6313119e-d916-4076-81f2-00e7cadbcc86 42 9 12 decimal

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Lambert W, principal branch

Lambert W is not a function: it is a "solution function". Don't scratch your head looking for it, you won't find. In other words: it is not a built-in aprroximation. You can fake a built-in like here in Smath. There is the good approximation "ProductLog" from Mathsoft Collaboratory.

Local time calculate: ........2 min........

<= solve ..... Compare Lambert ..... => roots

y W x x exp * y - 0 ≡ x 1 e / - e solve :

x Lambert x 1 e / - ≥ w w exp * x - 0 ≡ w 0 roots NO real roots -> granular if :

myW e :

myW W 1

Lambert W [solve conventional]

x W myW myW W + 20 red 1 5 mat 2 1 sys

Lambert [roots conventional]

x Lambert

2 W 0.85260550201352

Maple classical/direct inverser

ηη

2 Lambert 0.852605502013724

x a b q 1 x b / - exp (a^- :

y x a b q ≡ x solve maple 1 y - ln b * a / -

y a b sol 1 y - ln b * a / - :

a 2 : b 1 : 12 mat

x middle b x * :

x a b q x a b sol x middle 3 1 sys

x α μ D μ x * α x * - exp * :

y x α μ D ≡ x solve maple α y * μ / - LambertW α / -

y x x exp * ≡ x solve maple y LambertW

---------- Visit Lambert ProductLog ----------

Smath "roots" Inverser

L H N xd N+1 pts L...H U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 51 line :

L 0 : H 8 : N 100 : 15 mat

U L H N xd :

x F x exp :

The ierative 'roots' Assign the 'root' to a range variable sol(w) for generalised use. Note that now we have to initialise more wisely.

==>

init 1 - :

<==

w sol x F w - 0 ≡ x init roots :

Note about the scalar plot: Generally it will/might fail WHY ? Whereas the scalar reseeds itself on 'init' this may cause numerical instabilty then drift to failure. Scalar supports no further scalar procesing [parsing] vg: d/dx

Sol i 1 N 1 + range sol i el U i el sol : for sol 21 line :

x Scalar xx F x - 0 ≡ xx init roots :

MySol U Sol augment :

Y 0.834 :

Y exp 2.3025

2.3025 Scalar 0.834

<= on-line calculation

track Y exp Y + 35 black 15 mat :

Title 21 - Smath Inverser 12 black 15 mat :

blue = function, red it's solution

x F MySol track Title 4 1 sys

Scalar plot of 'roots'

x Scalar 0 2 1 sys

Sanity check

Y exp 2.3025

Y exp ln 0.834

Secret maths under the hood

x Pnorm 12 π * sqrt / x 2^- 2 / exp * :

No matter how much you want q(x) the quantile plot, cnorm(x) is a built-in approximation, not otherwise interpretable, i.e: not diffe- rentiable. Mathcad 11 does not suffer this. What's Mathcad 'cnorm' ? Most probably Hart.

x cnorm x CDF.Normal :

# Error, unable to evaluate sign ans ≡

x Pnorm x cnorm 2 1 sys x ICDF.Normal

In lieu, here is a very good approximation of 'cnorm' that will 'root' for the 'qnorm' quantiles. The accuracy of cn(x) < ± 0.000025 in the ± 4 domain smaller outside. Note Smath 'linterp, ainterp, cinterp' are interpolating functions but not scalar functions for subuse [Derivative(s), Integral ...].

x Cn 11 0.07056 x 3^* 1.5976 x * + (- exp + / :

init 0 :

x q xx Cn x - 0 ≡ xx init roots :

x Pnorm x cnorm x q 31 sys

x Pnorm x Cn x q 3 1 sys

Using the Maple code to generate the inverse function for export.

x c d Q 1 c x d^* - exp - :

y c d Inverse y x c d Q ≡ x solve maple : 1 y - ln c / ln d / exp

y c d Inverse 1 y - ln c / ln d / exp

c 1 :

d 0.5 - :

Given 'y' from Q(,,,)=> read X in Q(,,,) that gives 'y' y 0.5 :

X y c d Inverse :

X 2.0814

X abs 2.0814

y c d Q abs 0.7569

ptInv X abs y + 20 magenta 15 mat :

InversePath 0 y X abs y 22 mat X abs y X abs 022 mat 21 sys :

blue is the function, red it's inverse [solution]

x c d Q x c d Inverse ptInv 3 1 sys

Solving for 'x' => given 'y'

InversePath

The task of inversing a function is generally difficult or simply not possible. Many inverse exist as built-in from approximation [asin, acos, atan ...]

x F x exp :

x Inv xx F x - 0 ≡ xx 0 roots :

This one is 'baby stuff'

x F x Inv 2 1 sys

A more difficult example, that you need to instruct wisely wrt the initial(s).

x F 12 π * sqrt / x 2^- 2 / exp * :

