INTRODUCTORY NUMERICAL ANALYSIS

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1 ITRODUCTORY UMERICL LYSIS Lecture otes y Mrce ndrecut Unversl Pulshers/UPUBLISHCOM Prlnd FL US

2 Introductory umercl nlyss: Lecture otes Copyrght Mrce ndrecut ll rghts reserved ISB: 877 Unversl Pulshers/uPUBLISHcom US wwwupulshcom/oos/ndrecuthtm To my fther nd my son

3 Prefce The m of ths oo s to provde smple nd useful ntroducton for the fresh students nto the vst feld of numercl nlyss Le ny other ntroductory course on numercl nlyss ths oo contns the sc theory whch n the present tet refers to the followng topcs: lner equtons nonlner equtons egensystems nterpolton ppromton of functons numercl dfferentton nd ntegrton stochstcs ordnry dfferentl equtons nd prtl dfferentl equtons lso the tet s restrcted to twelve lectures nd one ppend coverng only one semester of techng hours/lecture nd hours/em Becuse the students need to qucly understnd why the numercl methods correctly wor the proofs of theorems were shorted s possle nsstng more on des thn on lot of lger mnpulton The ncluded emples re presented wth mnmum of complctons emphszng the steps of the lgorthms The numercl methods descred n ths oo re llustrted y computer progrms wrtten n C Our gol ws to develop very smple progrms whch re esly to red nd understnd y students lso the progrms should run wthout modfcton on ny compler tht mplements the SI C stndrd Becuse our ntenton ws to esly produce screen nputoutput usng scnf nd prntf n cse of WIDOWS vsul progrmmng envronments le Vsul C Mcrosoft nd Borlnd C Bulder the proect should e consolepplcton Ths wll e not prolem for DOS nd LIU complers If ths mterl s used s techng d n clss I would pprecte f under such crcumstnces the nstructor of such clss would send me note t the ddress elow nformng me f the mterl s useful lso I would pprecte ny suggestons or constructve crtcsm regrdng the content of these lecture notes Dr M ndrecut Fculty of Physcs BesBoly Unversty Clupoc Romn eml: ndrecut@physucluro

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5 TBLE OF COTETS Lecture Ect Methods For Lner Systems 7 Lecture Itertve Methods For Lner Systems Lecture Sclr onlner Equtons Lecture Systems of onlner Equtons Lecture Egensystems 6 Lecture 6 Trdgonl Systems 77 Lecture 7 Polynoml Interpolton 9 Lecture 8 Best ppromton of Functons 7 Lecture 9 umercl Dfferentton nd Integrton Lecture Stochstcs Lecture Ordnry Dfferentl Equtons 7 Lecture Prtl Dfferentl Equtons 77 ppend Some Theorems on Dfferentle Functons 9 References 99

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7 Lecture Ect Methods for Lner Systems Introducton Let us consder the prolem of solvng the lner system of equtons: B Here the rsed dot denotes mtr product [ ] s # mtr wth det $ B [ ] s the rghthnd sde wrtten s column vector nd [ ] s the unnown vector: 7 B The forml soluton of ths lner system cn e wrtten s B where s the nverse of the mtr The nverse of # mtr s defned to e mtr such tht I where I [ ] s the # dentty mtr f ; f $ The squre mtr s nvertle ff det $ The nverse s then unque nd to fnd we cn solve I Ths corresponds to solvng lner system for rghthnd sdes The ect methods for solvng lner systems gves the ect soluton n the sence of roundoffs fter fnte numer of rthmetc nd logcl opertons

8 Guss Elmnton Method Guss elmnton s the most frequently used ect method for solvng systems of lner equtons Let us strt wth the smple emple of three equtons wth three unnowns: On the ssumpton tht $ the frst equton of the system s dvded y Then from ech remnng equton we sutrct the frst equton multpled y the pproprte coeffcent s result of these opertons the system tes the form: where / The frst unnown ws elmnted from ll equtons ecept the frst ow on the ssumpton tht $ we dvde the second equton y the coeffcent nd elmnte the second unnown from the thrd equton resultng where 8

9 9 / Fnlly the thrd equton s dvded y whch for nonsngulr mtr must e nonzero $ The resulted equvlent system s where The soluton s otned y cwrd susttuton / One cn see tht the fnl mtr s uppertrngulr wth the elements on the mn dgonl equl to s onus the determnnt of mtr s gven y det Let us return to the generl cse of equtons wth unnowns fter elmnton steps we hve to elmnte the unnown The system s gven y # #

10 where / Fnlly the equvlent system s reduced to the uppertrngulr form # # Usng the c susttuton the soluton vector s otned s follows / Ovously the determnnt of the mtr s gven y det In order to vod the effect of computtonl error correspondng to dvson y possle zero dgonl element we must use the Gussn method wth pvotng The pvotng procedure conssts n nterchngng rows prtl pvotng or rows nd columns full pvotng so s to put prtculrly desrle element the pvot n the dgonl poston In prctce the pvot corresponds to the lrgest n mgntude vlle element

