7da3149c-dd90-4981-b2ac-966bc067190d 69 15 12 decimal

&[DATE] &[TIME] - &[FILENAME]

Implicit 'roots' & 'solve'

"implicit" has meaning that if not otherwise specified, their difference is = 0 ... if not so: q(x)-g(x)≡user. Both: 'roots', 'solve' substantiate your project(s).

x f x 2^4 - 2 x * 4 x 2^* - 21 mat :

R x f 1 el x f 2 el - x 1 - roots x f 1 el x f 2 el - x 1 roots 21 mat :

x f 1 el x f 2 el R 1 el R 1 el f 1 el + 15 black R 2 el R 2 el f 1 el + 15 black 2 5 mat 3 1 sys

R 0.716515138991168 - 1.11651513899117 21 mat

roots ... high accuracy

R 1 el f 1 el R 1 el f 2 el - R 2 el f 1 el R 2 el f 2 el - 21 mat 021014 -^* 21 mat

Solving a system

U y 3 ≡ 3 x * y + 5 ≡ 21 mat :

v x y 21 mat :

U v roots 0.6667 321 mat

A 013122 mat :

v 3521 mat :

A 1 -^v * 0.6667 321 mat

=====================================================================================

Reconciliate the solvers:

x q 5 x 3 nthroot - x 3^+ :

x g x 2^:

1. 'roots' solve for crossing fncts the functions are set in vector. 2. 'solve' => for zero(s) and datum(x) of ONE function over total span. Individual zero(s) may be refined. 3. 'solve' => for crossing of 2 fnct may be refined. 4. "Dichotomy" does not solve datum(x) ====== wrt: example q(x), g(x ===== 1. QuickPlot does not take q(x)-g(x) takes eval(,)...X_Y rejects eval(,) 2. D(Φ),a,b,ε) => rejects eval(,) 3. 'solve' => rejects eval(,) 4. 'RootSecant' compatible eval(,)

is not a 'roots' application

sol x q x g - x 1.6 - 1.5 solve :

sol 1.55306859139485 -

sol q sol g - 31014 -^*

sol x q x g - x 1.556 - 1.55 - solve :

sol q sol g - 31014 -^*

sol 1.55306859139485 -

Note that "FndRoots" is a much different animal. It has nothing to do in this document. About it,more can be read in other Smath5346 UNofficial release.

Collapse "RootSecant" in utilities Φ a b ε X typical RootSecant u 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : u i el u i 1 - el - abs ε ≥ u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : i i 1 + : 21 line while u 2 u rows 11 submatrix 61 line :

x q 5 x 3 nthroot - x 3^+ :

x g x 2^:

Recast the project for QuickPlot and suitable for "RootSecant". Note that X_Y does not support ..... f(x) ... eval(,)

x f 5 x 3 nthroot - x 3^+ x 2^- eval :

x q x g x f 5 1.553 - 10 1.553 - 10 - 2 2 mat 5 1 sys

a 1.6 - : b 1.5 - : ε 1014 -^: 13 mat

x Φ x f :

sol Φ a b ε X :

sol 1.55165595373858 - 1.5560899618159 - 1.55306617737349 - 1.55306858727188 - 1.55306859139486 - 1.55306859139484 - 1.55306859139485 - 71 mat

Not compatible eval(,) x f 0 ≡ x 1.6 - 1.5 - solve 1.553069 -

=====================================================================================

Collapse  "Dichotomy"  in  "Utilities" # Φ a b ε D RootDichotomy xn b a + (2 / : n 0 : xn Φ abs ε > xn Φ a Φ * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 51 line :

x f x 2^2 x * - 5 - :

ε 1013 -^:

x datum π :

R x f 1 - 2 - ε D x f 34 ε D 21 mat : 1.44948974278316 - 3.44948974278319 21 mat

Sanity check R 1 el f R 2 el f 21 mat 8.4 1014 -^* - 1.4 1014 -^* 21 mat

"Dichotomy" likes asymmetric initials. This case is natural from visual guesses. The module is fast and does not care much your "good guesses".

r x f 0 ≡ x 2 - 4 solve : 1.44948974278318 - 3.44948974278318 21 mat

x f x datum 2.024 - π . 15 black 4.024 π . 15 black 2 5 mat 3 1 sys

r x f x datum - 0 ≡ x 2.5 - 1.5 - solve x f x datum - 0 ≡ x 35 solve 21 mat : 2.024 - 4.024 21 mat

r 1 el f r 2 el f 21 mat 3.141593 3.141593 21 mat

Newton-Raphson Derivative-Root method

Collapse "Newton-Raphson" in utilities f q init ε Zero Newton-Raphson i 1 : x 1 el init : x 2 el x i el x i el f x i el q / - : x i 1 + el x i el - abs ε > i i 1 + : x i 1 + el x i el x i el f x i el q / - : 21 line while x 61 line :

