. Thus, . (lhs) Note that since , or . Further, . Combining the two inequalities gives us the lhs. QED Theorem 3.7: Suppose and satisfy , where and are perturbations. Let be x x x x A nonsingular and be so small that . Then for we have A . Note: Theorem 3.7 implies that when is small, small relative changes in and x f cause small changes in A x. Iterative Improvement
. Thus, . (lhs) Note that since , or . Further, . Combining the two inequalities gives us the lhs. QED Theorem 3.7: Suppose x and x satisfy , where x and x are perturbations. Let A be nonsingular and A be so small that . Then for we have . Note: Theorem 3.7 implies that when x is small, small relative changes in f and A cause small changes in x. Iterative Improvement
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