for loop help - Newton's method

Hi, I am trying to find the root/s of a polynomial using Newton's method, underneath is what I got till now. I wish to improve this code by somehow modifying the for loop to keep iterating until a fixed root is found i.e., the answer converges to a specified accuracy. Any ideas? Thanks !!

DP = 16;
DP_old = digits;
digits(DP);
DP_current = digits;

syms x

f_x = 2*x^5 - 6*x^4 -12*x^3 + 20*x^2 + 42*x + 18; % roots = [-1 -1 -1 3 3]
d_f_x = diff(f_x);

n = 77;
x_k = zeros(1, n);

for i = 1:n %keep looping until for example x_k(i+1) = -1
    disp(i);
    x_k(i+1) = x_k(i) - (subs(f_x, x_k(i))/subs(d_f_x, x_k(i)));
    disp(vpa(x_k(i+1)));
end

root = round(min(x_k));
fprintf("root = %d\n\n", root);

Answer 1

[David Hill]

I would avoid symbolic unless you need to.

f = [2,-6,-12,20,42,18];
df = polyder(f);
tol=1e-10;
error=1;
x=[-2,4];%initial root guesses
for i = 1:length(x)
    while error>tol
      a = x(i) - polyval(f,x(i))/polyval(df,x(i));
      error=abs(a-x(i));
      x(i)=a;
    end
    error=1;
end

[Joshua Scicluna]

Thanks @David

Answer 2

[John D'Errico]

I agree with much of what David has written (except for one thing). If you wish, feel free to accept his answer, as long as you read my answer and make an effort to understand what I have said.

The only real exception I have is his use of error as a variable. Since error is itself already a useful function in MATLAB, while you CAN name a variable error, it is a good idea to learn to NOT do so. Naming variables with the same name as useful and important function names in MATLAB will cause you a great deal of grief one day. Maybe not today, maybe not tomorrow, but one day you will be happy you learned this lesson. And if you do learn the lesson today, then you will never need to experience the frustration of having to post a question on Answers, like "Why does the function error not work for me?"

But the point of not needing symbolic at all is an important one. Symbolic tools will only slow down your code. Use of subs is a poor and terribly inefficient way to evaluate an expression. It is especially poor in the code written in the original question, because there the numbers will become nastier and nastier, with more an more digits per iteration.

syms x
f_x = 2*x^5 - 6*x^4 -12*x^3 + 20*x^2 + 42*x + 18;
d_f_x = diff(f_x);

x_k(1) = sym(0);
for i = 1:10
  x_k(i+1) = x_k(i) - (subs(f_x, x_k(i))/subs(d_f_x, x_k(i)));
end
x_k =
[0, -3/7, -9/14, -771/1001, -4621311/5436431, -74926899848343/83125166507383, -18581774083263136027457247627/19877527069259662209650910667, -1103319036541955809733816496515862383910895524139551341307/1153152790849676866377462019365820183982571984391568595067, -3805003198411584763071384595557020036783999204314124440401876921207856379118208705553806362983095031784762708934827/3917451661060113742522646647537708639977123244673765496048702056639847645722154249423498286801610805686924121201067, -44620265102425008438012206333772824282049114426102429636348887463032675153767570323339066109358111926782720359725693430762620655429210467799918876780596522930457686880362040431435938995030976223217915710812997971094098647247299307/45488643279245262836365573480880936426081773497468026618864429885112829609473977128504444926981915594681060887104619555566363613432127462604242164604262149425041580599335897332318096132621625271163335574355553292820283224955717867, -6080024711187465087473536404281988463194668261279871712733491242335920152554876810216579819091334458537201335450796854363122897508965813858632386672195213087533854500292501701587998151562657451181909984926226403588546123069244963775319411517264603307627835295994600646346393260488898527391752951803412155442625175070416889083851494965630801249701089202160553280055752428007624314384061908291074693265082479777068343378520120334306774741352817239549602215725227/6158273320763203257907091406734885630732908549023632199869057264774673048456502605509733573253289180413405123173766306495486437618081563402435561978259653128334804244002915150914911636257590561691019017778968217664655243265656012379563580437164996154041745088913796570501905564349152154717100965794733827363104495692525414965370636687387470968971689448613634324494392506212052025146558761887229941739996068878113661326381376079458104500058125060640454276074667]
 
double(x_k')
ans =
                         0
        -0.428571428571429
        -0.642857142857143
         -0.77022977022977
        -0.850063396371627
        -0.901374433237233
        -0.934813169591886
        -0.956784777608697
        -0.971295507289527
        -0.980910000513986
        -0.987293742011756

