By Gilberto E. Urroz, September 2004

Transcription

1 Solutio o o-liear equatios By Gilberto E. Urroz, September 004 I this documet I preset methods or the solutio o sigle o-liear equatios as well as or systems o such equatios. Solutio o a sigle o-liear equatio Equatios that ca be cast i the orm o a polyomial are reerred to as algebraic equatios. Equatios ivolvig more complicated terms, such as trigoometric, hyperbolic, epoetial, or logarithmic uctios are reerred to as trascedetal equatios. The methods preseted i this sectio are umerical methods that ca be applied to the solutio o such equatios, to which we will reer, i geeral, as o-liear equatios. I geeral, we will we searchig or oe, or more, solutios to the equatio, ( = 0. We will preset the Newto-Raphso algorithm, ad the secat method. I the secat method we eed to provide two iitial values o to get the algorithm started. I the Newto-Raphso methods oly oe iitial value is required. Because the solutio is ot eact, the algorithms or ay o the methods preseted herei will ot provide the eact solutio to the equatio ( = 0, istead, we will stop the algorithm whe the equatio is satisied withi a allowed tolerace or error, ε. I mathematical terms this is epressed as ( R < ε. The value o or which the o-liear equatio (=0 is satisied, i.e., = R, will be the solutio, or root, to the equatio withi a error o ε uits. The Newto-Raphso method Cosider the Taylor-series epasio o the uctio ( about a value = o : (= ( o +'( o (- o +("( o /!(- o +. Usig oly the irst two terms o the epasio, a irst approimatio to the root o the equatio ( = 0 ca be obtaied rom ( = 0 ( o +'( o ( - o

2 Such approimatio is give by, = o - ( o /'( o. The Newto-Raphso method cosists i obtaiig improved values o the approimate root through the recurret applicatio o equatio above. For eample, the secod ad third approimatios to that root will be give by ad = - ( /'(, 3 = - ( /'(, respectively. This iterative procedure ca be geeralized by writig the ollowig equatio, where i represets the iteratio umber: i+ = i - ( i /'( i. Ater each iteratio the program should check to see i the covergece coditio, amely, ( i+ <ε, is satisied. The igure below illustrates the way i which the solutio is oud by usig the Newto- Raphso method. Notice that the equatio ( = 0 ( o +'( o ( - o represets a straight lie taget to the curve y = ( at = o. This lie itersects the -ais (i.e., y = ( = 0 at the poit as give by = o - ( o /'( o. At that poit we ca costruct aother straight lie taget to y = ( whose itersectio with the -ais is the ew approimatio to the root o ( = 0, amely, =. Proceedig with the iteratio we ca see that the itersectio o cosecutive taget lies with the -ais approaches the actual root relatively ast.

3 The Newto-Raphso method coverges relatively ast or most uctios regardless o the iitial value chose. The mai disadvatage is that you eed to kow ot oly the uctio (, but also its derivative, '(, i order to achieve a solutio. The secat method, discussed i the ollowig sectio, utilizes a approimatio to the derivative, thus obviatig such requiremet. The programmig algorithm o ay o these methods must iclude the optio o stoppig the program i the umber o iteratios grows too large. How large is large? That will deped o the particular problem solved. However, ay Newto-Raphso, or secat method solutio that takes more tha 000 iteratios to coverge is either ill-posed or cotais a logical error. Debuggig o the program will be called or at this poit by chagig the iitial values provided to the program, or by checkig the program's logic. A MATLAB uctio or the Newto-Raphso method The uctio ewto, listed below, implemets the Newto-Raphso algorithm. It uses as argumets a iitial value ad epressios or ( ad '(. uctio [,iter]=ewto(0,,p % ewto-raphso algorithm N = 00; eps =.e-5; % deie ma. o. iteratios ad error maval = ; % deie value or divergece = 0; while (N>0 = -(/p(; i abs((<eps =;iter=00-n; retur; i abs((>maval disp(['iteratios = ',umstr(iter]; error('solutio diverges'; break; N = N - ; = ; error('no covergece'; break; % ed uctio We will use the Newto-Raphso method to solve or the equatio, ( = = 0. The ollowig MATLAB commads deie the uctio 00( ad its derivative, 0p(:»00 = ilie('.^3-*.^+','' 00 = Ilie uctio: 00( =.^3-*.^+ 3