The 'while loop' automates the prescribed accuracy of the root(s).

x f 4 - π sqrt / x 2^- exp * 8 π sqrt / x 2^* x 2^- exp * + :

x q 24 π sqrt / x * x 2^- exp * 16 π sqrt / x 3^* x 2^- exp * - :

Zero f q 0.5 - 1014 -^Zero :

X Zero Zero rows el : 0.707107 -

u 1 el 0.5 : i 15 range u i 1 + el u i el u i el f u i el q / - : for 21 line

x f x q X 2 X 2 - 2 2 mat 3 1 sys

Zero 0.5 - 0.7 - 0.707072135785007 - 0.707106780337931 - 0.707106781186549 - 0.707106781186545 - 61 mat

u 0.5 0.7 0.707072135785007 0.707106780337931 0.707106781186549 0.707106781186545 61 mat

x D1 2 π / x 2^- exp * : 1rst derivative erf(x) ≡

x deriv 2 π / x 2^- exp * 4 x * x 2^- exp * π / - 4 x 2^- exp * 12 x 2^* - (* π / - 8 x * x 2^- exp * 32 x 2^* - (* π / 41 mat :

The centered XY system x y axes x y / - :

x erf vectorize x deriv 1 el x deriv 2 el x deriv 3 el x deriv 4 el axes 61 sys

x deriv 1 el x deriv 2 el x deriv 3 el x deriv 4 el axes 5 1 sys

A remarkable symbolic expansion x D1 x 1 diff x D1 x 2 diff x D1 x 3 diff 31 mat 4 x * x 2^- exp * π / - 4 x 2^- exp * 12 x 2^* - (* π / - 8 x * x 2^- exp * 32 x 2^* - (* π / 31 mat

x erf vectorize

x f 4 - π sqrt / x 2^- exp * 8 π sqrt / x 2^* x 2^- exp * + 24 π sqrt / x * x 2^- exp * 16 π sqrt / x 3^* x 2^- exp * - (21 mat :

'roots' of two crossing functions

R = abcsissae of the crossing functions

x f 1 el x f 2 el 2 1 sys

R x f 1 el x f 2 el - x 1 - roots x f 1 el x f 2 el - x 0 roots x f 1 el x f 2 el - x 1 roots 31 mat : 1.4279 - 0.1608 - 1.0887 31 mat

x D1 2 π / x 2^- exp * :

x y XY x y / - :

x erf x D1 x D1 x 1 diff x D1 x 2 diff x D1 x 3 diff 5 1 sys erf x abs ( sqrt 2 x * sin 2 ^ XY 3 1 sys

x f 10 x sin * :

x q x :

x f x q 2 1 sys

SmathSolve x f x q - 0 ≡ x 10 - 10 solve :

SmathSolve 8.42320393236049 - 7.06817435809582 - 2.8523418944501 - 0 2.85234189445009 7.06817435809582 8.42320393236049 71 mat

SmathSolve x f x q - 0 ≡ x 8.44 - 8.4 - solve :

SmathSolve 8.42320393236049 -

Collapse  "Dichotomy"  in  "Utilities" # Φ a b ε D RootDichotomy xn b a + (2 / : n 0 : xn Φ abs ε > xn Φ a Φ * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 51 line :

x F x f x q - eval :

ε 1015 -^:

R x F 8.44 - 8.4 - ε D 11 mat : 8.42320393236049 - 11 mat

R R :

Sanity check 8.42320393236049 - f R - 21015 -^* - 11 mat

Collapse "RootSecant" in utilities Φ a b ε X u 0 : (clear the solver ≡ u 1 el 1015 -^: u 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : u i el u i 1 - el - abs ε ≥ u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : i i 1 + : 21 line while u 2 u rows 11 submatrix 71 line :

x q x Gamma :

x g 2 :

x q x g 0.375 5 0.375 5 - 2 2 mat 3 1 sys

a 0.4 : b 0.5 : ε 105 -^: 13 mat

x Φ x Gamma x g - :

solΓ Φ a b ε X : 0.448948 0.448948 21 mat

a 2.875 : b 3.125 : ε 109 -^: 13 mat

solΓ Φ a b ε X : 2.989466 2.989466 21 mat

Mathcad 11 x Γ 2 ≡ x 0.442877396484727 ≡ Smath Γ(x)is ~ ± 0.0001 41 line

x Γ 2 π * x / sqrt 1 e / x 112 x * 110 x * / - / + (* (x^* :