∫Adaptive(f,a,b,N,ε) [Carlos]

343b27a8-40ac-481d-a92f-d7a569adabef 140 4 5 decimal

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

ABOUT solve productive . . .

This document is a guide line visit solving some projects.

Inclusive of Lambert two branches . . . illustrated.

ISA 1976 is technical . . . from source.

Does not include local landing compansasions . . . kPa, °C.

Lambert is NOT a function . . . rather a function solution.

. . . maple RootOf exemplified/exploited.

. . . undocumented 'root ' exemplified.

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4 appVersion 0.98.6179.21440 ≡

Do NOT ask maple to solve NOT algebraic. maple will pedal in butter, frozen dead. Shut the PC, restart Smath from download...

NOT algebraic 4 x * 19 + x 1.123^-

NOT productive
Only algebraic φ sin β / α sin R / - 0 ≡ α solve maple φ sin β / α φ + sin 1 m + (R * / - 0 ≡ β solve maple 21 mat sinφ R * β / asin sinφ 1 m + (* R * α φ + sin / 21 mat

1. Solve productive wrt 'x' unknown Forum

x f1 φ sin β / α sin R / - x ≡ α solve maple : R sinφ - x β * + (* β / asin -

x f2 φ sin β / α φ + sin 1 m + (R * / - x ≡ β solve maple : sinφ 1 m + (* R * x R * 1 m + (* α φ + sin + /

α 1 : β 15 : R 0.5 : m 0.5 : φ 17.7975 : 15 mat

x plot R φ sin - x β * + (* β / asin - φ sin 1 m + (* R * x R * 1 m + (* α φ + sin + / 21 sys :

x plot 1.2676 - 0.6496 + 20 black 1 5 mat 1.2678 0.7247 - + 20 black 1 5 mat 3 1 sys

Invisible red vertical can't be solved accurately x plot 1 el x plot 2 el - 0 ≡ x 1.3 - 1.3 solve 1.2676 - 0.8011 1.2678 31 mat

X 1.2676 - :

φ sin 1 m + (* R * X R * 1 m + (* α φ + sin + / 0.6496

approx ± ∞ 0.0694 plot 2 el 37038 -

So much from so little ▲ ▼

X 1.2678 :

φ sin 1 m + (* R * X R * 1 m + (* α φ + sin + / 0.7247 -

2. Solve productive Thermodynamic

Maple style LambertW two branches ... x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : x W w w exp * x - 0 ≡ w 1 e / - roots : 31 sys

NO solvers are compatible Lambert Alternately: discretize each branch spline/solve ...

T ln b K ln - (c / + (T ln b K ln - c / (+ exp * x ≡ T solve maple b K ln - c x LambertW * - c / - exp

T ln b K ln - (c / + (T ln b K ln - c / (+ exp * x ≡ K solve maple T ln c * b + c x LambertW * - exp

u 0 x ≤ : b 0.75 : c 1 : K 15 : T 10 : 14 mat x equilibrium_T b K ln - c x W * - c / - exp : x equilibrium_K T ln c * b + c x W * - exp : 41 sys

x equilibrium_T u / x equilibrium_K u / 0.946 12.25 + 40 black 1 5 mat 3 1 sys

x 0.945965 :

x equilibrium_T x equilibrium_K - 0.0000

x equilibrium_K 12.2474

0 equilibrium_K 21.17

x Clear 1

Integrate certified ∫Adaptive

∫Adaptive(f,a,b,N,ε) [Carlos]

f a b N ε ∫Adaptive sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 41 line eval :

x f x equilibrium_K eval :

lower limit
upper limit TOL 106 -^: iter 100 : a TOL : b 6 : 41 sys

Simpson does NOT
recognize Lambert x equilibrium_K x a b int #

TOL 1013 -^≤

f a b iter TOL ∫Adaptive eval 9917

Export the cumulative integration

Σ X a b 0.1 + (a 0.1 + range : j 1 X rows range vy j el f a X j el iter 0.01 ∫Adaptive eval : for X vy 0 round vectorize augment 31 line :

isolate/store Σ Σ eval :

Σ Σ rows row 6991712 mat

evaluate ... copy the export Σ 11 mat

vx Σ 1 col : vy Σ 2 col : 12 mat x Σ vx vy x ainterp : 21 mat

u 3.5 :

track 10204822048 u u Σ 32 mat :

Cumulative integration ...

