Calculus Numericus: And its computational implications (Mathematical Graduate Books from The Scottish Science Society)
Kindle Edition or EPUB + Converted PDF نویسندگان: Richard Murdoch Montgomery
جزئیات
فرمت: Kindle Edition or EPUB + Converted PDF ناشر: Scottish Science Society تاریخ انتشار نسخه الکترونیکی : August 3, 2026
؟
لینک: https://www.amazon.com/dp/B0HCR9QZKP
2,297,000 تومان
زمان تحویل: حداکثر 24 ساعت
توضیحات
Why Numerical Mathematics Is a Science of Judgement There exists a widespread and rather comfortable belief that a computer, being incapable of opinion, must therefore be incapable of error. The reader who has opened this volume presumably harbours doubts on the matter, and the author hastens to confirm them: the machine is scrupulously honest and habitually wrong, in the precise sense that it answers, with impeccable speed, a question subtly different from the one we asked. The entire discipline assembled in these eight hundred pages may be described as the art of measuring that difference — and, where possible, of forgiving it. The thesis of this treatise is easily stated. Numerical mathematics is not a warehouse of recipes but a science of judgement: every computed number is a claim, every claim carries a debt of justification, and the currency in which such debts are paid is error analysis. The antithesis, held with sincerity by many otherwise reasonable people, is that modern software has made such scruples obsolete — that one presses the key marked solve and receives the truth, as one presses a switch and receives light. The synthesis, which the following chapters develop at leisure, is that software has not abolished judgement but concentrated it: the fewer decisions the user visibly makes, the graver the consequences of the decisions made invisibly on the user's behalf. A person who cannot audit an approximation has not been spared the mathematics; he has merely agreed not to witness it. The plan of the work follows the natural biography of a computed number. Part I establishes what it means for a finite machine to do analysis at all — arithmetic, error, conditioning, stability, approximation. Parts II and III supply the classical instruments: equations, interpolation, quadrature, and the linear algebra upon which everything else secretly rests. Parts IV and V treat differential equations, ordinary and partial, which is to say the mathematics of things that change; Part VI treats optimisation, inversion, and uncertainty, which is to say the mathematics of things we wish to change or cannot quite see. Part VII brings the account to the present day — automatic differentiation, parallel computation, reliable software, scientific machine learning — and Part VIII assembles the whole apparatus into three extended studios where the methods are made to earn their living on complete problems. Every algorithm in this book is presented through one invariable intellectual sequence: problem, derivation, algorithm, error analysis, implementation, application, verification. Nothing falls from the sky. Where a formula appears, its derivation precedes it; where a computation is displayed, its keystrokes or its code accompany it; where a graph is drawn, the reader is told exactly why it should — or, more instructively, should not — be believed. The author has also permitted himself the occasional aside to the reader, in the conviction that a treatise, like a dissection, proceeds better when the students are awake. A word on tone. The subject is exact; its practice is not; and the gap between the two is best surveyed with a certain composure. If the reader detects, here and there, a smile at the corner of the page, let it be taken as directed never at the mathematics, which deserves our reverence, but at ourselves, who compute — and who believe our computations with an enthusiasm that the following chapters will attempt, gently, to moderate. R. M. M. — Edinburgh and São Paulo, 2026.
2,297,000 تومان
زمان تحویل: حداکثر 24 ساعت