roots(#,#)... NO gain timing, NOT 1/e [-1/e+0.0000000001]

c9090444-46c0-4242-88db-2653aaec7b39 54 4 5 decimal

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

LambertW & ProductLog . . .

W0(x) W1(x) => to resume Smath/Maple symbolic

ProductLog(x) approximation => canvas/discrete

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3 appVersion 0.98.6179 ≡

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

Maple
Maple
approximation # LambertW 1 - # LambertW x W # ProductLog ≡ 31 sys

X 0.25 - :

The product log function gives the solution for W ... in z= W*e^W [Mathematica 3.2.10]

e W 1.000000000000000

Lambert Smath two principal branches

x W0 x W1 1 e / - 1 1 e / - 5 - 2 2 mat X 1 X 3 - 2 2 mat 0.5 - 1 - 0 1 - 2 2 mat 6 1 sys

x ProductLog 15 decimals u x e > x ln x ln ln - x 1 e / + sqrt 1 e / sqrt - if 11 line : w 2 u * : u w - abs u abs / 1012 -^≥ w u : u u 2^x e u -^* + 1 u + / eval : 21 line while u 51 line :

X W0 X W1 21 mat 0.357403 - 2.153292 - 21 mat

10 W 10 ProductLog 21 mat 1.7455280027407 1.7455280027407 21 mat

Except for the very low range => they are both equally accurate, rated 15 decimals

13 / - W 13 / - ProductLog 21 mat 0.619061286735947 - 0.619061286735950 - 21 mat

1 e / - 1010 -^+ W 1 e / - 1010 -^+ ProductLog 21 mat 0.999976683865137 - 0.999976683771404 - 21 mat

Most preferred discretizer N data set L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

u 1 e / - 0.000000001 + e 200 xd : Lambert u u ProductLog vectorize augment : 21 sys

Lambert

ω 1 e / - x ≤ (x e ≤ (* : λ e x ≤ (x 10 ≤ (* : 21 sys

x ProductLog ω / x ProductLog λ / e 1 + 40 black 1 5 mat 3 1 sys

z T z - W - :

x T x - W1 - 1 e / 2 1 e / 1 - 2 2 mat 3 1 sys

The tree generating function for counting distinct oriented trees is related to the ProductLog function T(z):=-W(-z)Mathematica ... for whatever it means ? flip vertical ... flip horizontal. In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree or singly connected network) is a directed acyclic graph whose underlying undirected graph is a tree. ... A polytree is an example of an oriented graph

x W0 x W1 1 e / - 1 1 e / - 2 - 2 2 mat 3 1 sys

image

roots(#,#)... NO gain timing, NOT 1/e [-1/e+0.0000000001]

un-forced root(,,) # # roots

x f w w exp * x - 0 ≡ w roots :

limit of the unforced roots 1 e / - 0.0000000001 + f 0.99998 -

1 e / - 0.0000000001 + 0.367879441 -

x c 11 e / - 0.0000000001 + x ≤ (x e ≤ (& cases :

t0 1 time :

x f x c *

1 time t0 - 2.9 s *

t0 1 time :

x W0 x c *

1 time t0 - 3.1 s *

Suggestive model fit as only resulting data set after fit

x W0 x W1 / x W1 x W0 / x W1 x W0 *

LambertW in discrete mode . . . ε min limit . . . 0.0000000001

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incomplete plot
from collected x Loop x x W1 * x x W0 * 21 sys :

data select ListCollect ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 31 line :

ε 0.0000000001 : U 1 e / - ε + ε - 50 xd : V 1 e / - ε + 1200 xd : 31 sys

L1 i 1 U rows range loop1 i el U i el U i el W1 * : for U U reverse : loop1 loop1 reverse : U loop1 augment 41 line :

L0 i 1 V rows range loop0 i el V i el V i el W0 * : for V loop0 augment 21 line :

Delete duplicate row [51]
Collect/export 'Loop' Lzero L0 2200 range ListCollect : Loop L1 Lzero stack : 21 sys

A most interesting escape loop

x x W1 * x x W0 * 1 e / - 1 e / + 40 black 1 5 mat 3 1 sys

Export the discrete loop

Loop 1 e / - 1 e / + 40 red 1 5 mat 2 1 sys

LambertW Applications:

Smath/Maple companions solve the most

useful Engineering/Scientific applications:

1. solve PDF [iPDF solve 'x' given 'y']

2. solve algebraic expressions,

3. many differential expressions result in

algebraic expression. Thus, the final solution

is expressible in term(s) of LambertW.