Σ track u u Σ + 20 black 1 5 mat 3 1 sys

Lambert ... two branches

x W0 x W1 1 e / - 1 - + 50 black 1 5 mat 3 1 sys

u u Σ 12 mat 4713712 mat

relevant Lambert calculus

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a 1 : b 2 : c 0.5 : 13 mat

x W x diff x W 1 x W + (x * / ≡

x W x int x - x x W / + x W x * + ≡

3 equivalent
formulations. f a b c b / a b / exp * W * - (- b / 1 b / a * exp b / c * W 1 b / a * - a b / - c b / a b / exp * W + 31 sys :

Lambert identity x W y W + x y * 1 x W / 1 y W / + (* W ≡

f 1 el f 2 el f 3 el 31 mat 0.1959 - 0.1959 - 0.1959 - 31 mat

x Clear 1

x ∫ x - x x W / + x W x * + :

Exploit ∫(x) profitably 1 e / - 0.000000001 + ∫ 1.1036

x D x W 1 x W + (x * / :

x ∫ x D 1 e / - 10 1 e / - 1 - 2 2 mat 0 1 + 500 black 1 5 mat 4 1 sys data Lambert ProductLog

data 0.368 - 1 - 0.352 - 0.734 - 0.337 - 0.637 - 0.321 - 0.566 - 0.306 - 0.509 - 0.29 - 0.46 - 0.275 - 0.417 - 0.259 - 0.379 - 0.244 - 0.344 - 0.228 - 0.312 - 0.213 - 0.282 - 0.197 - 0.254 - 0.182 - 0.228 - 0.166 - 0.204 - 0.151 - 0.181 - 0.135 - 0.158 - 0.12 - 0.137 - 0.104 - 0.117 - 0.089 - 0.098 - 0.073 - 0.079 - 0.058 - 0.061 - 0.042 - 0.044 - 0.027 - 0.027 - 0.011 - 0.011 - 0.004 0.004 0.02 0.019 0.035 0.034 0.051 0.048 0.066 0.062 0.082 0.076 0.097 0.089 0.113 0.102 0.128 0.114 0.144 0.127 0.159 0.139 0.175 0.15 0.19 0.162 0.206 0.173 0.221 0.184 0.237 0.195 0.252 0.206 0.268 0.216 0.283 0.226 0.299 0.236 0.314 0.246 0.33 0.256 0.346 0.265 0.361 0.274 0.377 0.284 0.392 0.293 0.408 0.301 0.423 0.31 0.439 0.319 0.454 0.327 0.47 0.336 0.485 0.344 0.501 0.352 0.516 0.36 0.532 0.368 0.547 0.376 0.563 0.383 0.578 0.391 0.594 0.399 0.609 0.406 0.625 0.413 0.64 0.42 0.656 0.428 0.671 0.435 0.687 0.442 0.702 0.448 0.718 0.455 0.733 0.462 0.749 0.469 0.764 0.475 0.78 0.482 0.795 0.488 0.811 0.494 0.826 0.501 0.842 0.507 0.857 0.513 0.873 0.519 0.888 0.525 0.904 0.531 0.919 0.537 0.935 0.543 0.95 0.549 0.966 0.555 0.981 0.56 0.997 0.566 1.012 0.572 1.028 0.577 1.043 0.583 1.059 0.588 1.074 0.593 1.09 0.599 1.105 0.604 1.121 0.609 1.136 0.615 1.152 0.62 1.167 0.625 1.183 0.63 1.198 0.635 1.214 0.64 1.229 0.645 1.245 0.65 1.26 0.655 1.276 0.66 1.292 0.665 1.307 0.669 1.323 0.674 1.338 0.679 1.354 0.683 1.369 0.688 1.385 0.693 1.4 0.697 1.416 0.702 1.431 0.706 1.447 0.711 1.462 0.715 1.478 0.72 1.493 0.724 1.509 0.728 1.524 0.733 1.54 0.737 1.555 0.741 1.571 0.745 1.586 0.75 1.602 0.754 1.617 0.758 1.633 0.762 1.648 0.766 1.664 0.77 1.679 0.774 1.695 0.778 1.71 0.782 1.726 0.786 1.741 0.79 1.757 0.794 1.772 0.798 1.788 0.802 1.803 0.806 1.819 0.81 1.834 0.813 1.85 0.817 1.865 0.821 1.881 0.825 1.896 0.828 1.912 0.832 1.927 0.836 1.943 0.839 1.958 0.843 1.974 0.847 1.989 0.85 2.005 0.854 2.02 0.857 2.036 0.861 2.051 0.864 2.067 0.868 2.082 0.871 2.098 0.875 2.113 0.878 2.129 0.882 2.144 0.885 2.16 0.888 2.175 0.892 2.191 0.895 2.207 0.898 2.222 0.902 2.238 0.905 2.253 0.908 2.269 0.912 2.284 0.915 2.3 0.918 2.315 0.921 2.331 0.925 2.346 0.928 2.362 0.931 2.377 0.934 2.393 0.937 2.408 0.94 2.424 0.943 2.439 0.947 2.455 0.95 2.47 0.953 2.486 0.956 2.501 0.959 2.517 0.962 2.532 0.965 2.548 0.968 2.563 0.971 2.579 0.974 2.594 0.977 2.61 0.98 2.625 0.983 2.641 0.986 2.656 0.988 2.672 0.991 2.687 0.994 2.703 0.997 2.718 12002 mat :

