Introduction to Scientific
Computing II
Gaussian Elimination
Dr. Miriam Mehl
Institut für
Informatik
Scientific Computing
In Computer Science
Typical SLE
• sparse• band structure
411
1411
11411
1141
114
Example
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]7[
]6[
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]2[
]1[
]9[
]8[
]7[
]6[
]5[
]4[
]3[
]2[
]1[
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141010000
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000100410
000010141
000001014
f
f
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f
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f
f
u
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u
u
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Gaussian Elimination
411
1411
141
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11411
****
141
****
114
Gaussian Elimination
411
1411
141
1411
*****
****
****
****
114
Gaussian Elimination
411
1411
141
****
****
****
****
****
114
Gaussian Elimination
411
1411
****
****
****
****
****
****
114
Gaussian Elimination
411
****
***
****
****
****
****
****
114
Gaussian Elimination
***
***
***
****
****
****
****
****
114
Gaussian Elimination
**
**
***
****
****
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****
114
Gaussian Elimination
*
**
***
****
****
****
****
****
114
Gaussian Elimination (LU)
411
1411
141
1411
11411
1141
141
1141
114
1
1
1
1
1
1
1
1
1
Gaussian Elimination (LU)
411
1411
141
1411
11411
****
141
****
114
1
1
1
1
1
1*
1
1*
1
Gaussian Elimination (LU)
411
1411
141
1411
*****
****
****
****
114
1
1
1
1
1*
1**
1*
1*
1
Gaussian Elimination (LU)
411
1411
141
****
****
****
****
****
114
1
1
1
1*
1**
1***
1*
1*
1
Gaussian Elimination (LU)
411
1411
****
****
****
****
****
****
114
1
1
1*
1**
1***
1***
1*
1*
1
Gaussian Elimination (LU)
411
****
***
****
****
****
****
****
114
1
1*
1**
1***
1***
1***
1*
1*
1
Gaussian Elimination (LU)
***
***
***
****
****
****
****
****
114
1*
1**
1***
1***
1***
1***
1*
1*
1
Gaussian Elimination (LU)
**
**
***
****
****
****
****
****
114
1**
1***
1***
1***
1***
1***
1*
1*
1
Gaussian Elimination (LU)
*
**
***
****
****
****
****
****
114
1***
1***
1***
1***
1***
1***
1*
1*
1
Gaussian Elimination – Costs
1
1
1
1
1
1
1
1
1
411
1411
141
1411
11411
1141
141
1141
114
Gaussian Elimination – Costs
O(1/h)2
411
1411
141
1411
11411
****
141
****
114
1
1
1
1
1
**
1
**
1
Gaussian Elimination – Costs
O(1/h)2*(1/h)2
*
**
***
****
****
****
****
****
114
****
****
****
****
****
****
**
**
1
*
**
***
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****
****
114
****
****
****
****
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****
**
**
1
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
*
**
***
****
****
****
****
****
114
****
****
****
****
****
****
**
**
1
Gaussian Elimination – Costs
O(1/h)2*(1/h)2 + O(1/h)*(1/h)2 + O(1/h)
*
**
***
****
****
****
****
****
114
Gaussian Elimination – Costs
O(1/h)2*(1/h)2 + O(1/h)*(1/h)2 + O(1/h)*(1/h)2
*
**
***
****
****
****
****
****
114
Gaussian Elimination – Costs
2D: O(1/h)4
3D: O(1/h)7
Gaussian Elimination – Costs 2D
halloh runtime (HLRB2, 62 TFlop/s)
2-7 0.02 sec
2-8 0.27 sec
2-9 4.4 sec
2-10 1 min 16 sec
2-11 18 min 55 sec
2-12 5 h 02 min 40 sec
2-13 3 d 8 h 37 min 15 sec
Gaussian Elimination – Costs 3D
hallo
h runtime (HLRB2, 62 TFlop/s)
2-6 4 min 44 sec
2-7 10 h 05 min 24 sec
2-8 53 d 19 h 21 min 17 sec
2-9 18 a 313 d 21 h 54 min 22 sec