4 » 0p = ilie('3*.^-','' 0p = Ilie uctio: 0p( = 3*.^- To have a idea o the locatio o the roots o this polyomial we'll plot the uctio usig the ollowig MATLAB commads:» = [-0.8:0.0:.0]';y=00(;» plot(,y;label('';ylabel('00(';» grid o ( We see that the uctio graph crosses the -ais somewhere betwee.0 ad 0.5, close to.0, ad betwee.5 ad.0. To activate the uctio we could use, or eample, a iitial value 0 = :» [,iteratios] = ewto(,00,0p =.680 iteratios = 39 The ollowig commad are aimed at obtaiig vectors o the solutios provided by uctio ewto.m or 00(=0 or iitial values i the vector 0 such that 0 < 0 < 0. The solutios oud are stored i variable s while the required umber o iteratios is stored i variable is.» 0 = [-0:0.5:0]; s = []; is = []; EDU» or i = :legth(0 [,ii] = ewto(0(i,00,0p; s = [s,]; is = [is,ii]; ed Plot o s vs. 0, ad o is vs. 0 are show et: 4

5 » igure(;plot(0,s,'o';hold;plot(0,s,'-';hold;» title('newto-raphso solutio';» label('iitial guess, 0';ylabel('solutio, s';.8 Newto-Raphso solutio solutio, s iitial guess, 0 This igure shows that or iitial guesses i the rage 0 < o <0, uctio ewto.m coverges maily to the solutio =.680, with ew istaces covertig to the solutios = - ad =.» igure(;plot(0,is,'+'; hold;plot(0,is,'-';hold;» title('newto-raphso solutio';» label('iitial guess, 0';ylabel('iteratios, is'; 70 Newto-Raphso solutio iteratios, is iitial guess, 0 This igure shows that the umber o iteratios required to achieve a solutio rages rom 0 to about 65. Most o the time, about 50 iteratios are required. The ollowig eample shows a case i which the solutio actually diverges: 5

6 » [,iter] = ewto(-0.75,00,0p iteratios = 6??? Error usig ==> ewto Solutio diverges NOTE: I the uctio o iterest is deied by a m-ile, the reerece to the uctio ame i the call to ewto.m should be placed betwee quotes. The Secat Method I the secat method, we replace the derivative '( i i the Newto-Raphso method with '( i (( i - ( i- /( i - i-. With this replacemet, the Newto-Raphso algorithm becomes ( i i+ i ( i ( ( = i i i To get the method started we eed two values o, say o ad, to get the irst approimatio to the solutio, amely, ( = ( 0. ( ( As with the Newto-Raphso method, the iteratio is stopped whe ( i+ <ε. Figure 4, below, illustrates the way that the secat method approimates the solutio o the equatio ( = 0. o. 6

7 A MATLAB uctio or the secat method The uctio secat, listed below, uses the secat method to solve or o-liear equatios. It requires two iitial values ad a epressio or the uctio, (. uctio [,iter]=secat(0,00, % ewto-raphso algorithm N = 00; eps =.e-5; % deie ma. o. iteratios ad error maval = ; % deie value or divergece = 0; = 00; while N>0 gp = ((-(/(-; = -(/gp; i abs((<eps =; iter = 00-N; retur; i abs((>maval iter=00-n; disp(['iteratios = ',umstr(iter]; error('solutio diverges'; abort; N = N - ; = ; = ; iter=00-n; disp(['iteratios = ',iter]; error('no covergece'; abort; % ed uctio Applicatio o secat method We use the same uctio 00( = 0 preseted earlier. the uctio secat.tt to obtai a solutio to the equatio: The ollowig commads call» [,iter] = secat(-0.0,-9.8,00 = iter = The ollowig commad are aimed at obtaiig vectors o the solutios provided by uctio Newto.m or 00(=0 or iitial values i the vector 0 such that 0 < 0 < 0. The solutios oud are stored i variable s while the required umber o iteratios is stored i variable is. 7

