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# Lec5 - Solving systems of linear equations Basic methods...

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Solving systems of linear equations Basic methods

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Lecture 5 2 Example: steady state mass balance c 2 c 3 c 5 c 4 c 1 01 5 Q = 01 10 c = 12 3 Q = 24 1 Q = 44 11 Q = 15 3 Q = 31 1 Q = 03 8 Q = 03 20 c = 54 2 Q = 34 8 Q = 55 2 Q = 25 1 Q = 23 1 Q = 01 01 31 3 15 1 12 1 Qc Qc Qc Qc +=+ Find the concentrations in each reactor: 5 equations, 5 unknowns mass balance @ 1 Q = flow (L/min) c = concentration
Lecture 5 3 The basic problem 11 1 12 2 1 1 21 1 22 2 2 2 11 2 2 ... ... ... nn n n n n ax b ax ax b b ++ += = MM M Solve n coupled linear equations with n unknowns Recast this as a matrix problem 11 12 1 1 1 21 22 2 2 2 12 ... ... ... n n n n aa a x b b b ⎛⎞ ⎜⎟ = ⎝⎠ MMO M

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Lecture 5 4 The basic problem Solve n coupled linear equations with n unknowns Recast this as a matrix problem 11 1 12 2 1 1 21 1 22 2 2 2 11 2 2 ... ... ... nn n n n n ax b ax ax b b ++ += = MM M 11 12 1 1 1 21 22 2 2 2 12 ... ... ... n n n n aa a x b b b ⎛⎞ ⎜⎟ = ⎝⎠ MMO M
Lecture 5 5 The basic problem Solve n coupled linear equations with n unknowns Recast this as a matrix problem [] { } { } Ax b = matrix equation 11 1 12 2 1 1 21 1 22 2 2 2 11 2 2 ... ... ... nn n n n n ax b ax ax b b ++ += = MM M 11 12 1 1 1 21 22 2 2 2 12 ... ... ... n n n n aa a x b b b ⎛⎞ ⎜⎟ = ⎝⎠ MMO M

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Lecture 5 6 The basic problem Solve n coupled linear equations with n unknowns Recast this as a matrix problem [] Ax { } { } b = matrix equation 11 1 12 2 1 1 21 1 22 2 2 2 11 2 2 ... ... ... nn n n n n ax ax ax b ++ += + = = MM M 11 12 1 1 1 21 22 2 2 2 12 ... ... ... n n n n aa a x b a x b a x b ⎛⎞ ⎜⎟ = ⎝⎠ O M
Lecture 5 7 Rules for solving systems of equations Rules: • can multiply any equation (row) by a constant • can subtract/add any equation (row) from/to another 11 1 12 2 1 1 21 1 22 2 2 2 11 2 2 ... ... ... nn n n n n ax ax ax b ++ += + = = MM M 11 12 1 1 1 21 22 2 2 2 12 ... ... ... n n n n aa a x b a x b a x b ⎛⎞ ⎜⎟ = ⎝⎠ O M

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Lecture 5 8 Naïve Gauss elimination multiply row 1 by 21 11 a a and subtract row 2 from it 11 1 12 2 1 1 21 1 22 2 2 2 11 2 2 ... ... ... nn n n n n ax ax ax b ++ += + = = MM M 11 12 1 1 1 21 22 2 2 2 12 ... ... ... n n n n aa a x b a x b a x b ⎛⎞ ⎜⎟ = ⎝⎠ O M (forward elimination)
Lecture 5 9 Naïve Gauss elimination multiply row 1 by 21 11 a a and subtract row 2 from it 21 21 21 21 1 12 2 1 1 11 11 11 21 1 22 2 2 2 21 21 21 12 22 2 1 2 1 2 11 11 11 ... ... ... nn n aa a ax b a ax b a x x b b a ++ += + = ⎛⎞ = ⎜⎟ ⎝⎠ 11 1 12 2 1 1 22 2 2 2 11 2 2 ...

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Lec5 - Solving systems of linear equations Basic methods...

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