11 Emple pply the Guss elmnton method wth prtl pvotng to the followng system: nterchnge row nd row ; row row # ; row row # ; nterchnge row nd row ; row row # ; Bcwrd susttuton gves

12 C progrm / Gussn Elmnton nd Pvotng UpperTrngulrzton Followed y Bc Susttuton To fnd the soluton of =B y reducng the mtr to uppertrngulr form nd performng the c susttuton / #nclude<stdoh> #nclude<stdlh> #nclude<mthh> // #defne m // The mmum numer of equtons // vod Gussnt doule [][m] nt p q t; // Loop counters nt row[m]; // Vector wth rownumers doule [m] sum m; // Solutonvector // Intlze the vector wth rownumers for=; <=; row[]=; // Strt uppertrngulrzton routne forp=; p<=; p for=p; <=; ffs[row[]][p]>fs[row[p]][p] t=row[p]; row[p]=row[]; row[]=t; f[row[p]][p]== prntf\n ERROR: the mtr s sngulr ; et; for=p; <=; m=[row[]][p]/[row[p]][p]; forq=p; q<=; q [row[]][q]=m[row[p]][q]; // End of uppertrngulrzton routne

13 // Strt the cwrd susttuton routne f[row[]][]== prntf\n Error: the mtr s sngulr ; et; []=[row[]][]/[row[]][]; for=; >=; sum=; forq=; q<=; q sum=[row[]][q][q]; []=[row[]][]sum/[row[]][]; // End of the cwrd susttuton routne // Soluton vector for=; <=; [][]=[]; // vod mnvod nt ; // Loop counters doule [m][m]; // =[ B] n =B nt ; // umer of equtons // Strt the mn routne // Intlze the ugmented mtr =[ B] prntf\n Input the numer of equtons <d = m; scnfd ; prntf Enter elements of =[ B] row y row:\n; for=; <=; for=; <=; prntf Input [d][d]= ; scnflf [][]; Guss ; prntf\n Soluton:; for=; <=; prntf\n [d]=lf [][]; // Remr By conventon the frst nde of n element denotes ts row the second ts column In mthemtcl notton ndces te the vlues n C notton the ndces te vlues

14 Emple Solve the lner system B where [ B] Soluton: [] = [] = [] = [] = LU Decomposton more effcent wy for solvng lner systems s the LU decomposton method whch mples lowerupper fctorzton of the gven mtr In other words we see to represent the mtr s product of two trngulr mtrces such tht wth l l L l l l # l L U u U u u u # u u The LU decomposton cn e used to solve the lner system B L U L U B y frst solvng for the vector Y the lner system L Y B nd then solvng U Y

15 The ulry vector Y s computed y forwrd susttuton: y l : 7 8 y l y l 9 6 The soluton vector s then otned y cwrd susttuton n the sme mnner s n the Guss elmnton method The system y u : 7 8 y u u 9 6 s equvlent to the followng L U l u equtons wth unnowns l u the dgonl eng represented twce In fct t s lwys possle to put l lso due to the trngulr structure of L nd U the summton nde wll not run over the whole ntervl [ ] We hve n fct the followng equtons: l u for ; l u for < Ths leds to the followng recursve procedure for computng l u : u u l u : 7 8 l lu u 9 6

16 6 s consequence of the LU decomposton the determnnt s clculted very esly s followng u u u U L U L det det det det Pvotng s lso n mportnt spect of the LU lgorthm ecuse of the dvson operton used n the ove equtons Here only prtl pvotng nterchnge of rows cn e mplemented It follows tht prctclly we decompose rowwse permutton of Includng the permutton mtr the orgnl system s trnsformed le tht B P U L P Emple pply the LU decomposton wth pvotng to the followng mtr: row Pvot new row l u 9 T new u l row Pvot

17 7 9 new row l u T new u l row Pvot new row l u T new u l row l u Hence U L P P L U

18 C progrm / LU Fctorzton wth Pvotng To fnd the soluton of the lner system =B y usng the followng steps: Fnd the mtrces P L U tht stsfy P=LU where P s the permutton mtr L s the lowertrngulr mtr U s the uppertrngulr mtr Fnd the soluton Y of the lowertrngulr system LY=PB Fnd the soluton of the uppertrngulr system U=Y The dgonl elements of L re ll these vlues re not stored The coeffcents of L nd U overwrte the mtr The soluton s returned n the lst column of ugmented mtr =[B] The determnnt of mtr s gven n det vrle / #nclude<stdoh> #nclude<stdlh> #nclude<mthh> // #defne m // The mmum numer of equtons // doule LU_fctorztonnt doule [][m] nt p q t; // Loop counters doule y[m] [m]; // Solutonvectors nt row[m]; // Vector wth rownumers doule sum d=; // Intlze the vector wth rownumers for=; <=; row[]=; // Strt the LU fctorzton routne forp=; p<=; p for=p; <=; ffs[row[]][p]>fs[row[p]][p] d=d; t=row[p]; row[p]=row[]; row[]=t; 8

