data 216

25be1940-a872-4dc7-b85e-e82ca925a7f8 240 15 12 decimal

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

3 appVersion 0.98.6179 ≡ range_215 14 min ≡ 21 mat

Iterate the GivenFind solve bloc

Predefine the solver invariants a b GivenFind a b FindRoot :

A 1011 : V 1500 : 12 mat

range 61 :

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 MCD 8 Pro ?

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i 1 range range idx i el i : d i el 2.9 i 10 / + : 21 line for

Acelerate demo: reduce range range [216] solvable range

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 :

init L 2 ≡ Lc 1 ≡ 21 mat :

t0 1 time :

Generalise an iterative collection of  solutions [L, Lc] j Iterate L Lc Find j system init GivenFind : d sol L Lc Find vectorize : 21 line :

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

1 time t0 - min 4.4

Explode/collect the solved system Plot n 1 range range L n el Solved n el 1 el eval : Lc n el Solved n el 2 el eval : 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

solve @ user [j]
disable "Solved" j 31 : User j Iterate : 21 sys

d j el User transpose 12 mat 6 58.57 22.67 12 mat 12 mat

header j d L Lc 14 mat : data header idx Plot augment stack : 21 sys

_L d j el User 11 el + 20 black 15 mat :_Lc d j el User 21 el + 20 black 15 mat : 21 line

data j d L Lc 13 110.07 110.26 2 3.1 108 103.91 3 3.2 105.8 98.93 4 3.3 103.61 94.29 5 3.4 101.45 89.87 6 3.5 99.32 85.64 7 3.6 97.22 81.6 8 3.7 95.16 77.72 9 3.8 93.14 74.01 10 3.9 91.15 70.45 114 89.2 67.04 124 mat

