3a3c6110-f27c-4a76-8c3a-670ffce65e14 113 4 5 decimal

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NIM algorithm (single zero)

NIM = Non Iterative Method

p'

zero of f(x)

x f x sin :

α 0.1 :

β 10 :

x F x f sign :

x f x F 2 1 sys

p' 12 / α β + α F z F z α β int * + (* :

x f x α β solve 3.1416 6.2832 9.4248 31 mat

p' 6.2792

p' f 0.0039 -

x f x 2 - (x 10^x + 1 + (* e x - 1 -^* :

α 0 :

β 5 :

x F x f sign :

x f x F 2 1 sys

p' 12 / α β + α F z F z α β int * + (* :

MULTIPOINT METHODS FOR SOLVING NONLINEAR EQUATIONS By Miodrag Petkovic, Beny Neta, Ljiljana Petkovic,

https://books.google.com/books?id=e2cx8hppYGYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

p' 2

p' f 0

x T.m m x f * tanh :

m 8 :

p' 12 / α β + α F z T.m z α β int * + (* :

p' 1.999

p' f 0.0494 -

NIM algorithm (single zero) approximated

x f x sin :

α 0.1 :

β 10 :

x F x f sign :

N 1000 :

q α β + 2 / :

δ β α - 2 N * / :

p.0 q δ α F * q 2 j * N - (δ * + F j 1 N 1 - sum * + eval :

x f x α β solve 3.1416 6.2832 9.4248 31 mat

p.0 6.28255

p.0 f 0.0006 -

using an iterative algorithm Φ(x), p0 may be used as initial guess

newton x Φ x x f z f' z f (z diff : x f' 21 line / - :

Ostrowski method of fourth order x Φ x u z f' z f (z diff : x f x f' / 21 line : x x u x x u - f x f - 2 x x u - f * x f - / * - eval 21 line :

SI algorithm

SI = Signum-based Iterations

ε.f 108 -^:

ε.x 108 -^:

x f α β N SI p α β + 2 / : δ β α - 2 N * / : x F x f sign : p p δ α F * p 2 j * N - (δ * + F j 1 N 1 - sum * + eval : k 0 : k 100 < r p : p x u x f z f' z f (z diff : x f' 21 line / : r r u r r u - f r f - 2 r r u - f * r f - / * - eval 21 line : p f abs ε.f < (p r - abs ε.x < (| break k k 1 + : if 31 line while p eval 71 line :

p r r f z f' z f (z diff : r f' 21 line / - eval :

p x u x f z f' z f (z diff : x f' 21 line / : r r u r r u - f r f - 2 r r u - f * r f - / * - eval 21 line :

x f x sin :

R x f 0.3 π * 3 π * 1000 SI : 6.2832

xmin xmax ymin ymax plotBox xmin ymin xmin ymax xmax ymax xmax ymin xmin ymin 52 mat 11 line :

x sin R R f . 20 red 1 5 mat 0.3 π * 3 π * 2 - 2 plotBox 3 1 sys

x f x 0.3 π * π solve #

SIM algorithm

SI = Signum-based Iterations for Mutiple roots

ε.f 108 -^:

x f a b N.0 SIM h b a - N.0 / eval : x F x f sign : j m 0 : (: p 01 matrix : a F abs ε.f < m m 1 + : p m el a : 21 line 0 if 31 line j N.0 < j j 1 + : α a j 1 - (h * + eval : β a j h * + eval : α F β F * 0 < m m 1 + : p m el x f α β 10 SI : 21 line β F abs ε.f < m m 1 + : p m el β : 21 line 0 if if 41 line while p rows 0 ≡ no roots found error p if 51 line :

R 1 f plotPts m 05 matrix : i 1 R rows range m m R i el R i el f . 20 red 15 mat stack : for m eval 31 line :

x f x ln :

R x f 0.1 1020 SIM : 111 mat

x f R X f plotPts 0.1 10 5 - 5 plotBox 3 1 sys

x f x sin :

R x f 04 π * 20 SIM : 0 3.1416 6.2832 9.4248 12.5664 51 mat

x f R X f plotPts 0 4 π * 2 - 2 plotBox 3 1 sys

x f.1 23 / 110 / x 11^- (e 2 x 2^-^* - :

x f.2 26 x π - (* sin * exp x + 1 - :

R.1 x f.1 1 - 120 SIM : 0.781151944 - 0.3208068825 - 0.3206916281 31 mat

R.2 x f.2 1.5 - 250 SIM : 0.9887140394 - 0.5610243641 - 3.92372 1010 -^* 0.6038168431 0.8458544308 51 mat

x f.1 x f.2 1.5 - 2 0.5 - 0.5 plotBox R.1 X f.1 plotPts R.2 X f.2 plotPts 5 1 sys

SIG algorithm

SIG = Signum-based Iteration for General cases

returns zeros (1st output) and  maximum/minimum points (2nd output) in a range x f a b N.0 SIG ε 107 -^: h b a - N.0 / eval : x f.ε ε x f (2^* x ε x f * + f x f - ε 3^+ / : x F.ε x f.ε sign : x invf.ε x ε x f * + f x f - ε x f (2^* / : j m n 0 : (: (: p q 01 matrix : (: a f 0 ≡ m m 1 + : p m el a : 21 line 0 if 41 line j N.0 < j j 1 + : α a j 1 - (h * + : β a j h * + : β F.ε α F.ε - 2 ≡ (β f.ε α f.ε - 2 h * < (& m m 1 + : p m el x f.ε α β 10 SI : 21 line 0 if β F.ε α F.ε - abs 2 ≡ (β f.ε α f.ε - abs 2 h * > (& n n 1 + : q n el x invf.ε α β 10 SI : 21 line 0 if 51 line while p q 21 mat 71 line :

ε 107 -^:

x f x sin :

x f.ε ε x f (2^* x ε x f * + f x f - ε 3^+ / :

<(--- ε2 for null difference

x f x f.ε x f.ε sign 3 1 sys

x f x sin :

x f x 02 π * solve 0 3.1416 21 mat

R x f 02 π * 20 SIG : 0 3.1416 6.2832 31 mat 1.5708 4.7124 21 mat 21 mat

x f 0 2 π * 2 - 2 plotBox R 1 el X f plotPts R 2 el X f plotPts 4 1 sys

x f 23 / 110 / x 11^- (e 2 x 2^-^* - :

R x f 1 - 125 SIG : 0.7811519469 - 0.3208068857 - 0.320691641 31 mat 0.6463055063 - 8.4855984 108 -^* - 21 mat 21 mat

x f 1 - 1 1 1 - plotBox R 1 el X f plotPts R 2 el X f plotPts 4 1 sys

x f 26 x π - (* sin * exp x + 1 - :

R x f 1.5 - 241 SIG : 0.9887140323 - 0.5610243645 - 4.19066 1010 -^* 0.6038168459 0.8458544288 51 mat 1.3926546143 - 0.7835185079 - 0.3454571158 - 0.2636789909 0.7017403839 1.3108764898 1.7489378816 71 mat 21 mat

x f 1.5 - 2 9 4 - plotBox R 1 el X f plotPts R 2 el X f plotPts 4 1 sys

Count zeros

x f a b N.0 NumOfZeros h b a - N.0 / : h.min π : h h h.min Min : k ξ a k h * + : x F x f sign : M 12 / 1 k 1 - ξ F k ξ F * - (k 1 N.0 sum * 0 round : M eval 71 line :

x sin 02 π * 10 NumOfZeros 2

x f.1 1 - 120 NumOfZeros 3

x f.2 1.5 - 250 NumOfZeros 5