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

Σ data Σ :

Finite differences Cumulative Integral

Σ data 1 e / - 0.25 1 e / - 1 - 2 2 mat 3 1 sys

Σ 2 col Min 0.1047 -

For whatever use RHS junction exists. vx data 1 col : vy data 2 col : 12 mat Vx Σ 1 col : Vy Σ 2 col : 12 mat x V vx vy x ainterp : x U Vx Vy x ainterp : 41 mat

sol_1 x V x U - 0 ≡ x 0.125 - 0 solve : sol_2 x V x U - 0 ≡ x 1.75 2 solve : 21 mat 0.091 - 1.843 21 mat

sol_1 U sol_2 U 21 mat 0.1 - 0.815 21 mat

range roots wisely
NormalPDF [μ=0, σ=1] x G 12 π * sqrt / x 2^- 2 / exp * :

x u Clear 1

x Inverse u G x - 0 ≡ u 1 - roots : x inverse u G x - 0 ≡ u 1 roots : 21 sys

sys(Scalar(x),-e,2,1)

x Inverse x inverse π 10 x * G * 1 2 π * sqrt / 2 1 2 π * sqrt / 2 - 2 2 mat 6 1 sys

12 π * sqrt / 0.3989

Example 3. Solving/Inversing a PDF

Maple LambertW two branches ... x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : 21 sys

Gumbel PDF μ σ x Gp 1 σ / x μ - x μ - (- σ / exp σ * + (- σ / exp * :

yy m σ x Gp - x solve maple 12 yy σ * - LambertW - ln * σ * - 2 /

μ 1 : σ 0.625 : 12 mat

solve Gumbel for two X's Ω μ σ x Gp 0.375 ≡ x 02 solve : 0.4873 1.7042 21 mat

Y μ σ 2 Gp : 0.264

X μ Y σ * - W0 - ln σ * - : 2

μ 0.375 σ * - W1 - ln σ * - μ 0.375 σ * - W0 - ln σ * - 21 mat 0.4873 1.7042 21 mat

μ σ Ω Gp vectorize 0.375 0.375 21 mat

μ x σ * - W0 - ln σ * - μ x σ * - W1 - ln σ * - 0.25 - 1 0.75 1 2 2 mat 3 1 sys μ σ x Gp X Y X 0 2 2 mat 2 1 sys

