[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 netic 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