Diagonal pivoting methods for solving tridiagonal systems without interchanges

Transcription

1 Diagonal pivoting methods for solving tridiagonal systems without interchanges Joseph Tyson, Jennifer Erway, and Roummel F. Marcia Department of Mathematics, Wake Forest University School of Natural Sciences, University of California, Merced Research supported by NSF-DMS and NSF-DMS SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 0

2 Summary Problem: Solve T x = b, where T is unsymmetric, tridiagonal, and nonsingular. Challenge: Some applications (e.g., bi-conjugate gradient) do not allow interchanges GEPP: stable but requires interchanges Approach: Use LBM T where L and M are lower triangular and B is block diagonal with 1 1 and 2 2 blocks. Can demonstrate backward stability. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 1

3 Background Symmetric tridiagonal: Bunch (1974) Bunch and Kaufman (1977) Bunch and RM (2005, 2006) Fang and O Leary (2006) Composite-step BiCG: Bank and Chan (1994) Stability analysis: Higham (lots) SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 2

4 Notation Let α 1 γ β 2 α 2 γ β 3 α γ n 0 0 β n α n and T max = max i,j T i,j. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 3

5 Tridiagonal matrices Partition d n d d n d B 1 T12 T T 21 T 22 T nonsingular B 1 nonsingular for d = 1 or 2: Note that B 1 = [α 1 ] or α 1 γ 2 β 2 α 2 Then α 1 = 0 and = α 1 α 2 β 2 γ 2 = 0 = β 2 = 0 or γ 2 = 0 = T has a zero column or row = T is singular. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 4

6 Tridiagonal matrices Partition d n d d n d B 1 T12 T T 21 T 22 T nonsingular B 1 nonsingular for d = 1 or 2. Thus I d 0 T 21 B 1 1 I n d B T 22 T 21 B 1 1 T 12 T I d B 1 1 T 12 T 0 I n d. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 4

7 Tridiagonal matrices Partition d n d d n d B 1 T12 T T 21 T 22 T nonsingular B 1 nonsingular for d = 1 or 2. Thus I d 0 T 21 B 1 1 I n d B T 22 T 21 B 1 1 T 12 T I d B 1 1 T 12 T 0 I n d. Schur complement T 22 T 21 B 1 1 T T 12 T 22 β 2γ 2 α 1 is tridiagonal: e (n 1) 1 e (n 1)T 1 or T 22 ( ) α1 β 3 γ 3 e (n 2) 1 e (n 2)T 1. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 4

8 Tridiagonal matrices Partition d n d d n d B 1 T12 T T 21 T 22 T nonsingular B 1 nonsingular for d = 1 or 2. Thus I d 0 T 21 B 1 1 I n d B T 22 T 21 B 1 1 T 12 T I d B 1 1 T 12 T 0 I n d. Compute recursively to obtain LBM T where L and M are lower triangular and B is block diagonal with 1 1 and 2 2 blocks. Question: How to choose size of pivot d? SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 4

9 Proposed pivoting strategy If d = 1, then 1 β 2 /α 1 L 2 α 1 B 2 1 γ 2 /α 1 M T 2 If d = 2, then 1 β 2β 3 1 α 1 β 3 L 2 α 1 γ 2 β 2 α 2 B 2 1 γ 2γ 3 1 α 1 γ 3 M T 2 where = α 1 α 2 β 2 γ 2. Choose pivot size that yields smaller elements in L and M. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 5

10 Proposed pivoting strategy If d = 1, then 1 β 2 /α 1 L 2 α 1 B 2 1 γ 2 /α 1 M T 2 If d = 2, then 1 β 2β 3 1 α 1 β 3 L 2 α 1 γ 2 β 2 α 2 B 2 1 γ 2γ 3 1 α 1 γ 3 M T 2 where = α 1 α 2 β 2 γ 2. Criterion #1: Choose a 1 1 pivot if { } { β2 max α 1, γ 2 β2 β 3 max κ, α 1β 3, γ 2γ 3, α } 1γ 3. α 1 SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 5

