<p> Stem Construction Utility </p>

80201438-7cce-479d-81b3-bd08c7e1ccdf 25 4 5 decimal

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Heron [Greek] sqrt(x) from the Babylonians

x sqrt x 0.5^≡ x 12 /^≡

"Heron" sqrt(x) as function x SQRT k 0 : ε 1012 -^: 12 mat r x 2 / : r 2^x - abs ε > r r 2 / x 2 r * / + : k k 1 + : 21 line while r 41 line :

x 1.23456789 : <-- plug your value ≡

buil-in vs algo demo x sqrt x SQRT 21 mat 1.11111110605556 1.11111110605556 21 mat

"Heron" sqrt(x) as function x ε SQRT k 0 : 11 mat r x 2 / : r 2^x - abs ε > r r 2 / x 2 r * / + : k k 1 + : 21 line while r k 21 mat 41 line :

x 10 : ε 109 -^: 12 mat

71 sqrt 647 + sqrt ≡

x sqrt x ε SQRT 1 el x ε SQRT 2 el 31 mat 3.16227766016838 3.16227766016837 531 mat

812 / 78 / * + 8.44

71 sqrt 8.4261

1 0.23456789 + sqrt 112 / 0.23456789 * + ≡

112 / 0.23456789 * + 1.1173

x 1.23456789 : <-- plug your value ≡

open style ......
r => result 
k => iterates k 0 : ε 1012 -^: 12 mat r x 2 / : r 2^x - abs ε > r r 2 / x 2 r * / + : k k 1 + : 21 line while r 41 line

x sqrt r k 31 mat 1.11111110605556 1.11111110605556 531 mat

some Legendre orthopoly x f x 157 x 2^* 10 - 9 x 2^* + (* + (* 8 / :

initialise the solver => a 0.6 - : b 0.5 - : xn b a + (2 / : ε 1014 -^: 41 sys

In mathematics, the root(s) is the abscissa at which the function crosses the X axis. Dichotomy is one of many methods, bracket the root region [a,b]. Closer the braketing less iterates.

Dichotomy iter 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : iter iter 1 + : 31 line while 31 line

x f xn 0 + 20 red 1 5 mat 2 1 sys

.. sanity ..

xn f 1.7 1015 -^*

xn iter 21 mat 0.538469310105684 - 4321 mat

c 0158 / 0354 / - 0638 / 61 mat :

If you have the coefficients of each monomials, you can obtain all roots at once from the universal "polyroots", available in all advanced CAS. Here is how it works in this example : the argument of "polyroots" is the vector of the coefficients of the monomials in ascending order. Under the hood, polyroots is a bit monkey business that some "grand savants" made work, sleep on that.

0 NO constant term ≡ 158 / coeff of x ≡ 0 NO monomial x^2 ≡ 354 / - coeff of x^3 ≡ 0 NO monomial x^4 ≡ 638 / coeff of x^5 ≡ 61 mat

Symmetric roots of f(x) c polyroots 0 0.538469310105683 - 0.538469310105683 0.906179845938664 - 0.906179845938664 51 mat

" 5th derivative erf(x)" x f 4 - π sqrt / x 2^- exp * 8 π sqrt / x 2^* x 2^- exp * + : (

root secant => accumulates iterated 'X' ε 1013 -^: x 1 el 0.5 : x 2 el 1 : 13 mat x 3 el x 2 el x 2 el f x 1 el x 2 el - (* x 1 el f x 2 el f - / eval - : i 3 : (iter counter ≡ x i 1 + el x i el x i el f x i 1 - el x i el - (* x i 1 - el f x i el f - / eval - : x i el x i 1 - el - abs ε > x i 1 + el x i el x i el f x i 1 - el x i el - (* x i 1 - el f x i el f - / eval - : i i 1 + : 21 line while X x : 61 line

x f X X rows el 0 + 20 red 1 5 mat 2 1 sys

idx 1 X rows range : iter idx X augment : 21 line

Sanity check X X rows el f 51015 -^*

iterates i 9

<p> Stem Construction Utility </p>

Core component of the stem style P v replicate k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 31 line :

Stem construction vy stem N vy length : i 1 N range q i el 3 : q N el 2 : 21 line for 11 line i 1 N range v i el i : for i 1 N range S 3 i * el vy i el : for u q v replicate : V u S augment : V V 3 V rows 12 submatrix : 71 line :

X stem 0.707 2 1 sys

iter 1 0.5 213 0.757104688859641 4 0.686846588913515 5 0.707868746524041 6 0.707117250350737 7 0.707106775536612 8 0.707106781186593 9 0.707106781186549 92 mat

0.5 sqrt 0.707106781186548