Are the iterations converging to -1? Well, it appears they are. A bit slowly. That is a characteristic of my about multiple roots in rootfinding problems. Rootfinders tend to be slowly convergent, if they converge at all for multiple roots.

But the point I want to make is EVERY iteration, we will find the iteration actually takes more time to compute, with no real gain. Do you see that each number in that sequence has become longer and longer. Lots more digits in the numerator and denominator, yet it is clearly slowly convergent. In fact, I was going to allow that iteration to iterate for 100 iterations, but I finally stopped it after 19 iterations, because nothing was happening on my computer. Do you want to see the 19th iterant?

x_k(19)
ans =
-2113500518605829274696263566312951920641824241697325989967877781200219390478310614680793557883465688487834534073860633497308087326957014914717705538532743064636884437783441096250913265777983618933153286249637059398975968740018468641816119078680559174235882629184367246540082905566454220247343300845059462579949432493821512247230014010730490775702072982527395766060588931998277028500012122338983514371669785411591943581009739238334800545621005947719969950635323387219016931363262658614217265556307409090664902562437732373749231322038556429749980994755943029041890493359154645111589549031122608043470012332502719403709489112445070519711308702165956930731486248347411680239951346586689051161164305342291114339383046837433704171516936347762885239693742718741103790560170311743913873535204262475248016049337785392670101697169560006506420866499669507399034609954695754328914054040400343511614272231605787413615275097370818940497516379947604027531314382176888803401322238725232136342087738229944758055548801316563874024591489066171063977429013807357612530823979863826338105217581318626810339216768993168633159161457130025074810527829386969223695675434228994243708014572404984941154002582483338009886684251552529897755139792910810245949950284676452638297154179146756140731232031276430324853561220212650930432958716503733143285886707593661245456965042954292270662024644557730533733028353951710354495508016714370873295223943582034606690271317285155479036456250015511286172293787128884348632017355242108738152745017121314189631418688729177361842998062701254016244716062269297211322751567120143437254557995315810495364425647685883126662826762719924605380130711174425614620145069012528262160511090346506564542844226472516034551563279330230133332331300556227982124106397562977786038857039854262706334727303773757490569932161148732831215540568096662258720901726901328126974729913878266840427210948374548616383010599881771946701938699730655989120266258171819572511053485088433124419563422874531705300609885512377951704306302808028881564007310749313975017704545637096154783835868536766580825666973077924222650027836270083398909575392732136942273465360980548625643704939532401450191254377841289418901267087796210481687089267808734708687983366000956349395016677917023390982317789270028744971217283849076131088610365168136959865780299901854403010185562074248897758754692711839554461598914529006609092564087957940805755498336263637023363518749318450554364318761832745516590205868723182977432635760339871297973620224183630902486792066030730351777899993626282891871623370321984598027689366041522575899748730674023729899201996165605049583107521650575037664145753101249631105343370663502272262370225848519201876466232376531374820105449272274318087684063032275929475439480908999375422302736158616294773069848180692206466597794279101342914062388467611663565218219106145858139561390199445143758906849650210523261774847036245056366312308214266499615126491984525078062620774139249561514192019863536977382561003599164925517881823413834660548143480111394302847140918707097804604556484136532005079735514817400179157220157756814191026943089725889617411486888182396745183822266122204149254713624059765643522862860401783094492035555689750779326043775697019939551561647598903912129372039952629947678687322606017782202881477902357811903674460035645303614173685461766499712478328263760525013722763192098507311113477410424916463201823784899977873235115442348924374787781051933776980552122333057958993804575866532374433211753459307804198892918668028231787534904695253085650231852275344036522504437342173594540195906915533924551768793790213969434755792518382435555515678605548265519945701482369534741825176898493332376073538312982971596309782737620553772163936380843495399900560494693515088407491331382212199454972896765418791794607330918648083917600281786857539811528256072403914917304876493959976970837382651039162992104251206669483430704302521422890234974490838838090341347739020624320933029298060363672080960643230715027058334709344392298883608264586608761983838198872472393370077599016931198538050785510952060275470273845496041637488289457296904886141891884460749779397015702445806664388357765001577898803213666374398646355088029439808111029333591500166267741002921375034908573773779100445572683795510102822768620448773765434114275784776964085957422814409413936240482871603580186457633855372725276570905996455820338591366510055925449874687060201150783720086621752716338945586931280240665170062368098061080150082080672298243900977162456572325745452315726874158701429000672244853534914791471831504055881026976792010954443260504715372530724310798036984528011465004994081832575151543560827768638286173879878292534471077015446170521102397522945722461959204054016968047845390099262757438028399166847658047096276274421854510633386156562302683718537032188641631053536132323678413838004233448125367057825809790887124451420029581756776128376003103861697118979026546694111692450923431085956061876565823773246904577596060993625822079106396803060958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Output truncated. Text exceeds maximum line length for Command Window display.

