Solving Equation by Fixed Point Iteration Method (FPI)

Equation: e^(-x)+ x^3-x-2=0

(2)g(x)= -ln⁡(-x^3+x+2)

First root: -0.48756103475
Number of steps: 15
x_0= -0.5
S = abs(-(1-3(r^2))/(-r^3+r+2)) = 0.17616267
Approximation rate: 0.176162171045838  Match with convergence rate above. However, it matches for up to 6 decimal places and only happen once.

FPI Code:
g = @(x) -log(- x^3 + x + 2);
numSteps = 15;
x = zeros(numSteps,1);
x(1) = -0.5;
for i=1:numSteps
    x(i+1) = g(x(i));
end
r = x
	Convergence Rate Code:
g = @(x) -log(- x^3 + x + 2);
numSteps = 15;
x = zeros(numSteps,1);
x(1) = -0.5;
for i=1:numSteps
    x(i+1) = g(x(i));
end
r = x(numSteps + 1); %best guess at root
e = abs(x-r); % array of errors
ratios = zeros(numSteps + 1,1);
for i=1:numSteps
 ratios(i) = e(i+1)/e(i);
end
ratios(numSteps+1) = 0;
ratios
	Matlab Session:
>> format long
>> FPI_P2

root =

  -0.500000000000000
  -0.485507815781701
  -0.487926450041332
  -0.487496780131671
  -0.487572357666484
  -0.487559040194681
  -0.487561386126457
  -0.487560972857440
  -0.487561045659910
  -0.487561032834829
  -0.487561035094129
  -0.487561034696125
  -0.487561034766238
  -0.487561034753887
  -0.487561034756063
  -0.487561034755679

>> fpi_error

ratios =

   0.165063486684809
   0.177971901820501
   0.175840000490639
   0.176219392449621
   0.176152672450803
   0.176164468417342
   0.176162171045838
   0.176163818967924
   0.176156465201077
   0.176197828573612
   0.175963200069280
   0.177297309389453
   0.169757954282590
   0.213874264478167
                   0
                   0

Second root: -4.451620296

Number of steps: 40
x_0= -4.45
S = abs(-(1-3(r^2))/(-r^3+r+2)) = 0.68151607
Approximation rate: 0.681515541009002  Match with convergence rate above. However, it matches for up to 5 decimal places and happens a couple of times.

FPI Code:
g = @(x) -log(- x^3 + x + 2);
numSteps = 15;
x = zeros(numSteps,1);
x(1) = -0.5;
for i=1:numSteps
    x(i+1) = g(x(i));
end
r = x
	Convergence Rate Code:
g = @(x) -log(- x^3 + x + 2);
numSteps = 15;
x = zeros(numSteps,1);
x(1) = -0.5;
for i=1:numSteps
    x(i+1) = g(x(i));
end
r = x(numSteps + 1); %best guess at root
e = abs(x-r); % array of errors
ratios = zeros(numSteps + 1,1);
for i=1:numSteps
 ratios(i) = e(i+1)/e(i);
end
ratios(numSteps+1) = 0;
ratios
	Matlab Session:
>> format long
>> FPI_P2

r =

  -4.450000000000000
  -4.450515837674487
  -4.450867496924248
  -4.451107208342674
  -4.451270598750689
  -4.451381962728406
  -4.451457864080729
  -4.451509594400246
  -4.451544850525814
  -4.451568878644365
  -4.451585254426693
  -4.451596414893949
  -4.451604020982121
  -4.451609204676858
  -4.451612737459011
  -4.451615145111884
  -4.451616785968364
  -4.451617904239521
  -4.451618666359797
  -4.451619185757252
  -4.451619539735076
  -4.451619780976705
  -4.451619945386776
  -4.451620057434893
  -4.451620133797492
  -4.451620185839833
  -4.451620221307526
  -4.451620245479329
  -4.451620261952803
  -4.451620273179739
  -4.451620280831078
  -4.451620286045587
  -4.451620289599360
  -4.451620292021312
  -4.451620293671912
  -4.451620294796823
  -4.451620295563467
  -4.451620296085948
  -4.451620296442027
  -4.451620296684700
  -4.451620296850086

