25be1940-a872-4dc7-b85e-e82ca925a7f8 92 3 5 auto

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

Iterate the GivenFind solve bloc

Inspired from Mathcad 8 Pro. It reproduces the solver 1/1. *****

Predefine the solver invariants

t0 1 time :

a b GivenFind a b FindRoot :

range 100 :

init L 2 ≡ Lc 1 ≡ 21 mat :

A 1011 : V 1500 : 12 mat

i 1 range range idx i el i : d i el 3 i 10 / + : 21 line for

Observe:

The solutions appear in the order of "init".

The negative 'Lc' has probably no meaning,

however: does not invalidate the solver.

The meaning of the system is unknown

from the Mathsoft originator ?

Smath 6179 => 2017_06_01

The open style construction is most EDU.

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Solutions [L, Lc] over variable 'd' wrt  'j'
Very robust, the solver supports long range. j system π d j el * L / Lc L - 2 / (2^d j el 2 / (2^+ sqrt * A L / - π d j el 2 / * - ≡ 23 / π L / * d j el 2 / (2^* Lc L - 2 / (* V L / - π d j el 2 / (2^* - ≡ 21 mat :

Generalise an iterative collection of the solutions [L, Lc] j Iterate L Lc Find j system init GivenFind : d sol L Lc Find : 21 line above => define the indexed formation d sol 31 line :

Collect the solver j 1 range range : Solved j Iterate vectorize : 21 line

Explode/collect the solved system Plot n 1 range range L n el Solved n el 1 el : Lc n el Solved n el 2 el : 21 line for index [1] Unests solutions d L Lc augment 11 mat 1 el 31 line :

Collect the plot solutions Plot Plot : L Plot 1 col Plot 2 col augment : Lc Plot 1 col Plot 3 col augment : 31 line

Extract indexed sol j 30 : User j Iterate : 21 line

1 time t0 - min 1.4

d j el User 12 mat 6 66.495 26.164 21 mat 12 mat

data j d L Lc 14 mat idx d Plot 2 col Plot 3 col augment stack :

_L d j el User 11 el + 15 black 15 mat :_Lc d j el User 21 el + 15 black 15 mat : 21 line

data j d L Lc 1 3.1 200.765 194.659 2 3.2 190.044 179.431 3 3.3 180.197 165.731 4 3.4 171.153 153.329 5 3.5 162.829 142.06 6 3.6 155.151 131.794 7 3.7 148.052 122.418 8 3.8 141.475 113.835 9 3.9 135.368 105.962 104 129.687 98.725 11 4.1 124.391 92.061 124 mat

L Lc _L _Lc 4 1 sys

Circle/line intersection

C 6 : Circle ≡

L 0 : Line ≡

Φ vector equalities Φ C x 2^y 2^+ ≡ L x y + ≡ 21 mat :

init x 1 ≡ y 1 ≡ 21 mat :

Collect solutions at kernel level x y sol Φ init FindRoot : 1.731985 - 1.731974 21 mat

Assign the solutions sol x y sol vectorize : 1.731985 - 1.731974 21 mat

C x 2 ^ - sqrt C x 2 ^ - sqrt - L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 2 5 mat 4 1 sys

C x 2^- sqrt C x 2^- sqrt - L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 25 mat 41 sys

Filling a statistical region [PDF]

t0 1 time :

Prepare the project

Construct the project

xo x f 1150 x xo - (2^* + / :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

sx 0.25 :

xo 0.5 :

N 1000 :

xmin sx :

xmax 1 sx - :

ymin 0 :

ymax 1 :

space_x xmax xmin - (:

PDF x 01 0.0025 range : i 1 x rows range y i el xo x i el f eval : 11 line for x y augment 31 line :

space_y ymax ymin - (:

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y xo x f ≤ (x xmax ≤ (* :

jmGiraud: 2015/12/31

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

a b c spec . 6 #808080 13 mat :

fill Pts a b c spec pts :

Domain

Xmin sx 1 sx 0.25 - 22 mat :

Xmax xmax 1 xmax 0.25 - 22 mat :

Xo xo 1 xo 022 mat :

Explain things::

These definitions [*] output nested elements. To get the value, Un-nest by indexing [1]

Filling a range of a PDF

PDF fill Xmin Xmax Xo 5 1 sys

1 Random space_x * xmin + 0.627817 11 mat

[*]

1 Random space_y * ymin + 0.112828 11 mat

[*]

1 Random space_x * xmin + 1 el 0.517472

1 Random space_y * ymin + 1 el 0.820471

In Mathcad, rnd(,) outputs a number, that can be fractional. In Smath, we make it fractional by multiplying [in this case]. In Smath, Random(,) creates a vector. We want a number, therefore Random(1)to make things simpler.

1 time t0 - 18.9 s *