<content> Solve Function Solution: </content>

c7901d9a-65cb-4ee6-a2e9-19305522d199 30 ioan92 4 5 false false 0 0 decimal

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4 appVersion 1.0.8348.30405

Input Data:

x T f x sinh T x * - :

x T df T - x cosh + :

x T df x T f x diff : T - x cosh +

Tin 1.00001 1.1 1.5 2551 mat :

<content> Solve Function Solution: </content>

T μex x T f 0 ≡ x 1010 -^5 solve :

k 1 Tin rows range Sol k el Tin k el μex : Check k el Sol k el Tin k el f : 21 line for

R.a Tin Sol Check augment : 1.00001 1.10619 1010 -^* 1.10618999999882 1015 -^* - 1.1 0.763400625604734 3.53920532880636 108 -^* - 1.5 1.62213100434004 2.41268161449426 107 -^* - 2 2.17731807769879 2.239116523 106 -^* - 5 3.57823408423261 9.356197335 106 -^* - 53 mat

<content> Alvaro's Solution + Initial value improvement (Ioan): </content>

T N Sol.Alvaro2 n 2 N range : X 1 el T : 12 mat X n el X n 1 - el X n 1 - el T f X n 1 - el T df / - eval : X N el X X N el T f 13 mat 31 line :

T N Sol.Alvaro2b n 2 N range : X 1 el 3.162 1 - 1 1.2 1 T - (* - sqrt + sqrt * eval : 12 mat X n el X n 1 - el X n 1 - el T f X n 1 - el T df / - eval : X N el X X N el T f 13 mat 31 line :

N 6 :

Sol Check Clear 1

k 1 Tin rows range ra Tin k el N Sol.Alvaro2 : rab Tin k el N Sol.Alvaro2b : Sol k el ra 1 el : Check k el ra 3 el : Solb k el rab 1 el : Checkb k el rab 3 el : 61 line for

R.b Tin Sol Check Solb Checkb augment : 1.00001 0.135693199060528 0.000415438365593 0.007745955073487 0 1.1 0.763400797564702 6.60449472889013 1013 -^* 0.763400797561493 0 1.5 1.62213121772442 1.77635683940025 1015 -^* - 1.62213121772442 1.77635683940025 1015 -^* - 2 2.17731898496531 7.99360577730113 1015 -^* 2.17731898496531 7.99360577730113 1015 -^* 5 3.57823516358779 4.588095514 106 -^* 3.57823480844736 1.8 1014 -^* 55 mat

Improved Ostrowskiroot-ﬁnding method, by Miquel Grau & Jose Luis Diaz-Barrero

Alvaro + ao Ti N R x fA x sinh Ti x * - : x fA' Ti - x cosh + : X ao : T 10200^: ε 1015 -^: 13 mat N 2100 range X fA abs ε < (X T - abs ε < (| break T X : Y X X fA X fA' / - : M X Y - X fA 2 Y fA * - / : Z Y M Y fA * - : X Z M Z fA * - : 51 line if for X X fA 12 mat 51 line :

k 1 Tin rows range ao R.b k 2 el : Checko k el ao Tin k el f : soo ao Tin k el N R : Sol k el soo 1 el : Check k el Sol k el Tin k el f : Ito k el N : 61 line for

Rb <-- Alvaro's solution

R.f Tin R.b 2 col Checko Sol Check Ito augment : 1.00001 0.135693199060528 0.000415438365593 0.00774595507355 07 1.1 0.763400797564702 6.60449472889013 1013 -^* 0.763400797561493 1.42728836229901 1016 -^* 3 1.5 1.62213121772442 1.77635683940025 1015 -^* - 1.62213121772442 032 2.17731898496531 7.99360577730113 1015 -^* 2.17731898496531 3.8252224549309 1015 -^* 65 3.57823516358779 4.588095514 106 -^* 3.57823480844737 1.2742741305808 1013 -^* 756 mat

<content> Newton's Solution by Ioan: </content>

T OneEq.Ioan ε 107 -^: k 1 : dk 2 ε * : 13 mat y 1 el 3.162 1 - 1 1.2 1 T - (* - sqrt + sqrt * eval : dk abs ε > eval dk y k el T f y k el T df / eval : y k 1 + el y k el dk - eval : k k 1 + : 12 mat 21 line while y k el k y 13 mat 41 line :

Sol Check Clear 1

k 1 Tin rows range so Tin k el OneEq.Ioan : Sol k el so 1 el : Check k el Sol k el Tin k el f : Ito k el so 2 el : 41 line for

R.c Tin Sol Check Ito augment : 1.00001 0.007745955073572 1.4210854715202 1018 -^* 3 1.1 0.763400797561493 03 1.5 1.62213121772442 1.77635683940025 1015 -^* - 42 2.17731898496531 7.99360577730113 1015 -^* 55 3.57823480844736 1.8 1014 -^* 554 mat

<content> 4th Deg. Polynome Ioan's Solution: </content>

μ sinh T μ * - 1 T - μ 2^6 / + μ 4^120 / + μ 6^5040 / + μ 8^362880 / +

Maclaurin aprox:

Poly 4th deg.   Ioan: T_μ4f R# 135862272522547200 T_* + eval : U# 483840 T_0.1 - (* (3^R# 2^+ sqrt eval : S# R# U# + 3 nthroot eval : T# R# U# - 3 nthroot eval : 12 mat ζ i 0.75 sqrt * : km ζ 0.5 + : kp ζ 0.5 - : 13 mat S# T# + kp S# * km T# * - km S# * - kp T# * + 31 mat 720 - sqrt vectorize sum eval 2 / 18 - sqrt eval Re 51 line :

Sol Check Clear 1

k 1 Tin rows range Sol k el Tin k el μ4f : Check k el Sol k el Tin k el f : 21 line for

R.d Tin Sol Check augment : 1.00001 0.00774595518006 2.13074002886061 1015 -^* 1.1 0.763400803830432 1.29027029061035 109 -^* 1.5 1.6221358289827 5.21385667839879 106 -^* 2 2.17737356574761 0.000134710927879 5 3.58084747317322 0.033814390246185 53 mat

<content> Dichotomy Solution by Ioan: (note! only one function evaluation for each iteration step) </content>

Dichotomy - Ioan 20220902 ab0 εv εf ItM Tv Dicot x Fon x sinh Tv x * - : ab ab0 : 12 mat k 0 : fab ab Fon vectorize eval : c 0.5 ab 1 el ab 2 el + (* : fc c Fon eval : 22 mat ab 2 el ab 1 el - abs εv > eval (k ItM ≤ (* fc abs εf > eval (* fc fab 1 el * 0 < ab 2 el c : fab 2 el fc : 12 mat ab 1 el c : fab 1 el fc : 12 mat if c 0.5 ab 1 el ab 2 el + (* eval : fc c Fon eval : k k 1 + : 13 mat 21 line while c ab fc k 14 mat 41 line :

Sol Check Clear 1

ab0 107 -^521 mat :

εv 1010 -^:

εf 1010 -^:

ItM 50 :

Fon Clear 1

k 1 Tin rows range soo ab0 εv εf ItM Tin k el Dicot : Sol k el soo 1 el : Check k el Sol k el Tin k el f : Ito k el soo 4 el : 41 line for

R.e Tin Sol Check Ito augment : 1.00001 0.007743935294341 4.03799060677556 1011 -^* - 16 1.1 0.763400797480975 1.65720770439748 1011 -^* - 31 1.5 1.62213121777326 5.52207168880161 1011 -^* 272 2.17731898496951 1.0374 1011 -^* 355 3.5782348084694 2.84757 1010 -^* 3654 mat