1 1              LU                          1 1 1 1                          1 1 1 1                                               1 1 1 1

BLU update 𝑊 = (𝑤 1 , 𝑤 2 … ) have replaced columns 𝑠 1 , 𝑠 2 , … of 𝐵 0 𝐵 0 𝑊 𝑧 = 𝑐 𝑨 𝐹 𝑈 0 0 columns of 𝐹 are columns 𝑠 1 , 𝑠 2 , … of the 𝑛 × 𝑛 identity

BLU update • Example:       x a a a a v v b 11 12 13 14 11 12 1 1       a a a a v v x b       21 22 23 24 21 22 2 2       a a a a v v x b   31 32 33 34 31 23   3   3   a a a a v v   x   b  41 42 43 44 41 24 4 4       1 x 0       5             1 x 0 6

BLU update • block-triangular factorization 𝐵 0 𝑊 𝑉 0 𝑍 𝑀 0 = 𝐹 𝑈 𝑎 𝑈 𝐷 𝐽 Where 𝑀 0 𝑍 = 𝑊,𝑎 𝑈 𝑉 0 = 𝐹 𝑈 , and 𝐷 = −𝑎 𝑈 𝑍

BLU update 𝑧 𝑀 0 𝑉 0 𝑍 = 𝑐 𝑨 𝑎 𝑈 0 𝐷 𝐽 𝑥 𝑀 0 𝑥′ = 𝑐 𝑎 𝑈 0 𝐽 Where 𝑀 0 𝑥 = 𝑐 , 𝑥 ′ = −𝑎 𝑈 𝑥

BLU update 𝑧 𝑥 𝑉 0 𝑍 = 𝑨 𝑥′ 𝐷 𝐷𝑨 = 𝑥′ , 𝑉 0 𝑧 = 𝑥 − 𝑍𝑨

BLU update • we can solve 𝐵𝑦 = 𝑐 by the following sequence:

BLU update The vectors in 𝑍 were often rather dense, but it did save time to store them and avoid double solves with 𝐵 0 On most problems, the BLU updates were found to be faster than LUSOL implementation of the Bartels-Golub-Reid update.

Rank-1 update

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