Pitfalls of Gauss Elimination ● Division by zero (avoid by pivoting) ● Round-off errors – Important with >100 eqns – Substitute answer into original eqn to check ● Ill conditioned systems – Small change in coefficients large change in sol'n – Determinant close to zero

Pivoting ● Rearrange equations (switch equations) – To avoid zero in pivot element – Partial pivoting ● Find largest coeff available in column below pivot element ● Switch rows so that largest element becomes pivot element – Complete pivoting ● Columns AND rows searched for largest element

Scaling ● Scaling (multiplying by a constant) ● In cases of coefficients with widely different magnitudes ● Can be used to determine if pivoting is necessary

Gauss-Jordan Elimination ● Results in identity matrix (no need for back- substitution) ● Achieved by Normalizing each row by dividing by pivot element before elimination ● Gauss-Jordan involves 50% more operations than Gauss Elimination

Elementary operations (matrix) ● Any equation can be multiplied (or divided) by a nonzero scalar ● Any equation can be added to (or subtracted from) another equation ● The positions of any two equations in the system can be interchanged