BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics

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1 Mohd Yusuf Yasi / BIBECHANA 8 (2012) : BMHSS, p. 31 BIBECHANA A Multidiscipliary Joural of Sciece, Techology ad Mathematics ISSN (olie) Joural homepage: Scietific Calculators ad the Skill of Efficiet Computatio Mohd Yusuf Yasi ECE Departmet, Itegral Uiversity, Luckow, UP, Idia mmyasi@rediffmail.com Article history: Received 12 May 2011; Accepted 21 Jue 2011 Abstract Calculators are part ad parcel of moder educatio. Ivolvemet of sciece ad egieerig i differet fields of kowledge is icreasig with each bit of time is passed by, ad they are playig a role i descriptio ad characterizatio of the delicate pheomea of ature arisig day by day. These fields of kowledge ad mathematics i particular, are ifluecig eve those distat braches of kowledge, which were so far imagied to be free of mathematics. Eve art is ot free of mathematics ad there eists mathematical art. Computatios are gettig legthy ad comple specially i desig ad aalysis of egieerig systems. Scietific calculators are hady tools. But a efficiet computatio is a skill that ca be developed. Keywords: scietific calculators; calculator techiques; efficiet computatio; efficiet use of calculators, computatio skills; umerical techiques 1. Itroductio Today s world is of sciece ad techology. Sciece has pervaded i other braches of kowledge, it has become difficult to thik a subject thoroughly eplaied without ivitig sciece to help. More isight ito the atural pheomea are required ad as a way, they demad computatios. Subjects specifically related to egieerig ad techology, are heavily computatioal. Fortuately, powerful computig devices are readily available ad i every body s reach. It has bee observed that usually studets buy scietific calculators. But it is a pity that most of these studets usually imagie the operatios of multiplyig ad dividig, ad computatios of fuctios like sie ad cosie is the all a scietific calculator ca do. Today s scietific calculators are highly powerful ad if their operatio is properly uderstood, a little iovative imagiatio ca do woders. Log calculatios are a ormal part of desig ad aalysis, where studets sped time like aythig, ad coclude the computatioal procedure i distaste. I this short paper efforts are made to poit out that this attitude eeds a rethikig o the process afresh to settle dow the matters favorably.

2 Mohd Yusuf Yasi / BIBECHANA 8 (2012) : BMHSS, p Prelimiaries A little kowledge is required before oe should start workig with a calculator. Every calculator is complemeted with a maual, which delieates the calculator capabilities. It eeds proper attetio to go through the maual. Scietific calculators are highly capable ad therefore are comple etities. Their fuctioig is grouped. Therefore a calculator eeds to be prepared for a proper fuctioig. A very commo mistake usually observed i this cotet is the computatio performed o fuctios cotaiig circular fuctios without a care of the proper mode of their argumets or computatios o fuctios cotaiig a log without a idea of the atural log or log o base 10. Such improper modes lead to wrog results ad may cause a failure to maitai iterest i the process of desig ad aalysis both. 3. Calculator Techiques Almost all scietific calculators posses a temporary register uder the key As, which automatically modifies after each computatio, ad holds the curret result. I desig or aalysis, this result is usually eeded by the et step. Still there are eamples where recursive computatios are ecessary to ascertai the fial results. Normally, calculators have a few more variable like A, B, C,, but presetly a descriptio is restricted o some simple tricks based upo the As key oly. I fig.1 the geeral scheme adopted here is preseted. Figure 1: Geeral flow graph of the computatioal procedure

3 Mohd Yusuf Yasi / BIBECHANA 8 (2012) : BMHSS, p. 33 I the followig sectios, two eamples are cosidered for the purpose of descriptio o how the calculator tricks are applicable to ease the process of approachig fial accurate result recursively. I both these eamples, due to their simplicity ad ecellet bidirectioal approach to the fial accurate solutio, the simple Newto Raphso method is employed. 3.1 Calculators Referred I this article, scietific calculators referred to as the casio make oly, like f-82ms, f-991ms or later versios. All these calculators have a distict feature of displayig full scietific equatios before ad after the calculatio is doe. A Scietific Calculator afterwards refers to a calculator possessig these specific features. Other brads of calculators with similar features ca also be used for the purpose of this article, possibly with a little or o modificatio i the computatioal scheme depedig upo their operatig priciples [1,2]. 3.2 Eample 1: Square root of a give umber I the first eample, computatio of the square root of a give umber a is cosidered. Let be 2 = the square root of the umber a Thus mathematically this ca be represeted as, ad the correspodig fuctioal equatio is f ( ) = 2 a. Accordig to the ordiary Newto = f ( ) ' f ( ) Raphso method, the square root is approached by the equatio, where + 1 ' is the first approimatio, + 1 is the ew improved approimatio ad f ( ) is the first derivative of the fuctio f ( ). Whe applied to the required square root of the give umber, this approimatig equatio ca be epressed as [3]: a = a (1) This method is particularly importat as it is quite fast i approachig the fial reasoable result i just a few iteratios. Further, the fial result is possible eve whe a absurd iitial approimatio is adopted, of course, i such a case, total umber of iteratios may be icreased. For applyig a scietific calculator, the above root approimatig equatio is cosidered for determiig a square root of the give umber 3, with a iitial approimatio, say, 1. The procedure ca be as follows. 1. Type the iitial approimatio, (say, 1 for this eample) ad press = (eter through the key = i casio calculators, refer, f-82ms, f-991ms). This step stores the 1 i the register associated with the key As. As every result is automatically stored uder the key As, therefore this key ca be used as a variable. Hece, the square root determiig equatio ca be re writte with As as variable, As= 2 As+ a As 1 (2) Here i equatio (2), the As iside the braces refers to ad cotai the last result, whereas the As outside refers to + 1 ad automatically modifies to the latest result. As

