[The following essay appeared in the November,1992 issue of SIAM News and the March, 1993 issue of the Bulletin of the Institute for Mathematics and Applications.] THE DEFINITION OF NUMERICAL ANALYSIS Lloyd N.Trefethen Dept.of Co nputer Science 1992 What is numerical analysis?I believe that this is more than a philosophical question.A certain wrong answer has taken hold among both outsiders to the field and insiders,distorting the image of a subject at the heart of the mathematical sciences. Here is the wrong answer: Numerical analysis is the study of rounding errors. (D1) that it v ould be hard to devis but the fundamental.If(D1)is a common perception,it is hardly surprising that numerical analysis is widely regarded as an unglamorous subject.In fact,mathematicians,physicists,and computer scient I tended to hold numerical analysis in low esteem for many years-a most asserts (DI)quite as But consider the following om some Isaacson&Keller(1966):1.Norms,arithmetic,and well-posed computations. Hamming (1971):1.Roundoff and function evaluation. Dahlquist&Bjorck (1974):1.Some general principles of numerical calculation. 2.How to obtain and estimate accuracy... Stor&Bulirsch(980：1.Error analysis, Conte&de Boor (1980):1.Number systems and errors. Atkinson (1987):1.Error:its sources,propagation,and analysis Kahaner,Moler Nash(1989):1.Introduction 2.Computer arithmetic and computational errors. comp metic on opening sucn b
❘✴✟ ❍✄✞ ✞✄❈✕✗✼ ✟ ✜ ✜ ✍✓ ✍ ✟ ✍✁✟✩ ✕✗ ✑✴✟ ☎✄✭✟✡✆ ✟✁ ✪ ⑩ ❶ ❶ ✂ ✕ ✜ ✜✺✟ ✄❍ ✁✄➼➴ ➳➧ ✲➭ ✍✗✩ ✑✴✟ ✸✍✁✏✴ ✪ ⑩ ❶ ❶ ➤ ✕ ✜ ✜✺✟ ✄❍ ✑✴✟ ➵➦ ➦➧ ➜➺➫ ➯➲ ➜➙ ➧ ✄➫➭ ➜➺ ➜➵➜ ➧ ➲➯ ➸ ➴➛➜➙ ➧➩➛➜➺ ➻ ➭ ➛➫ ★ ➼✥✥ ➦➺ ➻➛ ➜➺ ➯➫➭ ✷ ✝ ✞✟✠ ☛✠✍✏➝✏✞✏✔➝ ✔✍ ➝✗✘✠✚✏✜✣✤ ✣➝✣✤✦★✏ ★ ②✞✄✓✩ ☎ ✷ ❘✁✟❍✟✑✴✟✗ ➐✟ ✑ ✷ ✄❍ ✄✡ ✺✑ ✟✁ ❱ ✏✕ ✟✗✏ ✟ ✄✁✗✟✞✞ ✣✗✭✕ ✟✁✜ ✕✑✓ ②☎❘✫✏ ✜ ✷ ✏ ✄✁✗✟✞ ✞ ✷ ✟✩✺ ⑩ ❶ ❶ ✂ ✟✴✍✑ ✕ ✜ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ❐ ❦ ✆ ✟✞ ✕ ✟✭✟ ✑✴✍✑ ✑✴✕ ✜ ✕ ✜ ✡✄✁✟ ✑✴✍✗ ✍ ✴✕ ✞✄ ✜ ✄ ✴✕ ✏ ✍✞ ➅✺✟ ✜ ✑ ✕ ✄✗ ✷ ➔ ✏ ✟✁✑ ✍✕✗ ❈✁✄✗✼ ✍✗✜❈✟✁ ✴✍✜ ✑ ✍❙✟✗ ✴✄✞✩ ✍✡✄✗✼ ✆ ✄✑✴ ✄✺✑ ✜ ✕✩✟✁ ✜ ✑ ✄ ✑✴✟ ♥✟✞✩ ✍✗✩ ✕✗✜ ✕✩✟✁✜ ✪ ✩✕ ✜ ✑ ✄✁✑ ✕✗✼ ✑✴✟ ✕✡✍✼✟ ✄❍ ✍ ✜✺✆ ✟ ✏✑ ✍✑ ✑✴✟ ✴✟ ✍✁✑ ✄❍ ✑✴✟ ✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞ ✜ ✏✕ ✟✗✏ ✟ ✜ ✷ ➶✟✁✟ ✕ ✜ ✑✴✟ ❈✁✄✗✼ ✍✗✜❈✟✁ ⑦ ➳➵➩➧ ➸➺ ➻➛➦ ➛➫ ➛ ➦➾➭ ➺ ➭ ➺ ➭ ➜➙ ➧ ➭ ➜➵ ★➾ ➯➲ ➸➯ ➵➫ ★➺➫✪ ➧ ➸➸➯ ➸➭ ✭ ➌➐ ⑩ ➎ ❘✴✟ ✁✟ ✍✩✟✁ ❈✕ ✞ ✞ ✍✼✁✟ ✟ ✑✴✍✑ ✕✑ ❈✄✺✞✩ ✆ ✟ ✴✍✁✩ ✑ ✄ ✩✟✭✕ ✜ ✟ ✍ ✡✄✁✟ ✺✗✕✗✭✕ ✑ ✕✗✼ ✩✟ ✜ ✏✁✕ ✑ ✕ ✄✗ ✄❍ ✍ ♥✟✞✩ ✷ ❿✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✍✁✟ ✕✗✟✭✕ ✑ ✍✆✞ ✟ ✪ ✓✟ ✜ ✪ ✆✺✑ ✑✴✟✓ ✍✁✟ ✏ ✄✡ ✞✕ ✏ ✍✑ ✟✩ ✍✗✩ ✑ ✟✩✕ ✄✺✜ ✍✗✩ ❜➫ ➯ ➜ ➲➵➫ ★ ➛➩➧ ➫➜ ➛ ➦✭ ❦❍ ➌➐ ⑩ ➎ ✕ ✜ ✍ ✏ ✄✡✡✄✗ ✟✁✏ ✟ ✑ ✕✄✗ ✪ ✕ ✑ ✕ ✜ ✴✍✁✩✞✓ ✜✺✁ ✁✕ ✜ ✕✗✼ ✑✴✍✑ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕ ✜ ❈✕✩✟✞✓ ✁ ✟✼✍✁✩✟✩ ✍✜ ✍✗ ✺✗✼✞ ✍✡✄✁✄✺✜ ✜✺✆ ✟ ✏✑ ✷ ❦✗ ❍✍✏✑ ✪ ✡✍✑✴✟✡✍✑ ✕ ✏✕ ✍✗✜ ✪ ✴✓✜ ✕ ✏✕ ✜ ✑ ✜ ✪ ✍✗✩ ✏✄✡ ✺✑ ✟✁ ✜ ✏✕ ✟✗✑ ✕ ✜ ✑ ✜ ✴✍✭✟ ✍✞ ✞ ✑ ✟✗✩✟✩ ✑ ✄ ✴✄✞✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕✗ ✞✄❈ ✟ ✜ ✑ ✟ ✟✡ ❍✄✁ ✡✍✗✓ ✓✟ ✍✁✜❜✍ ✡✄ ✜ ✑ ✺✗✺✜✺✍✞ ✏ ✄✗✜ ✟✗✜✺✜ ✷ ❍ ✏ ✄✺✁✜ ✟ ✗✄✆ ✄ ✩✓ ✆ ✟✞✕ ✟✭✟ ✜ ✄✁ ✍✜ ✜ ✟✁✑ ✜ ➌➐ ⑩ ➎ ➅✺✕✑ ✟ ✍✜ ✆ ✍✞✩✞✓ ✍✜ ❈✁✕ ✑ ✑ ✟✗ ✷ ➄✺✑ ✏ ✄✗✜ ✕✩✟✁ ✑✴✟ ❍✄✞ ✞✄❈✕✗✼ ✄ ✟✗✕✗✼ ✏✴✍ ✑ ✟✁ ✴✟ ✍✩✕✗✼✜ ❍✁✄✡ ✜ ✄✡✟ ✜ ✑ ✍✗✩✍✁✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✑ ✟✌✑ ✜ ⑦ ✭✆✜✜✦✆✓★ ✮ ✰✁✕✕ ✁✡ ✫✭ ✯ ✱ ✱✴ ✵ ⑩ ✷ ☎✄✁✡✜ ✪ ✍✁✕✑✴✡✟✑ ✕ ✏ ✪ ✍✗✩ ❈✟✞ ✞ ✐ ✄ ✜ ✟✩ ✏ ✄✡ ✺✑ ✍✑ ✕✄✗✜ ✷ ❦✜❑❑✛★✙ ✫✭ ✯ ✰✭ ✴ ✵ ⑩ ✷ ❿✄✺✗✩✄◆ ✍✗✩ ❍✺✗✏✑ ✕ ✄✗ ✟✭✍✞✺✍✑ ✕ ✄✗ ✷ ✣✜✻✕❄ ✿✛✆ ✞ ✮ ✵✷✓✹✡✦♦ ✫✭ ✯ ✰✻✴ ✵ ⑩ ✷ ❱ ✄✡✟ ✼✟✗✟✁✍✞ ✁✕✗✏✕ ✞ ✟ ✜ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✏ ✍✞ ✏✺✞ ✍✑ ✕✄✗ ✷ ✂ ✷ ➶✄❈ ✑ ✄ ✄✆✑ ✍✕✗ ✍✗✩ ✟ ✜ ✑ ✕✡✍✑ ✟ ✍✏ ✏✺✁ ✍✏✓ 2 2 2 ✷ ✽✞ ✓ ✁✡ ✮ ✵✿✕✛✡✆✦✻ ✫✭ ✯❞❛✴ ✵ ⑩ ✷ ❏✁✁✄✁ ✍✗✍✞✓✜ ✕ ✜ ✷ ✑✓★✞ ✁ ✮ ❁✁ ✵✓ ✓✡ ✫✭ ✯❞❛✴ ✵ ⑩ ✷ ☎✺✡✆ ✟✁ ✜✓✜ ✑ ✟✡✜ ✍✗✩ ✟✁✁✄✁ ✜ ✷ ✐✞♦✛★✆✓★ ✫✭ ✯❞ ✰✴ ✵ ⑩ ✷ ❏✁✁✄✁ ⑦ ✕✑ ✜ ✜ ✄✺✁✏ ✟ ✜ ✪ ✁✄ ✍✼✍✑ ✕ ✄✗ ✪ ✍✗✩ ✍✗✍✞✓✜ ✕ ✜ ✷ ✰✜✻ ✜★ ✁✡❤ ✾✓✕✁✡ ✮ ✍✜✆✻ ✫✭ ✯❞ ✯✴ ✵ ⑩ ✷ ❦✗✑ ✁✄ ✩✺✏✑ ✕ ✄✗ ✷ ✂ ✷ ✄✡ ✺✑ ✟✁ ✍✁✕✑✴✡✟✑ ✕ ✏ ✍✗✩ ✏ ✄✡ ✺✑ ✍✑ ✕✄✗✍✞ ✟✁✁✄✁✜ ✷ ♦❏✁✁✄✁ ♣ 2 2 2 ♦✁✄✺✗✩✄◆♣ 2 2 2 ♦ ✏✄✡ ✺✑ ✟✁ ✍✁✕ ✑✴✡✟✑ ✕ ✏ ♣ ❜ ✑✴✟ ✜ ✟ ✍✁✟ ✑✴✟ ❈✄✁✩✜ ✑✴✍✑ ❙✟ ✟ ✁ ✟ ✍ ✟ ✍✁✕✗✼ ✷ ✟✴✍✑ ✕✡ ✁✟ ✜ ✜ ✕ ✄✗ ✩✄ ✟ ✜ ✍✗ ✕✗➅✺✕ ✜ ✕ ✑ ✕✭✟ ✏ ✄✞✞ ✟✼✟ ✜ ✑✺✩✟✗✑ ✼✟✑ ✺ ✄✗ ✄ ✟✗✕✗✼ ✜✺✏✴ ✆ ✄ ✄❙✜ ❐ ✁ ✏✄✗✜ ✕✩✟✁ ✑✴✟ ✩✟♥✗✕✑ ✕ ✄✗✜ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕✗ ✜ ✄✡✟ ✩✕ ✏✑ ✕✄✗✍✁✕ ✟ ✜ ⑦ ⑩
Webster's New Collegiate Dictionary (1973):"The study of quantitative approxi matotheofhcoemofth errors and bounds to the errors involved. Chambers20th Century Dictionary(1983):"The study of methods of approximation and their accuracy,etc. Themer Dct()or roximate solutions to mathematical problems,taking into account the extent of p T+ ne that these definitions would serve most effectively to deter the curious from investigating further The singular value decomposition (SVD)affords another example of the perception of nu- merical analysis as the science of rounding errors.Although the roots of the SVD go back more than 100 years,it is mainly since the 1960s,through the work of Gene Golub and other ental an ide t has ax e e Its pre of pro D is a ll kinds of c nd av Yet today.thirty vears later.most mathematical scientists and even many applied mathe maticians do not have a working knowledge of the SVD.Most of them have heard of it,but the impress on se ems to be widespread that the SVD is just a too for combating rounding othTs a I am convinced that consciously or unconsciously,many people think that(D1)is at least half true.In actuality,it is a very small part of the truth.And although there are historical explanations for the influence of(D1)in the past,it is a less appropriate definition today and is destined to become still less appropriate in the future. I propose the following alternative definition with which to enter the new century: Num rical analysis is the study of algorithms for the problems of continuous mathematics (D2 na nition an be perfect.But it The