L Lc 3 200 3 50 - 2 2 mat _L _Lc 5 1 sys

Search nearest in col ... export this section

data 216

data j d L Lc 13 110.07 110.26 2 3.1 108 103.91 3 3.2 105.8 98.93 4 3.3 103.61 94.29 5 3.4 101.45 89.87 6 3.5 99.32 85.64 7 3.6 97.22 81.6 8 3.7 95.16 77.72 9 3.8 93.14 74.01 10 3.9 91.15 70.45 114 89.2 67.04 12 4.1 87.3 63.77 13 4.2 85.43 60.64 14 4.3 83.61 57.64 15 4.4 81.83 54.77 16 4.5 80.08 52.03 17 4.6 78.38 49.4 18 4.7 76.72 46.89 19 4.8 75.1 44.49 20 4.9 73.52 42.19 215 71.98 39.99 22 5.1 70.47 37.88 23 5.2 69.01 35.87 24 5.3 67.58 33.95 25 5.4 66.19 32.11 26 5.5 64.83 30.36 27 5.6 63.51 28.68 28 5.7 62.22 27.07 29 5.8 60.97 25.54 30 5.9 59.75 24.07 316 58.57 22.67 32 6.1 57.41 21.33 33 6.2 56.29 20.05 34 6.3 55.19 18.83 35 6.4 54.13 17.66 36 6.5 53.09 16.54 37 6.6 52.08 15.47 38 6.7 51.1 14.45 39 6.8 50.14 13.47 40 6.9 49.21 12.53 417 48.3 11.64 42 7.1 47.42 10.79 43 7.2 46.56 9.97 44 7.3 45.72 9.19 45 7.4 44.9 8.44 46 7.5 44.11 7.73 47 7.6 43.33 7.04 48 7.7 42.58 6.39 49 7.8 41.85 5.76 50 7.9 41.13 5.17 518 40.43 4.59 52 8.1 39.75 4.05 53 8.2 39.09 3.52 54 8.3 38.44 3.02 55 8.4 37.81 2.55 56 8.5 37.2 2.09 57 8.6 36.6 1.65 58 8.7 36.01 1.23 59 8.8 36.51 0.96 60 8.9 34.89 0.45 619 34.34 0.08 62 9.1 33.81 0.27 - 63 9.2 33.29 0.6 - 64 9.3 32.79 0.92 - 65 9.4 32.29 1.23 - 66 9.5 31.81 1.52 - 67 9.6 31.34 1.8 - 68 9.7 30.88 2.07 - 69 9.8 30.43 2.33 - 70 9.9 29.99 2.57 - 7110 29.56 2.8 - 72 10.1 29.14 3.02 - 73 10.2 28.72 3.24 - 74 10.3 28.32 3.44 - 75 10.4 27.93 3.63 - 76 10.5 27.54 3.82 - 77 10.6 27.16 3.99 - 78 10.7 26.79 4.16 - 79 10.8 26.43 4.32 - 80 10.9 26.08 4.47 - 8111 25.73 4.61 - 82 11.1 25.39 4.75 - 83 11.2 25.06 4.88 - 84 11.3 24.73 5 - 85 11.4 24.41 5.12 - 86 11.5 24.1 5.23 - 87 11.6 23.79 5.33 - 88 11.7 23.48 5.43 - 89 11.8 23.19 5.53 - 90 11.9 22.9 5.61 - 9112 22.61 5.7 - 92 12.1 22.33 5.78 - 93 12.2 22.06 5.85 - 94 12.3 21.79 5.92 - 95 12.4 21.52 5.98 - 96 12.5 21.26 6.04 - 97 12.6 21 6.1 - 98 12.7 20.75 6.15 - 99 12.8 20.5 6.2 - 100 12.9 20.26 6.24 - 10113 20.02 6.28 - 102 13.1 19.79 6.32 - 103 13.2 19.55 6.35 - 104 13.3 19.33 6.38 - 105 13.4 19.1 6.4 - 106 13.5 18.88 6.43 - 107 13.6 18.66 6.45 - 108 13.7 18.45 6.46 - 109 13.8 18.24 6.48 - 110 13.9 18.03 6.49 - 11114 17.83 6.5 - 112 14.1 17.63 6.5 - 113 14.2 17.43 6.5 - 114 14.3 17.23 6.5 - 115 14.4 17.04 6.5 - 116 14.5 16.85 6.5 - 117 14.6 16.66 6.49 - 118 14.7 16.47 6.48 - 119 14.8 16.29 6.47 - 120 14.9 16.11 6.45 - 12115 15.93 6.44 - 122 15.1 15.75 6.42 - 123 15.2 15.58 6.4 - 124 15.3 15.41 6.38 - 125 15.4 15.24 6.35 - 126 15.5 15.07 6.32 - 127 15.6 14.91 6.3 - 128 15.7 14.74 6.26 - 129 15.8 14.58 6.23 - 130 15.9 14.42 6.2 - 13116 14.26 6.16 - 132 16.1 14.1 6.12 - 133 16.2 13.95 6.08 - 134 16.3 13.79 6.04 - 135 16.4 13.64 6 - 136 16.5 13.49 5.95 - 137 16.6 13.34 5.9 - 138 16.7 13.19 5.86 - 139 16.8 13.05 5.8 - 140 16.9 12.9 5.75 - 14117 12.76 5.7 - 142 17.1 12.61 5.64 - 143 17.2 12.47 5.59 - 144 17.3 12.33 5.53 - 145 17.4 12.19 5.47 - 146 17.5 12.05 5.4 - 147 17.6 11.91 5.34 - 148 17.7 11.78 5.27 - 149 17.8 11.64 5.21 - 150 17.9 11.51 5.14 - 15118 11.37 5.07 - 152 18.1 11.24 5 - 153 18.2 11.11 4.92 - 154 18.3 10.97 4.85 - 155 18.4 10.84 4.77 - 156 18.5 10.71 4.69 - 157 18.6 10.58 4.61 - 158 18.7 10.45 4.53 - 159 18.8 10.32 4.44 - 160 18.9 10.2 4.36 - 16119 10.07 4.27 - 162 19.1 9.94 4.18 - 163 19.2 9.81 4.09 - 164 19.3 9.69 4 - 165 19.4 9.56 3.9 - 166 19.5 9.43 3.8 - 167 19.6 9.31 3.71 - 168 19.7 9.18 3.6 - 169 19.8 9.06 3.5 - 170 19.9 8.93 3.4 - 17120 8.81 3.29 - 172 20.1 8.68 3.18 - 173 20.2 8.56 3.07 - 174 20.3 8.43 2.96 - 175 20.4 8.3 2.84 - 176 20.5 8.18 2.72 - 177 20.6 8.05 2.6 - 178 20.7 7.92 2.48 - 179 20.8 7.8 2.35 - 180 20.9 7.67 2.22 - 18121 7.54 2.09 - 182 21.1 7.41 1.96 - 183 21.2 7.29 1.82 - 184 21.3 7.16 1.68 - 185 21.4 7.03 1.54 - 186 21.5 6.89 1.39 - 187 21.6 6.76 1.24 - 188 21.7 6.63 1.09 - 189 21.8 6.49 0.93 - 190 21.9 6.36 0.77 - 19122 6.22 0.6 - 192 22.1 6.08 0.43 - 193 22.2 5.94 0.26 - 194 22.3 5.8 0.08 - 195 22.4 5.66 0.11 196 22.5 5.51 0.3 197 22.6 5.36 0.5 198 22.7 5.21 0.7 199 22.8 5.06 0.91 200 22.9 4.9 1.13 20123 4.74 1.36 202 23.1 4.57 1.59 203 23.2 4.4 1.84 204 23.3 4.23 2.1 205 23.4 4.05 2.37 206 23.5 3.86 2.66 207 23.6 3.66 2.96 208 23.7 3.46 3.29 209 23.8 3.24 3.64 210 23.9 3.01 4.01 21124 2.76 4.43 212 24.1 2.49 4.89 213 24.2 2.18 5.42 214 24.3 1.82 6.07 215 24.4 1.35 6.93 216 24.5 0.25 9.06 2174 mat :