Example 4. Lambert family ... from solve

1 - x / y ln - 1 y / - C + ≡ y solve maple 11 C x * + x / - exp - LambertW / -

x C DEsol_W0 11 C x * + x / - exp - W0 / - : x C DEsol_W1 11 C x * + x / - exp - W1 / - : 21 sys

x 1 DEsol_W0 x 0.875 DEsol_W0 x 0.75 DEsol_W0 x 1 DEsol_W1 x 0.875 DEsol_W1 x 0.75 DEsol_W1 x 1.0625 DEsol_W0 x 1.0625 DEsol_W1 81 sys

x 1 DEsol_W0 x 0.875 DEsol_W0 x 0.75 DEsol_W0 x 1 DEsol_W1 x 0.875 DEsol_W1 x 0.75 DEsol_W1 x 1.125 DEsol_W0 x 1.125 DEsol_W1 8 1 sys x 1.25 DEsol_W0 x 1.25 DEsol_W1 2 1 sys

x 1 DEsol_W0 x 0.875 DEsol_W0 x 0.75 DEsol_W0 31 sys x 1 DEsol_W1 x 0.875 DEsol_W1 x 0.75 DEsol_W1 31 sys 21 sys

x 1 DEsol_W0 x 0.875 DEsol_W0 x 0.75 DEsol_W0 3 1 sys x 1 DEsol_W1 x 0.875 DEsol_W1 x 0.75 DEsol_W1 3 1 sys 2 1 sys

ISA 1976 atmosphere . . . 2 regions . . . m(kPa)

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m R Clear 1

m ISA11 po To λ m * + To / (g - R λ * /^* : m ISA25 p1 g 1 m z1 - (* * R T1 * / - exp * : 21 sys

kPa m ISA11 - m solve maple To 1 - kPa po / ln R * λ * g / - exp + (* λ /

kPa m ISA25 - m solve maple g z1 * - kPa p1 / ln R * T1 * + g / -

ISA 1976 g 9.806645 : R 287.0528 : po 101.325 : p1 22.6321 : z1 11000 : To 288.15 : T1 216.65 : λ 0.0065 - : 81 sys

kPa alt_11 To 1 - kPa po / ln R * λ * g / - exp + (* λ / :

kPa alt_25 g z1 * - kPa p1 / ln R * T1 * + g / - :

x Lo 1 x 101.325 ≤ (22.632 x ≤ (& cases :

x Hi 1 x 22.632 ≤ (2.489 x ≤ (& cases :

Plot x alt_11 x Lo * x alt_25 x Hi * 6.41 19000 + 100 green 15 mat 22.63 11000 + 40 black 15 mat 3222000 Global 7500 12 green 15 mat 3213500 Commercial 12 black 15 mat 61 sys :

altitue(m) vs pressure(kPa) ... canvas

Plot

lowest/highest local barometric pressures recorded

lowest local kPa Typhoon
ISA datum
highest local kPa Mongolia 87 alt_11 101.325 alt_11 108.57 alt_11 31 mat 12670586 - 31 mat

ISA 1976 m(kPa)... SI-Units MSL Mean Sea Level g 9.806645 : (acceleration of gravity m/sec² ≡ R 287.0528 : (realgas constant air m²/°Ksec² ≡ po 101.325 : (datum sea level kPa ≡ p1 22.6321 : (Tropopause kPa 11000 m ≡ z1 11000 : (Tropopause altitude [m] ≡ To 288.15 : (sea level °K ≡ T1 216.65 : (°K @ Tropopause 11000 m ≡ λ 0.0065 - : (temperature laps °C/m ≡ 81 sys

kPa(m) 11000 ISA11 11000 ISA25 25000 ISA25 0 ISA11 19000 ISA25 51 mat 22.632 22.632 2.489 101.325 6.41 51 mat ≡

56.5 15 + (11000 / °C m / (* - 0.0065 °C m / * -

Landing corrections kPa, °C available for interested visitors.

Maple RootOf . . .

for solving reluctant Frechet PDF

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α β x w Clear 1

Fréchet distribution
α > 0 ... β > 0 α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

w α β x Fr - x solve maple w β * α β_Z / (1 α +^* β_Z / (α^- exp * - RootOf

Fr 51 x Fr :

recast RootOf

α 5 : β 1 : Y 0.75 : 13 mat w t ipdf w β * α β t / (1 α +^* β t / (α^- exp * - : w t Sol w t ipdf 0 ≡ t 1 - 10 solve : X Y t Sol : 41 sys

plot Fr 1 - Y 1.5 Y 22 mat X 1 el Y . 14 black 15 mat X 2 el Y . 14 black 15 mat 41 sys :

10

x c 1 0.001 x ≤ (x 1.874278 ≤ (& cases :

t0 1 time :