4 Mohd Yusuf Yasi / BIBECHANA 8 (2012) : BMHSS, p. 34 per the data hadlig scheme of the calculators, the equatio (2) is read as a As 2 As+ As 1, where the calculatio results o the right had side are trasferred to the register attached to the As key, 2. Secod step is to type the equatio i accordace with the right had side of the above equatio assumig the variable key As ad do the required computatio by just pressig the = key as may umber of times as required. The sequece of keys pressed o a calculator may be as follows: 1 =; ( As + a / As ) / 2 = = = (3) I this costruct, each key is separated by multiple spaces for the sake of clarity oly, the spaces are ot a part of the equatio. As is evidet from the calculatio of the eamples below, for three iteratios, the total umber of key presses are just 14 i each case. Covergece of 3 to it s fial umerical value is as follows: As 1.0 As 2.0 As 1.75 As As As (4) The value calculated from the calculator is 3= , thereby showig o error i the two results. However, 3= is ormally the accepted approimate value, ad is available withi third iteratio. 3.3 Root of a equatio Agai, cosider to fid the itersectio of the curves [4] y= cos ad y=. (5) It is a sigle root problem. A solutio eists as show i Fig. 2. Aalytically, the solutio is obtaied as follows. Applyig the ordiary Newto Raphso method, equatio for solutio is obtaied as f ( ) = cos from eq. (5) above. The correspodig approimatio equatio to it s root is + 1 = cos 1+ si si + cos 1+ si =. (6)

5 Mohd Yusuf Yasi / BIBECHANA 8 (2012) : BMHSS, p. 35 The computatioal equatio for a calculator is writte as give i equatio (7) As As sias+ cosas 1+ sias = (7) Covergece patter of the root of the above equatio with 1.0 as iitial approimatio, ad for radias as the mode of calculatio, is Figure 2: Graphical solutio of the equatios of sectio 3.3 [4] As 1 As As As As (8) A root of third decimal place accuracy is agai achieved just withi three iteratios. It is importat to cosider that the cocered equatios cotai circular fuctio, ad therefore the selectio of the mode RADIAN becomes importat. The results preseted above are obtaied i RADIAN mode. Purpose of the problem i equatio (5) is to brig out the importace of cosideratio as metioed earlier. I the solutio of equatio (6), Deg mode selectio will produce a look alike root, , which ufortuately is ot a root but merely a approimatio of the equatio (5) for the argumet of the circular fuctio tedig 0 ( cos ; 0 ). This situatio arises due to a iappropriate mode selectio. 4. Discussios The above eamples are commo yet simple ad are readily available i tet books o Advaced Mathematics for Egieers ad Scietists, Numerical Techiques etc. here the poit of emphasis is how to make the computatio effectively. The effective computatioal process for eample i sectio 3.2 for a sigle digit umber as per the flow graph, eeds, all i all, 14 judicial key presses

6 Mohd Yusuf Yasi / BIBECHANA 8 (2012) : BMHSS, p. 36 o a already ON calculator. However, the egieerig studets have ofte bee observed hagglig for hours over solvig such problems ad with absurd or at least, iaccurate come outs. 5. Coclusio For fields like umerical methods, where usually computatios are huge but are iterative i ature, it is importat to optimize computatioal efforts. Such efforts are based o computatioal skills. These are some of the calculator tricks preseted here which ca be foud quite useful. Scietific calculators are quite powerful, but their real power remais utapped. A little practice ca help develop the capability to perform computatios i techical maer. Refereces [1] Users Guide, f-82ms/83ms/85ms/270ms/300ms/350ms, [2] Users Guide, f-95ms/100ms/115ms/(912ms)/570ms/991ms, [3] Erwi Kreyszig, Advaced Egieerig Mathematics, 5 th editio, Wiley Easter Limited, 1989, pp 764 [4] Graphical solutio for the equatios give i eq. (5)