## Group Castrillon

Group Description ```Counter-example matrix (denormal in A+alpha*I), J. Castrillon, Teledyne The problem is x=(A+alpha*I)\b as alpha varies from 0 to 100. However as this occurs, performance suffers because of the many denormals that appear below the diagonal in L. %------------------------------------------------------------------------------- % consider the following code: %------------------------------------------------------------------------------- Problem = UFget ('Castrillon/denormal') ; K = Problem.A ; fprintf ('nnz (K-K'') = %d\n', nnz(K-K')) ; fprintf ('nnz (diag (K)): %d\n', nnz (diag (K))) ; p = amd (K) ; [count, h, parent, post, Lpattern] = symbfact (K(p,p), 'sym', 'lower') ; fprintf ('nnz(K) %d nnz(L) %d\n', nnz(K), nnz(Lpattern)) ; n = size (K,1) ; rand ('state',0) ; b = rand (n,1) ; I = speye (n) ; spparms ('spumoni',0) ; maxNumCompThreads (1) ; for alpha = [0 .01 .02 .1 .3 1 10 100] fprintf ('\n\n --------------------- alpha = %g:\n', alpha) ; A = K + alpha*I ; for threads = 1:2 maxNumCompThreads (threads) ; fprintf ('threads: %d\n', threads) ; tic [L,ignore,p] = chol (A, 'lower') ; t = toc ; fprintf (' chol time: %g nnz(L) %d (%d dropped)\n', ... t, nnz (L), nnz(Lpattern)-nnz(L)) ; [i j s] = find (L) ; fprintf (' min(abs(L)): %g\n', min (abs (s))) ; clear L i j s tic x = A\b ; t = toc ; fprintf (' backslash time: %g\n', t) ; end end %------------------------------------------------------------------------------- % the output is below %------------------------------------------------------------------------------- diary off tryK --------------------------------------------------------------------------- MATLAB Version 7.6.0.324 (R2008a) MATLAB License Number: 523006 Operating System: Linux 2.6.24-19-generic #1 SMP Wed Jun 18 14:43:41 UTC 2008 i686 Java VM Version: Java 1.6.0_01 with Sun Microsystems Inc. Java HotSpot(TM) Client VM mixed mode, sharing --------------------------------------------------------------------------- MATLAB Version 7.6 (R2008a) Optimization Toolbox Version 4.0 (R2008a) Symbolic Math Toolbox Version 3.2.3 (R2008a) Name Size Bytes Class Attributes K 89400x89400 14232292 double sparse nnz (K-K') = 0 nnz (diag (K)): 89400 nnz(K) 1156224 nnz(L) 8231641 --------------------- alpha = 0: threads: 1 chol time: 3.04069 nnz(L) 8231641 (0 dropped) min(abs(L)): 2.3735e-15 backslash time: 2.93038 threads: 2 chol time: 2.18102 nnz(L) 8231641 (0 dropped) min(abs(L)): 2.3735e-15 backslash time: 2.0675 --------------------- alpha = 0.01: threads: 1 chol time: 3.06346 nnz(L) 8231641 (0 dropped) min(abs(L)): 7.68315e-168 backslash time: 2.95286 threads: 2 chol time: 2.19825 nnz(L) 8231641 (0 dropped) min(abs(L)): 7.68315e-168 backslash time: 2.08248 --------------------- alpha = 0.02: threads: 1 chol time: 3.58369 nnz(L) 8211304 (20337 dropped) min(abs(L)): 7.90408e-200 backslash time: 3.4712 threads: 2 chol time: 2.48563 nnz(L) 8211304 (20337 dropped) min(abs(L)): 7.90408e-200 backslash time: 2.39297 --------------------- alpha = 0.1: threads: 1 chol time: 8.60043 nnz(L) 8231641 (0 dropped) min(abs(L)): 1.32989e-295 backslash time: 8.49517 threads: 2 chol time: 5.83973 nnz(L) 8231641 (0 dropped) min(abs(L)): 1.32989e-295 backslash time: 5.74531 --------------------- alpha = 0.3: threads: 1 chol time: 14.6864 nnz(L) 8221018 (10623 dropped) min(abs(L)): 9.88131e-324 backslash time: 14.5832 threads: 2 chol time: 9.6974 nnz(L) 8221018 (10623 dropped) min(abs(L)): 9.88131e-324 backslash time: 9.56342 --------------------- alpha = 1: threads: 1 chol time: 21.5543 nnz(L) 8161666 (69975 dropped) min(abs(L)): 4.94066e-324 backslash time: 21.4453 threads: 2 chol time: 13.3063 nnz(L) 8161666 (69975 dropped) min(abs(L)): 4.94066e-324 backslash time: 13.218 --------------------- alpha = 10: threads: 1 chol time: 29.1026 nnz(L) 7957149 (274492 dropped) min(abs(L)): 4.94066e-324 backslash time: 29.0284 threads: 2 chol time: 19.0662 nnz(L) 7957149 (274492 dropped) min(abs(L)): 4.94066e-324 backslash time: 19.5602 --------------------- alpha = 100: threads: 1 chol time: 31.3377 nnz(L) 7698212 (533429 dropped) min(abs(L)): 4.94066e-324 backslash time: 31.1925 threads: 2 chol time: 20.2513 nnz(L) 7698212 (533429 dropped) min(abs(L)): 4.94066e-324 backslash time: 20.1174```
Displaying 1 collection matrix
Id Name Group Rows Cols Nonzeros
1892 denormal Castrillon 89,400 89,400 1,156,224