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Example 2. Solving a DE

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 : 21 sys

dy 1 x 2^/ C 1 y - y 2^/ (* + ≡

x f 1 x 2^/ :

y g 1 y - y 2^/ :

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

x f x int maple 1 x / -

y g y int maple 1 y ln y * + y / -

1 - x / 1 y ln y * + 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 1.125 DEsol_W0 x 1.125 DEsol_W1 8 1 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 81 sys

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 m yy σ * - LambertW - ln σ * -

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 * - W - 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

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

relevant Lambert calculus

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

x W x diff x W 1 x W + (x * / ≡

x W x int x - x x W / + x W x * + ≡

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

x Inverse x inverse π 10 x * G * 3 1 sys x W 1 x W + ( x * / x - x x W / + x W x * + 1 e / - 10 1 e / - 2 - 2 2 mat 3 1 sys

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

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

Example: solving a DE

L 0 : H 10 : N 200 : 13 mat vu L H N xd : 21 line

t C t diff Vm t C * - Km t C + / ≡

C0 1 :

Vm 2 :

Km 3 :

solution borrowed from Mathematica "ProductLog" DE i 1 N range p i el vu i el Vm Km / * (- C0 C0 Km / exp * ln Km / + exp eval : d i el Km p i el W0 * : 21 line for d 21 line :

DEsol vu DE augment :

Title 1.5 1.85 DE solution in Pharmacology 12 red 15 mat :

DEsol Title 2 1 sys

more algebraic Lambert

collapsed in there

]]> More algebraic [12 sec]

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

more algebraic u 2^e u^- 0 ≡ a b v * + c 10 v -^* ≡ a b v * + c e v -^* ≡ 3 x^x 2^≡ α α exp * 1 e / - ≡ a x^b x * + c + 0 ≡ 2 t^5 t * - 0 ≡ x t^z - 0 ≡ z z b log * a - 0 ≡ 91 sys

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

υ 2^e υ^- 0 ≡ υ solve maple 212 / - LambertW * - 212 / LambertW * - 21 mat

a b v * + c e v -^* ≡ v solve maple c a b / exp * b / LambertW b * - a + b / -

a b v * + c 10 v -^* ≡ v solve maple 10 ln a * b 10 ln c * 10 ln a * b / exp * b / LambertW * - 10 ln b * / -

a xx^b xx * + c + 0 ≡ xx solve maple a ln c a ln * b / - exp * b / LambertW b * c a ln * + a ln b * / -

3 xx^xx 2^≡ xx solve maple 23 ln 2 / - LambertW * 3 ln / - 23 ln 2 / LambertW * 3 ln / - 21 mat

a Clear 1

a a exp * 1 e / - ≡ a solve maple 1 -

2 tt^5 tt * - 0 ≡ tt solve maple 2 ln 5 / - LambertW 2 ln / - 1 - 2 ln 5 / - LambertW 2 ln / - 21 mat

xx tt^z - 0 ≡ tt solve maple z ln xx ln /

xx tt^z - 0 ≡ xx solve maple z ln tt / exp

xx tt^z - 0 ≡ z solve maple xx tt^

z z log * a - 0 ≡ z solve maple a a LambertW /

MCD 11 a 10 ln a 10 ln * W / *

a 3 :

a a W / a 10 ln a 10 ln * W / * 21 mat 2.857390784 4.555535705 21 mat

a b c x Clear 1

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 / 0.947 12.25 + 40 black 1 5 mat 3 1 sys

Most preferred discretizer N data set L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 0 : H e : N 25 : 13 mat vx L H N xd : 21 mat

NOTE ... blue/red intersection is complex. No method was found to solve [blue/red]. Option: transit/isolate discrete via file.

CWD C:\SmathFile CurrentDirectory :

vx vx equilibrium_T vectorize augment CWD Tequilibrium wfile 1

vx vx equilibrium_K vectorize augment CWD Kequilibrium wfile 1

Teq CWD Tequilibrium rfile :

Keq CWD Kequilibrium rfile :

x f1 Teq 1 col Teq 2 col x linterp : x f2 Keq 1 col Keq 2 col x linterp : X x f1 x f2 - 0 ≡ x 0.75 1.5 solve : 31 sys

X X f1 21 mat 0.947 12.25 21 mat

b c K T Clear 1

T ln b K ln - (c / + (T ln b K ln - c / (+ exp * 0 ≡ T solve maple b K ln - c / - exp

T ln b K ln - (c / + (T ln b K ln - c / (+ exp * 0 ≡ K solve maple T ln c * b + exp