>> fpi_error

ratios =

   0.681640018950950
   0.681600499565625
   0.681573536077006
   0.681555120306377
   0.681542513645649
   0.681533840860771
   0.681527811649525
   0.681523528889349
   0.681520355261449
   0.681517818485699
   0.681515541009002
   0.681513183925514
   0.681510396343762
   0.681506763332523
   0.681501744151592
   0.681494591611086
   0.681484240733670
   0.681469150291914
   0.681447072413142
   0.681414717841083
   0.681367262133698
   0.681297625967863
   0.681195407146443
   0.681045308385230
   0.680824818921794
   0.680500746829058
   0.680024038085926
   0.679321965208885
   0.678286225840684
   0.676754279600852
   0.674479867818450
   0.671084007739061
   0.665971057838265
   0.658174162490739
   0.646051232063440
   0.626621431466552
   0.593911793699313
   0.534012597301309
   0.405298693609964
                   0
                   0

(3)g(x)= √(3&〖-e〗^(-x)+x+2)

Third root: -1.80657027
Number of steps: 55
x_0= -2
S = abs((1+e^(-x))/(3(e^(-x)+x+2)^(2/3))) = 0.72407947
Approximation rate: 0.724079440171620 Match with convergence rate above. However, it matches for up to 7 decimal places and only happen once.

	FPI Code:
g = @(x) nthroot(x+2-exp(-x),3);
numSteps = 55;
x = zeros(numSteps,1);
x(1) = -2;
for i=1:numSteps
 x(i+1) = g(x(i));
end
r = x 
	Convergence Rate Code:
g = @(x) nthroot(x+2-exp(-x),3);
numSteps = 15;
x = zeros(numSteps,1);
x(1) = -0.5;
for i=1:numSteps
    x(i+1) = g(x(i));
end
r = x(numSteps + 1); %best guess at root
e = abs(x-r); % array of errors
ratios = zeros(numSteps + 1,1);
for i=1:numSteps
 ratios(i) = e(i+1)/e(i);
end
ratios(numSteps+1) = 0;
ratios
	Matlab Session:
>> format long

>> FPI_P2

r =

  -2.000000000000000
  -1.947734041054676
  -1.909327443839455
  -1.881244454642133
  -1.860775674961654
  -1.845887833520614
  -1.835074295841679
  -1.827227423000708
  -1.821536970924202
  -1.817412161733315
  -1.814423162445708
  -1.812257691450433
  -1.810689095220473
  -1.809552982074631
  -1.808730176289082
  -1.808134310875333
  -1.807702810594613
  -1.807390345829074
  -1.807164083796965
  -1.807000245446301
  -1.806881609971286
  -1.806795706630932
  -1.806733504827357
  -1.806688465275962
  -1.806655852798046
  -1.806632238634203
  -1.806615140030535
  -1.806602759244666
  -1.806593794551870
  -1.806587303391407
  -1.806582603269892
  -1.806579200005515
  -1.806576735770135
  -1.806574951467092
  -1.806573659489471
  -1.806572723994780
  -1.806572046622163
  -1.806571556150495
  -1.806571201009998
  -1.806570943860037
  -1.806570757663020
  -1.806570622841577
  -1.806570525220136
  -1.806570454534453
  -1.806570403352400
  -1.806570366292526
  -1.806570339458232
  -1.806570320028070
  -1.806570305959089
  -1.806570295772028
  -1.806570288395786
  -1.806570283054801
  -1.806570279187504
  -1.806570276387273
  -1.806570274359683
  -1.806570272891547

