maple1

r1 := x*exp(x)-2 = 0;
                        x exp(x) - 2 = 0
plot(lhs(r1), x = 0 .. 1);

r2 := x = ln(2/x);
                                 /2\
                           x = ln|-|
                                 \x/
plot([abs(diff(rhs(r2), x)), 1], x = 0 .. 1);

eps := 10^(-8);
                               1
                           ---------
                           100000000
xs := 1.0;
                              1.0
xn := 2/exp(xs);
                          0.7357588824
i := 0;
                               0
while abs(xn-xs) > eps do i := i+1; xs := xn; xn := 2/exp(xs) end do;
                               1
                          0.7357588824
                          0.9582834174
                               2
                          0.9582834174
                          0.7671014354
                               3
                          0.7671014354
                          0.9287141770
                               4
                          0.9287141770
                          0.7901227258
                               5
                          0.7901227258
                          0.9075782006
                               6
                          0.9075782006
                          0.8070004768
                               7
                          0.8070004768
                          0.8923888630
                               8
                          0.8923888630
                          0.8193518466
                               9
                          0.8193518466
                          0.8814344286
                               10
                          0.8814344286
                          0.8283767236
                               11
                          0.8283767236
                          0.8735153792
                               12
                          0.8735153792
                          0.8349627228
                               13
                          0.8349627228
                          0.8677813106
                               14
                          0.8677813106
                          0.8397642092
                               15
                          0.8397642092
                          0.8636246576
                               16
                          0.8636246576
                          0.8432620820
                               17
                          0.8432620820
                          0.8606090854
                               18
                          0.8606090854
                          0.8458088378
                               19
                          0.8458088378
                          0.8584201128
                               20
                          0.8584201128
                          0.8476623182
                               21
                          0.8476623182
                          0.8568305216
                               22
                          0.8568305216
                          0.8490108262
                               23
                          0.8490108262
                          0.8556758574
                               24
                          0.8556758574
                          0.8499917146
                               25
                          0.8499917146
                          0.8548369466
                               26
                          0.8548369466
                          0.8507050812
                               27
                          0.8507050812
                          0.8542273518
                               28
                          0.8542273518
                          0.8512238248
                               29
                          0.8512238248
                          0.8537843416
                               30
                          0.8537843416
                          0.8516010092
                               31
                          0.8516010092
                          0.8534623684
                               32
                          0.8534623684
                          0.8518752460
                               33
                          0.8518752460
                          0.8532283496
                               34
                          0.8532283496
                          0.8520746242
                               35
                          0.8520746242
                          0.8530582516
                               36
                          0.8530582516
                          0.8522195726
                               37
                          0.8522195726
                          0.8529346112
                               38
                          0.8529346112
                          0.8523249480
                               39
                          0.8523249480
                          0.8528447376
                               40
                          0.8528447376
                          0.8524015528
                               41
                          0.8524015528
                          0.8527794078
                               42
                          0.8527794078
                          0.8524572418
                               43
                          0.8524572418
                          0.8527319190
                               44
                          0.8527319190
                          0.8524977252
                               45
                          0.8524977252
                          0.8526973980
                               46
                          0.8526973980
                          0.8525271546
                               47
                          0.8525271546
                          0.8526723040
                               48
                          0.8526723040
                          0.8525485482
                               49
                          0.8525485482
                          0.8526540626
                               50
                          0.8526540626
                          0.8525640998
                               51
                          0.8525640998
                          0.8526408024
                               52
                          0.8526408024
                          0.8525754054
                               53
                          0.8525754054
                          0.8526311628
                               54
                          0.8526311628
                          0.8525836236
                               55
                          0.8525836236
                          0.8526241560
                               56
                          0.8526241560
                          0.8525895976
                               57
                          0.8525895976
                          0.8526190624
                               58
                          0.8526190624
                          0.8525939406
                               59
                          0.8525939406
                          0.8526153594
                               60
                          0.8526153594
                          0.8525970976
                               61
                          0.8525970976
                          0.8526126678
                               62
                          0.8526126678
                          0.8525993924
                               63
                          0.8525993924
                          0.8526107112
                               64
                          0.8526107112
                          0.8526010608
                               65
                          0.8526010608
                          0.8526092886
                               66
                          0.8526092886
                          0.8526022736
                               67
                          0.8526022736
                          0.8526082546
                               68
                          0.8526082546
                          0.8526031550
                               69
                          0.8526031550
                          0.8526075032
                               70
                          0.8526075032
                          0.8526037958
                               71
                          0.8526037958
                          0.8526069566
                               72
                          0.8526069566
                          0.8526042618
                               73
                          0.8526042618
                          0.8526065594
                               74
                          0.8526065594
                          0.8526046006
                               75
                          0.8526046006
                          0.8526062708
                               76
                          0.8526062708
                          0.8526048466
                               77
                          0.8526048466
                          0.8526060610
                               78
                          0.8526060610
                          0.8526050254
                               79
                          0.8526050254
                          0.8526059084
                               80
                          0.8526059084
                          0.8526051556
                               81
                          0.8526051556
                          0.8526057974
                               82
                          0.8526057974
                          0.8526052502
                               83
                          0.8526052502
                          0.8526057168
                               84
                          0.8526057168
                          0.8526053188
                               85
                          0.8526053188
                          0.8526056582
                               86
                          0.8526056582
                          0.8526053690
                               87
                          0.8526053690
                          0.8526056154
                               88
                          0.8526056154
                          0.8526054054
                               89
                          0.8526054054
                          0.8526055844
                               90
                          0.8526055844
                          0.8526054318
                               91
                          0.8526054318
                          0.8526055620
                               92
                          0.8526055620
                          0.8526054508
                               93
                          0.8526054508
                          0.8526055456
                               94
                          0.8526055456
                          0.8526054650
                               95
                          0.8526054650
                          0.8526055336
                               96
                          0.8526055336
                          0.8526054750
                               97
                          0.8526054750
                          0.8526055248
                               98
                          0.8526055248
                          0.8526054828
                               99
                          0.8526054828
                          0.8526055184
                              100
                          0.8526055184
                          0.8526054882
                              101
                          0.8526054882
                          0.8526055136
                              102
                          0.8526055136
                          0.8526054922
                              103
                          0.8526054922
                          0.8526055104
                              104
                          0.8526055104
                          0.8526054948
                              105
                          0.8526054948
                          0.8526055082
                              106
                          0.8526055082
                          0.8526054968
                              107
                          0.8526054968
                          0.8526055064

r1;
                        x exp(x) - 2 = 0
r2;
                                 /2\
                           x = ln|-|
                                 \x/
r3 := (r2-x*(diff(r2, x)))/(1-(diff(r2, x)));
                               /2\
                             ln|-| + 1
                               \x/
                         0 = ---------
                               1
                               - + 1
                               x
r4 := x = rhs(r3);
                               /2\
                             ln|-| + 1
                               \x/
                         x = ---------
                               1
                               - + 1
                               x
plot([abs(diff(rhs(r4), x)), 1], x = 0 .. 1);

xs := 1.0;
                              1.0
xn := (ln(2/xs)+1)/(1/xs+1);
                          0.8465735905

i := 0;
                               0
while abs(xn-xs) > eps do i := i+1; xs := xn; xn := (ln(2/xs)+1)/(1/xs+1) end do;
                               1
                          0.8465735905
                          0.8525939199
                               2
                          0.8525939199
                          0.8526055019
                               3
                          0.8526055019
                          0.8526055018