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✁✄✭✕✗✼ ✟✌✕ ✜ ✑ ✟✗✏ ✟ ✑✴✟✄✁✟✡✜ ✷ ➤