<p> MCD Given/Maximize/minimize </p>

b8d23853-9e7d-4927-8f0c-59a6fdd1703d 64 4 5 decimal

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

Minimize & Maximize subject to "Lagrangian"

The Lagrangian system
recast from collaboration
Alvaro/Jean ...  date ? a b c 13 mat 25 sqrt 513 mat : Φ x 2^y 2^+ z 2^+ : φ x 2^a 2^/ y 2^b 2^/ + z 2^c 2^/ + 1 - : Ψ x y + z - : L Φ λ φ * + μ Ψ * + : 51 sys

Determine the objective "Lagrangian"

fx L x diff : fy L y diff : fz L z diff : fλ L λ diff : fμ L μ diff : 51 sys x 4 λ + (* 2 μ * + 2 / 2 y * 5 λ + (* 5 μ * + 5 / 2 z * 25 λ + (* 25 μ * - 25 / 554 - x 2^+ (* 4 y 2^* + (* 4 z 2^* + 100 / x y + z - 51 sys

local definition V X 15 range el vectorize 11 line :

◄ Vectors of guesses ► min 21 - 00051 mat :

max 1 1.5 2.5 0051 mat :

The objective function to be solved X F x X 1 el : y X 2 el : z X 3 el : λ X 4 el : μ X 5 el : 15 mat fx fy fz fλ fμ 15 mat transpose 21 line :

Each of Maximize/Minimize 3 first solutions meet Mathcad [8, 11] Given/Maximize/Minimize. However, Mathcad is more robust wrt initials.

Minimize X F V min roots : 1.5738 1.3771 - 0.1967 4.4118 - 0.324 51 mat

Maximize X F V max roots : 1.026 1.539 2.5649 10 - 3.0779 51 mat

subMinimize Minimize 13 range 11 range el :

In short ... The two sets of solved [x, y, z] sanity {x^2}/{a^2}+{y^2}/{b^2}+{z^2}/{c^2}≡1

x y z 31 mat subMinimize : 1.5738 1.3771 - 0.1967 31 mat

x y z 31 mat norme 2.1

subMaximize Maximize 13 range 11 range el :

x 2^a 2^/ y 2^b 2^/ + z 2^c 2^/ + 1

x y z 31 mat subMaximize : 1.026 1.539 2.5649 31 mat

x y z 31 mat norme 3.162

x 2^a 2^/ y 2^b 2^/ + z 2^c 2^/ + 1

x y z Clear 1

<p> MCD Given/Maximize/minimize </p>

3 ways solving a system ...

U y 3 ≡ 3 x * y + 5 ≡ 21 mat :

v x y 21 mat :

U v roots 0.666666666666667 321 mat

01331523 mat rref maple 10 0.6667 01323 mat

A 013122 mat :

v 3521 mat :

A 1 -^v * 0.666666666666667 321 mat

=======================================================

x f x 2^4 - 2 x * 4 x 2^* - 21 mat :

R x f 1 el x f 2 el - x 1 - roots x f 1 el x f 2 el - x 1 roots 21 mat :

x f 1 el x f 2 el R 1 el R 1 el f 1 el + 15 black R 2 el R 2 el f 1 el + 15 black 2 5 mat 3 1 sys

R 0.716515138991168 - 1.11651513899117 21 mat

x f 1 el x f 2 el R 1 el R 1 el f 1 el + 15 black R 2 el R 2 el f 1 el + 15 black 25 mat 31 sys

'roots init' => high accuracy R 1 el f 1 el R 1 el f 2 el - R 2 el f 1 el R 2 el f 2 el - 21 mat 021014 -^* 21 mat

Dichotomy recovers 15 decimals

Global approximate solutions x f 1 el x f 2 el - 0 ≡ x 1 - 1.5 solve 0.716515179692222 - 1.11651505356483 21 mat

More accurate x f 1 el x f 2 el - 0 ≡ x 0.72 - 0.7 - solve 0.716515138290328 -

x f x f 1 el x f 2 el - :

a 0.72 - : b 0.7 - : xn b a + (2 / : 31 line

ε 1015 -^:

Dichotomy n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 0.716515138991168 -

xn f 7.3 1016 -^*

n 76

===================================================

a 1 : b 1.2 : xn b a + (2 / : 31 line

Dichotomy n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 1.11651513899117

xn f 3.6 1016 -^*

n 51

Smath rref solving conventional invertible matrix M rref U M 1 M rows range 1 M rows range el : V M M cols col : sol U 1 -^V * : out U rows identity sol augment : 41 line :

U 1545423945518725426262571955 mat :

v 8749851 mat :

S U 1 -^v * (: 1.9787 1.3394 0.0266 0.1054 0.3273 - 51 mat

'U' is invertible

M 154548239457518725442626925719856 mat :

M rref 10000 1.9787 01000 1.3394 00100 0.0266 00010 0.1054 00001 0.3273 - 56 mat

U S * 8.000000000000000 7.000000000000000 4.000000000000000 9.000000000000000 8.000000000000000 51 mat

x y g x 2^- 3 x * + 2 y 2^* - 34 / y * - 100 + :

x y axes x y / - :

y 5 - :

