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

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

Matlab code

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

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

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

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

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

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

Recall ill conditioned system from graphical methods

Can write as Since slopes are almost equal

becomes

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

Multiply our set of equations by 100

However, graphically No change

Scaling Make maximum element in any row =1

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

Calculating determinant using Gauss elimination Given then

How to avoid pitfalls higher precision pivoting scaling

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

Example: Exchange rows 2 and 3 And now eliminate

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

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