Stem(data)

Stem plot for QuickPlot  ... given paired data data Stem stem 0 : stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - (range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - (range V V stem j el stack : for V V data data rows row stack : 81 line :

project data 2 data rows range 14 range el : L project 2 col project 3 col augment : Lc project 2 col project 4 col augment : 31 sys

S data 61194 range 14 range el : neg S 2 col S 4 col augment : 21 sys

data data rows row 216 24.5 0.25 9.06 14 mat

Lc L 3 200 3 50 - 2 2 mat 9 0 + 150 black 1 5 mat neg Stem 22.3 0 + 50 black 1 5 mat 6 1 sys

data data 210 range 14 range el :

Scan down for nearest lower V threshold FindIdx j 1 : V j el abs threshold ≥ j j 1 + : while j 1 - V j el 21 mat 31 line :

data 13 110.07 110.26 2 3.1 108 103.91 3 3.2 105.8 98.93 4 3.3 103.61 94.29 5 3.4 101.45 89.87 6 3.5 99.32 85.64 7 3.6 97.22 81.6 8 3.7 95.16 77.72 9 3.8 93.14 74.01 94 mat

V data 3 col :

idx V 100 FindIdx : 5 99.32 21 mat

data idx 1 el row 5 3.4 101.45 89.87 14 mat

Circle/line intersection

D 4 : (Circle ≡

L 0.5 : (Line ≡

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

init x 1 ≡ y 0.5 ≡ 21 mat :

Collect solutions at kernel level x y sol Φ init FindRoot :

Assign the solutions sol x y sol vectorize : 1.643038 1.143038 - 21 mat

x y 21 mat x y sol vectorize : 1.643038 1.143038 - 21 mat

D x 2 ^ - sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 2 5 mat 3 1 sys

D x 2^- sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 25 mat 31 sys

x y Clear 1

D 4 : (Circle ≡

L 0 : (Line ≡

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

init x 1 ≡ y 0.5 ≡ 21 mat :

Collect solutions at kernel level x y sol Φ init FindRoot :

Assign the solutions sol x y sol vectorize : 1.416072 1.416072 - 21 mat

x y 21 mat x y sol vectorize : 1.416072 1.416072 - 21 mat

D x 2 ^ - sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 2 5 mat 3 1 sys

D x 2^- sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 25 mat 31 sys

jmGiraud: 2015/12/31

Solving for a vectorised function

In Mathematics, an InverseFunction is the solution of this function. f(x) is the model of difficulty, documented in Mathematica 4.0. The seeded solve scanner is of no gain versus the direct "Sol". The 'roots' can't be applied wrt the derivative that does not take discrete vector of values. In lieu, the 'solve' block concludes pretty well.. a bit slow. _____________________________ Note: the function starts existing around 'x = 0.02, f(0.02) = 0.0564' our choice to solve is wise , down to lower possible limit. The upper limit 0.99 is practical. In fact, at 1, the solution is infinite. Pushing a bit more, 'solve' cries to murder.