8 » 0 = [-0:0.5:0]; 00 = ; s = []; is = [];» or i = :legth(0 [,ii] = secat(0(i,00(i,00; s = [s, ]; is = [is, ii]; ed Plot o s vs. 0, ad o is vs. 0 are show et:» igure(;plot(0,s,'o';hold;plot(0,s,'-';hold;» title('secat method solutio';» label('irst iitial guess, 0';ylabel('solutio, s'; Secat method solutio.5 solutio, s irst iitial guess, 0 This igure shows that or iitial guesses i the rage 0 < o <0, uctio ewto.m coverges to the three solutios. Notice that iitial guesses i the rage 0 < 0 < -, coverge to = ; those i the rage < <, coverge to = ; ad, those i the rage <<0, coverge to =.680.» igure(;plot(0,is,'o';hold;plot(0,is,'-';hold;» label('irst iitial guess, 0';ylabel('iteratios, is';» title('secat method solutio'; 5 Secat method solutio 0 iteratios, is irst iitial guess, 0 8

9 9 This igure shows that the umber o iteratios required to achieve a solutio rages rom 0 to about 5. Notice also that, the closer the iitial guess is to zero, the less umber o iteratios to covergece are required. NOTE: I the uctio o iterest is deied by a m-ile, the reerece to the uctio ame i the call to ewto.m should be placed betwee quotes. Solvig systems o o-liear equatios Cosider the solutio to a system o o-liear equatios i ukows give by (,,, = 0 (,,, = 0... (,,, = 0 The system ca be writte i a sigle epressio usig vectors, i.e., ( = 0, where the vector cotais the idepedet variables, ad the vector cotais the uctios i (:. ( ( (,...,, (,...,, (,...,, ( (, = = = M M M Newto-Raphso method to solve systems o o-liear equatios A Newto-Raphso method or solvig the system o liear equatios requires the evaluatio o a matri, kow as the Jacobia o the system, which is deied as:. ] [ / / / / / / / / /,...,, (,...,, ( j i = = = L M O M M L L J I = 0 (a vector represets the irst guess or the solutio, successive approimatios to the solutio are obtaied rom + = - J - ( = -, with = + -.

10 A covergece criterio or the solutio o a system o o-liear equatio could be, or eample, that the maimum o the absolute values o the uctios i ( is smaller tha a certai tolerace ε, i.e., ma ( < ε. i i Aother possibility or covergece is that the magitude o the vector ( be smaller tha the tolerace, i.e., ( < ε. We ca also use as covergece criteria the dierece betwee cosecutive values o the solutio, i.e., ma ( ( < ε., or, i i + i = + - < ε. The mai complicatio with usig Newto-Raphso to solve a system o o-liear equatios is havig to deie all the uctios i / j, or i,j =,,,, icluded i the Jacobia. As the umber o equatios ad ukows,, icreases, so does the umber o elemets i the Jacobia,. MATLAB uctio or Newto-Raphso method or a system o o-liear equatios The ollowig MATLAB uctio, ewtom, calculates the solutio to a system o oliear equatios, ( = 0, give the vector o uctios ad the Jacobia J, as well as a iitial guess or the solutio 0. uctio [,iter] = ewtom(0,,j % Newto-Raphso method applied to a % system o liear equatios ( = 0, % give the jacobia uctio J, with % J = del(,,...,/del(,,..., % = [;;...;], = [;;...;] % 0 is a iitial guess o the solutio N = 00; % deie ma. umber o iteratios epsilo = e-0; % deie tolerace maval = ; % deie value or divergece = 0; % load iitial guess while (N>0 JJ = eval(j,; i abs(det(jj<epsilo error('ewtom - Jacobia is sigular - try ew 0'; abort; = - iv(jj*eval(,; 0