R x y g x roots eval : 5.9833 -

Given the ordinate 'y' on the contour "roots" solves for unique abscissa 'R'. "solve" will complement the inscribed remaining system. Bracket wisely 'solve' wrt 'y'

sanity solution R y g 11013 -^* -

X x y g 0 ≡ x 810 solve : 8.9833

U X y g 0 ≡ y 46 solve : 4.625

plot g R 10 R 10 - 22 mat 10 - y 12 y 22 mat X 10 X 10 - 22 mat R y R U X U X y R y 52 mat axes 61 sys :

plot

system R y R U X U X y R y 52 mat : 5.983 - 5 - 5.983 - 4.625 8.983 4.625 8.983 5 - 5.983 - 5 - 52 mat

x system 1 col : y system 2 col : 21 sys

Area Polygon wrt vertices coordinates A 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * abs :

Perimeter  Polygon wrt vertices coordinates P x i 1 + el x i el - (2^y i 1 + el y i el - (2^+ sqrt i 1 x rows 1 - sum :

Center of gravity Xo 16 A * / x i el x i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : Yo 16 A * / y i el y i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : 21 mat

A P 21 mat 144.0538 49.1833 21 mat

Xo abs Yo 21 mat 1.5 0.1875 21 mat

x y f x 2^- y 2^- 2 + : x y g x 2^y + 4 - : x y Z x y f x y f x y g ≤ x y g cases : 31 sys

Z g f

x Clear 1

Goat function
courtesy Carlos x Goat 21 x 2^2 / - asin * 2 x 2^* x 2 / asin * + x 2^4 x 2^- (* sqrt + :

"Goat radius" ► Mathematica certifies 9 decimals ...
Mathcad Given/Find ►  1.15872847301812 r x Goat π x 2^* - 0 ≡ x 1.5 roots x Goat π x 2^* - 0 ≡ x 1 roots 21 mat : 1.15872847301812 1.15872847301813 21 mat

function var target roots

Accurate roots(3) searches recursively around the target here, superior to 'solve' that needs bracketing wisely.

Algebraic implicit solve

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

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

sanity x y z f 0.000000000000028 0.000000000000031 - 0.00000000000002 - 31 mat

'FindRoot' solving coefficients for Frobenius ODE

Vector of the initials V C 1 el 1 ≡ C 2 el 1 ≡ C 3 el 1 ≡ C 4 el 1 ≡ C 5 el 1 ≡ C 6 el 1 ≡ C 7 el 1 ≡ C 8 el 1 ≡ C 9 el 1 ≡ 19 mat transpose :

The vector F(C) is collected from the Frobenius DE solver equal power as components ... not offeredf here. Vector F(C) is implicit of the terms [2 ...9] ≡0

C F C 1 el 1 ≡ 3 C 2 el * C 1 el + 10 C 3 el * 6 C 2 el * + 112 / C 1 el * + 15 C 3 el * 21 C 4 el * + 92 / C 2 el * + 36 C 5 el * 28 C 4 el * + 232 / C 3 el * + 55 C 6 el * 532 / C 4 el * + 45 C 5 el * + 78 C 7 el * 66 C 6 el * + 992 / C 5 el * + 91 C 7 el * 105 C 8 el * + 1612 / C 6 el * + 136 C 9 el * 120 C 8 el * + 2392 / C 7 el * + 91 mat :

C C F V FindRoot (: 1 0.3333 - 0.35 - 0.3214 0.1382 - 0.0418 - 0.1231 0.0746 - 0.0423 - 91 mat

C C 6 round vectorize eval :

FindRoot("1:function", "2:condition") - Find root(s) of function(s) "1:function", giving an initial guess "2:condition" for each variable; ... at least 4 decimal places precision.

sanity Mathcad 11 C transpose 13333331000000 / - 720 / - 3214291000000 / 69097500000 / - 20901500000 / - 1230711000000 / 37307500000 / - 66115625 / - 19 mat

Solving 'd,h' of the tomatoes can

data surface area A volume V 21 mat :

A 10 m 2^* :

V 2 m 3^* :

Bekannte Werte definieren

Objective values

f A 2 π 4 / d 2^* (* π d * h * + ≡ V π 4 / d 2^* h * ≡ 21 mat :

Gleichungssystem

Equations of the system Area vs Volume

(embedded assignment) f d 2 m * ≡ h 1 m * ≡ 21 mat FindRoot 1.931 m * 0.6829 m * 21 mat

Startwerte mit passenden Maßeinheiten vorgeben und Gleichungssystem auflösen.