pivotal word is algorithms.Where was this word in those chapter headings and dictionary definitions?Hidden between the lines,at best,and yet surely this is the center of numerical analysis:devising and analyzing algorithms to solve a certain class of problems. at real or complex aretheamanofotherenptlemoiproblems,whiealg rithms for disc te proble Let us consider the implications of (D2).First of all it is clear that since real and complex numbers cannot be represented exactly on computers,(D2)implies that part of the business umerical analysis must be t o approximate them.This is where the rounding errors come 2
✁ ✄✆ ✞ ✁✡ ☞✆ ✍✁✏ ✑✓✕✕✁✙✛✜ ✞ ✁ ✣✛✦✞✛✓★ ✜✡✪ ✫✭ ✯ ✰✲✴ ✵ ✷✹✻ ✁ ✆ ✞ ✿ ❁✪ ✓❃ ❄ ✿ ✜★✞✛ ✞ ✜ ✞✛ ❊✁ ✜❋❋✡✓❍✛❏ ❑✜ ✞✛✓★✆ ✞ ✓ ✞✻ ✁ ✆✓✕ ✿ ✞✛✓★✆ ✓❃ ❑✜ ✞✻ ✁❑✜ ✞✛✦✜✕ ❋✡✓ ✄✕✁❑✆ ✛★ ✦✕✿ ❁✛★✙ ✦✓★✆✛❁✁✡✜ ✞✛✓★ ✓❃ ✞✻ ✁ ✁✡✡✓✡✆ ✜★ ❁ ✄ ✓ ✿★ ❁✆ ✞ ✓ ✞✻ ✁ ✁✡✡✓✡✆ ✛★❊✓✕❊✁❁❭ ❪ ✑✻ ✜❑✄ ✁✡✆ ❴❛✞✻ ✑✁★✞ ✿✡✪ ✣✛✦✞✛✓★ ✜✡✪ ✫✭ ✯❞✲✴ ✵ ✷✹✻ ✁ ✆ ✞ ✿ ❁✪ ✓❃❑✁ ✞✻ ✓ ❁✆ ✓❃ ✜❋❋✡✓❍✛❑✜ ✞✛✓★ ✜★ ❁ ✞✻ ✁✛✡ ✜✦✦✿✡✜✦✪❤ ✁ ✞ ✦❭ ❪ ✹✻ ✁ ✐❑✁✡✛✦✜★ ❦✁✡✛ ✞ ✜✙✁ ✣✛✦✞✛✓★ ✜✡✪ ✫✭ ✯ ✯❴✴ ✵ ✷✹✻ ✁ ✆ ✞ ✿ ❁✪ ✓❃ ✜❋❋✡✓❍✛❑✜ ✞ ✁ ✆✓✕ ✿ ✞✛✓★✆ ✞ ✓ ❑✜ ✞✻ ✁❑✜ ✞✛✦✜✕ ❋✡✓ ✄✕✁❑✆❤ ✞ ✜♦✛★✙ ✛★✞ ✓ ✜✦✦✓ ✿★✞ ✞✻ ✁ ✁❍✞ ✁★✞ ✓❃ ❋ ✓✆✆✛ ✄✕✁ ✁✡✡✓✡✆ ❭ ❪ ♦➔✁✄✌✕✡✍✑ ✕ ✄✗✜ ♣ s s s ♦ ✍✏ ✏✺✁ ✍✏✓♣ s s s ♦ ✟✁✁✄✁✜ ♣ ✍✼✍✕✗ ✷ ❦ ✑ ✜ ✟ ✟✡✜ ✑ ✄ ✡✟ ✑✴✍✑ ✑✴✟ ✜ ✟ ✩✟♥✗✕ ✑ ✕ ✄✗✜ ❈✄✺✞✩ ✜ ✟✁✭✟ ✡✄ ✜ ✑ ✟◆✟ ✏✑ ✕✭✟✞✓ ✑ ✄ ✩✟✑ ✟✁ ✑✴✟ ✏✺✁✕✄✺✜ ❍✁✄✡ ✕✗✭✟ ✜ ✑ ✕ ✼✍✑ ✕✗✼ ❍✺✁✑✴✟✁ ✷ ❘✴✟ ✜ ✕✗✼✺✞ ✍✁ ✭✍✞✺✟ ✩✟ ✏✄✡ ✄ ✜ ✕ ✑ ✕✄✗ ➌ ❱✡➐➎ ✍◆✄✁✩✜ ✍✗✄✑✴✟✁ ✟✌✍✡✞ ✟ ✄❍ ✑✴✟ ✟✁✏ ✟✑ ✕✄✗ ✄❍ ✗✺✐ ✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✍✜ ✑✴✟ ✜ ✏✕ ✟✗✏ ✟ ✄❍ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✷ ➔✞✑✴✄✺✼✴ ✑✴✟ ✁✄ ✄✑ ✜ ✄❍ ✑✴✟ ❱✡➐ ✼✄ ✆ ✍✏❙ ✡✄✁✟ ✑✴✍✗ ⑩ → → ✓✟ ✍✁✜ ✪ ✕ ✑ ✕ ✜ ✡✍✕✗✞✓ ✜ ✕✗✏ ✟ ✑✴✟ ⑩ ❶ ✝ → ✜ ✪ ✑✴✁✄✺✼✴ ✑✴✟ ❈✄✁❙ ✄❍ ❢✟✗✟ ❢✄✞✺✆ ✍✗✩ ✄✑✴✟✁ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ✪ ✑✴✍✑ ✕ ✑ ✴✍✜ ✍✏✴✕ ✟✭✟✩ ✕ ✑ ✜ ✁ ✟ ✜ ✟✗✑ ✩✟ ✼✁✟ ✟ ✄❍ ✁✄✡✕✗✟✗✏ ✟ ✷ ❘✴✟ ❱✡➐ ✕ ✜ ✍✜ ❍✺✗✩✍✡✟✗✑ ✍✞ ✍✗ ✕✩✟ ✍ ✍✜ ✑✴✟ ✟✕✼✟✗✭✍✞✺✟ ✩✟ ✏✄✡ ✄ ✜ ✕ ✑ ✕ ✄✗ ✢ ✕ ✑ ✕ ✜ ✑✴✟ ✗✍✑✺✁✍✞ ✞ ✍✗✼✺✍✼✟ ❍✄✁ ✩✕ ✜ ✏✺✜ ✜ ✕✗✼ ✍✞ ✞ ❙✕✗✩✜ ✄❍ ➅✺✟ ✜ ✑ ✕✄✗✜ ✄❍ ✗✄✁✡✜ ✍✗✩ ✟✌✑ ✁✟✡✍ ✕✗✭✄✞✭✕✗✼ ✗✄✗✜✓✡✡✟✑ ✁✕ ✏ ✡✍✑ ✁✕ ✏ ✟ ✜ ✄✁ ✄ ✟✁✍✑ ✄✁ ✜ ✷ ✍✟✑ ✑ ✄ ✩✍✓✪ ✑✴✕✁✑✓ ✓✟ ✍✁ ✜ ✞✍✑ ✟✁ ✪ ✡✄ ✜ ✑ ✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞ ✜ ✏✕ ✟✗✑ ✕ ✜ ✑ ✜ ✍✗✩ ✟✭✟✗ ✡✍✗✓ ✍✞✕ ✟✩ ✡✍✑✴✟✐ ✡✍✑ ✕ ✏✕ ✍✗✜ ✩✄ ✗✄✑ ✴✍✭✟ ✍ ❈✄✁❙✕✗✼ ❙✗✄❈✞✟✩✼✟ ✄❍ ✑✴✟ ❱✡➐ ✷ ✸✄ ✜ ✑ ✄❍ ✑✴✟✡ ✴✍✭✟ ✴✟ ✍✁✩ ✄❍ ✕✑ ✪ ✆✺✑ ✑✴✟ ✕✡✁ ✟ ✜ ✜ ✕✄✗ ✜ ✟ ✟✡✜ ✑ ✄ ✆ ✟ ❈✕✩✟ ✜✁ ✟ ✍✩ ✑✴✍✑ ✑✴✟ ❱✡➐ ✕ ✜ z ✺✜ ✑ ✍ ✑ ✄ ✄✞ ❍✄✁ ✏ ✄✡✆ ✍✑ ✕✗✼ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✷ ➔ ✼✞ ✍✗✏ ✟ ✍✑ ✍ ❍✟❈ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✑ ✟✌✑✆ ✄ ✄❙✜ ✜✺✼✼✟ ✜ ✑ ✜ ❈✴✓✷ ❦✗ ✄✗✟ ✏ ✍✜ ✟ ✍❍✑ ✟✁ ✍✗✐ ✄✑✴✟✁ ✪ ✑✴✟ ❱✡➐ ✕ ✜ ✆✺✁✕ ✟✩ ✩✟ ✟ ✕✗ ✑✴✟ ✆ ✄ ✄❙ ✪ ✑✓✕ ✏ ✍✞ ✞✓ ✕✗ ✍✗ ✍✩✭✍✗✏ ✟✩ ✜ ✟ ✏✑ ✕ ✄✗ ✄✗ ✁✍✗❙✐ ✩✟♥✏✕ ✟✗✑ ✞ ✟ ✍✜ ✑ ✐ ✜➅✺✍✁✟ ✜ ✁✄✆✞✟✡✜ ✪ ✍✗✩ ✁✟ ✏✄✡✡✟✗✩✟✩ ✡✍✕✗✞✓ ❍✄✁ ✕✑ ✜ ✜ ✑ ✍✆✕ ✞ ✕✑✓ ✁✄ ✟✁✑ ✕ ✟ ✜ ✷ ❦ ✍✡ ✏✄✗✭✕✗✏ ✟✩ ✑✴✍✑ ✏✄✗✜ ✏✕ ✄✺✜ ✞✓ ✄✁ ✺✗✏ ✄✗✜ ✏✕✄✺✜ ✞✓✪ ✡✍✗✓ ✟✄✞ ✟ ✑✴✕✗❙ ✑✴✍✑ ➌➐ ⑩ ➎ ✕ ✜ ✍✑ ✞ ✟ ✍✜ ✑ ✴✍✞❍ ✑ ✁✺✟ ✷ ❦✗ ✍✏✑✺✍✞✕ ✑✓✪ ✕✑ ✕ ✜ ✍ ✭✟✁✓ ✜✡✍✞ ✞ ✍✁✑ ✄❍ ✑✴✟ ✑ ✁✺✑✴ ✷ ➔✗✩ ✍✞✑✴✄✺✼✴ ✑✴✟✁✟ ✍✁✟ ✴✕ ✜ ✑ ✄✁✕ ✏ ✍✞ ✟✌✞✍✗✍✑ ✕✄✗✜ ❍✄✁ ✑✴✟ ✕✗✜✺✟✗✏ ✟ ✄❍ ➌➐ ⑩ ➎ ✕✗ ✑✴✟ ✍✜ ✑ ✪ ✕ ✑ ✕ ✜ ✍ ✞ ✟ ✜ ✜ ✍✁✄✁✕ ✍✑ ✟ ✩✟♥✗✕ ✑ ✕ ✄✗ ✑ ✄ ✩✍✓ ✍✗✩ ✕ ✜ ✩✟ ✜ ✑ ✕✗✟✩ ✑ ✄ ✆ ✟ ✏✄✡✟ ✜ ✑ ✕ ✞ ✞ ✞ ✟ ✜ ✜ ✍✁✄✁✕ ✍✑ ✟ ✕✗ ✑✴✟ ❍✺✑✺✁ ✟ ✷ ❦ ✁✄ ✄ ✜ ✟ ✑✴✟ ❍✄✞ ✞✄❈✕✗✼ ✍✞✑ ✟✁✗✍✑ ✕✭✟ ✩✟♥✗✕ ✑ ✕ ✄✗ ❈✕✑✴ ❈✴✕ ✏✴ ✑ ✄ ✟✗✑ ✟✁ ✑✴✟ ✗✟❈ ✏ ✟✗✑✺✁✓ ⑦ ➳➵➩➧ ➸➺ ➻➛ ➦ ➛➫ ➛ ➦➾➭ ➺ ➭ ➺ ➭ ➜➙ ➧ ➭ ➜➵ ★➾ ➯➲ ➛➦✪ ➯ ➸➺ ➜➙➩➭ ➲➯ ➸ ➜➙ ➧ ✥➸➯ ✁ ➦➧➩➭ ➯➲ ➻➯➫➜➺➫➵ ➯ ➵➭ ➩➛ ➜➙ ➧➩➛ ➜➺ ➻ ➭ ✭ ➌➐✂ ➎ ➄✄✺✗✩✍✁✕ ✟ ✜ ✆ ✟✑❈✟ ✟✗ ♥✟✞✩✜ ✍✁✟ ✍✞❈✍✓✜ ❍✺✘ ✘✓ ✢ ✗✄ ✩✟♥✗✕✑ ✕ ✄✗ ✏ ✍✗ ✆ ✟ ✟✁❍✟ ✏✑ ✷ ➄✺✑ ✕✑ ✜ ✟ ✟✡✜ ✑ ✄ ✡✟ ✑✴✍✑ ➌➐✂ ➎ ✕ ✜ ✍✜ ✜✴✍✁ ✍ ✏✴✍✁ ✍✏✑ ✟✁✕ ✘ ✍✑ ✕ ✄✗ ✍✜ ✓✄✺ ✏✄✺✞✩ ✏ ✄✡✟ ✺ ❈✕ ✑✴ ❍✄✁ ✡✄ ✜ ✑ ✩✕ ✜ ✏✕✞ ✕✗✟ ✜ ✷ ❘✴✟ ✕✭✄✑ ✍✞ ❈✄✁✩ ✕ ✜ ➛ ➦✪ ➯ ➸➺ ➜➙➩➭ ✭ ✟✴✟✁✟ ❈✍✜ ✑✴✕ ✜ ❈✄✁✩ ✕✗ ✑✴✄ ✜ ✟ ✏✴✍✑ ✟✁ ✴✟ ✍✩✕✗✼✜ ✍✗✩ ✩✕ ✏✑ ✕✄✗✍✁✓ ✩✟♥✗✕ ✑ ✕ ✄✗✜ ❐ ➶✕✩✩✟✗ ✆ ✟✑❈✟ ✟✗ ✑✴✟ ✞ ✕✗✟ ✜ ✪ ✍✑ ✆ ✟ ✜ ✑ ✪ ✍✗✩ ✓✟✑ ✜✺✁✟✞✓ ✑✴✕ ✜ ✕ ✜ ✑✴✟ ✏ ✟✗✑ ✟✁ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ⑦ ✩✟✭✕ ✜ ✕✗✼ ✍✗✩ ✍✗✍✞✓✘ ✕✗✼ ✍✞✼✄✁✕ ✑✴✡✜ ✑ ✄ ✜ ✄✞✭✟ ✍ ✏ ✟✁✑ ✍✕✗ ✏✞✍✜ ✜ ✄❍ ✁✄✆✞✟✡✜ ✷ ❘✴✟ ✜ ✟ ✍✁✟ ✑✴✟ ✁✄✆✞ ✟✡✜ ✄❍ ➻➯➫➜➺➫➵ ➯ ➵➭ ➩➛ ➜➙ ➧➩➛ ➜➺ ➻ ➭ ✭ ♦t✄✗✑ ✕✗✺✄✺✜ ♣ ✡✟ ✍✗✜ ✑✴✍✑ ✁✟ ✍✞ ✄✁ ✏ ✄✡✞✟✌ ✭✍✁✕ ✍✆✞ ✟ ✜ ✍✁✟ ✕✗✭✄✞✭✟✩ ✢ ✕✑ ✜ ✄ ✄ ✜ ✕ ✑ ✟ ✕ ✜ ♦✩✕ ✜ ✏✁✟✑ ✟ ✷ ♣ ➔ ✩✄ ✘ ✟✗ ➅✺✍✞✕♥✏ ✍✑ ✕ ✄✗✜ ✍✜ ✕✩✟ ✪ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✐ ✞✓✜ ✑ ✜ ✍✁✟ ✆✁✄ ✍✩✞✓ ✏ ✄✗✏ ✟✁✗✟✩ ❈✕ ✑✴ ✏✄✗✑ ✕✗✺✄✺✜ ✁✄✆✞ ✟✡✜ ✪ ❈✴✕ ✞✟ ✍✞✼✄✁✕ ✑✴✡✜ ❍✄✁ ✩✕ ✜ ✏✁✟✑ ✟ ✁✄✆✞ ✟✡✜ ✍✁✟ ✑✴✟ ✏ ✄✗✏ ✟✁✗ ✄❍ ✄✑✴✟✁ ✏ ✄✡✺✑ ✟✁ ✜ ✏✕ ✟✗✑ ✕ ✜ ✑ ✜ ✷ ② ✟✑ ✺✜ ✏ ✄✗✜ ✕✩✟✁ ✑✴✟ ✕✡✞ ✕ ✏ ✍✑ ✕ ✄✗✜ ✄❍ ➌➐✂ ➎ ✷ ➇✕✁ ✜ ✑ ✄❍ ✍✞ ✞ ✕ ✑ ✕ ✜ ✏✞ ✟ ✍✁ ✑✴✍✑ ✜ ✕✗✏ ✟ ✁ ✟ ✍✞ ✍✗✩ ✏ ✄✡✞✟✌ ✗✺✡✆ ✟✁✜ ✏ ✍✗✗✄✑ ✆ ✟ ✁✟✁✟ ✜ ✟✗✑ ✟✩ ✟✌✍✏✑ ✞✓ ✄✗ ✏ ✄✡✺✑ ✟✁✜ ✪ ➌➐✂ ➎ ✕✡✞ ✕ ✟ ✜ ✑✴✍✑ ✍✁✑ ✄❍ ✑✴✟ ✆✺✜ ✕✗✟ ✜ ✜ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✡✺✜ ✑ ✆ ✟ ✑ ✄ ✍✁✄✌✕✡✍✑ ✟ ✑✴✟✡✷ ❘✴✕ ✜ ✕ ✜ ❈✴✟✁✟ ✑✴✟ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✏✄✡✟ ✕✗ ✷ ☎✄❈ ❍✄✁ ✍ ✏ ✟✁✑ ✍✕✗ ✜ ✟✑ ✄❍ ✁✄✆✞ ✟✡✜ ✪ ✗✍✡✟✞✓ ✑✴✟ ✄✗✟ ✜ ✑✴✍✑ ✍✁✟ ✜ ✄✞✭✟✩ ✆✓ ✍✞✼✄✁✕✑✴✡✜ ✑✴✍✑ ✑ ✍❙✟ ✍ ♥✗✕ ✑ ✟ ✗✺✡✆ ✟✁ ✄❍ ✜ ✑ ✟✜ ✪ ✑✴✍✑ ✕ ✜ ✍✞ ✞ ✑✴✟✁ ✟ ✕ ✜ ✑ ✄ ✕✑ ✷ ❘✴✟ ✁✟✡✕ ✟✁ ✟✌✍✡✞ ✟ ✕ ✜ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✂
for solving a linear system of equations Ar=b.To understand Gaussian elimination,you have to understand computer science issues such as operation counts and machine architectures, and you have to understand the propagation ling errorsstability.That's all you to unde erstand,an a more on er wrong wit "ut most problems of continaous mathematies cannot be solued bu finite alaorithms!t nlike Ar =b and unlike the discrete problems of computer science most of the problems of numer ical analysis could not be solved exactly even if we could work in exact arithmetic.Numerical analysts know this,and mention it along with a few words about Abel and Galois when they ms r or computing matrix