Even direct algorithms have been affected by the new manner of omputing Thanks to the work of Skeel and others,it has been noticed that the expense of making a direct method say,of pivoting in Gaussian elimination-may in certain contexts be cost-ineffective. Instead,skip that step-solve the problem directly but unstably,then do one or two steps of iterative refinement."Exact"Gaussian elimination becomes iust another preconditioner! Other problems besides Az=bhave undergone analogous changes,and the famous example is linear programming. decades that them by a the simplex metho od ntire occupied towar s I believe that the existence of finite algorithms for certain problems.together with other historical forces,has distracted us for decades from a balanced view of numerical analysis Rounding errors and instability are important,and numerical analysts will always be the experts in these subjects and at pains to ensure that the unwary are not tripped up by them.But our central mission is to compute quant ties that are typically uncomputable ieinadton it with lig ion ard and Rokhin's()m ltipole algorithm for particle simulations or the exponential conver solving certain PDEs-or the convergence in O(1)iteration achieved by multigrid methods for many kinds of problems-or even Borwein and Borwein's magical AGM iteration for determining 1,000,000 digits of in the blink of an eye.That is the heart of numerical analysis. Notes us to n c执r你ea30weeo,品 i do not claim that any of the ideas expr constructive methods in mathematical analysis."Others have expressed similar views;Joseph Traub(Communications of the ACM,1972),for example,defined numerical analysis as "the analysis of continuous algorithms."For that matter,both the Random House and the Oxford English dictionaries offer better definitions than the three quoted here. ￾✁✄✆✞✟✡ ✟✌✍✏✑ ✞✓ ✕✗ ✄✗✟ ￾ ✍✜ ✜ ✢ ✕✗✜ ✑ ✟ ✍✩ ✪ ✜ ✄✞✭✟ ✕✑ ✍￾￾✁✄✌✕✡✍✑ ✟✞✓✪ ✑✴✟✗ ✕ ✑ ✟✁ ✍✑ ✟ ✷ ✸✺✞✑ ✕✼✁✕✩ ✡✟✑✴✄ ✩✜ ✪ ￾ ✟✁✴✍￾✜ ✑✴✟ ✡✄ ✜ ✑ ✕✡￾ ✄✁✑ ✍✗✑ ✩✟✭✟✞✄￾✡✟✗✑ ✕✗ ✗✺✡✟✁✕ ✏ ✍✞ ✏✄✡￾✺✑ ✍✑ ✕✄✗ ✕✗ ✑✴✟ ￾ ✍✜ ✑ ✑❈✟✗✑✓ ✓✟ ✍✁✜ ✪ ✍✁✟ ✆ ✍✜ ✟✩ ✄✗ ✍ ✁ ✟ ✏✺✁✜ ✕✭✟ ✍￾￾✞ ✕ ✏ ✍✑ ✕ ✄✗ ✄❍ ✑✴✕ ✜ ✕✩✟ ✍ ✷ ❏✭✟✗ ✩✕✁✟ ✏✑ ✍✞✼✄✁✕✑✴✡✜ ✴✍✭✟ ✆ ✟ ✟✗ ✍◆✟ ✏✑ ✟✩ ✆✓ ✑✴✟ ✗✟❈ ✡✍✗✗✟✁ ✄❍ ✏✄✡￾✺✑ ✕✗✼ ✷ ❘✴✍✗❙✜ ✑ ✄ ✑✴✟ ❈✄✁❙ ✄❍ ❱❙✟ ✟✞ ✍✗✩ ✄✑✴✟✁✜ ✪ ✕✑ ✴✍✜ ✆ ✟ ✟✗ ✗✄✑ ✕ ✏ ✟✩ ✑✴✍✑ ✑✴✟ ✟✌￾ ✟✗✜ ✟ ✄❍ ✡✍❙✕✗✼ ✍ ✩✕✁✟ ✏✑ ✡✟✑✴✄ ✩ ✜ ✑ ✍✆✞✟❜✜ ✍✓✪ ✄❍ ￾✕✭✄✑ ✕✗✼ ✕✗ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗❜✡✍✓ ✕✗ ✏ ✟✁✑ ✍✕✗ ✏ ✄✗✑ ✟✌✑ ✜ ✆ ✟ ✏✄ ✜ ✑ ✐ ✕✗✟◆✟ ✏✑ ✕✭✟ ✷ ❦✗✜ ✑ ✟ ✍✩ ✪ ✜❙✕￾ ✑✴✍✑ ✜ ✑ ✟￾❜✜ ✄✞✭✟ ✑✴✟ ￾✁✄✆✞ ✟✡ ✩✕✁ ✟ ✏✑ ✞✓ ✆✺✑ ✺✗✜ ✑ ✍✆✞✓✪ ✑✴✟✗ ✩✄ ✄✗✟ ✄✁ ✑❈✄ ✜ ✑ ✟￾✜ ✄❍ ✕ ✑ ✟✁✍✑ ✕✭✟ ✁ ✟♥✗✟✡✟✗✑ ✷ ♦❏✌✍✏✑ ♣ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✆ ✟ ✏ ✄✡✟ ✜ q ✺✜ ✑ ✍✗✄✑✴✟✁ ￾✁✟ ✏ ✄✗✩✕ ✑ ✕✄✗✟✁ r s✑✴✟✁ ￾✁✄✆✞✟✡✜ ✆ ✟ ✜ ✕✩✟ ✜ t✉ ✈ ✇ ✴✍✭✟ ✺✗✩✟✁✼✄✗✟ ✍✗✍✞✄✼✄✺✜ ✏✴✍✗✼✟ ✜ ✪ ✍✗✩ ✑✴✟ ❍✍✡✄✺✜ ✟✌✍✡￾✞ ✟ ✕ ✜ ✞ ✕✗✟ ✍✁ ￾✁✄✼✁ ✍✡✡✕✗✼ ✷ ②✕✗✟ ✍✁ ￾✁✄✼✁✍✡✡✕✗✼ ￾✁✄✆✞ ✟✡✜ ✍✁✟ ✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞✞✓ ♥✗✕✑ ✟ ✪ ✍✗✩ ❍✄✁ ✩✟ ✏ ✍✩✟ ✜ ✪ ￾ ✟✄￾✞✟ ✜ ✄✞✭✟✩ ✑✴✟✡ ✆✓ ✍ ♥✗✕✑ ✟ ✍✞✼✄✁✕ ✑✴✡⑦ ✑✴✟ ✜ ✕✡￾✞ ✟✌ ✡✟✑✴✄ ✩ ✷ ❘✴✟✗ ⑨✍✁✡✍✁❙✍✁ ✍✗✗✄✺✗✏ ✟✩ ✕✗ ⑩ ❶ ❷ ❸ ✑✴✍✑ ✕ ✑ ✟✁ ✍✑ ✕✭✟ ✪ ✕✗♥✗✕ ✑ ✟ ✍✞✼✄✁✕✑✴✡✜ ✍✁✟ ✜ ✄✡✟✑ ✕✡✟ ✜ ✆ ✟✑ ✑ ✟✁ ✷ ❘✴✟ ✁✟ ✜✺✞✑ ✴✍✜ ✆ ✟ ✟✗ ✏ ✄✗✑ ✁✄✭✟✁✜✓✪ ✕✗✑ ✟✞✞ ✟ ✏✑✺✍✞ ✟✌✏✕✑ ✟✡✟✗✑ ✪ ✍✗✩ ✍ ￾ ✟✁✏ ✟￾✑ ✕✆✞ ✟ ✜✴✕❍✑ ✄❍ ✑✴✟ ✟✗✑ ✕✁✟ ♥✟✞✩ ✄❍ ✞ ✕✗✟ ✍✁ ￾✁✄✼✁✍✡✡✕✗✼ ✍❈✍✓ ❍✁✄✡ ✑✴✟ ✁ ✍✑✴✟✁ ✍✗✄✡✍✞✄✺✜ ￾ ✄ ✜ ✕ ✑ ✕✄✗ ✕ ✑ ✴✍✜ ✑ ✁✍✩✕ ✑ ✕✄✗✍✞✞✓ ✄ ✏ ✏✺￾✕ ✟✩ ✑ ✄❈✍✁✩✜ ✑✴✟ ✡✍✕✗✜ ✑ ✁ ✟ ✍✡ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✏✄✡￾✺✑ ✍✑ ✕ ✄✗ ✷ ❦ ✆ ✟✞✕ ✟✭✟ ✑✴✍✑ ✑✴✟ ✟✌✕ ✜ ✑ ✟✗✏ ✟ ✄❍ ♥✗✕ ✑ ✟ ✍✞✼✄✁✕✑✴✡✜ ❍✄✁ ✏ ✟✁✑ ✍✕✗ ￾✁✄✆✞✟✡✜ ✪ ✑ ✄✼✟✑✴✟✁ ❈✕✑✴ ✄✑✴✟✁ ✴✕ ✜ ✑ ✄✁✕ ✏ ✍✞ ❍✄✁✏ ✟ ✜ ✪ ✴✍✜ ✩✕ ✜ ✑ ✁✍✏✑ ✟✩ ✺✜ ❍✄✁ ✩✟ ✏ ✍✩✟ ✜ ❍✁✄✡ ✍ ✆ ✍✞ ✍✗✏ ✟✩ ✭✕ ✟❈ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✷ ❿✄✺✗✩✕✗✼ ✟✁✁✄✁ ✜ ✍✗✩ ✕✗✜ ✑ ✍✆✕ ✞ ✕ ✑✓ ✍✁✟ ✕✡￾ ✄✁✑ ✍✗✑ ✪ ✍✗✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ❈✕ ✞✞ ✍✞❈✍✓✜ ✆ ✟ ✑✴✟ ✟✌￾ ✟✁✑ ✜ ✕✗ ✑✴✟ ✜ ✟ ✜✺✆q ✟ ✏✑ ✜ ✍✗✩ ✍✑ ￾ ✍✕✗✜ ✑ ✄ ✟✗✜✺✁✟ ✑✴✍✑ ✑✴✟ ✺✗❈✍✁✓ ✍✁✟ ✗✄✑ ✑ ✁✕￾￾ ✟✩ ✺￾ ✆✓ ✑✴✟✡✷ ➄✺✑ ✄✺✁ ✏ ✟✗✑ ✁✍✞ ✡✕ ✜ ✜ ✕ ✄✗ ✕ ✜ ✑ ✄ ✏ ✄✡￾✺✑ ✟ ➅✺✍✗✑ ✕ ✑ ✕ ✟ ✜ ✑✴✍✑ ✍✁ ✟ ✑✓￾✕ ✏ ✍✞ ✞✓ ✺✗✏ ✄✡￾✺✑ ✍✆✞ ✟ ✪ ❍✁✄✡ ✍✗ ✍✗✍✞✓✑ ✕ ✏ ✍✞ ￾ ✄✕✗✑ ✄❍ ✭✕ ✟❈✪ ✍✗✩ ✑ ✄ ✩✄ ✕✑ ❈✕ ✑✴ ✞ ✕✼✴✑✗✕✗✼ ✜￾ ✟ ✟✩ ✷ ➇✄✁ ✼✺✕✩✍✗✏ ✟ ✑ ✄ ✑✴✟ ❍✺✑✺✁✟ ❈✟ ✜✴✄✺✞✩ ✜ ✑✺✩✓ ✗✄✑ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✍✗✩ ✕ ✑ ✜ ✆ ✟✼✺✕ ✞ ✕✗✼ ✜ ✑ ✍✆✕ ✞ ✕✑✓ ￾✁✄￾ ✟✁✑ ✕ ✟ ✜ ✪ ✆✺✑ ✑✴✟ ✩✕ ✍✆ ✄✞✕ ✏ ✍✞✞✓ ❍✍✜ ✑ ✏✄✗q ✺✼✍✑ ✟ ✼✁ ✍✩✕ ✟✗✑ ✕✑ ✟✁✍✑ ✕ ✄✗❜✄✁ ❢✁✟ ✟✗✼✍✁✩ ✍✗✩ ❿✄❙✴✞✕✗ ➉ ✜ ➋ ➌➍➎ ✡✺✞✑ ✕￾ ✄✞ ✟ ✍✞✼✄✁✕ ✑✴✡ ❍✄✁ ￾ ✍✁✑ ✕ ✏✞ ✟ ✜ ✕✡✺✞ ✍✑ ✕ ✄✗✜❜✄✁ ✑✴✟ ✟✌￾ ✄✗✟✗✑ ✕ ✍✞ ✏✄✗✭✟✁✼✟✗✏ ✟ ✄❍ ✜￾ ✟ ✏✑ ✁✍✞ ✡✟✑✴✄ ✩✜ ❍✄✁ ✜ ✄✞✭✕✗✼ ✏ ✟✁✑ ✍✕✗ ➏➐❏✜❜✄✁ ✑✴✟ ✏✄✗✭✟✁✼✟✗✏ ✟ ✕✗ ➋ ➌ ⑩ ➎ ✕ ✑ ✟✁ ✍✑ ✕ ✄✗ ✍✏✴✕ ✟✭✟✩ ✆✓ ✡✺✞✑ ✕✼✁✕✩ ✡✟✑✴✄ ✩✜ ❍✄✁ ✡✍✗✓ ❙✕✗✩✜ ✄❍ ￾✁✄✆✞✟✡✜❜✄✁ ✟✭✟✗ ➄✄✁❈✟✕✗ ✍✗✩ ➄✄✁❈✟✕✗ ➉ ✜ ✡✍✼✕ ✏ ✍✞ ➔❢✸ ✕ ✑ ✟✁ ✍✑ ✕ ✄✗ ❍✄✁ ✩✟✑ ✟✁✡✕✗✕✗✼ ⑩ ✪ → → → ✪ → → → ✩✕✼✕ ✑ ✜ ✄❍ ➣ ✕✗ ✑✴✟ ✆✞ ✕✗❙ ✄❍ ✍✗ ✟✓✟ ✷ ↔➙ ➛ ➜ ✕ ✜ ✑✴✟ ✴✟ ✍✁✑ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✷ ➝➞➟ ➠➡ ✸✍✗✓ ￾ ✟ ✄￾✞ ✟ ✪ ✑ ✄ ✄ ✗✺✡✟✁✄✺✜ ✑ ✄ ✗✍✡✟ ✪ ￾✁✄✭✕✩✟✩ ✏ ✄✡✡✟✗✑ ✜ ✄✗ ✩✁✍❍✑ ✜ ✄❍ ✑✴✕ ✜ ✟ ✜ ✜ ✍✓✷ ❘✴✟✕✁ ✜✺✼✼✟ ✜ ✑ ✕✄✗✜ ✞✟✩ ✡✟ ✑ ✄ ✡✍✗✓ ￾✺✆✞ ✕ ✏ ✍✑ ✕ ✄✗✜ ✑✴✍✑ ❦ ❈✄✺✞✩ ✄✑✴✟✁❈✕ ✜ ✟ ✗✄✑ ✴✍✭✟ ❍✄✺✗✩ ✷ ❦ ✩✄ ✗✄✑ ✏✞ ✍✕✡ ✑✴✍✑ ✍✗✓ ✄❍ ✑✴✟ ✕✩✟ ✍✜ ✟✌￾✁✟ ✜ ✜ ✟✩ ✴✟✁✟ ✍✁✟ ✟✗✑ ✕✁✟✞✓ ✗✟❈✷ ❦✗ ❍✍✏✑ ✪ ➤ → ✓✟ ✍✁✜ ✍✼✄ ✪ ✕✗ ✴✕ ✜ ➥➦➧➩➧ ➫➜➭ ➯➲ ➳➵➩➧ ➸➺ ➻➛ ➦ ➼➫ ➛ ➦➾➭ ➺ ➭ ➪ ➏✟✑ ✟✁ ➶✟✗✁✕ ✏✕ ✩✟♥✗✟✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✍✜ ♦✑✴✟ ✑✴✟ ✄✁✓ ✄❍ ✏ ✄✗✜ ✑ ✁✺✏✑ ✕✭✟ ✡✟✑✴✄ ✩✜ ✕✗ ✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✷ ♣ s✑✴✟✁✜ ✴✍✭✟ ✟✌￾✁✟ ✜ ✜ ✟✩ ✜ ✕✡✕ ✞ ✍✁ ✭✕ ✟❈✜ ✢ ➹ ✄ ✜ ✟￾✴ ❘✁ ✍✺✆ ➌ ➘➯➩➩➵➫➺ ➻➛ ➜➺ ➯➫➭ ➯➲ ➜➙ ➧ ➼ ➘➴➪ ➷ ➬ ➮➱➎ ✪ ❍✄✁ ✟✌✍✡￾✞ ✟ ✪ ✩✟♥✗✟✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✍✜ ♦✑✴✟ ✍✗✍✞✓✜ ✕ ✜ ✄❍ ✏ ✄✗✑ ✕✗✺✄✺✜ ✍✞✼✄✁✕✑✴✡✜ ✷ ♣ ➇✄✁ ✑✴✍✑ ✡✍✑ ✑ ✟✁ ✪ ✆ ✄✑✴ ✑✴✟ ❿✍✗✩✄✡ ➶✄✺✜ ✟ ✍✗✩ ✑✴✟ s✌❍✄✁✩ ❏✗✼✞ ✕ ✜✴ ✩✕ ✏✑ ✕✄✗✍✁✕ ✟ ✜ ✄◆✟✁ ✆ ✟✑ ✑ ✟✁ ✩✟♥✗✕ ✑ ✕✄✗✜ ✑✴✍✗ ✑✴✟ ✑✴✁ ✟ ✟ ➅✺✄✑ ✟✩ ✴✟✁✟ ✷ ➔✗✩ ✜✴✄✺✞✩ ✑✴✟ ♥✟✞✩ ✆ ✟ ✏ ✍✞✞ ✟✩ ♦✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✪ ♣ ♦ ✜ ✏✕ ✟✗✑ ✕♥✏ ✏ ✄✡￾✺✑ ✕✗✼ ✪ ♣ ✄✁ ✜ ✄✡✟✑✴✕✗✼ ✟✞ ✜ ✟ ✟✗✑ ✕✁✟✞✓❐ ➌ ♦✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞ ✟✗✼✕✗✟ ✟✁✕✗✼❐ ♣ ➎ ✷ ❘✴✍✑ ✕ ✜ ✍✗✄✑✴✟✁ ✟ ✜ ✜ ✍✓✷ ❒