Introduction to Scientific Computing II Institut für Informatik Scientific Computing In Computer Science Introduction to Scientific Computing II Gaussian Elimination Dr. Miriam Mehl

Typical SLE sparse band structure

Example

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination – Costs

Gaussian Elimination – Costs O(1/h)2

Gaussian Elimination – Costs O(1/h)2*(1/h)2

Gaussian Elimination – Costs O(1/h)2*(1/h)2 + O(1/h)

Gaussian Elimination – Costs O(1/h)2*(1/h)2 + O(1/h)*(1/h)2

Gaussian Elimination – Costs O(1/h)2*(1/h)2 + O(1/h)*(1/h)2 + O(1/h)

Gaussian Elimination – Costs O(1/h)*(1/h)2 + O(1/h)*(1/h)2 + O(1/h)*(1/h)2

Gaussian Elimination – Costs 2D: O(1/h)4 3D: O(1/h)7

Gaussian Elimination – Costs 2D h runtime (33 TFlop/s) 2-7 0.03 sec 2-8 0.5 sec 2-9 8.3 sec 2-10 2 min 23 sec 2-11 35 min 32 sec 2-12 9 h 28 min 38 sec 2-13 6 d 7 h 28 min 10 sec hallo

Gaussian Elimination – Costs 3D hallo h runtime (33 TFlop/s) 2-6 8 min 53 sec 2-7 18 h 57 min 26 sec 2-8 100 d 26 h 10 min 53 sec 2-9 35 a 156 d 57 h 53 min 04 sec

hallo

Relaxation Methods – Gauss-Seidel

Relaxation Methods – Jacobi