Guess wisely with appropriate units and solve for the unknowns [d, h].

solved variables d h 21 mat 1.931 m * 0.6829 m * 21 mat

Sanity check of the "Given/Find Cylinder" data 2 π 4 / d 2^* (* π d * h * + π 4 / d 2^* h * 21 mat 12 mat surface area A volume V 21 mat 10 m 2^* 2 m 3^* 21 mat 12 mat

x f 2 x * sin :

x g 0.75 x * :

x f x g 2 1 sys

x y z Clear 1

X x f x g - 0 ≡ x 2 - 2 solve : 1.09137374321273 - 0 1.0913722737593 31 mat

x f x g - 0 ≡ x 1 roots 1.09137289803883

Collect/solve inifinite length system

x y z Clear 1

V list Collect collect list ct 1 : i list a ct el V i 1 el : ct ct 1 + : 21 line for a 41 line :

LGS x 2 y * - z + 4 - 2 x * - 4 y * + z - 2 + y z - 3 - y - z - e - y - z - 2 sqrt - 51 sys :

collect a triplet list 12513 mat :

assign symbolic x y z LGS LGS list Collect :

Implicit solve x y z 31 mat x y z LGS x y z 31 mat roots : 16.828427 - 7.414214 - 631 mat

sanity x y z LGS 0031016 -^* - 31 mat

Multi-solve algo style ...

collapse in utilities ... @ top of document m b collect i 1 b cols range collect i el m b i el solve : for collect transpose 21 line :

Plug algo in utilities,
collapsed top document. U V solve x y z 31 mat U 1 -^V * : 11 line 11 sys :

u 12 - 12 - 41 - 011 - 33 mat :

v 4 - 23 - 31 mat :

solutions are in ...
lexicographic order. x y z 31 mat u v solve : 16 - 9 - 6 - 31 mat

m 12 - 12 - 41 - 011 - 33 mat :

b 4 - 23 - 31 mat 4.56789 - 2.987654321 2 sqrt - 31 mat 4.56789 2.987654321 e sqrt - 31 mat 13 mat :

The solved triplet sol m b collect (: 16 - 9 - 6 - 31 mat 13.5444 - 7.5623 - 6.1481 - 31 mat 13.3939 10.4747 12.1234 31 mat 13 mat

j sanity m sol 1 j el * :

1 sanity 2 sanity 3 sanity 13 mat 4 - 23 - 31 mat 4.56789 - 2.987654321 1.414213562 - 31 mat 4.56789 2.987654321 1.648721271 - 31 mat 13 mat

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clear the compactum for re-use U U Clear : V V Clear : 12 mat [w,x,y,z] solve compactum ........ U 107 8.1 7.2 7.08 5.04 658 5.98 9.89 9 6.99 4.99 9 9.98 44 mat : V 3223333141 mat : wxyz U 1 -^V * : 15 mat 41 line

[w,x,y,z] solutions and sanity wxyz U wxyz * 12 mat 81 - 13734 - 2241 mat 3223333141 mat 12 mat

Use the compactum wisely, rename the variables [w,x,y,z] solve compactum ........ u 107 8.1 7.201 7.08 5.04 658 5.98 9.89 9 6.99 4.99 9 9.98 44 mat : v 3223333141 mat : sol u 1 -^v * : 15 mat 21 line

[w,x,y,z] solutions sol 121.168757 - 203.704132 51.24021 - 32.329448 41 mat

So much from so little !

[w,x,y,z]  solutions/sanity sol u sol * 12 mat 121.168757 - 203.704132 51.24021 - 32.329448 41 mat 3223333141 mat 12 mat

M => tridiagonal matrix of the "linear cubic spline"
'b' => the vector of derivatives
s => vector coefficients of the linear cubic spline M 100000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00000166 mat :

b 0 0.6782 1.44 - 0.74 0.052 - 061 mat :

s M 1 -^b * : 0 2.4192 3.2851 - 2.0809 0.5983 - 061 mat

[x,y,z] solve compactum ........ A 03534 - 050633 mat : b 1.20736 2.34066 - 0.329193 - 31 mat : A b algo A 1 -^b * : 15 mat 21 line

u A 1 -^b * : 0.142856086956522 0.69230706521739 0.173912239130435 - 31 mat

A b algo 0.142856086956522 0.692307065217392 0.173912239130435 - 31 mat

x A u * 1 el : y A u * 2 el : z A u * 3 el : 31 mat 1.20736 2.34066 - 0.329193 - 31 mat

A u * b - 41015 -^* - 71015 -^* 21015 -^* 31 mat

Solve implicit NON-algebraic . . .

A difficult task as many sets of solutions may result

depending upon the initial. The rule is trial/error

checking sanity wrt seed . . . include ε in the bloc.

]]>

x y z Clear 1

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

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" roots" does not support inequalities

x f 10 x ≤ (x 1.2 ≤ (& 2 1.2 x ≤ (x 2.4 ≤ (& 3 cases :

x Line 1.0625 x * :

x λ 10 x ≤ (x 4 ≤ (& cases :

"roots" does not support inequalities x f x Line - x 03 solve 0.9412 1.8824 1.8824 2.8235 2.8235 51 mat

x f x λ * x Line 2 1 sys

Not sanity checked. x f x 04 int 8.3733