Design a discrete algo inverser  in the L, H range , N points f L H N Inverse dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x 0.01 40 solve : for ω sol augment 51 line :

[0.01,40] bracket solver to avoid "No real roots"

x f x 2 π * / 2 π * x / tanh * sqrt :

Sol f 0.06 0.99 200 Inverse :

Magnify the function to see

50 x f * Sol 2 1 sys

The native function

x f

algo solve roots ... solves intersection

NOT needed as isolated document x Clear 1

'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

x f x Φ 2 1 sys

Roots solve on the canvas

NOT neede on isolated document x Clear 1

x F x tan x - :

x sol u F x - 0 ≡ u 1 roots :

x Sol u F x - 0 ≡ u 1 - roots :

IntersecX u F u sol - 0 ≡ u 1 1.2 solve : 1.165561

IntersecX F 1.165561

roots solve for quick plot

x F x sol x Sol x IntersecX IntersecX F + 50 black 1 5 mat 5 1 sys

0.5 sol 0.75 - Sol 21 mat 0.975 1.0679 - 21 mat

uu 1.16556118520721 :

uu F uu sol - 51015 -^* -

plotG(vx,vy,a,b,c)

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

Maple code "root" generatrix

x t w Clear 1

This dual solver is most precious for Engineering/Scientific works. 1. given the independent, it solves for the dependent. 2. given the dependent, it solves for the independent. Just a matter of coding the respective root range appropriately. The AntColony model was developped before and we use the result. sol(w):=root[f(t,β,colony)-w,t] recast the model in the Maple symbolic executable format that Smath loves crunching. This format is "general" wrt 't', but it can be recast in term of either solving for 't' or 'w' by equating to 0 and specifying the variable to slove for and its respective bracketed root range.

maple will NOT solve 
any and all model ... x μ colony f colony μ 1 el 11 μ 2 el / 1 - (μ 3 el x μ 4 el - (* - exp * + / + :

y Inverse y x μ colony f ≡ x solve maple : μ 3 el μ 4 el * μ 2 el y - colony + μ 1 el + (* y 1 - μ 2 el + (* colony 1 μ 2 el - (* + / ln - μ 3 el /

a native Maple code, not documented # # root

1. The general coding to be adapted for each solver

w sol t μ colony f w - t root :

w sol colony μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + (* μ 1 el μ 2 el * + w μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + (* - μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + / t root

evaluate symbolic the numerator to enable generalise the "root solve" sol(w).
Doing this transaction will enable to recast the solver for solving for  ...
... 'w' or for 't' ... after passing in the solve bloc "solve" equated to 0
and respecify for either 'w' or 't' within their respective range. Num colony μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + (* μ 1 el μ 2 el * + w μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + (* - :

Num 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + (colony w - (* μ 1 el μ 2 el * +

plot x Inverse 50055002 - 22 mat 1150010115002 - 22 mat 31 sys :

Verhulst fit to 'ant' data μ 11000 0.76 1.35 5.037 41 mat : colony 500 : 12 mat

plot

2. Solving for 'w' [dependent] given 't' independent

superflous => strictly equivalent to f(x,μ,colony) w t Sol 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + (colony w - (* μ 1 el μ 2 el * + 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + / 0 ≡ w 012000 solve :

w 5 Sol 8758

3. Solving for 't' [independent] given 'w' dependent

inverse solver

This is the "inverser" we are interested in ... solving for 'x' given: y:=f(x) w t Sol 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + (colony w - (* μ 1 el μ 2 el * + 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + / 0 ≡ t 010 solve :

8758 t Sol 5

4. At this point, recast the dual solver .

Next, we recast the original model from the symbolic 1. above and make 'w=0', that will plot and interpolates the dependent.

equate 'w' 0

t AntColony 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + (colony 0 - (* μ 1 el μ 2 el * + 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + / :

x c 10 x ≤ (x 10 ≤ (& cases :

ant 15002110032050 3.5 38004540058700610550711200811100911900102 mat :

X ant 1 col : Y ant 2 col : 12 mat data X Y . 12 black plotG : 21 line

user 5000 :

given Y from Y:=f(x) => solve for 'x' user t Sol 3.91

x AntColony x c * user t Sol user + 40 red 1 5 mat data 3 1 sys

0 AntColony 1 AntColony 10 AntColony 31 mat 5396481149631 mat

user user t Sol 21 mat 5000 3.91 21 mat

1 time t0 - min 5.6

t0 1 time :

Sol_x w 55011500650 range : i 1 w rows range sol i el w i el t Sol eval : for w sol augment 31 line :

Ant Sol_x 50045002 - 22 mat 1150010115002 - 22 mat 041 sys :

Ant

1 time t0 - min 1.1

Smath . . . doing the muffed American Scientific

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t0 1 time :