Fréchet distribution ... α >0 ... β > 0

plot x t Sol 1 el x c * x t Sol 2 el x c * 1.874 0.9642 + 100 black 1 5 mat 3 1 sys

1 time t0 - min 1.7

Dual RootOf solver

X α β X Fr vectorize 21 mat 0.785 1.315 21 mat 0.75 0.75 21 mat 21 mat

0.5 t Sol 0.75 t Sol 1 t Sol 31 mat 0.76 1.427 21 mat 0.785 1.315 21 mat 0.809 1.234 21 mat 31 mat

NOTE ... Canvas Sol(x,t) is wrt [α, β]. For other Frechet [α, β], consider ranging the solve bloc to suit the PDF as well as finding optimum c(x)... using Golden ratio.

They both pedal
solving ... 
on canvas pixels. x t Sol 1 el x t Sol 2 el 21 mat

Max/Min Golden ratio

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 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

x λ 10 x ≤ (x 3 ≤ (& cases :

x f 51 x Fr :

X f 0.75 1.25 y1 y2 ≥ 1012 -^Extrema 1 el :

X X f 12 mat 0.964192 1.874278 12 mat

x f x λ * X X f + 40 red 1 5 mat 2 1 sys

Solve on running canvas . . .

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γ 1.4 : (Air isentropic ≡ k 0.88128485 : (Pitot constant ≡ p static pressure ≡ qc dynamic pressure ≡ x Pitot dynamic ratio qc/P ≡ x qc p / ≡ (supersonic Pitot ratio ≡ 61 line

Observe the canvas solutions:

as 'x' in Supersonic(x) runs on the plot canvas,

the 'roots' solves and plots each canvas pixel result ...

each result on the basis of 96 ppi [pixel per inch].

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Plot limit x c 10 x ≤ (x 5 ≤ (& cases :

x Subsonic 2 γ 1 - / (x 1 + (γ 1 - γ /^1 - (* sqrt 0 x ≤ (x k ≤ (& Supersonic cases : x Supersonic U x 1 + (117 U 2^* / - (2.5^* sqrt / k - 0 ≡ U 1 roots k x ≤ Subsonic cases : 21 line

x Subsonic x Supersonic x c * k 1 + 20 black 1 5 mat 2 0 Pitot Dynamic 14 red 1 5 mat 4 1 sys

Mach 1 :

supersonic

X 1.25 :

X Supersonic Mach 1.14

subsonic

Short revisit Lambert ...

kph 1 :

X Supersonic 1225.044 * kph 1401.605

y 1 υ e υ^* + ≡ υ solve maple y υ υ e υ^* + υ υ e υ^* - / ≡ υ solve maple y 1 υ e υ^* + 1 υ e υ^* - / ≡ υ solve maple 31 mat 1 - y + LambertW 1 - y + 1 y + / ln 1 - y + 1 y + / LambertW 31 mat

X Supersonic 1225.044 * km hr / * 389.335 m s / *

in empty place holder m/s ...  paste kn/hr X Supersonic 1225.044 * km hr / * km hr / 1401.605

km/hr NOT SI ... silent unwanted subprocessed X Supersonic 1225.044 *'km'hr / (* 1401.605'km'hr / *

Do NOT ask maple to solve NOT algebraic. maple will pedal in butter, frozen dead. Shut the PC, restart Smath from download.

ISA 1976 is SI units compatible λ 0.0065 - : (temperature laps °C/m ≡

NOT algebraic 4 x * 19 + x 1.123^-

56.5 15 + (11000 / °C m / (* - 0.0065 °C m / * -

recast the solve bloc . . .

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Circular segment  ... source ChemEng. x C 1 π / 12 x * - acos 12 x * - (4 x * 2 x * (2^- sqrt * - (* :

Recast solve wrt 't',  assign sol(x) x sol t C x - 0 ≡ t 1 - 1 solve :

x C x sol 2 1 sys

Solutions are plotted on the canvas. Wider size the canvas is, longer it takes to plot. Lot of Smath maths are compatible to canvas plot.