eigenval 10 en they forget to me on tha in itn name it. nun id he deeper bus These points are sometimes overlooked by enthusiasts of symbolic computing.especially recent converts,who are apt to think that the existence of Maple or Mathematica renders Matlab and Fortran obsolete.It is true that rounding errors can be made to vanish in the sense that in principle,any finite sequence of algebraic operations can be represented exactly on a which point the a antities one is working with may have bec extraordinarily cumbe Floating-point arithmetic is a name for numerical analysts'habit of doing their pruning at every step along the way of a calculation rather than in a single act at the end.Chichever way one proceeds onvergentalgoritor symbolically,the main problem of finding a rapidly ysis is con ned with rounding e pbyariousnansncatiodiectiration,iteration.ofeSreOne2iadcdooetomake (D2)more explicit by adding words to describe these approximations and errors.But once to be added it is hard to know where to stop,for (D2)also fails to mention words b Important matters:that t s are I ters,who ms and others and most important,that all of this work is applied,applied daily and successfully to thousands of applications on millions of computers around the world. "The problems of continuous mathematics"are the problems that scnd gineering are buil pon;without nu merica ds,scie nce a as pra 1a to a As much as any pure mathematicians,nu merical analysts are the heirs to the great tradition of Euler,Lagrange,Gauss and the rest.If Euler were alive today,he wouldn't be proving existence theorems. 3
❍✄✁ ✜ ✄✞✭✕✗✼ ✍ ✞ ✕✗✟ ✍✁ ✜✓✜ ✑ ✟✡ ✄❍ ✟➅✺✍✑ ✕ ✄✗✜ '✉ ✈ ✇ ✷ ❘✄ ✺✗✩✟✁ ✜ ✑ ✍✗✩ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✪ ✓✄✺ ✴✍✭✟ ✑ ✄ ✺✗✩✟✁✜ ✑ ✍✗✩ ✏✄✡B✺✑ ✟✁ ✜ ✏✕ ✟✗✏ ✟ ✕ ✜ ✜✺✟ ✜ ✜✺✏✴ ✍✜ ✄B ✟✁ ✍✑ ✕✄✗ ✏ ✄✺✗✑ ✜ ✍✗✩ ✡✍✏✴✕✗✟ ✍✁✏✴✕ ✑ ✟ ✏✑✺✁✟ ✜ ✪ ✍✗✩ ✓✄✺ ✴✍✭✟ ✑ ✄ ✺✗✩✟✁ ✜ ✑ ✍✗✩ ✑✴✟ B✁✄B ✍✼ ✍✑ ✕ ✄✗ ✄❍ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁ ✜❜✜ ✑ ✍✆✕ ✞✕ ✑✓✷ ❘✴✍✑ ➉ ✜ ✍✞✞ ✓✄✺ ✴✍✭✟ ✑ ✄ ✺✗✩✟✁ ✜ ✑ ✍✗✩ ✪ ✍✗✩ ✕❍ ✜ ✄✡✟✆ ✄ ✩✓ ✏✞ ✍✕✡✜ ✑✴✍✑ ➌➐✂ ➎ ✕ ✜ c ✺✜ ✑ ✍ ✡✄✁✟ B ✄✞ ✕✑ ✟ ✁✟ ✜ ✑ ✍✑ ✟✡✟✗✑ ✄❍ ➌➐ ⑩ ➎ ✪ ✓✄✺ ✏ ✍✗ ➉ ✑ B✁✄✭✟ ✴✕✡ ✄✁ ✴✟✁ ❈✁✄✗✼ ❈✕ ✑✴ ✑✴✟ ✟✌✍✡B✞✟ ✄❍ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✷ ➵➜ ➩➯ ➭ ➜ ✥➸➯ ✁ ➦➧➩➭ ➯➲ ➻➯➫➜➺➫➵ ➯ ➵➭ ➩➛ ➜➙ ➧➩➛ ➜➺ ➻ ➭ ➻➛➫➫ ➯ ➜ ✁➧ ➭ ➯ ➦☎ ➧ ★ ✁ ➾ ✞➫➺ ➜ ➧ ➛ ➦✪ ➯ ➸➺ ➜➙➩➭ ✠ ✣✗✞✕❙✟ '✉ ✈ ✇ ✪ ✍✗✩ ✺✗✞✕❙✟ ✑✴✟ ✩✕ ✜ ✏✁✟✑ ✟ B✁✄✆✞✟✡✜ ✄❍ ✏ ✄✡B✺✑ ✟✁ ✜ ✏✕ ✟✗✏ ✟ ✪ ✡✄ ✜ ✑ ✄❍ ✑✴✟ B✁✄✆✞ ✟✡✜ ✄❍ ✗✺✡✟✁✐ ✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✏✄✺✞✩ ✗✄✑ ✆ ✟ ✜ ✄✞✭✟✩ ✟✌✍✏✑ ✞✓ ✟✭✟✗ ✕❍ ❈✟ ✏ ✄✺✞✩ ❈✄✁❙ ✕✗ ✟✌✍✏✑ ✍✁✕ ✑✴✡✟✑ ✕ ✏ ✷ ☎✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ❙✗✄❈ ✑✴✕ ✜ ✪ ✍✗✩ ✡✟✗✑ ✕ ✄✗ ✕ ✑ ✍✞✄✗✼ ❈✕✑✴ ✍ ❍✟❈ ❈✄✁✩✜ ✍✆ ✄✺✑ ➔✆ ✟✞ ✍✗✩ ❢✍✞✄✕ ✜ ❈✴✟✗ ✑✴✟✓ ✑ ✟ ✍✏✴ ✍✞✼✄✁✕ ✑✴✡✜ ❍✄✁ ✏ ✄✡B✺✑ ✕✗✼ ✡✍✑ ✁✕✌ ✟✕✼✟✗✭✍✞✺✟ ✜ ✷ ❘✄ ✄ ✄❍✑ ✟✗ ✑✴✟✓ ❍✄✁✼✟✑ ✑ ✄ ✡✟✗✑ ✕✄✗ ✑✴✍✑ ✑✴✟ ✜ ✍✡✟ ✏ ✄✗✏✞✺✜ ✕✄✗ ✟✌✑ ✟✗✩✜ ✑ ✄ ✭✕✁✑✺✍✞ ✞✓ ✍✗✓ B✁✄✆✞✟✡ ❈✕✑✴ ✍ ✗✄✗✞✕✗✟ ✍✁ ✑ ✟✁✡ ✄✁ ✍ ✩✟✁✕✭✍✑ ✕✭✟ ✕✗ ✕✑❜✘ ✟✁✄♥✗✩✕✗✼ ✪ ➅✺✍✩✁✍✑✺✁✟ ✪ ✩✕◆✟✁✟✗✑ ✕ ✍✞ ✟➅✺✍✑ ✕✄✗✜ ✪ ✕✗✑ ✟✼✁✍✞ ✟➅✺✍✑ ✕ ✄✗✜ ✪ ✄B✑ ✕✡✕ ✘ ✍✑ ✕ ✄✗ ✪ ✓✄✺ ✗✍✡✟ ✕ ✑ ✷ ➥☎ ➧ ➫ ➺➲ ➸➯➵➫ ★➺➫✪ ➧ ➸➸➯ ➸➭ ☎ ➛➫➺ ➭ ➙ ➧★➪ ➫➵➩➧ ➸➺ ➻➛ ➦ ➛➫ ➛ ➦➾➭ ➺ ➭ ✲➯➵➦★ ➸➧➩➛➺➫ ✭ ➔BB✁✄✌✕✡✍✑ ✕✗✼ ✡✟✁✟ ✗✺✡✆ ✟✁✜ ✪ ✑✴✟ ✑ ✍✜❙ ✄❍ ✜✄ ✍✑ ✕✗✼✐B ✄✕✗✑ ✍✁✕✑✴✡✟✑ ✕ ✏ ✪ ✕ ✜ ✕✗✩✟ ✟✩ ✍ ✁✍✑✴✟✁ ✜✡✍✞ ✞ ✑ ✄B✕ ✏ ✍✗✩ ✡✍✓✆ ✟ ✟✭✟✗ ✍ ✑ ✟✩✕ ✄✺✜ ✄✗✟ ✷ ❘✴✟ ✩✟ ✟B ✟✁ ✆✺✜ ✕✗✟ ✜ ✜ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕ ✜ ✍BB✁✄✌✕✡✍✑ ✕✗✼ ✺✗❙✗✄❈✗✜ ✪ ✗✄✑ ❙✗✄❈✗✜ ✷ ❿✍B✕✩ ✏ ✄✗✭✟✁✼✟✗✏ ✟ ✄❍ ✍BB✁✄✌✕✡✍✑ ✕ ✄✗✜ ✕ ✜ ✑✴✟ ✍✕✡✪ ✍✗✩ ✑✴✟ B✁✕✩✟ ✄❍ ✄✺✁ ♥✟✞✩ ✕ ✜ ✑✴✍✑ ✪ ❍✄✁ ✡✍✗✓ B✁✄✆✞ ✟✡✜ ✪ ❈✟ ✴✍✭✟ ✕✗✭✟✗✑ ✟✩ ✍✞✼✄✁✕✑✴✡✜ ✑✴✍✑ ✏✄✗✭✟✁✼✟ ✟✌✏ ✟ ✟✩✕✗✼✞✓ ❍✍✜ ✑ ✷ ❘✴✟ ✜ ✟ B ✄✕✗✑ ✜ ✍✁✟ ✜ ✄✡✟✑ ✕✡✟ ✜ ✄✭✟✁✞✄ ✄❙✟✩ ✆✓ ✟✗✑✴✺✜ ✕ ✍✜ ✑ ✜ ✄❍ ✜✓✡✆ ✄✞ ✕ ✏ ✏ ✄✡B✺✑ ✕✗✼ ✪ ✟ ✜B ✟ ✏✕ ✍✞✞✓ ✁✟ ✏ ✟✗✑ ✏ ✄✗✭✟✁✑ ✜ ✪ ❈✴✄ ✍✁✟ ✍B✑ ✑ ✄ ✑✴✕✗❙ ✑✴✍✑ ✑✴✟ ✟✌✕ ✜ ✑ ✟✗✏ ✟ ✄❍ ✸✍B✞✟ ✄✁ ✸✍✑✴✟✡✍✑ ✕ ✏ ✍ ✁✟✗✩✟✁✜ ✸✍✑ ✞ ✍✆ ✍✗✩ ➇✄✁✑ ✁✍✗ ✄✆✜ ✄✞ ✟✑ ✟ ✷ ❦ ✑ ✕ ✜ ✑ ✁✺✟ ✑✴✍✑ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✏ ✍✗ ✆ ✟ ✡✍✩✟ ✑ ✄ ✭✍✗✕ ✜✴ ✕✗ ✑✴✟ ✜ ✟✗✜ ✟ ✑✴✍✑ ✕✗ B✁✕✗✏✕B✞✟ ✪ ✍✗✓ ♥✗✕ ✑ ✟ ✜ ✟➅✺✟✗✏ ✟ ✄❍ ✍✞✼✟✆✁✍✕ ✏ ✄B ✟✁ ✍✑ ✕ ✄✗✜ ✏ ✍✗ ✆ ✟ ✁✟B✁✟ ✜ ✟✗✑ ✟✩ ✟✌✍✏✑ ✞✓ ✄✗ ✍ ✏✄✡B✺✑ ✟✁ ✆✓ ✡✟ ✍✗✜ ✄❍ ✍BB✁✄B✁✕ ✍✑ ✟ ✜✓✡✆ ✄✞✕ ✏ ✄B ✟✁✍✑ ✕ ✄✗✜ ✷ ✣✗✞ ✟ ✜ ✜ ✑✴✟ B✁✄✆✞ ✟✡ ✆ ✟✕✗✼ ✜ ✄✞✭✟✩ ✕ ✜ ✍ ♥✗✕ ✑ ✟ ✄✗✟ ✪ ✴✄❈✟✭✟✁ ✪ ✑✴✕ ✜ ✄✗✞✓ ✩✟❍✟✁✜ ✑✴✟ ✕✗✟✭✕ ✑ ✍✆✞ ✟ ✍BB✁✄✌✕✡✍✑ ✕✄✗✜ ✑ ✄ ✑✴✟ ✟✗✩ ✄❍ ✑✴✟ ✏ ✍✞✏✺✞ ✍✑ ✕✄✗ ✪ ✆✓ ❈✴✕ ✏✴ B ✄✕✗✑ ✑✴✟ ➅✺✍✗✑ ✕ ✑ ✕ ✟ ✜ ✄✗✟ ✕ ✜ ❈✄✁❙✕✗✼ ❈✕✑✴ ✡✍✓ ✴✍✭✟ ✆ ✟ ✏✄✡✟ ✟✌✑ ✁ ✍✄✁✩✕✗✍✁✕ ✞✓ ✏✺✡✆ ✟✁✜ ✄✡✟ ✷ ➇✞✄ ✍✑ ✕✗✼✐B ✄✕✗✑ ✍✁✕✑✴✡✟✑ ✕ ✏ ✕ ✜ ✍ ✗✍✡✟ ❍✄✁ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ➉ ✴✍✆✕ ✑ ✄❍ ✩✄✕✗✼ ✑✴✟✕✁ B✁✺✗✕✗✼ ✍✑ ✟✭✟✁✓ ✜ ✑ ✟B ✍✞✄✗✼ ✑✴✟ ❈✍✓ ✄❍ ✍ ✏ ✍✞✏✺✞ ✍✑ ✕✄✗ ✁ ✍✑✴✟✁ ✑✴✍✗ ✕✗ ✍ ✜ ✕✗✼✞✟ ✍✏✑ ✍✑ ✑✴✟ ✟✗✩ ✷ ✟✴✕ ✏✴✟✭✟✁ ❈✍✓ ✄✗✟ B✁✄ ✏ ✟ ✟✩✜ ✪ ✕✗ ✜✄ ✍✑ ✕✗✼✐B ✄✕✗✑ ✄✁ ✜✓✡✆ ✄✞ ✕ ✏ ✍✞ ✞✓✪ ✑✴✟ ✡✍✕✗ B✁✄✆✞ ✟✡ ✄❍ ♥✗✩✕✗✼ ✍ ✁✍B✕✩✞✓ ✏ ✄✗✭✟✁✼✟✗✑ ✍✞✼✄✁✕ ✑✴✡ ✕ ✜ ✑✴✟ ✜ ✍✡✟ ✷ ❦✗ ✜✺✡✡✍✁✓✪ ✕ ✑ ✕ ✜ ✍ ✏ ✄✁✄✞ ✞✍✁✓ ✄❍ ➌➐✂ ➎ ✑✴✍✑ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕ ✜ ✏ ✄✗✏ ✟✁✗✟✩ ❈✕ ✑✴ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✍✗✩ ✍✞ ✜ ✄ ❈✕ ✑✴ ✑✴✟ ✩✟ ✟B ✟✁ ❙✕✗✩✜ ✄❍ ✟✁✁✄✁✜ ✍✜ ✜ ✄ ✏✕ ✍✑ ✟✩ ❈✕✑✴ ✏✄✗✭✟✁✼✟✗✏ ✟ ✄❍ ✍BB✁✄✌✕✡✍✑ ✕✄✗✜ ✪ ❈✴✕ ✏✴ ✼✄ ✆✓ ✭✍✁✕ ✄✺✜ ✗✍✡✟ ✜ ➌ ✑ ✁✺✗✏ ✍✑ ✕ ✄✗ ✪ ✩✕ ✜ ✏✁✟✑ ✕ ✘ ✍✑ ✕ ✄✗ ✪ ✕ ✑ ✟✁ ✍✑ ✕✄✗➎ ✷ ❍ ✏ ✄✺✁✜ ✟ ✄✗✟ ✏ ✄✺✞✩ ✏✴✄ ✄ ✜ ✟ ✑ ✄ ✡✍❙✟ ➌➐✂ ➎ ✡✄✁✟ ✟✌B✞ ✕ ✏✕ ✑ ✆✓ ✍✩✩✕✗✼ ❈✄✁✩✜ ✑ ✄ ✩✟ ✜ ✏✁✕✆ ✟ ✑✴✟ ✜ ✟ ✍BB✁✄✌✕✡✍✑ ✕ ✄✗✜ ✍✗✩ ✟✁✁✄✁✜ ✷ ➄✺✑ ✄✗✏ ✟ ❈✄✁✩✜ ✆ ✟✼✕✗ ✑ ✄ ✆ ✟ ✍✩✩✟✩ ✕✑ ✕ ✜ ✴✍✁✩ ✑ ✄ ❙✗✄❈ ❈✴✟✁✟ ✑ ✄ ✜ ✑ ✄B ✪ ❍✄✁ ➌➐✂ ➎ ✍✞ ✜ ✄ ❍✍✕ ✞ ✜ ✑ ✄ ✡✟✗✑ ✕✄✗ ✜ ✄✡✟ ✄✑✴✟✁ ✕✡B ✄✁✑ ✍✗✑ ✡✍✑ ✑ ✟✁✜ ⑦ ✑✴✍✑ ✑✴✟ ✜ ✟ ✍✞✼✄✁✕ ✑✴✡✜ ✍✁✟ ✕✡B✞ ✟✡✟✗✑ ✟✩ ✄✗ ✏✄✡B✺✑ ✟✁✜ ✪ ❈✴✄ ✜ ✟ ✍✁✏✴✕✑ ✟ ✏✑✺✁✟ ✡✍✓ ✆ ✟ ✍✗ ✕✡B ✄✁✑ ✍✗✑ B ✍✁✑ ✄❍ ✑✴✟ B✁✄✆✞✟✡✢ ✑✴✍✑ ✁✟✞ ✕ ✍✆✕ ✞ ✕✑✓ ✍✗✩ ✟✌✏✕ ✟✗✏✓ ✍✁✟ B ✍✁ ✍✡✄✺✗✑ ✼✄ ✍✞ ✜ ✢ ✑✴✍✑ ✜ ✄✡✟ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ❈✁✕✑ ✟ B✁✄✼✁ ✍✡✜ ✍✗✩ ✄✑✴✟✁ ✜ B✁✄✭✟ ✑✴✟ ✄✁✟✡✜ ✢ ✍✗✩ ✡✄ ✜ ✑ ✕✡B ✄✁✑ ✍✗✑ ✪ ✑✴✍✑ ✍✞ ✞ ✄❍ ✑✴✕ ✜ ❈✄✁❙ ✕ ✜ ➛✥✥ ➦➺ ➧★➪ ✍BB✞✕ ✟✩ ✩✍✕ ✞✓ ✍✗✩ ✜✺✏ ✏ ✟ ✜ ✜ ❍✺✞ ✞✓ ✑ ✄ ✑✴✄✺✜ ✍✗✩✜ ✄❍ ✍BB✞ ✕ ✏ ✍✑ ✕ ✄✗✜ ✄✗ ✡✕ ✞ ✞✕ ✄✗✜ ✄❍ ✏✄✡B✺✑ ✟✁✜ ✍✁✄✺✗✩ ✑✴✟ ❈✄✁✞✩ ✷ ♦❘✴✟ B✁✄✆✞✟✡✜ ✄❍ ✏ ✄✗✑ ✕✗✺✄✺✜ ✡✍✑✴✟✡✍✑ ✕ ✏ ✜ ♣ ✍✁✟ ✑✴✟ B✁✄✆✞✟✡✜ ✑✴✍✑ ✜ ✏✕ ✟✗✏ ✟ ✍✗✩ ✟✗✼✕✗✟ ✟✁✕✗✼ ✍✁✟ ✆✺✕ ✞✑ ✺B ✄✗ ✢ ❈✕ ✑✴✄✺✑ ✗✺✡✟✁✕ ✏ ✍✞ ✡✟✑✴✄ ✩✜ ✪ ✜ ✏✕ ✟✗✏ ✟ ✍✗✩ ✟✗✼✕✗✟ ✟✁✕✗✼ ✍✜ B✁✍✏✑ ✕ ✏ ✟✩ ✑ ✄ ✩✍✓ ❈✄✺✞✩ ✏ ✄✡✟ ➅✺✕ ✏❙✞✓ ✑ ✄ ✍ ✴✍✞✑ ✷ ❘✴✟✓ ✍✁✟ ✍✞ ✜ ✄ ✑✴✟ B✁✄✆✞ ✟✡✜ ✑✴✍✑ B✁✟ ✄ ✏ ✏✺B✕ ✟✩ ✡✄ ✜ ✑ ✡✍✑✴✟✡✍✑ ✕ ✏✕ ✍✗✜ ❍✁✄✡ ✑✴✟ ✑ ✕✡✟ ✄❍ ☎✟❈✑ ✄✗ ✑ ✄ ✑✴✟ ✑❈✟✗✑ ✕ ✟✑✴ ✏ ✟✗✑✺✁✓✷ ➔✜ ✡✺✏✴ ✍✜ ✍✗✓ B✺✁✟ ✡✍✑✴✟✡✍✑ ✕ ✏✕ ✍✗✜ ✪ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ✍✁✟ ✑✴✟ ✴✟✕✁✜ ✑ ✄ ✑✴✟ ✼✁✟ ✍✑ ✑ ✁✍✩✕✑ ✕ ✄✗ ✄❍ ❏✺✞✟✁ ✪ ② ✍✼✁ ✍✗✼✟ ✪ ❢✍✺✜ ✜ ✍✗✩ ✑✴✟ ✁✟ ✜ ✑ ✷ ❦❍ ❏✺✞ ✟✁ ❈✟✁✟ ✍✞✕✭✟ ✑ ✄ ✩✍✓✪ ✴✟ ❈✄✺✞✩✗ ➉ ✑ ✆ ✟ B✁✄✭✕✗✼ ✟✌✕ ✜ ✑ ✟✗✏ ✟ ✑✴✟✄✁✟✡✜ ✷ ➤
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❘✟✗ ✓✟ ✍✁✜ ✍✼✄ ✪ ❦ ❈✄✺✞✩ ✴✍✭✟ ✜ ✑ ✄%% ✟✩ ✍✑ ✑✴✕ ✜ % ✄✕✗✑ ✷ ➄✺✑ ✑✴✟ ✟✭✄✞✺✑ ✕ ✄✗ ✄❍ ✏ ✄✡%✺✑ ✕✗✼ ✕✗ ✑✴✟ % ✍✜ ✑ ✩✟ ✏ ✍✩✟ ✴✍✜ ✼✕✭✟✗ ✑✴✟ ✩✕◆✟✁ ✟✗✏ ✟ ✆ ✟✑❈✟ ✟✗ ➌➐ ⑩ ➎ ✍✗✩ ➌➐✂ ➎ ✍ ✗✟❈ ✑ ✄%✕ ✏ ✍✞ ✕ ✑✓✷ ② ✟✑ ✺✜ ✁✟✑✺✁✗ ✑ ✄ Z✉ ✈ ✇ ✷ ✸✺✏✴ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✏✄✡%✺✑ ✍✑ ✕✄✗ ✩✟% ✟✗✩✜ ✄✗ ✞ ✕✗✟ ✍✁ ✍✞✼✟✆✁ ✍ ✪ ✍✗✩ ✑✴✕ ✜ ✴✕ ✼✴✞✓ ✩✟✭✟✞✄% ✟✩ ✜✺✆d ✟ ✏✑ ✴✍✜ ✆ ✟ ✟✗ ✑✴✟ ✏✄✁ ✟ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✜ ✕✗✏ ✟ ✑✴✟ ✆ ✟✼✕✗✗✕✗✼ ✷ ☎✺✐ ✡✟✁✕ ✏ ✍✞ ✞ ✕✗✟ ✍✁ ✍✞✼✟✆✁✍ ✜ ✟✁✭✟✩ ✍✜ ✑✴✟ ✜✺✆d ✟ ✏✑ ❈✕✑✴ ✁✟ ✜% ✟ ✏✑ ✑ ✄ ❈✴✕ ✏✴ ✑✴✟ ✗✄❈ ✜ ✑ ✍✗✩✍✁✩ ✏✄✗✏ ✟%✑ ✜ ✄❍ ✜ ✑ ✍✆✕ ✞ ✕✑✓✪ ✏✄✗✩✕✑ ✕✄✗✕✗✼ ✪ ✍✗✩ ✆ ✍✏❙❈✍✁✩ ✟✁✁✄✁ ✍✗✍✞✓✜ ✕ ✜ ❈✟✁ ✟ ✩✟♥✗✟✩ ✍✗✩ ✜✴✍✁% ✟✗✟✩ ✪ ✍✗✩ ✑✴✟ ✏ ✟✗✑ ✁ ✍✞ ♥✼✺✁✟ ✕✗ ✑✴✟ ✜ ✟ ✩✟✭✟✞✄%✡✟✗✑ ✜ ✪ ❍✁✄✡ ✑✴✟ ⑩ ❶ ❒ → ✜ ✑ ✄ ✴✕ ✜ ✩✟ ✍✑✴ ✕✗ ⑩ ❶ ❷ ✝ ✪ ❈✍✜ ➹✕✡ ✟✕ ✞❙✕✗✜ ✄✗ ✷ ❦ ✴✍✭✟ ✡✟✗✑ ✕ ✄✗✟✩ ✑✴✍✑ Z✉ ✈ ✇ ✴✍✜ ✑✴✟ ✺✗✺✜✺✍✞ ❍✟ ✍✑✺✁✟ ✑✴✍✑ ✕ ✑ ✏ ✍✗ ✆ ✟ ✜ ✄✞✭✟✩ ✕✗ ✍ ♥✗✕ ✑ ✟ ✜ ✟➅✺✟✗✏ ✟ ✄❍ ✄% ✟✁ ✍✑ ✕ ✄✗✜ ✷ ❦✗ ❍✍✏✑ ✪ Z✉ ✈ ✇ ✕ ✜ ✡✄✁✟ ✺✗✺✜✺✍✞ ✑✴✍✗ ✑✴✍✑ ✪ ❍✄✁ ✑✴✟ ✜ ✑ ✍✗✩✍✁✩ ✍✞✼✄✁✕✑✴✡ ❍✄✁ ✜ ✄✞✭✕✗✼ ✕✑ ✪ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞ ✕✡✕✗✍✑ ✕ ✄✗ ✪ ✑✺✁✗✜ ✄✺✑ ✑ ✄ ✴✍✭✟ ✟✌✑ ✁✍✄✁✩✕✗✍✁✕ ✞✓ ✏✄✡%✞ ✕ ✏ ✍✑ ✟✩ ✜ ✑ ✍✆✕ ✞ ✕✑✓ %✁✄% ✟✁✑ ✕ ✟ ✜ ✷ ✡✄✗ ☎✟✺✡✍✗✗ ❈✁✄✑ ✟ ⑩ ❷ → % ✍✼✟ ✜ ✄❍ ✡✍✑✴✟✡✍✑ ✕ ✏ ✜ ✄✗ ✑✴✕ ✜ ✑ ✄%✕ ✏ ✢ ❘✺✁✕✗✼ ❈✁✄✑ ✟ ✄✗✟ ✄❍ ✴✕ ✜ ✡✍d ✄✁ % ✍% ✟✁✜ ✢ ✟✕ ✞❙✕✗✜ ✄✗ ✩✟✭✟✞✄% ✟✩ ✍ ✑✴✟✄✁✓ ✑✴✍✑ ✼✁✟❈ ✕✗✑ ✄ ✑❈✄ ✆ ✄ ✄❙✜ ✍✗✩ ✍ ✏ ✍✁ ✟ ✟✁ ✷ ✍✟✑ ✑✴✟ ❍✍✏✑ ✁✟✡✍✕✗✜ ✑✴✍✑ ❍✄✁ ✏ ✟✁✑ ✍✕✗ ✏ ✒ ✏ ✡✍✑ ✁✕ ✏ ✟ ✜ ✪ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞ ✕✡✕✗✍✑ ✕ ✄✗ ❈✕ ✑✴ % ✍✁✑ ✕ ✍✞ %✕✭✄✑ ✕✗✼ ✍✡%✞ ✕♥✟ ✜ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✆✓ ✍ ❍✍✏✑ ✄✁ ✄❍ ✄✁✩✟✁ ✂ ✖ ✪ ✡✍❙✕✗✼ ✕✑ ✍ ✺✜ ✟✞✟ ✜ ✜ ✍✞✼✄✁✕ ✑✴✡ ✕✗ ✑✴✟ ❈✄✁✜ ✑ ✏ ✍✜ ✟ ✷ ❦ ✑ ✜ ✟ ✟✡✜ ✑✴✍✑ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞ ✕✡✕✗✍✑ ✕ ✄✗ ❈✄✁❙✜ ✕✗ %✁✍✏✑ ✕ ✏ ✟ ✆ ✟ ✏ ✍✺✜ ✟ ✑✴✟ ✜ ✟✑ ✄❍ ✡✍✑ ✁✕ ✏ ✟ ✜ ❈✕ ✑✴ ✜✺✏✴ ✆ ✟✴✍✭✕ ✄✁ ✕ ✜ ✭✍✗✕ ✜✴✕✗✼✞✓ ✜✡✍✞ ✞ ✪ ✆✺✑ ✑ ✄ ✑✴✕ ✜ ✩✍✓✪ ✗✄✆ ✄ ✩✓ ✴✍✜ ✍ ✏ ✄✗✭✕✗✏✕✗✼ ✟✌%✞ ✍✗✍✑ ✕ ✄✗ ✄❍ ❈✴✓ ✑✴✕ ✜ ✜✴✄✺✞✩ ✆ ✟ ✜ ✄ ✷ ❦✗ ✡✍✗✕❍✄✞✩ ❈✍✓✜ ✪ ✑✴✟✗ ✪ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞ ✕✡✕✗✍✑ ✕ ✄✗ ✕ ✜ ✍✑✓%✕ ✏ ✍✞ ✷ ➇✟❈ ✗✺✡✟✁✕ ✏ ✍✞ ✍✞✼✄✁✕✑✴✡✜ ✴✍✭✟ ✜✺✏✴ ✜✺✆✑ ✞✟ ✜ ✑ ✍✆✕ ✞ ✕✑✓ %✁✄% ✟✁✑ ✕ ✟ ✜ ✪ ✍✗✩ ✏ ✟✁✑ ✍✕✗✞✓ ✗✄ ✄✑✴✟✁ ❈✍✜ ✜ ✏✁✺✑ ✕✗✕ ✘ ✟✩ ✕✗ ✜✺✏✴ ✩✟%✑✴ ✆✓ ✭✄✗ ☎✟✺✡✍✗✗ ✪ ❘✺✁✕✗✼ ✪ ✍✗✩ ✟✕ ✞❙✕✗✜ ✄✗ ✷ ❘✴✟ ✟◆✟ ✏✑ ❐ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✪ ❈✴✕ ✏✴ ✜✴✄✺✞✩ ✴✍✭✟ ✆ ✟ ✟✗ ✍ ✜ ✕✩✟ ✜✴✄❈✪ ✞ ✕✗✼✟✁✟✩ ✕✗ ✑✴✟ ✜% ✄✑ ✞ ✕ ✼✴✑ ❈✴✕ ✞ ✟ ✄✺✁ ♥✟✞✩ ❈✍✜ ✓✄✺✗✼ ✍✗✩ ✼✁✟❈ ✕✗✑ ✄ ✑✴✟ ✏ ✍✗✄✗✕ ✏ ✍✞ ✍✞✼✄✁✕ ✑✴✡ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✷ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✜ ✟✑ ✑✴✟ ✍✼✟✗✩✍ ✪ ✟✕ ✞❙✕✗✜ ✄✗ ✜ ✟✑ ✑✴✟ ✑ ✄✗✟ ✪ ✍✗✩ ✑✴✟ ✩✕ ✜ ✑ ✁✟ ✜ ✜ ✕✗✼ ✁✟ ✜✺✞✑ ✴✍✜ ✆ ✟ ✟✗ ➌➐ ⑩ ➎ ✷ ❍ ✏ ✄✺✁✜ ✟ ✑✴✟✁✟ ✕ ✜ ✡✄✁✟ ✑✴✍✗ ✑✴✕ ✜ ✑ ✄ ✑✴✟ ✴✕ ✜ ✑ ✄✁✓ ✄❍ ✴✄❈ ➌➐ ⑩ ➎ ✍✏➅✺✕✁✟✩ ✏✺✁✁✟✗✏✓✷ ❦✗ ✑✴✟ ✟ ✍✁✞✓ ✓✟ ✍✁✜ ✄❍ ✏ ✄✡%✺✑ ✟✁✜ ✪ ✕ ✑ ❈✍✜ ✕✗✟✭✕✑ ✍✆✞✟ ✑✴✍✑ ✍✁✕✑✴✡✟✑ ✕ ✏ ✕ ✜ ✜✺✟ ✜ ❈✄✺✞✩ ✁✟ ✏ ✟✕✭✟ ✏ ✄✗✏ ✟✁✑ ✟✩ ✍✑ ✑ ✟✗✐ ✑ ✕ ✄✗ ✷ ➇✕✌✟✩✐% ✄✕✗✑ ✏✄✡%✺✑ ✍✑ ✕ ✄✗ ✁✟➅✺✕✁ ✟✩ ✏ ✍✁ ✟❍✺✞ ✑✴✄✺✼✴✑ ✍✗✩ ✗✄✭✟✞ ✴✍✁✩❈✍✁✟ ✢ ✜✄ ✍✑ ✕✗✼✐% ✄✕✗✑ ✏ ✄✡%✺✑ ✍✑ ✕ ✄✗ ✍✁✁✕✭✟✩ ✍✜ ✍ ✜ ✟ ✏✄✗✩ ✁ ✟✭✄✞✺✑ ✕ ✄✗ ✍ ❍✟❈ ✓✟ ✍✁✜ ✞ ✍✑ ✟✁ ✷ ✣✗✑ ✕ ✞ ✑✴✟ ✜ ✟ ✡✍✑ ✑ ✟✁ ✜ ❈✟✁✟ ❈✟✞ ✞ ✺✗✩✟✁ ✜ ✑ ✄ ✄ ✩ ✕✑ ❈✍✜ ✗✍✑✺✁✍✞ ✑✴✍✑ ✍✁✕ ✑✴✡✟✑ ✕ ✏ ✕ ✜ ✜✺✟ ✜ ✜✴✄✺✞✩ ✆ ✟ ✍ ✏ ✟✗✑ ✁ ✍✞ ✑ ✄%✕ ✏ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞ ✐ ✓✜ ✕ ✜ ✪ ✍✗✩ ✪ ✆ ✟ ✜ ✕✩✟ ✜ ✑✴✕ ✜ ✪ ✍✗✄✑✴✟✁ ❍✄✁✏ ✟ ❈✍✜ ✍✑ ❈✄✁❙ ✷ ❘✴✟✁✟ ✕ ✜ ✍ ✼✟✗✟✁ ✍✞ %✁✕✗✏✕%✞ ✟ ✄❍ ✏✄✡%✺✑ ✕✗✼ ✑✴✍✑ ✜ ✟ ✟✡✜ ✑ ✄ ✴✍✭✟ ✗✄ ✗✍✡✟ ⑦ ➜➙ ➧ ➲➛➭ ➜ ➧ ➸ ➜➙ ➧ ➻➯➩✥➵➜ ➧ ➸➪ ➜➙ ➧ ➩➯ ➸➧ ➺➩✥➯ ➸➜ ➛➫➜ ➜➙ ➧ ➭✥➧ ➧ ★ ➯➲ ➛ ➦✪ ➯ ✬ ➸➺ ➜➙➩➭ ✭ ❦✗ ✑✴✟ ✟ ✍✁✞✓ ✓✟ ✍✁ ✜ ✪ ❈✕✑✴ ✑✴✟ ✟ ✍✁✞✓ ✏ ✄✡%✺✑ ✟✁✜ ✪ ✑✴✟ ✩✍✗✼✟✁✜ ✄❍ ✕✗✜ ✑ ✍✆✕ ✞✕ ✑✓ ❈✟✁✟ ✗✟ ✍✁✞✓ ✍✜ ✼✁ ✟ ✍✑ ✍✜ ✑✴✟✓ ✍✁✟ ✑ ✄ ✩✍✓✪ ✍✗✩ ❍✍✁ ✞✟ ✜ ✜ ❍✍✡✕ ✞ ✕ ✍✁ ✷ ❘✴✟ ✼✍%✜ ✆ ✟✑❈✟ ✟✗ ❍✍✜ ✑ ✍✗✩ ✜ ✞✄❈ ✍✞✼✄✁✕✑✴✡✜ ✪ ✴✄❈✟✭✟✁ ✪ ❈✟✁✟ ✗✍✁✁✄❈✟✁ ✷ ➔ ✩✟✭✟✞✄%✡✟✗✑ ✴✍✜ ✄ ✏ ✏✺✁✁✟✩ ✕✗ ✁✟ ✏ ✟✗✑ ✓✟ ✍✁✜ ✑✴✍✑ ✁✟✜✟ ✏✑ ✜ ✴✄❈ ❍✍✁ ❈✟ ✴✍✭✟ ✏✄✡✟ ❍✁✄✡ ✑✴✍✑ ✑ ✕✡✟ ✷ ❦✗✜ ✑ ✍✗✏ ✟ ✜ ✴✍✭✟ ✆ ✟ ✟✗ ✍✏ ✏✺✡✺✞✍✑ ✕✗✼ ✕✗ ❈✴✕ ✏✴ ✪ ✟✭✟✗ ✑✴✄✺✼✴ ✍ ♥✗✕ ✑ ✟ ✍✞✼✄✁✕ ✑✴✡ ✟✌✕ ✜ ✑ ✜ ❍✄✁ ✍ %✁✄✆✞ ✟✡✪ ✍✗ ✕✗♥✗✕ ✑ ✟ ✍✞✼✄✁✕ ✑✴✡ ✡✍✓ ✆ ✟ ✆ ✟✑ ✑ ✟✁ ✷ ❘✴✟ ✩✕ ✜ ✑ ✕✗✏✑ ✕ ✄✗ ✑✴✍✑ ✜ ✟ ✟✡✜ ✍✆✜ ✄✞✺✑ ✟ ❍✁✄✡ ✍ ✞✄✼✕ ✏ ✍✞ % ✄✕✗✑ ✄❍ ✭✕ ✟❈ ✑✺✁✗✜ ✄✺✑ ✑ ✄ ✴✍✭✟ ✞ ✕✑ ✑ ✞✟ ✕✡% ✄✁✑ ✍✗✏ ✟ ✕✗ %✁ ✍✏✑ ✕ ✏ ✟❜✍✗✩ ✕✗ ❍✍✏✑ ✪ ➔✆ ✟✞ ✍✗✩ ❢✍✞✄✕ ✜ ✗✄✑❈✕ ✑✴✜ ✑ ✍✗✩✕✗✼ ✪ ✞✍✁✼✟✐ ✜ ✏ ✍✞✟ ✡✍✑ ✁✕✌ ✟✕✼✟✗✭✍✞✺✟ %✁✄✆✞✟✡✜ ✍✁✟ ✍✆ ✄✺✑ ✍✜ ✟ ✍✜✓ ✑ ✄ ✜ ✄✞✭✟ ✕✗ %✁ ✍✏✑ ✕ ✏ ✟ ✍✜ ✞✕✗✟ ✍✁ ✜✓✜ ✑ ✟✡✜ ✄❍ ✟➅✺✍✑ ✕✄✗✜ ✷ ➇✄✁ Z✉ ✈ ✇ ✪ ✕ ✑ ✟✁ ✍✑ ✕✭✟ ✡✟✑✴✄ ✩✜ ✍✁✟ ✆ ✟ ✏ ✄✡✕✗✼ ✡✄✁✟ ✍✗✩ ✡✄✁✟ ✄❍✑ ✟✗ ✑✴✟ ✡✟✑✴✄ ✩✜ ✄❍ ✏✴✄✕ ✏ ✟ ✍✜ ✏ ✄✡%✺✑ ✟✁✜ ✼✁✄❈ ❍✍✜ ✑ ✟✁ ✪ ✡✍✑ ✁✕ ✏ ✟ ✜ ✼✁✄❈ ✞ ✍✁✼✟✁ ✍✗✩ ✞ ✟ ✜ ✜ ✜% ✍✁✜ ✟ ➌✆ ✟ ✏ ✍✺✜ ✟ ✄❍ ✑✴✟ ✍✩✭✍✗✏ ✟ ❍✁✄✡ ✂➐ ✑ ✄ ➤➐ ✜ ✕✡✺✞ ✍✑ ✕ ✄✗✜ ➎ ✪ ✍✗✩ ✑✴✟ ➋ ➌➍✱ ➎ ✄% ✟✁ ✍✑ ✕✄✗ ✏✄✺✗✑ ✜ ✄❍ ✑✴✟ ✺✜✺✍✞ ✩✕✁ ✟ ✏✑ ➌✈ ♥✗✕ ✑ ✟ ➎ ✍✞✼✄✁✕ ✑✴✡✜ ✆ ✟ ✏ ✄✡✟ ✟✭✟✁ ✡✄✁✟ % ✍✕✗❍✺✞ ✷ ❘✴✟ ✗✍✡✟ ✄❍ ✑✴✟ ✗✟❈ ✼ ✍✡✟ ✕ ✜ ➺ ➜ ➧ ➸➛ ➜➺ ➯ ➫ ✲➺ ➜➙ ✥➸➧ ➻➯ ➫ ★➺ ➜➺ ➯ ➫➺➫✪ ✭ ❦✗✏✁✟ ✍✜ ✕✗✼✞✓ ✄❍✑ ✟✗ ✕ ✑ ✕ ✜ ✗✄✑ ✄%✑ ✕✡✍✞ ✑ ✄ ✑ ✁✓ ✑ ✄ ✜ ✄✞✭✟ ✍ ❸
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 ✷ ❘✴✍✑ ✕ ✜ ✍✗✄✑✴✟✁ ✟ ✜ ✜ ✍✓✷ ❒