11 Proposed pivoting strategy Criterion #1: Equivalently, choose a 1 1 pivot if max{ β 2, γ 2 } α 1 max κ{ β 2 β 3, α 1 β 3, γ 2 γ 3, α 1 γ 3 } (Intuition: a 2 2 pivot is closer to being singular than a 1 1 pivot is to 0) Criterion #2: Choose a 1 1 pivot if α 1 α 2 κ β 2 γ 2. (LBM T reduces to the LDM T factorization if T is positive definite). Constant: κ = ( 5 1)/ (Bunch (1974)) SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 6

12 Proposed pivoting strategy Algorithm 1. κ = ( 5 1)/ = α 1 α 2 β 2 γ 2 if max{ β 2, γ 2 } α 1 max κ{ β 2 β 3, α 1 β 3, γ 2 γ 3, α 1 γ 3 } or α 1 α 2 κ β 2 γ 2 d I = 1 else d I = 2 end SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 7

13 Alternative pivoting strategy Algorithm 2. κ = ( 5 1)/ σ 1 = max{ α 2, γ 2, β 2, γ 3, β 3 } if α 1 σ 1 κ β 2 γ 2, d II = 1 else d II = 2 end α 1 γ β 2 α 2 γ β 3 α γ n 0 0 β n α n (cf. Bunch-Kaufman strategy for symmetric tridiagonal matrices different from the more general Bunch-Kaufman LBL T ). SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 8

14 Pivoting strategies Lemma. If d I = 1, then d II = 1. If d II = 2, then d I = 2. And, if d I = 2 and d II = 1, then the subsequent pivot size d II is 1. Summary: The two pivoting strategies differ only when the first strategy chooses a 2 2 pivot while the alternative chooses two 1 1 pivots. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 9

15 Main theorem Theorem. Let T ˆL ˆB ˆM T be the computed LBM T factorization of the unsymmetric tridiagonal matrix T IR n n using Algorithm 1 or 2. Assume that all linear systems Ey = f involving 2 2 pivots E are solved using the explicit inverse. Then T ˆL ˆB ˆM T 1, and (T + T 2 )ˆx = b, where T i max cu T max + O(u 2 ), for i = 1, 2, where c is a constant. Summary: ˆx is the exact solution to a nearby problem. Proof: Long and not particularly enlightening. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 9

16 Numerical results Obtained 16 unsymmetric tridiagonal matrices from Dhillon (1998) and Hargreaves (2004) on estimating condition numbers of tridiagonal matrices. Matrix type Description 1 Random elements (uniformly distributed on [ 1, 1]). 2 Preassigned condition number with one very small singular value: gallery( randsvd,n,1e15,2,1,1). 3 Preassigned condition number with geometric distribution of singular values: gallery( randsvd,n,1e15,3,1,1). 7 Ill-conditioned, tridiagonal: gallery( dorr,n,1e-4). 12 Toeplitz: main diagonal zero, each off-diagonal a random number (uniformly distributed on [ 1, 1]). 15 Tridiagonal with main diagonal zero and known eigenvalues: gallery( clement,n,0). SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 10

17 Numerical results Generated matrices of each type and computed Aˆx b 2 : Matrix Type Algorithm 1 Algorithm 2 GEPP Cond. Num e e e e e e e e e e e e e e e e e e e e e e e e+015 Conclusion: GEPP performs slightly better than LBM T, but results are comparable. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 11

18 Summary Problem: Solve T x = b, where T is unsymmetric, tridiagonal, and nonsingular. Approach: Use LBM T where L and M are lower triangular and B is block diagonal with 1 1 and 2 2 blocks. Can demonstrate backward stability. Future work: Extension to unsymmetric Lanczos / bi-conjugate gradient method. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 12

19 References A backward stability analysis of diagonal pivoting methods for solving unsymmetric tridiagonal systems without interchanges, Jennifer Erway and RM, Accepted for publication in Numerical Linear Algebra with Applications. Solving unsymmetric tridiagonal systems without interchanges, Joseph Tyson, Jennifer Erway, and RM, In preparation. google: roummel SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 13

20 . Thank you. SIAM Applied Linear Algebra 2009: Diagonal pivoting for tridiagonal systems without interchanges Slide 14