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Recapitulate the most common solving techniques

time 16 sec * ≡

Pseudo Solving a vector system

This is not the solution to Quaternion, just something that in some way produces rotation of a 3D solid in the 2D plot from Andrey

α 0 :

β 0.00001 :

q 1 :

r 20 :

p 0.5 - :

Quaternion solve system

a p q r - p 2^q 2^r 2^α p * β r * 033 mat :

b 05031 mat :

v a 1 -^b * :

v 0.5128 - 0 0.0128 31 mat

'v' is the vector solution wrt the initial parameters. For Quaternion, further processing 'v' is required. Visit: Rotate Breather [Quaternion]

q v p stack :

q 0.5128 - 0 0.0128 0.5 - 41 mat

q 0.5128 - 0 0.0128 0.5 - 41 mat

q q q norme / :

<= this is the 'Unit q' you feed in the 'quat' converter.

Solving for intersecting functions

x f x 20 - :

x q 5 :

Error message  'No real roots' sol x f x - 5 ≡ x 2030 solve :

"No real roots" simply means "something coco in there". Smath 'solve', finds the abcsissa(s) of at least two intersecting functions within the bracketed region. Write x=5 as a function q(x):= 5 ,,, bingo ! Maple has more generalised understanding. Smath needs specific guidance, particularly about the bracketing of the region the solution(s) exist.

MapleSolve x f 5 ≡ solve maple :

MapleSolve 25

SmathSolve x f x q - 0 ≡ x 2030 solve :

SmathSolve 25

Sanity check

MapleSolve f SmathSolve f 21 mat 5521 mat

MapleSolve q SmathSolve q 21 mat 5521 mat

sol SmathSolve SmathSolve f + 30 black 15 mat :

plot x f x q sol 31 sys :

plot

Maple code inverser

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 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 1.0407 1014 -^* i * -

X abs 2.0814

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

y c d Q abs 0.7569

ptInv X abs y + 20 magenta 15 mat :

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

Maple code "root" generatrix

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.

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

maple("symbolicExpression") - evaluates symbolic expression.

# # root

a native Maple code, not documented

1. The general coding to be adapted for each solver

w sol t μ colony f w - t root : 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 + / t root

μ 11000 0.76 1.35 5.037 41 mat :

colony 500 :

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

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 011000 solve :

w 5 Sol 8758

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

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, the dual solver is done for use in-situ.

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

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 + / :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx i * dx - (+ : for U 21 line 31 line :

L 0 : H 10 : N 200 : 13 mat

U L H N xd :

Ant i 1 U rows range AntCol i el U i el AntColony : for Ant U AntCol augment : 21 line :

AntColony symbolic model

Ant colony 2 1 sys

Root solve

Convenient Polyroots

x f 0.001 x 2^* 0.00001 x * + 0.005 - :

RootsRange for both [-100, 100] Roots must be within range, otherwise "NO real roots"

Polyroots is often useful, not so robust as it claims from ACM "American Computing Machinery" in the public domain.

v 210 - 0141 mat :

x f x 2^802^- :

210 x * - x 3^+

r x f x 100 - 100 solve :

r 80 - 8021 mat

v polyroots 0.20081 3.057087 3.257897 - 31 mat

2 10 x * - x 3 ^ +

ρ1 r 1 el 0 + 20 red 15 mat :

ρ2 r 2 el 0 + 20 red 15 mat :

x f ρ1 ρ2 3 1 sys

8 i * - 4 - 6 i * + (x * + 32 i * + (x 2^* + x 3^+

Don't get confused: NOT explored .. just "as is" 8 i * - 4 - 6 i * + (32 i * + (141 mat mwabs polyroots 0.803 - 1.832 i * + 0.803 - 1.832 i * - 2 - 31 mat

Solving analytics

β 1031.1298 4.2853 0.2817 15.8683 - 41 mat :

x β f β 1 el 1 β 2 el β 3 el x * - exp + / β 4 el + :

x D x β f x diff :

x D2 x D x diff :

r x D2 0 ≡ x 1218 solve :

r 15.2123

r D 72.62

peak r 100 r 25 - 22 mat :

r β f 499.7

PDF of the XY sampled data.