11 i abs(eval(,<epsilo =; iter = 00-N; retur; i abs(eval(,>maval iter = 00-N; disp(['iteratios = ',umstr(iter]; error('solutio diverges'; abort; N = N - ; = ; error('no covergece ater 00 iteratios.'; abort; % ed uctio The uctios ad the Jacobia J eed to be deied as separate uctios. To illustrate the deiitio o the uctios cosider the system o o-liear equatios: whose Jacobia is (, = = 0, (, = - 5 = 0, J = =. We ca deie the uctio as the ollowig user-deied MATLAB uctio : uctio [] = ( % ( = 0, with = [(;(] % represets a system o o-liear equatios = (^ + (^ - 50; = (*( -5; = [;]; % ed uctio The correspodig Jacobia is calculated usig the user-deied MATLAB uctio jacob: uctio [J] = jacob( % Evaluates the Jacobia o a % system o o-liear equatios J(, = *(; J(, = *(; J(, = (; J(, = (; % ed uctio

12 Illustratig the Newto-Raphso algorithm or a system o two o-liear equatios Beore usig uctio ewtom, we will perorm some step-by-step calculatios to illustrate the algorithm. We start by deiig a iitial guess or the solutio as:» 0 = [;] 0 = Let s calculate the uctio ( at = 0 to see how ar we are rom a solutio:» (0 as = Obviously, the uctio ( 0 is ar away rom beig zero. Thus, we proceed to calculate a better approimatio by calculatig the Jacobia J( 0 :» J0 = jacob(0 J0 = 4 The ew approimatio to the solutio,, is calculated as:» = 0 - iv(j0*(0 = Evaluatig the uctios at produces: ( as = Still ar away rom covergece. Let s calculate a ew approimatio, :» = -iv(jacob(*( =

13 Evaluatig the uctios at idicates that the values o the uctios are decreasig:» ( as = A ew approimatio ad the correspodig uctio evaluatios are:» 3 = - iv(jacob(*( 3 = E» (3 as = The uctios are gettig eve smaller suggestig covergece towards a solutio. Solutio usig uctio ewtom Net, we use uctio ewtom to solve the problem postulated earlier. A call to the uctio usig the values o 0,, ad jacob is:» [,iter] = ewtom(0,'','jacob' = iter = 6 The result shows the umber o iteratios required or covergece (6 ad the solutio oud as = ad = Evaluatig the uctios or those solutios results i:» ( as =.0e-00 *

14 The values o the uctios are close eough to zero (error i the order o 0 -. Secat method to solve systems o o-liear equatios I this sectio we preset a method or solvig systems o o-liear equatios through the Newto-Raphso algorithm, amely, + = - J - (, but approimatig the Jacobia through iite diereces. This approach is a geeralizatio o the secat method or a sigle o-liear equatio. For that reaso, we reer to the method applied to a system o o-liear equatios as a secat method, although the geometric origi o the term ot loger applies. The secat method or a system o o-liear equatios ree us rom havig to deie the uctios ecessary to deie the Jacobia or a system o equatios. Istead, we approimate the partial derivatives i the Jacobia with i j i (,, L, j +, L, i (,, L, j, L,, where is a small icremet i the idepedet variables. Notice that i / j represets elemet J ij i the jacobia J = (,,, /(,,,. To calculate the Jacobia we proceed by colums, i.e., colum j o the Jacobia will be calculated as show i the uctio jacobfd (jacobia calculated through Fiite Diereces listed below: uctio [J] = jacobfd(,,del % Calculates the Jacobia o the % system o o-liear equatios: % ( = 0, through iite diereces. % The Jacobia is built by colums [m ] = size(; or j = :m = ; (j = (j + del; J(:,j = ((-(/del; % ed uctio Notice that or each colum (i.e., each value o j we deie a variable which is irst made equal to, ad the the j-th elemet is icremeted by del, beore calculatig the j-th colum o the Jacobia, amely, J(:,j. This is the MATLAB implemetatio o the iite dierece approimatio or the Jacobia elemets J ij = i / j as deied earlier. 4

