Find stable zero of f(x) = 1- (1+d)x^2 -e^{-x} Iteration x = x +dt*f(x) for dt=1, d = 0 iter,t,x 0 0 0.6321205588285577 iter,t,x 1 1 0.701080552548214 iter,t,x 2 2 0.7135176043103406 iter,t,x 3 3 0.7144924068105107 iter,t,x 4 4 0.7145525220089581 iter,t,x 5 5 0.7145561518154666 iter,t,x 6 6 0.714556370698249 iter,t,x 7 7 0.7145563838961609 iter,t,x 8 8 0.7145563846919478 iter,t,x 9 9 0.7145563847399308 iter,t,x 10 10 0.714556384742824 iter,t,x 11 11 0.7145563847429985 iter,t,x 12 12 0.7145563847430091 iter,t,x 13 13 0.7145563847430096 iter,t,x 14 14 0.7145563847430096 iter,t,x 15 15 0.7145563847430096 iter,t,x 16 16 0.7145563847430096 iter,t,x 17 17 0.7145563847430096 iter,t,x 18 18 0.7145563847430096 iter,t,x 19 19 0.7145563847430096 Iteration x = x +dt*f(x) for dt=1, d = 1 iter,t,x 0 0 0.3934693402873666 iter,t,x 1 1 0.4091210930591159 iter,t,x 2 2 0.4101271621373348 iter,t,x 3 3 0.4101527195741639 iter,t,x 4 4 0.4101533073344211 iter,t,x 5 5 0.41015332081567646 iter,t,x 6 6 0.41015332112487246 iter,t,x 7 7 0.4101533211319639 iter,t,x 8 8 0.41015332113212655 iter,t,x 9 9 0.4101533211321302 iter,t,x 10 10 0.4101533211321303 iter,t,x 11 11 0.4101533211321303 iter,t,x 12 12 0.4101533211321303 iter,t,x 13 13 0.4101533211321303 iter,t,x 14 14 0.4101533211321303 iter,t,x 15 15 0.4101533211321303 iter,t,x 16 16 0.4101533211321303 iter,t,x 17 17 0.4101533211321303 iter,t,x 18 18 0.4101533211321303 iter,t,x 19 19 0.4101533211321303 Iteration x = x +dt*f(x) for dt=1, d = 2 iter,t,x 0 0 0.2834686894262108 iter,t,x 1 1 0.28923849191862944 iter,t,x 2 2 0.28942818295220135 iter,t,x 3 3 0.2894306038934476 iter,t,x 4 4 0.2894306332214391 iter,t,x 5 5 0.2894306335764845 iter,t,x 6 6 0.2894306335807825 iter,t,x 7 7 0.2894306335808346 iter,t,x 8 8 0.2894306335808354 iter,t,x 9 9 0.2894306335808354 iter,t,x 10 10 0.2894306335808354 iter,t,x 11 11 0.2894306335808354 iter,t,x 12 12 0.2894306335808354 iter,t,x 13 13 0.2894306335808354 iter,t,x 14 14 0.2894306335808354 iter,t,x 15 15 0.2894306335808354 iter,t,x 16 16 0.2894306335808354 iter,t,x 17 17 0.2894306335808354 iter,t,x 18 18 0.2894306335808354 iter,t,x 19 19 0.2894306335808354 Iteration x = x +dt*f(x) for dt=1, d = 3 iter,t,x 0 0 0.22119921692859512 iter,t,x 1 1 0.22392586198579334 iter,t,x 2 2 0.22398029911856832 iter,t,x 3 3 0.22398071980842305 iter,t,x 4 4 0.22398072295797566 iter,t,x 5 5 0.22398072298154936 iter,t,x 6 6 0.22398072298172578 iter,t,x 7 7 0.2239807229817271 iter,t,x 8 8 0.2239807229817271 iter,t,x 9 9 0.2239807229817271 iter,t,x 10 10 0.2239807229817271 iter,t,x 11 11 0.2239807229817271 iter,t,x 12 12 0.2239807229817271 iter,t,x 13 13 0.2239807229817271 iter,t,x 14 14 0.2239807229817271 iter,t,x 15 15 0.2239807229817271 iter,t,x 16 16 0.2239807229817271 iter,t,x 17 17 0.2239807229817271 iter,t,x 18 18 0.2239807229817271 iter,t,x 19 19 0.2239807229817271 