>> fpi_error

ratios =

   0.729793554865435
   0.727928789979321
   0.726705309826425
   0.725892146380041
   0.725343953324356
   0.724969262947089
   0.724709987263902
   0.724528696047858
   0.724400854357602
   0.724310096586489
   0.724245326469991
   0.724198911383240
   0.724165536866324
   0.724141465009127
   0.724124044423006
   0.724111380162923
   0.724102107423232
   0.724095233783219
   0.724090027365721
   0.724085935942462
   0.724082526803060
   0.724079440171620
   0.724076350802992
   0.724072933482670
   0.724068828664892
   0.724063604572003
   0.724056711706485
   0.724047424982366
   0.724034767347444
   0.724017406776880
   0.723993515712993
   0.723960578024288
   0.723915122495205
   0.723852354567014
   0.723765645506867
   0.723645822787801
   0.723480181829357
   0.723251103629130
   0.722934113925429
   0.722495139807702
   0.721886600108629
   0.721041769035122
   0.719866531306552
   0.718227075687898
   0.715931075395360
   0.712697932682268
   0.708109817757045
   0.701526959444763
   0.691931717342620
   0.677618592545310
   0.655514539615575
   0.619482352538149
   0.555233482074148
   0.419980372665284
                   0
                   0

Fourth root 1.481974814587466
Number of steps: 25
x_0=1.5
S = abs((1+e^(-x))/(3(e^(-x)+x+2)^(2/3))) = 0.1862552
Approximation rate: 0.186255194701434, match with convergence rate above. However, it matches for up to 6 decimal places and only happen once.

	FPI Code:
g = @(x) nthroot(x+2-exp(-x),3);
numSteps = 25;
x = zeros(numSteps,1);
x(1) = 1.5;
for i=1:numSteps
 x(i+1) = g(x(i));
end
r = x 
	Convergence Rate Code:
g = @(x) nthroot(x+2-exp(-x),3);
numSteps = 25;
x = zeros(numSteps,1);
x(1) = 1.5;
for i=1:numSteps
    x(i+1) = g(x(i));
end
r = x(numSteps + 1); %best guess at root
e = abs(x-r); % array of errors
ratios = zeros(numSteps + 1,1);
for i=1:numSteps
 ratios(i) = e(i+1)/e(i);
end
ratios(numSteps+1) = 0;
ratios

	Matlab Session:
>> format long

>> FPI_P2

r =

   1.500000000000000
   1.485318979020808
   1.482597228565602
   1.482090726683878
   1.481996403272099
   1.481978835573305
   1.481975563516332
   1.481974954079339
   1.481974840568552
   1.481974819426578
   1.481974815488775
   1.481974814755339
   1.481974814618733
   1.481974814593289
   1.481974814588550
   1.481974814587667
   1.481974814587503
   1.481974814587472
   1.481974814587467
   1.481974814587466
   1.481974814587466
   1.481974814587466
   1.481974814587466
   1.481974814587466
   1.481974814587466
   1.481974814587466

>> fpi_error

ratios =

   0.185527325062480
   0.186119429993050
   0.186229905631342
   0.186250488965471
   0.186254322950300
   0.186255037025190
   0.186255170055531
   0.186255194075115
   0.186255194701434
   0.186255174955528
   0.186255062035339
   0.186254604614866
   0.186251464687711
   0.186220307316887
   0.186117936117936
   0.185918591859186
   0.183431952662722
   0.161290322580645
   0.200000000000000
                   0
                 NaN
                 NaN
                 NaN
                 NaN
                 NaN
                   0

Conclusion:

I found 4 roots of the given equation by the following: {-0.48756103475, -4.451620296, -1.80657027, 1.481974814587466}
I derive the original equation into 3 different form of x = g(x). However, I only use 2 of them since I tried to work with the other one but it doesn’t work (not working MATLAB session is not included)
(1) x= e^(-x)+x^3-2
Not used, derive from isolating x
(2) x=-ln⁡(-x^3+x+2)
Used, derive from isolation x^3 and solve for x
(3) x= ∛(-e^(-x)+x+2 )
Used, derive from isolation x^3 and solve for x
First root took the least steps (15 steps) while third root took the most steps (55 steps).
I am satisfied about the comparison between convergence rate calculated by using g prime verse generated by MATLAB’s code.
Final conclusion: the closer your initial x to the actual root, the higher chance your “guess” root will match the actual root of given equation.