Biomechanics_2

# Biomechanics_2 - Lecture 3 BMEn 3001 Biomechanics 8...

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Lecture 3 BMEn 3001 Biomechanics 8 September 2008

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Last Lecture • Dimensional vs. Dimensionless Numbers • Start Math Review – Scalars, Vectors, Matrices, and Tensors • Notation • Basic operations
Topics for Today • Gauss Elimination • LU Decomposition • Pivoting • Remember: Lab starts tomorrow!

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The Linear Algebra Problem a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 a 33 x 1 + a 32 x 2 + a 33 x 3 = b 3 Ax = b , A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 , x = x 1 x 2 x 3 , b = b 1 b 2 b 3
Gauss Elimination • Let’s solve the linear system at right 2 x 1 + x 2 + 3x 3 = 12 4 x 1 + x 2 + 8x 3 + x 4 = 23 6 x 1 + 5x 2 + 6 x 3 3x 4 = 37 x 2 + 4 x 3 + x 4 = 1

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Gauss Elimination • Here it is in matrix form 2 1 3 0 4 1 8 1 6 5 6 3 0 1 4 1 x = 12 23 37 1
Gauss Elimination • We then construct a so-called "augmented matrix." 2 1 3 0 12 4 1 8 1 23 6 5 6 3 37 0 1 4 1 1

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Gauss Elimination • Step 1: Subtract 2 × ROW 1 from ROW 2 2 1 3 0 12 0 1 2 1 1 6 5 6 3 37 0 1 4 1 1
Gauss Elimination • Step 2: Subtract 3 × ROW 1 from ROW 3 2 1 3 0 12 0 1 2 1 1 0 2 3 3 1 0 1 4 1 1

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Gauss Elimination • Step 3: Subtract 0 × ROW 1 from ROW 4 (easy!) 2 1 3 0 12 0 1 2 1 1 0 2 3 3 1 0 1 4 1 1
Gauss Elimination • Step 4: Subtract (-2) × ROW 2 from ROW 3 2 1 3 0 12 0 1 2 1 1 0 0 1 1 1 0 1 4 1 1

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Gauss Elimination • Step 5: Subtract 1 × ROW 2 from ROW 4 2 1 3 0 12 0 1 2 1 1 0 0 1 1 1 0 0 2 0 0
Gauss Elimination • Step 6: Subtract 2 × ROW 3 from ROW 4 2 1 3 0 12 0 1 2 1 1 0 0 1 1 1 0 0 0 2 2

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Gauss Elimination • Works well, gets us to the answer • What if we had to solve a problem with the same matrix but a different RHS? • What if this happened many times, as it often does? • Can we take reuse some information so that we don't have to start from scratch every time?

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• Gauss elimination has two steps: – A series of manipulations to put the equation in a convenient form, and – A series of solutions to get the x vector. • The Frst series, which is the more costly
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## This note was uploaded on 09/16/2008 for the course BMEN 3001 taught by Professor Victorbarocas during the Spring '08 term at Minnesota.

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Biomechanics_2 - Lecture 3 BMEn 3001 Biomechanics 8...

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