The code reads: for F(xx) running over xx find 'x' that makes F(xx)-x=0 wrt to initial ± 2 assign scalar 'x' to a function for recovery in in the canvas scalar plot. It goes this way only at time it plots on the canvas, it does not create a function internally.

x Inverse xx F x - 0 ≡ xx 2 - roots :

x inverse xx F x - 0 ≡ xx 2 roots :

Two complementary branches

x Inverse x inverse 0 3 1 sys x F 0 2 1 sys

Solving each branch

By including the roots range in the solve block, we get independence from roots (range) options.

x F 12 π * sqrt / x 2^- 2 / exp * :

x sol t F x - 0 ≡ t 5 - 0 solve :

x Sol t F x - 0 ≡ t 05 solve :

Scalar inverse on the canvas

x sol x Sol 0.3989 0 + 20 green 1 5 mat 3 1 sys

Mother function to inverse

x F 0.398942280401435 4.97 - 0 . 15 red 1 5 mat 4.97 0 . 15 red 1 5 mat 4 1 sys

0.25 sol 0.25 Sol 21 mat 0.9668 - 0.9668 21 mat

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx i * dx - (+ : for U 41 line :

L 0.000002 : H 0.3989 : N 100 : 13 mat

U L H N xd :

i 1 U rows range invXFR i el t F U i el ≡ t 05 solve : for

INV_XFR U invXFR augment :

Discrete inverse positive branch

INV_XFR 0 4.97 . 15 red 1 5 mat 0.4 0 . 15 red 1 5 mat 3 1 sys

The negative "No real roots", but we don't need. We search for the solving range of the + branch.

0.00000169 Sol 4.9743

0.398942280401435 Sol #

In term of the range of the mother F(x), it solves

x F x 0.00000169 ≥ (x 0.398942280401435 ≤ (& (≡

More example

Maple 11 does not find the analytical "InverseFnct". The "roots method" in example above fails because the Smath 'roots' is not robust enough. For this example, Smath will do in two ways: 1. Solving scalar on the graph canvas [within the real roots domain]. 2. Solving discrete on same range than scalar. Both ways via the Samth 'solve block' [lot more robust than 'roots']. Neither 1, 2 come out clicking the fingers. Observe the scalar construct, here again we include the solve range in the solve block to get independence from the "Options" No sweat, no blod ... does not scratch Smath a bit ! sol(x) optimiz symbolic invXFR optimiz numeric One last observation and not the least: both scalar and discrete solve and plot within the real range of the solution domain, i.e: ~ ± 4.5 ,,, and that's it: done, maybe not exhausted.

α β γ t F α β t * - γ t * / ln :

α 1 :

β 1 :

γ 1 :

Bracketed Roots range 1 ± ≡

x sol α β γ tt F x - 0 ≡ tt 1 - 1 solve :

r1 0 4.5 + 15 red 15 mat :

r2 1 4.5 - + 15 red 15 mat :

Scalar inverse on the canvas

x sol 4.5 - 1 + 15 red 1 5 mat 4.5 0 + 15 red 1 5 mat 3 1 sys

Mother function to inverse

α β γ x F r1 r2 3 1 sys

4.588 - sol 0.9899

4.588 sol 0.0101

4.588 - sol 4.588 sol 21 mat 0.989929 0.010071 21 mat

<= Why symmetric end point solutions ?

Construct a complete discrete inverser ... typical for this example L H N Inverse U 0 : dx H L - N / : i 1 N 1 + range U i el L dx i * dx - (+ : for i 1 U rows range invXFR i el α β γ t F U i el ≡ t 1 - 1 solve : for U invXFR augment 51 line :

INV_XFR 4.588 - 4.588 100 Inverse :

Discrete inverse

INV_XFR

Rework this example with Maple

t α β γ F α β t * - γ t * / ln :

y α β γ Inverse y t α β γ F ≡ t solve maple : α β y exp γ * + /

x α β γ myInv α β x exp γ * + / :

α 1 :

β 1 :

γ 1 :

0 x α β γ myInv 2 1 sys 0 x α β γ F 2 1 sys

Get more > plotted

1012 -^α β γ F 27.63

0.999 α β γ F 6.907 -

Relevant calculus facts

Lambert derivative x D x Lambert 1 x Lambert + (x * / :

Lambert integral x Int x 11 - x Lambert + (x Lambert * + (* x Lambert / :

x D 1 e / - 10 1 e / - 0 2 2 mat 2 1 sys x Int 1 e / - 2 1 e / - 0 2 2 mat 2 1 sys

x F x tan x - :

x sol xx F x - 0 ≡ xx 1 roots :

x Sol xx F x - 0 ≡ xx 1 - roots :

roots solve for quick plot

x F x sol x Sol 1.16556115 1.16556115 o 10 black 1 5 mat 4 1 sys

1012 -^sol 0.000144

x 1.16556115 :

x F x sol - 0

x F 1.165561