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

Update LU factors

10/11/2013 Zack

Update LU Factors Replace one column (or one row) Add a rank-1 matrix 𝜏𝑤𝑥𝑈 𝐵 = 𝐵 + ( 𝑏 − 𝑏)𝑓𝑠

𝑈

𝐵 = 𝐵 + 𝜏𝑤𝑥𝑈

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

BGR update

𝐵 = 𝐵 + 𝑏 − 𝑏 𝑓𝑠

𝑈

= 𝑀 𝑉 + 𝑣 − 𝑣 𝑓𝑠

𝑈

= 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 𝑠th 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 1 1 1 1 1 1 1 1 1 1 1 1 LU                                                                                                                                                                                      

BLU update

𝐵0 𝑊 𝐹𝑈 𝑧 𝑨 = 𝑐

𝑊 = (𝑤1, 𝑤2 … ) have replaced columns 𝑠

1, 𝑠 2, … of 𝐵0

columns of 𝐹 are columns 𝑠

1, 𝑠 2, … of the 𝑛 × 𝑛 identity

BLU update

• Example:

1 11 12 13 14 11 12 1 2 21 22 23 24 21 22 2 3 31 32 33 34 31 23 3 4 41 42 43 44 41 24 4 5 6

1 1 x a a a a v v b x a a a a v v b x a a a a v v b x a a a a v v b x x                                                             

BLU update

• block-triangular factorization

𝐵0 𝑊 𝐹𝑈 = 𝑀0 𝑎𝑈 𝐽 𝑉0 𝑍 𝐷 Where 𝑀0𝑍 = 𝑊,𝑎𝑈𝑉0 = 𝐹𝑈 , and 𝐷 = −𝑎𝑈𝑍

BLU update

𝑀0 𝑎𝑈 𝐽 𝑉0 𝑍 𝐷 𝑧 𝑨 = 𝑐 𝑀0 𝑎𝑈 𝐽 𝑥 𝑥′ = 𝑐

Where 𝑀0𝑥 = 𝑐, 𝑥′ = −𝑎𝑈𝑥

BLU update

𝑉0 𝑍 𝐷 𝑧 𝑨 = 𝑥 𝑥′ 𝐷𝑨 = 𝑥′, 𝑉0𝑧 = 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 𝐵0 On most problems, the BLU updates were found to be faster than LUSOL implementation of the Bartels-Golub-Reid update.

Rank-1 update

Thank you