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Unformatted text preview: Lecture 11 Accuracy, Condition Numbers and Pivoting In this lecture we will discuss two separate issues of accuracy in solving linear systems. The first, pivoting , is a method that ensures that Gaussian elimination proceeds as accurately as possible. The second, condition number , is a measure of how bad a matrix is. We will see that if a matrix has a bad condition number, the solutions are unstable with respect to small changes in data. The effect of rounding All computers store numbers as finite strings of binary floating point digits. This limits numbers to a fixed number of significant digits and implies that after even the most basic calculations, rounding must happen. Consider the following exaggerated example. Suppose that our computer can only store 2 significant digits and it is asked to do Gaussian elimination on: parenleftbigg . 001 1 3 1 2 5 parenrightbigg . Doing the elimination exactly would produce: parenleftbigg . 001 1 3 − 998 − 2995 parenrightbigg , but rounding to 2 digits, our computer would store this as: parenleftbigg . 001 1 3 − 1000 − 3000 parenrightbigg . Backsolving this reduced system gives: x 1 = 0 , x 2 = 3 . This seems fine until you realize that backsolving the unrounded system gives: x 1 = − 1 , x 2 = 3 . 001 . Row Pivoting A way to fix the problem is to use pivoting, which means to switch rows of the matrix. Since switching rows of the augmented matrix just corresponds to switching the order of the equations, 39 40 LECTURE 11. ACCURACY, CONDITION NUMBERS AND PIVOTING no harm is done:...
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