'f' dummy algo style f Solve f x solve :

x f x 2^5 x * - 3 + :

x Φ x 3^5 x * - 3 + :

x Φ Solve 2.4909 - 0.6566 1.8342 31 mat

x f Solve 0.6972 4.3028 21 mat

x f x Φ 2 1 sys

algo style solve intersection(s) f Φ x L H Intersection f Φ - 0 ≡ x L H solve :

x f x Φ x 0.5 - 1.5 Intersection 0121 mat

'undocumented 'root dual solve bloc . . .

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In the case maple(solve(,)) results RootOf it may be doctored from "allvalues" ... if not => a job of great attention, not easy. maple(solve(,)) may result in complicated relationship with functions not available in native Smath.

σ x Clear 1

Verhulst Logistic fitted from 
published data American Scientific . x σ ant Φ ant σ 1 el 11 σ 2 el / 1 - (σ 3 el x σ 4 el - (* - exp * + / + :

Commutative ... y≡Φ(x,σ,ant) ... Φ(x,σ,ant)≡y y x σ ant Φ ≡ x solve maple σ 3 el σ 4 el * σ 2 el y - ant + σ 1 el + (* y 1 - σ 2 el + (* ant 1 σ 2 el - (* + / ln - σ 3 el /

Reserved for the inverse plot below x inverse σ 3 el σ 4 el * σ 2 el x - ant + σ 1 el + (* x 1 - σ 2 el + (* ant 1 σ 2 el - (* + / ln - σ 3 el / :

w sol t σ ant Φ w - t root : ant σ 2 el σ 2 el - 1 + (σ 3 el t σ 4 el - (* - exp * + (* σ 1 el σ 2 el * + w σ 2 el σ 2 el - 1 + (σ 3 el t σ 4 el - (* - exp * + (* - σ 2 el σ 2 el - 1 + (σ 3 el t σ 4 el - (* - exp * + / t root

simplified argument 1 σ 2 el - (σ 3 el t σ 4 el - (* - exp * σ 2 el + (ant w - (* σ 1 el σ 2 el * + 1 σ 2 el - (σ 3 el t σ 4 el - (* - exp * σ 2 el + /

Verhulst fit to 'ant' data
from American Scientific.
Install the parameters ... σ 11000 0.76 1.35 5.037 41 mat : ant 0 : 12 mat

Recast the solving  dual argument [w, t] w t argument 1 σ 2 el - (σ 3 el t σ 4 el - (* - exp * σ 2 el + (ant w - (* σ 1 el σ 2 el * + 1 σ 2 el - (σ 3 el t σ 4 el - (* - exp * σ 2 el + / :

Solving for (t), given (w)... Roots(range) t,0,12 w t Sol w t argument 0 ≡ t 012 solve :

Update the companion plot Y 5000 :

Y t Sol 4.0481

Solving for (w), given (t)... Roots(range) w,0,12000 w t sol w t argument 0 ≡ w 012000 solve :

X 5 :

Given variate X  ... read Y w X sol 8258.483

Sanity the system ... illustrate via companion plot

x λ 10 x ≤ (x 10 ≤ (& cases :

x f x σ ant Φ :

X x f Y - 0 ≡ x 012 solve : 4.0481

track 0 Y X Y X 1000 - 32 mat :

showY 0.75 - Y 1000 + Y 14 red 15 mat : showX X 0 X 14 red 15 mat : 21 mat

plot x f x λ * track showY showX 41 sys :

plot

Y X 12 mat 5000 4.048 12 mat

x λ 1 ant x ≤ (x 11000 ant + ≤ (& cases :

U 5000 :

U U inverse Re 12 mat 5000 4.0481 12 mat

The inverse Verhulst will solve for X given Y

x inverse Re x λ * 11000 ant + 16 11000 ant + 4 - 2 2 mat 2 1 sys

Solve intersection of two orthogonal hyperbolas

t0 1 time :

r a b ρ x y Clear 1

r 0.25 - : a 1 : b 1 - : ρ 2 : 14 mat x y π1 a x 2^(* b y 2^(* + r - : x y π2 2 a * x 2^(* 0.5 b * y 2^(* + ρ r * + : 31 sys

