Lecture 7 Conditioning of Problems Stability of Algorithms AMath 352 Mon Apr 12 1 11 Usually interested in relative error in computed value y y y If answer is a vector use norms cid 107 x cid 126 x cid 107 cid 107 cid 126 x cid 107 Conditioning of Problems Stability of Algorithms Errors of many sorts in scienti c computations 1 Replace physical problem by mathematical model 2 Replace mathematical model by one that is suitable for numerical solution e g truncate a Taylor series 3 Input data may come from inexact measurements 4 Rounding errors We will deal with 2 4 2 11 Conditioning of Problems Stability of Algorithms Errors of many sorts in scienti c computations 1 Replace physical problem by mathematical model 2 Replace mathematical model by one that is suitable for numerical solution e g truncate a Taylor series 3 Input data may come from inexact measurements 4 Rounding errors We will deal with 2 4 Usually interested in relative error in computed value y y y If answer is a vector use norms cid 107 x cid 126 x cid 107 cid 107 cid 126 x cid 107 2 11 Absolute condition no y y C x x x y y f x f x x x f cid 48 x C x f cid 48 x Relative condition no y y y x x x x y y f x f x x x f cid 48 x x x xf cid 48 x y f x f x x f x x cid 12 cid 12 cid 12 cid 12 xf cid 48 x f x cid 12 cid 12 cid 12 cid 12 Conditioning of Problems Want to evaluate y f x If x is close to x is y f x close to y Note that this is a question about the problem not about any algorithm used to solve the problem 3 11 Conditioning of Problems Want to evaluate y f x If x is close to x is y f x close to y Note that this is a question about the problem not about any algorithm used to solve the problem Absolute condition no y y C x x x y y f x f x x x f cid 48 x C x f cid 48 x Relative condition no y y y x x x x y y y f x f x f x x x f cid 48 x f x x x x xf cid 48 x f x x cid 12 cid 12 cid 12 cid 12 xf cid 48 x f x cid 12 cid 12 cid 12 cid 12 3 11 2 f x x f cid 48 x 1 2 x 1 2 x 0 C x 1 2 x 1 2 Ill conditioned in absolute sense near x 0 For example if x 10 16 and x 1 21 10 16 then f x 10 8 and f x 1 1 10 8 so f x f x cid 0 1 2 108 cid 1 x x cid 12 cid 12 cid 12 x 1 2 x 1 2 x cid 12 cid 12 cid 12 1 x sense 2 Well conditioned in relative Examples 1 f x 2x f cid 48 x 2 C x f cid 48 x 2 Well conditioned in absolute sense x 2x 1 Well conditioned in relative sense cid 12 cid 12 2x cid 12 xf cid 48 x f x cid 12 cid 12 cid 12 4 11 Examples 1 f x 2x f cid 48 x 2 2 f x C x f cid 48 x 2 Well conditioned in absolute sense x 2x 1 Well conditioned in relative sense cid 12 cid 12 2x cid 12 xf cid 48 x f x cid 12 cid 12 cid 12 x f cid 48 x 1 2 x 1 2 2 x 1 2 x 0 C x 1 Ill conditioned in absolute sense near x 0 For example if x 10 16 and x 1 21 10 16 then f x 10 8 and f x 1 1 10 8 so f x f x 2 108 cid 1 x x cid 0 1 cid 12 cid 12 cid 12 2 Well conditioned in relative x 1 2 x 1 2 x cid 12 cid 12 1 cid 12 x sense 4 11 3 Solve A cid 126 y cid 126 b where cid 126 b and maybe A are input cid 107 y cid 126 y cid 107 cid 107 cid 126 y cid 107 cid 126 b cid 107 b cid 126 b cid 107 cid 107 cid 126 b cid 107 Examples 3 f x sin x f cid 48 x cos x C x cos x 1 Well conditioned in absolute sense x cos x sin x x sense if x is near 2 cid 12 cid 12 cid 12 x cot x Ill conditioned in relative cid 12 cid 12 cid 12 5 11 Examples 3 f x sin x f cid 48 x cos x C x cos x 1 Well conditioned in absolute sense x cos x sin x x sense if x is near 2 cid 12 cid 12 cid 12 x cot x Ill conditioned in relative cid 12 cid 12 cid 12 3 Solve A cid 126 y cid 126 b where cid 126 b and maybe A are input cid 107 y cid 126 y cid 107 cid 107 cid 126 y cid 107 cid 126 b cid 107 b cid 126 b cid 107 cid 107 cid 126 b cid 107 5 11 Forward error analysis How much does the computed value di er from the exact solution Backward error analysis Is the computed value the exact solution to a nearby problem If problem is ill conditioned cannot expect to get close to the exact solution the exact solution with rounded input values might be very di erent from that with the true input values But if algorithm delivers the exact solution to a problem with nearby input values then this is the best one can do Stability of Algorithms Suppose we have a well conditioned problem and an algorithm for solving it Will our algorithm give the answer to the expected no of places when implemented in nite precision arithmetic 6 11 Backward error analysis Is the computed value the exact solution to a nearby problem If problem is ill conditioned cannot expect to get close to the exact solution the exact solution with rounded input values might be very di erent from that with the true input values But if algorithm delivers the exact solution to a problem with nearby input values then this is the best one can do Stability of Algorithms Suppose we have a well conditioned problem and an algorithm for solving it Will our algorithm give the answer to the expected no of places when implemented in nite precision arithmetic Forward error analysis How much does the computed value di er from the exact solution 6 11 Stability of Algorithms …