Concise Guide to Numerical Algorithmics: The Foundations and Spirit of Scientific Computing

Numerical Algorithmic Science and Engineering (NAS&E), or more compactly, Numerical Algorithmics, is the theoretical and empirical study and the practical implementation and application of algorithms for solving finite-dimensional problems of a numeric nature. The variables of such problems are either discrete-valued, or continuous over the reals, or, and as is often the case, a combination of the two, and they may or may not have an underlying network/graph structure. This re-emerging discipline of numerical algorithmics within computer science is the counterpart of the now well-established discipline of numerical analysis within mathematics, where the latter's emphasis is on infinite-dimensional, continuous numerical problems and their finite-dimensional, continuous approximates.

A discussion of the underlying rationale for numerical algorithmics, its foundational models of computation, its organizational details, and its role, in conjunction with numerical analysis, in support of the modern modus operandi of scientific computing, or computational science & engineering, is the primary focus of this short monograph. It comprises six chapters, each with its own bibliography. Chapters 2, 3 and 6 present the book's primary content. Chapters 1, 4, and 5 are briefer, and they provide contextual material for the three primary chapters and smooth the transition between them. Mathematical formalism has been kept to a minimum, and, whenever possible, visual and verbal forms of presentation are employed and the discussion enlivened through the use of motivating quotations and illustrative examples.

The reader is expected to have a working knowledge of the basics of computer science, an exposure to basic linear algebra and calculus (and perhaps some real analysis), and an understanding of elementary mathematical concepts such as convexity of sets and functions, networks and graphs, and so on. Although this book is not suitable for use as the principal textbook for a course on numerical algorithmics (NAS&E), it will be of value as a supplementary reference for a variety of courses. It can also serve as the primary text for a research seminar. And it can be recommended for self-study of the foundations and organization of NAS&E to graduate and advanced undergraduate students with sufficient mathematical maturity and a background in computing.

When departments of computer science were first created within universities worldwide during the middle of the twentieth century, numerical analysis was an important part of the curriculum. Its role within the discipline of computer science has greatly diminished over time, if not vanished altogether, and specialists in that area are now to be found mainly within other fields, in particular, mathematics and the physical sciences. A central concern of this monograph is the regrettable, downward trajectory of numerical analysis within computer science and how it can be arrested and suitably reconstituted. Resorting to a biblical metaphor, numerical algorithmics (NAS&E) as envisioned herein is neither old wine in new bottles, nor new wine in old bottles, but rather this re-emerging discipline is a decantation of an age-old vintage that can hopefully find its proper place within the larger arena of computer science, and at what appears now to be an opportune time.

數值算法科學與工程（NAS&E），或更簡潔地說，數值算法，是對解決數值性的有限維問題的算法進行理論和實證研究，以及實際實現和應用的學科。這些問題的變量可以是離散值，也可以是連續的實數，或者通常情況下是兩者的組合，它們可能具有或不具有底層的網絡/圖結構。這個在計算機科學中重新出現的數值算法學科是數學中已經確立的數值分析學科的對應物，後者的重點是無窮維的連續數值問題及其有限維的連續近似。

本短篇論文的主要焦點是對數值算法學科的基本模型、組織細節以及與數值分析相結合在科學計算或計算科學與工程中的作用的基本原理的討論。它由六章組成，每章都有自己的參考文獻。第2、3和6章介紹了本書的主要內容。第1、4和5章較短，它們為三個主要章節提供了背景材料，並平滑了它們之間的過渡。數學形式主義被保持在最低限度，並且在可能的情況下，使用視覺和口語形式進行演示，並通過引人入勝的引言和實例來進行討論。

讀者應該具備計算機科學基礎知識，了解基本的線性代數和微積分（也許還有一些實分析），以及對凸集和函數、網絡和圖等基本數學概念的理解。雖然本書不適合作為數值算法（NAS&E）課程的主要教材，但它對於各種課程的補充參考資料具有價值。它還可以作為研究研討會的主要教材。並且可以推薦給具有足夠數學成熟度和計算背景的研究生和高年級本科生自學NAS&E的基礎和組織。

當計算機科學系在二十世紀中葉首次在全球各大學成立時，數值分析是課程的重要組成部分。隨著時間的推移，它在計算機科學學科中的地位大大降低，甚至完全消失，該領域的專家現在主要在其他領域，特別是數學和物理科學中找到。本論文的一個核心問題是數值分析在計算機科學中的可惜下降趨勢，以及如何阻止並適當重建它。以聖經的隱喻來說，這裡所設想的數值算法（NAS&E）既不是新瓶裝舊酒，也不是舊瓶裝新酒，而是這個重新出現的學科是一種古老的陳年佳釀的倒瓶，希望它能在計算機科學的更大領域中找到合適的位置，並且現在似乎是一個有利的時機。