<p> Plot data utility </p>

8af48fd3-20c2-49f6-8994-f2e130b5dd53 63 4 5 decimal

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

T P char size color plotG n T length : plot T 1 el P 1 el char size color augment : i 2 n range plot plot T i el P i el char size color augment stack : for plot 41 line :

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 21 line 41 line :

L 0 :

H 8 :

# # # range

N 100 :

# # range

U L H N xd :

x F x exp :

x F

Assign the 'root' to a specific solution(w) for local use.

w 5 :

init 0 :

sol x F w - 0 ≡ x init roots :

sol 1.6094

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

In mathematics, an inverse function is a function that "reverses" another function. i.e: if 'f' is a function mapping x to y, then the inverse function of 'f' maps y back to x. Solve always tries to give you explicit formulas for the solutions to equations. However, it is a basic mathematical result that, for sufficiently complicated equations, explicit algebraic formulas cannot be given. If you have an algebraic equation in one variable, and the highest power of the variable is at most four, then Mathematica can always give you formulas for the solutions. However, if the highest power is five or more, it may be mathematically impossible to give explicit algebraic formulas for all the solutions. Smath will inverse functions as far as Maple can do. When no explict inversion may be found, you may rescue the project via the iterative solver as exemplified above. There are numerical limit wrt multiple solutions that the iterative solver can manage

a a :

b b :

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 :

x a b q x a b sol 2 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 ----------

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 :

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 1.0407 1014 -^* i * -

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

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

x cnorm x CDF.Normal :

# Error, unable to evaluate sign ans ≡

init 2 :

x Pnorm x cnorm 2 1 sys

x q xx cnorm x - 0 ≡ xx init roots :

x q

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 q xx Cn x - 0 ≡ xx init roots :

x Pnorm x Cn x q 3 1 sys