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ConditionNumber

ConditionNumber - The Condition Number for a Matrix James...

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The Condition Number for a Matrix James Keesling 1 Condition Numbers In the section we outline the general idea of a condition number . The condition number is a means of estimating the accuracy of a result in a given calculation. The simplest way to convey the idea is to do an example. Suppose that numbers on the computer are given with a certain accuracy ± . So, a given number x is represented roughtly by x + ± . If the number x is represented to machine accuracy on a computer, then the IEEE standards require that | ± | | x | 2 - 52 . Suppose that we are evaluating a function f at the point x . Let us assume that we evaluate the function f completely accurately with the only error being the representation of the number x . Let y = f ( x ) be the exact value of f at the true value of x and let y + δ = f ( x + ± ). Then we have f ( x + ± ) - f ( x ) ± δ ± f 0 ( x ) . We can say that | f 0 ( x ) | is a condition number for the absolute error in the computation of f since | δ | ≈ | f 0 ( x ) |·| ± | . We can also obtain a condition number for the relative error in the computation. From the above inequality we have the following. | δ | ≈ | f 0 ( x ) | · | ± | | δ | | f ( x ) | | f 0 ( x ) | | f ( x ) | | x | · | ± | | x | So, the condition number for the absolute error in computing f at x is | f 0 ( x ) | and the condition number for the relative error in computing f at x is | f 0 ( x ) | | f ( x ) | · | x | . 2 Vector and Matrix Norms To apply this to matrix computations, we will need a measure of the error. This is done by means of vector norms and matrix norms which we now describe. 1

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Let V be an n - dimensional vector space. Let || x || be a function from V to the non- negative real numbers, || · || : V [0 , ). Then || x || is a norm for x V if it satisﬁes the following axioms. (1) || x || = 0 if and only if x = 0 V . (2) || x + y || ≤ || x || + || y || for all x,y V . (3) || λ · x || = | λ | · || x || for all λ R and x V . The norm on an
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ConditionNumber - The Condition Number for a Matrix James...

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