lecture6 - Numerical Methods in Engineering ENGR 391 System...

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Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University System of linear equations
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Pitfalls of Gauss Elimination Methods Division by zero It is possible that during both elimination and back-substitution phases a division by zero can occur. Here we have a division by zero if we replace in the above formula for Gauss elimination (the same thing will happen if we use a very small number). The pivoting technique has been developed to avoid this problem. ° ¯ ° ® ­ ± ± ± ± ± ± 7 5 1 2 5 1 3 7 8 5 4 0 3 2 1 3 2 1 3 2 1 x x x x x x x x x Numerical problems
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Pitfalls of Gauss Elimination Methods Round-off errors The large number of equations to be solved induces a propagation of the errors. A rough rule of thumb is that round-off error may be important when dealing with 100 or more equations. Always substitute your answers back into the original equations to check whenever a substantial errors has occurred
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ill-conditioned systems Systems where small changes in coefficients result in large changes in the solution. It happens when two or more equations are nearly identical, resulting a wide ranges of answers to approximately satisfy the equations.
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