<content> Import Integral Curve </content>

08eb20a4-178c-45ac-b3d5-524c03ee55e5 21 4 5 false false 0 0 decimal

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

Hyper fast/accurate integration ...

x 0.5 1 -^x - exp * x int maple π sqrt x sqrt erf *

a 1015 -^:

recast for vectorize π sqrt x sqrt erf vectorize * a e at 1.7375020114801

in-situ command line 1 x / x int maple a 2 at 35.2319235754706

avoid gross mistake 1 x / x a 2 int 3.3333 1010^*

avoid gross mistake x 0.5 1 -^x - exp * x a e int 1434.3831

<content> Import Integral Curve </content>

x ∫ x 5083589401604030000000000000002566556759757630 - 2566556759757640 x 2^* + (* 366593321287399000000000000000 x * - (* 3914196222755740000000000000000000000000000000 / :

r 0.5774 - 0.5774 21 mat :

x ∫ r 1 el r 2 el at 0.256600116783998 -

algo style ... command line

# f a b ∫ x Ω x f x int maple : x Ω a b at 21 line :

x f 1 x / :_a 1015 -^:_b 2 : 15 mat

x f a b ∫ 35.2319235754706

x f x sqrt exp :

x f x int maple 2 x sqrt exp * 1 - x sqrt + (*

x f x 0 0.5 int 0.811957740

x f 0 0.5 ∫ 0.811957749802971

x f x ln :

x f x int maple x 1 - x ln + (*

x q x 1 - x ln + (* :

e q 0

x f x q x 2 ^ 3 - 2 x ln * + ( * 4 / e 2 e 2 - 2 2 mat 4 1 sys

NFG x f x a 1 int 1.0004 -

Kaput x f x a e int 0.001 -

x f x 0 e int e 2^4 / - ≡ (1.84726402473266 - ≡

x f x ln :_a 1015 -^:_b 2 : 15 mat

x f a 1 ∫ 0.999999999999964 -

x f a e ∫ 41014 -^*

x f x q :_a 1015 -^:_b 2 : 15 mat

x f a e ∫ 1.84726402473266 -

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

x b 0 x 2 - ≤ 16 / 2 x + (3^* 2 - x ≤ (x 1 - ≤ (& 16 / 2 x + (3^* 23 / 1 x + (3^* - 1 - x ≤ (x 0 ≤ (& 23 / 1 x - (3^* - 16 / 2 x - (3^* + 0 x ≤ (x 1 ≤ (& 16 / 2 x - (3^* 1 x ≤ (x 2 ≤ (& 0 cases :

exact by design of b_spline. x b x 2 - 2 int 1.00000000000000

x λ 12 - x ≤ (x 2 ≤ (& cases :

x b 2 - x ≤ ( x 2 ≤ ( & /

0 b 23 /

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

x f x 4^2 x 3^* - 7 x 2^* - 2 - :

x m x f x int maple : x 354 x 3^+ (* - 2 x 4^* + (* 70 x 2^* - (* 30 /

on failure , increase TOL TOL 1013 -^:

rootsDichotomy local command line style x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 41 line :

r 2.5 - 1.5 - TOL D 3.5 4 TOL D 21 mat : 1.922518473317620 - 3.852153430325420 21 mat

plot x f 0.5 x m * r 1 el 0.14 black 15 mat r 2 el 0.14 black 15 mat 41 sys :

plot

x f r 1 el r 2 el ∫ 89.876889346430900 -

x f x r 1 el r 2 el int 89.876889346431200 -

x m x 0 roots 0.000000000000000

Golden-Ratio search [min/max]

<content> Golden ratio min/max </content>

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / 71 line :

a b + 2 / iter 21 mat

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

x f x 4^2 x 3^* - 7 x 2^* - 2 - eval :

ε 1015 -^:

By setting f(x):=eval(,)by default, it will cope with all "_not defined"
As well, you may have to use local definition.
Some functions will ONLY accept higher 'ε' ... 10^{-14} ... 10^{-13}

rootsDichotomy local command line style x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 41 line :

X f 1.5 - 1 - y1 y2 ≤ ε Extrema f 0.5 - 0.5 y1 y2 ≥ ε Extrema f 2.5 3 y1 y2 ≤ ε Extrema 31 mat :

x y axes x y / - :

on failure , increase TOL TOL 1013 -^:

r 2.5 - 1.5 - TOL D 3.5 4 TOL D 21 mat : 1.922518473318 - 3.852153430325 21 mat

plot x f X 1 el X 1 el f . 5 red X 2 el X 2 el f . 5 red X 3 el X 3 el f . 5 red 35 mat r 1 el 0.5 black 15 mat r 2 el 0.5 black 15 mat axes 51 sys :

r f vectorize 51014 -^* - 91014 -^* 21 mat

X 1.26556446000827 - 2.81605475000001 108 -^* - 2.76556437660309 31 mat

plot

X f vectorize 6.5923 - 2 - 39.3452 - 31 mat

Stand alone ... algo command line

# f a b ∫ x Ω x f x int maple : x Ω a b at 21 line :

WolframAlpha style # # # at

x f x sqrt exp 1 - :_a 0 :_b 2 : 15 mat

15 D accuracy x f a b ∫ 3.407528184656320

xx a b 0.01 range :

xx Q xx f a xx ∫ :

Quat xx xx Q vectorize augment :

∫cumulative

x sp Quat 1 col Quat 2 col x cinterp :

jnct x f x sp - x 1.5 2 solve : 1.8082

jnct jnct f 12 mat 1.8082 2.837 12 mat

plot x f Quat jnct jnct f + 25 black 15 mat 31 sys :

plot

x f x 0 b int 3.4075281

jmG: 20230825