Mathematical Background
Numerical Examples






In all the experiments presented in this section we are solving the linear system

Click on each example below to view the results:

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9



In the second set of experiments we have used matrices from real-life problems taken from Matrix-Market collection ( http://math.nist.gov/MatrixMarket/ ). Numerical solution of large scale linear systems is based on two ingredients:

1.) Accelerator
2.) Preconditioner

Preconditioning is mandatory. There is almost no chance to solve a real-life linear system without preconditioning. The numerical results presented here are intended to show the capabilities of our accelerator ( RISOLV ). Hence we do not use any preconditioning. For sure, any problem presented here can be solved faster by using the appropriate preconditioner. As discribed in Mathematical Background, when 0 is surrounded by the domain of eigenvalues, there is no polynomial algorithm which will achieve convergence. Therefore, for matrices where this is the case, we have used a shift. It means that we have solved Ax = b where



The numerical results are presented in the following table:


Linear System Method mat-vecs inner-products
e40r5000
α=3.4 		risolv
gmres		 412
2919
 1384
16054
e40r5000
α=3.3 		risolv
gmres		 425
-		 1417
-
add20
α=0 		risolv
gmres		 512
801		 1907
4405
nnc1374
α=-7 		risolv
gmres		 340
2252		 1219
12386
nnc1374
α=-3 		risolv
gmres		 1102
-		 3974
-
orsirr_1
α=-7 		risolv
gmres		 1101
-		 4017
-
gemat11
α=5 		risolv
gmres		 1348
-		 4341
-
gemat12
α=5 		risolv
gmres		 324
-		 1026
-
bp_1600
α=14 		risolv
gmres		 1589
-		 5118
-
sherman4
α=0  		risolv
gmres		 243
758		 931
4169
sherman5
α=140 		risolv
gmres		 339
-		 1239
-
orani678
α=1.3 		risolv
gmres		 626
-		 2026
-
mahindas
α=11.49 		risolv
gmres		 73
2497		 230
13733
mahindas
α=11 		risolv
gmres		 72
-		 235
-
orsreg_1
α=0 		risolv
gmres		 325
1064		 4.7560e-006
1161
5852


In the next experiment we solved the above matrix-market linear systems with different right hand side vector. We took



Using information on the domain of eigenvalues we have gathered in the first time we have solved a system, the number of inner-products is reduced significantly.
remark: The saving of inner-products is applicable for general non-symmetric linear systems. It is not applicable yet for the symmetric case.


Linear System α Method mat-vecs inner-products
e40r5000 3.4 risolv 421 94
e40r5000 3.3 risolv 440 116
add20 0 risolv 533 70
ncc1374 -7 risolv 344 25
orsirr_1 -7 risolv 1121 70
gemat11 5 risolv 1320 19
gemat12 5 risolv 353 151
bp_1600 14 risolv 1607 193
sherman4 0 risolv 262 65
sherman5 140 risolv 341 10
orani678 1.3 risolv 604 2
mahindas 11.49 risolv 71 27
orserg_1 0 risolv 339 69