condition - EE103(Fall 2009-10 7 The condition number •...

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Unformatted text preview: EE103 (Fall 2009-10) 7. The condition number • Ax = b when A is singular • condition of a set of linear equations • matrix norm • condition number 7-1 Linear equations with singular coefficient matrix if A is nonsingular, then Ax = b has a unique solution for every b if A is singular, then Ax = b has zero or infinitely many solutions: • for some righthand sides b , the equation Ax = b has no solution for example, we can take a nonzero vector b in the nullspace of A T Ax = b = ⇒ 0 = b T Ax = b T b > hence Ax = b is unsolvable • if Ax = b is solvable, then it has infinitely many solutions: A ( x + tv ) = b for all scalar t , and all nonzero vectors v in the nullspace of A The condition number 7-2 Example A = bracketleftbigg 1 − 1 − 2 2 bracketrightbigg is singular: Ax = 0 for x = (1 , 1) • the equation Ax = b has no solution for b = (1 , 0) • the equation Ax = b has infinitely many solutions for b = (2 , − 4) : bracketleftbigg x 1 x 2 bracketrightbigg = bracketleftbigg 2 + t t bracketrightbigg is a solution, for any t The condition number 7-3 Condition of a set of linear equations assume A is nonsingular and Ax = b if we change b to b + Δ b , the new solution is x + Δ x with A ( x + Δ x ) = b + Δ b the change in x is Δ x = A − 1 Δ b ‘condition’ of the equations: a technical term used to describe how sensitive the solution is to changes in the righthand side...
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