15 Illustratig the secat algorithm or a system o two o-liear equatios To illustrate the applicatio o the secat algorithm we use agai the system o two o-liear equatios deied earlier through the uctio. We choose a iitial guess or the solutio as 0 = [;3], ad a icremet i the idepedet variables o = 0.: 0 = [;3] 0 = 3 EDU» d = 0. d = Variable J0 will store the Jacobia correspodig to 0 calculated through iite diereces with the value o deied above:» J0 = jacobfd('',0,d J0 = A ew estimate or the solutio, amely,, is calculated usig the Newto-Raphso algorithm:» = 0 - iv(j0*(0 = The iite-dierece Jacobia correspodig to gets stored i J:» J = jacobfd('',,d J = Ad a ew approimatio or the solutio ( is calculated as:» = - iv(j*( =

16 The et two approimatios to the solutio ( 3 ad 4 are calculated without irst storig the correspodig iite-dierece Jacobias:» 3 = - iv(jacobfd('',,d*( 3 = EDU» 4 = 3 - iv(jacobfd('',3,d*(3 4 = To check the value o the uctios at = 4 we use:» (4 as = The uctios are close to zero, but ot yet at a acceptable error (i.e., somethig i the order o 0-6. Thereore, we try oe more approimatio to the solutio, i.e., 5 :» 5 = 4 - iv(jacobfd('',4,d*(4 5 = The uctios are eve closer to zero tha beore, suggestig a covergece to a solutio.» (5 as = MATLAB uctio or secat method to solve systems o o-liear equatios To make the process o achievig a solutio automatic, we propose the ollowig MATLAB user -deied uctio, secatm: uctio [,iter] = secatm(0,d, % Secat-type method applied to a % system o liear equatios ( = 0, 6

17 % give the jacobia uctio J, with % The Jacobia built by colums. % = [;;...;], = [;;...;] % 0 is the iitial guess o the solutio % d is a icremet i,,... variables N = 00; epsilo =.0e-0; maval = ; % deie ma. umber o iteratios % deie tolerace % deie value or divergece i abs(d<epsilo error('d = 0, use dieret values'; break; = 0; % load iitial guess [ m] = size(0; while (N>0 JJ = [,;,3]; = zeros(,; or j = : = ; (j = (j + d; = eval(,; = eval(,; % Estimatig % Jacobia by % iite JJ(:,j = (-/d; % diereces % by colums i abs(det(jj<epsilo error('ewtom - Jacobia is sigular - try ew 0,d'; break; p = - iv(jj*; p = eval(,p; i abs(p<epsilo =p; disp(['iteratios: ', umstr(00-n]; retur; i abs(p>maval disp(['iteratios: ', umstr(00-n]; error('solutio diverges'; break; N = N - ; = p; error('no covergece'; break; % ed uctio 7

18 Solutio usig uctio secatm To solve the system represeted by uctio, we ow use uctio secatm. The ollowig call to uctio secatm produces a solutio ater 8 iteratios:» [,iter] = secatm(0,d,'' iteratios: 8 = iter = 8 Solvig equatios with Matlab uctio zero Matlab provides uctio zero or the solutio o sigle o-liear equatios. Use» help zero to obtai additioal iormatio o uctio zero. Also, read Chapter 7 (Fuctio Fuctios i the Usig Matlab guide. For the solutio o systems o o-liear equatios Matlab provides uctio solve as part o the Optimizatio package. Sice this is a add-o package, uctio solve is ot available i the studet versio o Matlab. I usig the ull versio o Matlab, check the help acility or uctio solve by usig:» help solve I uctio solve is ot available i the Matlab istallatio you are usig, you ca always use uctio secatm (or ewtom to solve systems o o-liear equatios. 8