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Gaussian Elimination. Scientific Computing In Computer Science. Institut für Informatik. Dr. Miriam Mehl. Introduction to Scientific Computing II. Typical SLE. sparse band structure. Example. Gaussian Elimination. Gaussian Elimination. Gaussian Elimination. Gaussian Elimination. - PowerPoint PPT Presentation
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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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]9[
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]7[
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]5[
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]3[
]2[
]1[
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000010141
000001014
f
f
f
f
f
f
f
f
f
u
u
u
u
u
u
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u
Gaussian Elimination
411
1411
141
1411
11411
****
141
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Gaussian Elimination
411
1411
141
1411
*****
****
****
****
114
Gaussian Elimination
411
1411
141
****
****
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****
****
114
Gaussian Elimination
411
1411
****
****
****
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****
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Gaussian Elimination
411
****
***
****
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****
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114
Gaussian Elimination
***
***
***
****
****
****
****
****
114
Gaussian Elimination
**
**
***
****
****
****
****
****
114
Gaussian Elimination
*
**
***
****
****
****
****
****
114
Gaussian Elimination (LU)
411
1411
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11411
1141
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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
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141
1411
11411
1141
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1141
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Gaussian Elimination – Costs
O(1/h)2
411
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11411
****
141
****
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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