19 f[row[p]][p]== prntferror: the mtr s sngulr \n; et; d=d[row[p]][p]; for=p; <=; [row[]][p]=[row[]][p]/[row[p]][p]; forq=p; q<=; q [row[]][q]=[row[]][p][row[p]][q]; d=d[row[]][]; // Strt the forwrd susttuton routne y[]=[row[]][]; for=;<=; sum=; forq=; q<=; q sum=[row[]][q]y[q]; y[]=[row[]][]sum; f[row[]][]== prntferror: the mtr s sngulr \n; et; // Strt the c susttuton routne []=y[]/[row[]][]; for=; >=; sum=; forq=; q<=; q sum=[row[]][q][q]; []=y[]sum/[row[]][]; // Soluton vector for=; <; [][]=[]; return d; // vod mnvod nt ; // =numer of equtons doule [m][m]; // =[B] n =B doule det; // det=det // Intlze the ugmented mtr =[B] 9

20 prntf\n Input the numer of equtons <d= m; scnfd ; prntf Enter elements of =[B] row y row:\n; for=; <=; for=; <=; prntf Input [d][d] = ; scnflf [][]; det=lu_fctorzton ; prntf\n Soluton:; for=; <=; prntf\n [d] = lf [][]; prntf\n det=lf det; // Emple Solve the lner system B where [ B] Soluton: [] = [] = [] = [] = det =

21 Lecture Itertve Methods for Lner Systems The Method of Smple Iterton The methods descred so fr re clled ect methods However n the ect methods the roundoff errors ccumulte nd f the mtr s ner sngulr the errors re mplfed n n nconvenent wy In such cses one cn use the tertve methods to fnd n mproved soluton Let e the ect soluton of the lner system B nd let e some slghtly wrong soluton such tht Insertng ths wrong soluton n the lner system we fnd B The rghthnd sde of ths equton s nown snce s the wrong soluton nd we cn use t to compute nd therefore ow we cn nterpret ths equton s n tertve formul B = used to mprove the soluton of the lner system dfferent pproch s sed on the replcement of the nverse of the mtr y n ppromton Q > nd defnng the resdul mtr? s? I Q Here I s the dentty mtr It follows tht Q Q Q Q Q Q I? Q I?? Q

22 Therefore the nverse lm Q n n@ s gven y Q n I?? Q The soluton vector s then gven y the followng tertve formuls n Q n B n wth Q n n Q B I Q Q B? > Here the most mportnt prolem s to estlsh the condtons for the convergence of the lmt lm Q n n@ n In order to do such nlyss we need to ntroduce few concepts regrdng the norm of vectors nd mtrces T Let B C D R then some emples of vector norms re the one norm the two norm or Euclden length m the nfnty norm or m ; ; The ove norms re prtculr cses of Vector norms re requred to stsfy: p : 8 9 p 7 6 p E F D R nd G ; H H F D R nd FH D R ; Y ; Y F Y D R

23 One cn esly prove tht ll of the emples gven ove stsfy these requrements The nturl norm of squre mtr s defned n terms of gven vector norm s followng m $ Thus mesures the mmum reltve stretchng n gven vector norm tht occurs when multplyng ll nonzero vectors D R y From the ove defnton t mmedtely results tht ; F D R Theorem ll mtr norm defned n terms of gven vector norm stsfy E nd G O zero mtr; v H H F H D R ; B ; B B ; B Proof: B m B m B $ $ ; B B ; m ; m m B $ $ $ v B B B B m ; m ; m B $ $ $

24 For specfc choces of vector norm t s convenent to epress the correspondng nduced mtr norm n terms of the elements of the mtr For emple n the cse of the nfnty norm m ; ; m ; ; ; m m ; ; lso for ny squre mtr there s lwys vector Y for whch Y ; m To show ths let e the row of for whch s mmum nd te the vector It follows tht T By y y C D R Y f E y / f I y Y ; m From the ove consdertons clerly results ; ; Smlrly one cn show tht m mmum solute row sum m mmum solute column sum ; ;

25 Emple Let us consder then m nd m6 ow we need to ntroduce few concepts regrdng the egenvlues nd the egenvectors of mtrces We shll consder how to compute the egenvlues of mtrces n other lecture For the present t s suffcent to now tht f there re nonzero vectors nd sclrs H whch together stsfy the equtons H where s squre the mtr nd solutons of the chrcterstc equton det H I One cn show tht # mtr then the constnts H re clled egenvlues of re the correspondng egenvectors The egenvlues re m m H $ ; ; where H re ndeed nonnegtve ecuse the mtr re the egenvlues of the mtr T these egenvlues Theorem For ny mtr norm we hve J ; T s rel nd symmetrc where J s the modulus of the egenvlue of gretest mgntude of mtr the spectrl rdus of mtr