SOLVING NON-LINEAR SYSTEMS BY NEWTON s METHOD USING SPREADSHEET EXCEL Tay Kim Gaik Universiti Tun Hussein Onn Malaysia
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1 SOLVING NON-LINEAR SYSTEMS BY NEWTON s METHOD USING SPREADSHEET EXCEL Tay Km Gak Unverst Tun Hussen Onn Malaysa <tay@uthmedumy> Kek Se Long Unverst Tun Hussen Onn Malaysa <slkek@uthmedumy> Rosmla Abdul-Kahar Unverst Tun Hussen Onn Malaysa <rosmla@uthmedumy> Abstract Tay (6 had proposed solvng numercal methods usng the Caso fx-57ms calculator to overcome the tedousness of dong recursve calculatons Here we present another alternatve that s solvng a non-lnear system usng Newton s method n Mcrosoft Offce Excel For ths we just make use of the MULT functon to do matrx multplcaton and MINVERSE functon to do the matrx nverse operaton The concept presented here can be developed nto a solver where the user just needs to nput the ntal vector X the correspondng formula of Jacoban matrx J( x y and non-lnear system vector F( x y The full solutons wll be dsplayed automatcally The solver s easy and user frendly for students and educator who needs ts full solutons quckly Introducton Many real-lfe problems are non-lnear A couple of non-lnear equatons form a system of non-lnear equatons Systems of non-lnear equatons can be solved numercally by Newton s method fxed-pont teraton method Solvng a system of non-lnear equatons by Newton s method s tedous Tay (6 proposed a step by step nstructon on how to use Caso fx-57ms n Numercal methods whle Guerrer- Garca and Santos-Polomo (8 presented a collecton of keystroke sequences helpng n solvng numercal methods Tay and Kek (8 developed a solver to solve the most domnant egenvalue by the power method through Mcrosoft Offce Excel On the other hand Kek and Tay (8 developed a solver to solve system of lnear equatons usng Excel too None of the lterature works dealt wth a system of non-lnear equatons Hence n ths paper we have developed a solver for solvng system of non-lnear equatons by usng Newton s method through Excel 3 Newton s Method Gven the contnuous functons f ( x y g( x y * * * * * * we would lke to fnd the values x x and y y such that f ( x y and g( x y Assume that an ntal approxmate soluton to equaton ( s known as ( x y Express f ( x y and g( x y n two-varables Taylor seres about ( x y we get f ( x y f ( x y f ( x y ( x x f ( x y( y y x y g( x y g( x y g( x y ( x x g( x y( y y x y f ( x y g( x y and because wll be the roots Truncatng Eq ( after the frst dervatves terms and rearrangng yelds f ( x y f ( x y x f ( x y y x y g( x y g( x y x g( x y y x y x x x y y y where Wrtng Eq (3 n matrx notaton yelds the small changes below: ( (3 (
2 where J( x y s the Jacoban matrx f ( x y f ( x y x y x f ( x y y g( x y g( x y g( x y x y J ( x y X F( x y J( x y F( x y X (4 Thus the new approxmaton soluton wll be or equvalently Procedure for solvng nonlnear system f ( x y x x x y y y x x x y y y X =X X g( x y by Newton s method f ( x y f ( x y x y Step : Obtan Jacoban matrx J( x y = g( x y g( x y x y f ( x y Step : Obtan F( x y = g( x y Step 3: Solve J ( x y X F( x y for X X =X X Step 4: Calculate Step 5: Repeat Steps 3 and 4 untl X X Step 6: Roots s X Numercal Example Gven the system of non-lnear equaton below: f ( x y x y g x y y x Ther graphs are shown n the Fgure below: (
3 ( Soluton Step Step Fgure Graph of f ( x y and g( x y Solve the system of non-lnear equaton above by usng Newton s method wth ntal guess ( - and f f x y x J ( x y h h y x y x y F( x y y x Step 3 By usng X =X X where J x y wth ntal guesses ( - and ( as n Tables and respectvely Table Intal guess ( - X ( F ( x y we obtan the numercal soluton X J(x y F(x y X E-6 6E E-5 8E E- E E- 4E Snce X6 X5 5 so one of the root s (
4 Table Intal guess ( X J(x y F(x y X E E-5-6E E-9-3E E-9-3E-9 Snce X4 X3 5 so the other root s (49 7 Steps In Usng The Solver For Solvng The System Of Non-Lnear Equaton Step : User nput the ntal vector X Step : User nput formula for ntal Jacoban matrx J ( x y Step 3: User nput formula for ntal vector F( x y Step 3: The results wll be automatcally dsplayed up to teratons Step 4: User needs to copy and drag the teraton f the stoppng crtera s not met yet Step 5: User needs to cut the approprate number of teratons f the number of teratons s less than teratons Fgure shows the solver n the Excel envronment: Fgure Excel Solver for Solvng the System of Non-Lnear Equatons 4
5 The Excel Command Involved In Buldng the Solver Step : We select cells F8:F9 and type n = -MMULT(MINVERSE(C8:D9 E8:E9 then press SHIFT CRTL and ENTER at the same tme to calculate X Step : We select cells B:B and type n = B8:B9+F8:F9 and then press SHIFT CRTL and ENTER at the same tme to calculate X Step 3:We then select cells C8:F9 and then copy t to F Step 4:Fnally we select A:F and then copy them tll F9 to get the full soluton as shown n Fgure Advantages and Shortcomes of The Solver Ths solver helps educators and students who need the full numercal solutons of a system of nonlnear equatons by nputtng the ntal vector X Jacoban matrx J ( x y and ntal vector F( x y Even though t s desgned for a system of two non-lnear equatons the EXCEL command nvolved s not that dffcult f compared to the MATHCAD MAPLE MATLAB or C Programmng Hence users can extend the concept presented here to a system of n non-lnear equatons One of the lmtatons here s the solver s not yet fully automated so that the users need to control the number of teratons themselves manually by draggng to ncrease the number of teratons or select only the desred number of teratons from the gven ten teratons However ths manual opton s sutable for the purpose of teachng and learnng n the class room and evaluaton n the examnaton hall where students need to know where to stop the teratons Concluson A solver s developed to get the full numercal solutons of a system of two non-lnear equatons by Newton s method through Excel Ths solver helps educators and students who need full numercal solutons of a system of two non-lnear equatons by just nputtng the ntal vector X Jacoban matrx J ( x y and ntal vector F( x y but the solver s not yet fully automated where user needs to control the number of teratons themselves manually So we ntend to study the Mcrosoft Excel VBA language to overcome ths shortage n future Acknowledgement(s We wsh to acknowledge regstrar of Unverst Tun Hussen Onn Malaysa for supportng the budget of the presentaton of ths paper n 3 rd CoSMED 9 n Penang References Guerrero-García P and Santos-Palomo Á (8: Squeezng the Most Out of the Caso fx-57ms for Electrcal/Electroncs Engneers Proceedngs of the Internatonal Conference on Computatonal and Mathematcal Methods n Scence Kek S L & Tay K G (8 Solver For System of Lnear Equatons Prosdng Semnar Kebangsaan Aplkas Sans dan Matematk 8 (SKASM 8 Batu Pahat: Penerbt UTHM pp Tay KG (6 How To Use Calculator Caso FX-57MS In Numercal Methods Batu Pahat: Penerbt KUTTHO Tay K G & Kek S L (8 Approxmatng The Domnant Egenvalue Usng Power Method Through Spreedsheet Excel Prosdng Semnar Kebangsaan Aplkas Sans dan Matematk 8 (SKASM 8 Batu Pahat: Penerbt UTHM Pg
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