"FindRoot" needs 
the two algorithms
in explicit form. eq_1 a x 2^(* b y 2^(* + r - : eq_2 2 a * x 2^(* 0.5 b * y 2^(* + ρ r * + : 21 sys

x y axes x y / - :

guess wisely ... init 0.75 0.75 - 21 mat 0.75 - 0.75 - 21 mat 0.75 - 0.75 21 mat 0.75 0.75 21 mat 14 mat :

roots sol 0 : j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el FindRoot : all j el sol j el transpose stack : ct ct 1 + : 31 line for all 51 line :

roots roots eval : 0.6455 0.8165 - 12 mat 0.6455 - 0.8165 - 12 mat 0.6455 - 0.8165 12 mat 0.6455 0.8165 12 mat 41 mat

◄ roots nested ......... ▲

π1 π2 roots axes 4 1 sys

π1 π2 roots axes 41 sys

1 time t0 - min 0.2

k t h u T1 Clear 1

random Engineering project . . .

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k t p k t * t T t diff * :

h T1 u q h u * T1 u T - (* :

EXPLAIN something in there ... 1. Smath likes being spooled to file. 2. Equate the functions, solve. 3. Invent/define the T(u) ... 4. For a given ordinate/solution of your project, the seeded scanner will return two possible solutions. 5. For the given Δ increment, N is limited 23. Above that limit the scan is above the solution ... thus, "no real roots" error message, ... brown datum in the plot.

CWD C:\SmathFile CurrentDirectory :

k t p CWD MapleDEp wfile 1

h T1 u q CWD MapleDEq wfile 1

k t P CWD MapleDEp rfile :

h T1 u Q CWD MapleDEq rfile :

k t P k t * t T t diff *

h T1 u Q h u * T1 u T - (*

k h t T1 u sol k t P h T1 u Q ≡ t T t diff solve maple :

k h t T1 u sol h u * T1 - u T + (* k t * / -

k h t T1 u f h u * T1 - u T + (* k t * / - :

Invent the silent function u T u sqrt :

k 1 :

h 0.5 :

t π :

T1 1 :

recast the functio/solution x f k h t T1 x T f :

0 f 0

0.1722 0.2249 21 mat

Seeded discrete inverser  in the L, Δ increment , N points f L Δ N Inverse i 1 N range ω i 1 + el L Δ i * + : ω 1 el L : 21 line for ω n 1 ω rows range Sol n el x f ω n el - 0 ≡ x 05 solve : n n 1 + : 21 line for ω Sol augment 41 line :

f 0 0.0001 235 Inverse

SOL f 0 0.001 23 Inverse :

SOL SOL rows row 0.023 0.1334 0.2767 21 mat 12 mat

SOL 16 row 0.015 0.0242 0.5933 21 mat 12 mat

evaluate/export SOL 11 mat

In the function-solution f(x) given some 'Y' values ... find productive 'X' variates.

x f 0.125 - 0.015 1 0.015 2 2 mat 0.0242 0.015 + 40 red 1 5 mat 0.5933 0.015 + 40 red 1 5 mat 0.0625 0.023 0.375 0.023 2 2 mat 0.1334 0.023 + 40 brown 1 5 mat 0.2767 0.023 + 40 brown 1 5 mat 8 1 sys Max/Min Golden ratio

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 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

x λ 10 x ≤ (x 2 ≤ (& cases :

X f 0.125 0.25 y1 y2 ≥ 1012 -^Extrema 1 el :

X X f 12 mat 0.197531 0.023579 12 mat

x f x λ * X X f + 40 red 1 5 mat 2 1 sys

x vx RowIdx ct 1 : i 1 vx rows range x vx i el ≡ idx ct el i : continue if ct ct 1 + : 21 line for idx 31 line :

0.015 SOL 1 col RowIdx 1611 mat

L H N # f n Algo U 0 : dx H L - N 1 - / : i 1 N range U i el L dx - (dx i * + : for j 1 U rows range fu j el U j el : fy j el U j el f eval : 21 line for fu n round vectorize fy n round vectorize augment 51 line :

data 01100 x f 4 Algo :

data data rows row 1012 mat

store/isolate data data eval :

data Σ Finite differences Cumul Integration Y 0 : vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 71 line :

Σ data Σ :

data Σ 2 1 sys

vx Σ 1 col : vy Σ 2 col : 12 mat x spline vx vy x ainterp : 21 mat

Σ Σ rows row 0.5 spline 21 mat 1 0.0151 12 mat 0.0104 21 mat