If you look at the end of that "number", it will tell you

... Output truncated. Text exceeds maximum line length for Command Window display.

How well has it done so far?

double(x_k(17:19))
ans =
          -0.9988877350343        -0.999258558784364        -0.999505736406519

As I said, slowly convergent. Right up there with watching grass grow and nails rust.

Far better to use numerical tools directly on this problem, as there is no need ot use symbolic computations just because you don't know the value of x. For example, using fzero, we see:

f = [2,-6,-12,20,42,18];
tol=1e-10;
[xroot,fval,exitflag,output] = fzero(@(x) polyval(f,x),[-2,0])

xroot =
        -0.999997431947064

fval =
     0

exitflag =
     1

output = 
  struct with fields:
    intervaliterations: 0
            iterations: 50
             funcCount: 52
             algorithm: 'bisection, interpolation'
               message: 'Zero found in the interval [-2, 0]'

So 50 ityerations, with 52 function evaluations. The default tolerance on x for fzero here was 2.22044604925031e-16, but note that you can only get a root to within the cuberoot of eps for triple roots. So setting a super high tolerance on x is meaningless here. Again, multiple roots are highly problematic.

Again though, we actually got a fast solution, far better than Newton's method would provide using syms as you wanted to run it.

f = [2,-6,-12,20,42,18];
df = polyder(f);
tol = 1e-10;
err = 1;
x_n = 0;%initial root guess
iter = 1;
maxiter = 100;
while (iter < maxiter) && (err>tol)
  x_n(iter+1) = x_n(iter) - polyval(f,x_n(iter))/polyval(df,x_n(iter));
  err = abs(x_n(iter)-x_n(iter+1));
  iter = iter + 1;
end

But even worse, look at what happened.

semilogy(x_n - -1,'o')

Around iteration 30, the estimate happend to jump across the root. And when it did, the NEXT iteration suddenly moved FAR away from the root.

x_n(29:35)
ans =
  Columns 1 through 4
        -0.999991384766257        -0.999994376401644        -0.999996716814727         -1.00000015003179
  Columns 5 through 7
        -0.998355413189684        -0.998903759174153        -0.999269239572119

What you need to understand is for this function, the Newton's iteration will probably NEVER converge to the tolerance you want to see.

And this is why we use better tools for rootfinding! Did you notice this algorithmic comment from fzero?

algorithm: 'bisection, interpolation'

fzero uses guarded methods, switching methods internally when a problem is discovered.

[Joshua Scicluna]

Hi John,

I understood you clearly and I thank-you for your time. I would also accept your answer if I could.

Thanks again, Joshua