Iteration x = x +dt*f(x) for dt=1, d = 4 iter,t,x 0 0 0.18126924692201823 iter,t,x 1 1 0.18276582773677813 iter,t,x 2 2 0.182785899019481 iter,t,x 3 3 0.1827860033138145 iter,t,x 4 4 0.182786003844351 iter,t,x 5 5 0.18278600384704963 iter,t,x 6 6 0.18278600384706328 iter,t,x 7 7 0.1827860038470634 iter,t,x 8 8 0.18278600384706328 iter,t,x 9 9 0.1827860038470634 iter,t,x 10 10 0.18278600384706328 iter,t,x 11 11 0.1827860038470634 iter,t,x 12 12 0.18278600384706328 iter,t,x 13 13 0.1827860038470634 iter,t,x 14 14 0.18278600384706328 iter,t,x 15 15 0.1827860038470634 iter,t,x 16 16 0.18278600384706328 iter,t,x 17 17 0.1827860038470634 iter,t,x 18 18 0.18278600384706328 iter,t,x 19 19 0.1827860038470634 Iteration x = x +dt*f(x) for dt=1, d = 5 iter,t,x 0 0 0.1535182751093859 iter,t,x 1 1 0.15442602058725122 iter,t,x 2 2 0.15443476231551326 iter,t,x 3 3 0.15443479499594373 iter,t,x 4 4 0.15443479511627414 iter,t,x 5 5 0.15443479511671712 iter,t,x 6 6 0.1544347951167188 iter,t,x 7 7 0.1544347951167188 iter,t,x 8 8 0.1544347951167188 iter,t,x 9 9 0.1544347951167188 iter,t,x 10 10 0.1544347951167188 iter,t,x 11 11 0.1544347951167188 iter,t,x 12 12 0.1544347951167188 iter,t,x 13 13 0.1544347951167188 iter,t,x 14 14 0.1544347951167188 iter,t,x 15 15 0.1544347951167188 iter,t,x 16 16 0.1544347951167188 iter,t,x 17 17 0.1544347951167188 iter,t,x 18 18 0.1544347951167188 iter,t,x 19 19 0.1544347951167188 Iteration x = x +dt*f(x) for dt=1, d = 6 iter,t,x 0 0 0.13312210024981846 iter,t,x 1 1 0.13371344109738476 iter,t,x 2 2 0.13371772870182214 iter,t,x 3 3 0.13371774079568782 iter,t,x 4 4 0.13371774082941384 iter,t,x 5 5 0.13371774082950788 iter,t,x 6 6 0.1337177408295082 iter,t,x 7 7 0.1337177408295081 iter,t,x 8 8 0.1337177408295082 iter,t,x 9 9 0.1337177408295081 iter,t,x 10 10 0.1337177408295082 iter,t,x 11 11 0.1337177408295081 iter,t,x 12 12 0.1337177408295082 iter,t,x 13 13 0.1337177408295081 iter,t,x 14 14 0.1337177408295082 iter,t,x 15 15 0.1337177408295081 iter,t,x 16 16 0.1337177408295082 iter,t,x 17 17 0.1337177408295081 iter,t,x 18 18 0.1337177408295082 iter,t,x 19 19 0.1337177408295081 Iteration x = x +dt*f(x) for dt=1, d = 7 iter,t,x 0 0 0.11750309741540454 iter,t,x 1 1 0.11790951648449677 iter,t,x 2 2 0.11791181526785699 iter,t,x 3 3 0.1179118203361984 iter,t,x 4 4 0.11791182034727454 iter,t,x 5 5 0.11791182034729875 iter,t,x 6 6 0.11791182034729886 iter,t,x 7 7 0.11791182034729875 iter,t,x 8 8 0.11791182034729886 iter,t,x 9 9 0.11791182034729875 iter,t,x 10 10 0.11791182034729886 iter,t,x 11 11 0.11791182034729875 iter,t,x 12 12 0.11791182034729886 iter,t,x 13 13 0.11791182034729875 iter,t,x 14 14 0.11791182034729886 iter,t,x 15 15 0.11791182034729875 iter,t,x 16 16 0.11791182034729886 iter,t,x 17 17 0.11791182034729875 iter,t,x 18 18 0.11791182034729886 iter,t,x 19 19 0.11791182034729875 Iteration x = x +dt*f(x) for