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It demonstrates limited (Taylor series) expansion, integration and verification by differentiation, alongside sample symbolic computations such as Taylor expansion, integration, differentiation, factorization, and simplification. It argues that what is often called “scientific calculation” hides ambiguity between numerical calculation and algebraic calculation by formulas, emphasizing that computer algebra remains relevant and requires well-structured systems with suitable representations, language tools, and effective algebraic function libraries.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/preface-an-example/326742/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/preface-an-example/326742.png","ImageObject",300,407,{"name":92,"@type":93},"Theodore","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-27","2026-09-21",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does the preface example demonstrate in MACSYMA?","Question",{"text":112,"@type":113},"It illustrates symbolic computation workflows: a Taylor series expansion, an integral computation, and a check that differentiating the result returns the original function.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What ambiguity does the text highlight about “scientific calculation”?",{"text":117,"@type":113},"The text notes that “scientific calculation” often gets treated as purely numerical computation, even though historically it blended numerical work with algebraic calculation by formulas.",{"name":119,"@type":110,"acceptedAnswer":120},"Why is algebraic calculation still considered relevant with computers?",{"text":121,"@type":113},"Because computing power does not eliminate the need to restate underlying formulas and can still face practical limits where numerical-only approaches become too costly, especially when higher precision or finer meshes are required.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},326742,1790452370,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},7971461740886,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Preface  \nAn example  \nThe example which follows is a work session on the computer, using the software MACSYMA. The user's commands are entered in the lines beginning with (ci) . Here we are dealing with a limited (or Taylor series) expansion, calculating an integral and checking that the derivative in the result is indeed the initial function.  \n(c2) taylor( sinh(sin(x)) -sin(sinh(x)) , x ,0 , 15);  \n7 11 15  \nx x 5699 x  \n(d2)/T/ -- - ---- - ---------- + . . .  \n45 1575 1277025750  \n(c3) primitive: integrate( 1/(x**2 + 1) ** 4 , x);  \n5 3  \n5 atan(x) 15 x + 40 x + 33 x (d3) --------- + ----------------------------  \n16 6 4 2  \n48 x + 144 x + 144 x + 48  \n(c4) derivative: diff(primitive, x);  \n4 2  \n75 x + 120 x + 33 (d4) --------------------------- -  \n6 4 2  \n48 x + 144 x + 144 x + 48  \n5 3 5 3  \n(15 x + 40 x + 33 x) (288 x + 576 x + 288 x) 5  \n- --------------------------- ---- ----- ---- ----- -- + -----------  \n6 4 2 2 2  \n(48 x + 144 x + 144 x + 48) 16 (x + 1)  \nv  \nvi Preface  \n(c5) factor(ratsimp(derivative)) ;  \n1  \n(d5) ---------  \n2 4  \n(x + 1)  \nAll these calculations can, in theory, be done by a 􀀌rst year student,  \nbut they are quite laborious and it is diÆcult to do them with pencil and paper without making a mistake.  \nIt is not suÆciently well-known that this type of calculation is as much within the range of computers as is numerical calculation. The purpose of this book is to demonstrate the existing possibilities, to show how to use them, and to indicate the principles on which systems of non numerical calculation are based, to show the diÆculties which the designers of these systems had to solve | diÆculties with which the user is soon faced.  \nBut before we try to describe how the authors hope to carry out this ambitious design, we should 􀀌rst try to de􀀌ne the subject better.  \nScienti􀀌c calculation and algebraic calculation  \nFrom the start of electronic calculation, one of the main uses of computers has been for numerical calculation. Very soon, applications to management began to dominate the scene, as far as the volume of calculation assigned to them is concerned. Nevertheless, scienti􀀌c applications are still the most prestigious, especially if we look at the performance required of the computer: the most powerful computers are usually reserved for scienti􀀌c  \ncalculation.  \nThis concept of \\scienti􀀌c calculation\" conceals an ambiguity, which it is important to note: before computers appeared on the scene, a calculation usually consisted of a mixture of numerical calculation and what we shall call \\algebraic calculation\", that is calculation by mathematical formulae. The only example of purely numerical calculation seems to have been the feats of calculating prodigies such as Inaudi: the authors of tables, especially of logarithms, did indeed carry out enormous numerical calculations, but these were preceded by a restatement of the algebraic formulae and methods which were essential if the work was to be within the bounds of what is humanly possible. For example, the famous large calculations of the 19th century include a large proportion of formula manipulation. The best known is certainly Le Verrier's calculation of the orbit of Neptune, which started from the disturbances of the orbit of Uranus, and which led to the discovery of Neptune. The most impressive calculation with pencil and paper is also in the 􀀌eld of astronomy: Delaunay took 10 years to calculate the orbit of the moon, and another 10 years to check it. The result is not numerical, because it consists for the most part of a formula which by itself occupies all the 128 pages of Chapter 4 of his book.  \nComputer Algebra vii  \nThe ambiguity mentioned above is the following: when computers came on the scene, numerical calculation was made very much easier and it became commonplace to do enormous calculations, which in some cases made it possible to avoid laborious algebraic manipulations. The result was that, for the public at large and e","cbCaiq36kn2X7X8P","https://ap.wps.com/l/cbCaiq36kn2X7X8P","pdf",1120069,313,"English","# Preface\n# Scientific calculation and algebraic calculation\n## Ambiguity between numerical and algebraic calculation\n## Relevance of algebraic calculation","[{\"question\":\"What does the preface example demonstrate in MACSYMA?\",\"answer\":\"It illustrates symbolic computation workflows: a Taylor series expansion, an integral computation, and a check that differentiating the result returns the original function.\"},{\"question\":\"What ambiguity does the text highlight about “scientific calculation”?\",\"answer\":\"The text notes that “scientific calculation” often gets treated as purely numerical computation, even though historically it blended numerical work with algebraic calculation by formulas.\"},{\"question\":\"Why is algebraic calculation still considered relevant with computers?\",\"answer\":\"Because computing power does not eliminate the need to restate underlying formulas and can still face practical limits where numerical-only approaches become too costly, especially when higher precision or finer meshes are required.\"}]","Preface - An example | PDF",1789975238,789]