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Algorithm for Gauss elimination 1) first eliminate for each eq. j, j=1 to n-1 for all eq.s k greater than j a) multiply eq. j by a kj /a jj b) subtract the result from eq. K This leads to upper triangular

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2) now backsubstitute a) determine x n from b) put x n into n-1 eq. c) solve for x n-1 d) repeat from b), moving back to n-2, n-3, etc. until all equations are solved

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Matlab code

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Operation counting Important as matrix gets large For Gauss elimination elimination routine uses on the order of operations backsubstitution uses

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Problems with Gauss elimination (as done last time) 1) division by zero 2) round off errors 3) ill conditioned systems

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Division by zero Using first eq. to eliminate x1 in second eq. means dividing by 0

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Pivoting developed to avoid this find row with largest absolute value under pivot element switch rows More later

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Round off errors With more than n 3 /3 operations, get a lot of chopping. More important - error is propagted More than 100 equations - round can be very important - system dependent

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Ill conditioned systems - small changes in coefficients lead to large changes in solution Round-off errors especially important in ill- conditioned systems

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Recall ill conditioned system from graphical methods

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Can write as Since slopes are almost equal

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becomes

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Determinant close to zero indicates ill- conditioned set of equations. How close? No clear answer Problem of scale

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Multiply our set of equations by 100

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However, graphically No change

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Scaling Make maximum element in any row =1

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Another problem - singular systems Two equations in the set are the same Determinant is 0.

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Calculating determinant using Gauss elimination Given then

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How to avoid pitfalls higher precision pivoting scaling

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Partial Pivoting Determine the largest coefficient in the column below pivot element Then switch rows (Compete pivot switches columns also, but is rarely used.)

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Example: Exchange rows 2 and 3 And now eliminate

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Algorithm for Gauss elimination using improvements 1) first eliminate for each eq. j, j=1 to n-1 first scale each equation k greater than j then pivot now a) multiply eq. j by a kj /a jj b) subtract the result from eq. k

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The Gauss Jordan method Major difference - eliminate unknowns from all rows, not just subsequent ones Normalize matrix so all entries are 1 Leads to identity matrix instead of upper triangular Backsubstitution is easy

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