dt=1, d = 8 iter,t,x 0 0 0.10516068318563015 iter,t,x 1 1 0.10545189239698266 iter,t,x 2 2 0.10545321321402573 iter,t,x 3 3 0.10545321555348314 iter,t,x 4 4 0.10545321555759751 iter,t,x 5 5 0.10545321555760473 iter,t,x 6 6 0.10545321555760484 iter,t,x 7 7 0.10545321555760484 iter,t,x 8 8 0.10545321555760484 iter,t,x 9 9 0.10545321555760484 iter,t,x 10 10 0.10545321555760484 iter,t,x 11 11 0.10545321555760484 iter,t,x 12 12 0.10545321555760484 iter,t,x 13 13 0.10545321555760484 iter,t,x 14 14 0.10545321555760484 iter,t,x 15 15 0.10545321555760484 iter,t,x 16 16 0.10545321555760484 iter,t,x 17 17 0.10545321555760484 iter,t,x 18 18 0.10545321555760484 iter,t,x 19 19 0.10545321555760484 And now Newton's method Iteration x = x -f(x)/f'(x) for d = 0 iter,x 1 0.7746003264394359 iter,x 2 0.7186480119813213 iter,x 3 0.7145783105037122 iter,x 4 0.714556385379744 iter,x 5 0.7145563847430098 iter,x 6 0.7145563847430096 iter,x 7 0.7145563847430098 Iteration x = x -f(x)/f'(x) for d = 1 iter,x 1 0.4235500512048121 iter,x 2 0.41055540602670126 iter,x 3 0.4101537062107809 iter,x 4 0.41015332113248426 iter,x 5 0.4101533211321303 iter,x 6 0.4101533211321303 iter,x 7 0.4101533211321303 Iteration x = x -f(x)/f'(x) for d = 2 iter,x 1 0.2944818644054742 iter,x 2 0.2895148471732734 iter,x 3 0.28943065779062366 iter,x 4 0.2894306335808373 iter,x 5 0.28943063358083543 iter,x 6 0.2894306335808353 iter,x 7 0.2894306335808352 Iteration x = x -f(x)/f'(x) for d = 3 iter,x 1 0.22641598301721733 iter,x 2 0.22400645260409932 iter,x 3 0.22398072591565574 iter,x 4 0.22398072298172716 iter,x 5 0.22398072298172716 iter,x 6 0.22398072298172716 iter,x 7 0.22398072298172716 Iteration x = x -f(x)/f'(x) for d = 4 iter,x 1 0.18414353617789705 iter,x 2 0.18279589004394872 iter,x 3 0.18278600437910153 iter,x 4 0.1827860038470634 iter,x 5 0.18278600384706328 iter,x 6 0.1827860038470634 iter,x 7 0.18278600384706328 Iteration x = x -f(x)/f'(x) for d = 5 iter,x 1 0.15526815496468785 iter,x 2 0.15443922826149176 iter,x 3 0.1544347952435153 iter,x 4 0.15443479511671884 iter,x 5 0.15443479511671873 iter,x 6 0.15443479511671884 iter,x 7 0.15443479511671873 Iteration x = x -f(x)/f'(x) for d = 6 iter,x 1 0.13426579810693623 iter,x 2 0.1337199628172235 iter,x 3 0.13371774086632982 iter,x 4 0.13371774082950807 iter,x 5 0.13371774082950807 iter,x 6 0.13371774082950807 iter,x 7 0.13371774082950807 Iteration x = x -f(x)/f'(x) for d = 7 iter,x 1 0.11829138093493027 iter,x 2 0.11791303176403337 iter,x 3 0.11791182035971808 iter,x 4 0.11791182034729884 iter,x 5 0.11791182034729873 iter,x 6 0.11791182034729873 iter,x 7 0.11791182034729873 Iteration x = x -f(x)/f'(x) for d = 8 iter,x 1 0.10572689140802792 iter,x 2 0.10545392093093507 iter,x 3 0.10545321556231488 iter,x 4 0.1054532155576048 iter,x 5 0.1054532155576048 iter,x 6 0.1054532155576048 iter,x 7 0.1054532155576048