Verhulst fit to 'ant' data μ 11000 0.76 1.35 5.037 41 mat : colony 500 : 12 mat

x ant colony μ 1 el 11 μ 2 el / 1 - (μ 3 el x μ 4 el - (* - exp * + / + :

w t Sol 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + (colony w - (* μ 1 el μ 2 el * + 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + / 0 ≡ t 010 solve :

user 6000 :

in-situ solve Ω user t Sol (: 4.18

x c 1500 x ≤ (x 12000 ≤ (& cases :

x Verhulst 10 x ≤ (x 10 ≤ (& cases :

x t Sol x c * user Ω + 40 red 1 5 mat 2 1 sys x ant x Verhulst * Ω user + 40 red 1 5 mat 2 1 sys

1 time t0 - min 2

Solving each branch

x y t Clear 1

t0 1 time :

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 * :

0 F 0.3989

x sol tt F x - 0 ≡ tt 5 - 0 solve : x Sol tt F x - 0 ≡ tt 05 solve : 21 sys

maple is generous: however,
may NOT solve any/all stuff,
rescue wisely via solve bloc y t F ≡ t solve maple 2 π * ln 2 y ln * + (- sqrt 2 π * ln 2 y ln * + (- sqrt - 21 mat

Scalar inverse on the canvas

x Sol x sol 0.3989 0 + 40 green 1 5 mat 3 1 sys

x m 2 π * ln 2 x ln * + (- sqrt ± :

1 time t0 - min 2.9

x F 1 - 0.4 1 0.4 2 2 mat 2 1 sys

Scalar inverse on the canvas

x m 0.3989 0 + 40 green 1 5 mat 2 1 sys

pairedX x F 0.25 - 0 ≡ x 2 - 2 solve : 0.967 - 0.967 21 mat

x Clear 1

f x init Inv u f x - 0 ≡ u init roots :

x f 12 π * sqrt / x 2^- 2 / exp * : x sol_1 f x 0 f Inv : x sol_2 f x 0 f - Inv : 31 line

0 f 0.398942280401433

x sol_1 x sol_2 0 f 0 + 50 green 1 5 mat 3 1 sys

More example ... solve canvas

α β γ t F α β t * - γ t * / ln :

α 1 :

β 1 :

γ 1 :

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

Bracketed Roots range 1 ± ≡

r1 0 4.5 + 20 red 15 mat :

r2 1 4.5 - + 20 red 15 mat :

Scalar inverse on the canvas

x sol 4.5 - 1 + 20 red 1 5 mat 4.5 0 + 20 red 1 5 mat 3 1 sys

Mother function to inverse

α β γ x F r1 r2 3 1 sys

Mach sub/super sonic

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 :

t0 1 time :

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 3 1 sys

1 time t0 - 70 s *

A very accurate global model has been designed, not given here to de-confused the intent "solve"

Many InverseFunctions are built-in Smath

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

x cnorm x CDF.Normal :

x Pnorm x cnorm 2 1 sys x ICDF.Normal

Range wisely ... "roots"

x Clear 1

f x init Inv u f x - 0 ≡ u init roots :

x f x exp : x sol f x 0 Inv : 21 line

x c 10 x ≤ (x 8 ≤ (& cases :

x f x sol x c * x ln 3 1 sys x abs sol f 2 x * abs sol f 0.5 x * abs sol f x abs ln exp - 2 x * abs ln exp - 0.5 x * abs ln exp - 6 1 sys

very accurate  "roots" 8 sol 8 ln - 11014 -^* - ≡

Implicit solve algebraic

x y z Clear 1

x y z f 5 x * 6 y z + (* - 8 - 3 x y - (* 2 z * + 6 + x y + z + 2 - 31 mat :

sol x y z f x y z 31 mat roots : 1.818182 2.363636 2.181818 - 31 mat

x y z 31 mat sol 1 el sol 2 el sol 3 el 31 mat :

sanity x y z f 0.000000000000031 0.00000000000003 - 0.00000000000002 - 31 mat

x y z Clear 1

Solve implicit NON-algebraic . . .

A difficult task as many sets of solutions are expected.

In this example, we get [3] sets quite easily. No known

rule(s) are suggested ... trial/error.

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Non-algebraic system f 2 x * y + 2 z 2^* + 5 - 0 ≡ y 3^4 z * + 4 - 0 ≡ x y * z + e z^- 0 ≡ 31 mat :

seed init x 1 ≡ y 1 ≡ z 0.1 ≡ 31 mat :

Accuracy ε 1014 -^:

x y z 31 mat f init ε FindRoot : 1.42246939034512 0.975388528087757 0.768008037465102 31 mat

MCD single set

sanity 2 x * y + 2 z 2^* + 5 - y 3^4 z * + 4 - x y * z + e z^- 31 mat : 71015 -^* - 11015 -^* - 11014 -^* - 31 mat