x D x D2 peak 15.2 72.6 . 15 red 1 5 mat 4 1 sys

Fitted sampled data

0 x ≤ x β f NOT negative if 15.2 500 + 15 red 1 5 mat 2 1 sys

Legendre "roots" ... Smath limit n≤32

357911131517 mat odd roots -> set the range -1..1 ≡

4681012162024324048648096114 mat even roots -> set the range -1..0 ≡

for n > 32, lower roots start losing sanity vs Abramowitz. We get symmetric odd roots up to n = 15 and the plot.

https://pomax.github.io/bezierinfo/legendre-gauss.html

Legendre  more profitable than Rodrigues n z L 1 - (k^2 n * 2 k * - (! (z n 2 k * -^* (* 2 n^k! (* n k - (! (n 2 k * - (! (* (* / (k 0 n 2 / Floor sum :

n 15 :

r n x L x 1 - 1 solve eval :

roots ii 1 r rows range p ii el . : for ii 1 r rows range zero ii el 0 : for r zero p augment 31 line :

r 0.98799 - 0.93727 - 0.84821 - 0.72442 - 0.57097 - 0.39415 - 0.20119 - 0 0.20119 0.39415 0.57097 0.72442 0.84821 0.93727 0.98799 151 mat

roots n x L 2 1 sys

Revisit the Maple code inverser

y Inv y x exp ≡ x solve maple : y ln

y Inverse y x ln ≡ x solve maple : y exp

This one is 'baby stuff'

x exp x Inv 2 1 sys

This one is 'baby stuff'

x ln x Inverse 2 1 sys x Inv exp x Inverse ln 2 1 sys

1. solve => lacks accuracy 2. worst: result depends upon the bracketing

x f x 3^2 x * (2^- 5 + :

a 3 :

b 4 :

r 1 - 1.3819660112501 3.61803398874989 31 mat :

x f 0 ≡ x 1.25 - 3.75 solve 1 - 1.38196601125124 3.61803398696977 31 mat

x Lagrange x r 1 el - (x r 2 el - (* x r 3 el - (* :

x f

'c' in ascending order of x^n

c 504 - 141 mat :

c polyroots 1 - 1.3819660112501 3.61803398874989 31 mat

Improve root accuracy

x f x 3^2 x * (2^- 5 + :

ε 1013 -^:

a 3.5 :

b 3.7 :

xn b a + (2 / :

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

.. sanity ..

xn 3.6180339887499

xn f 5.8 1014 -^*

n 40

Implicit solve

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

x f 1 el x f 2 el - x 1 - 2 solve implicit of ≡ x f 1 el x f 2 el - ≡ 0 ≡

x f 1 el x f 2 el x f 1 el x f 2 el - 3 1 sys

r x f 1 el x f 2 el - x 1 - 1.25 solve : 0.716515138721672 - 1.11651513876761 21 mat

Alternately, more accurate roots

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 :

R 0.716515138991168 - 1.11651513899117 21 mat

solve ... low accuracy

roots ... high accuracy

r 1 el f 1 el r 1 el f 2 el - 2.5 109 -^* -

R 1 el f 1 el R 1 el f 2 el - 0

r 2 el f 1 el r 2 el f 2 el - 2109 -^* -

R 2 el f 1 el R 2 el f 2 el - 1.7 1014 -^*

Solving for either root intersection

Система уравнений Persamaan nonlinear Equations set x f x 2^4 - 2 x * 2 x 2^* - 21 mat :

1 - 121 mat

init 1 - :

<= plug either 'init' <=

X x f 1 el x f 2 el - 0 ≡ x init roots :

X 0.8685 -

Y X f 1 el :

XY X Y + 20 black 15 mat :

XY 0.87 - 3.25 - + 20 black 15 mat

either root of intersection

x f 1 el x f 2 el XY 3 1 sys

From the example above: 'X' is very accurate.

Solving complex cubic

Special application in Laplace modeling

x cubic 204 x * - 6 x 2^* + x 3^+ :

Solving a complex cubic Z 204 - 6141 mat polyroots : 0.491489 1.619428 i * + 0.491489 1.619428 i * - 6.982977 - 31 mat

x cubic 20 0.49 20 + 20 black 1 5 mat 3 1 sys

u 0.491489 1.619428 i * + abs : 1.692368

x cubic 19.6 20 0.49 20 + 100 black 1 5 mat 0.49 19.6 . 15 black 1 5 mat 5 1 sys

x q 204 x * - 6 x 2^* + x 3^+ :

0.491489 q 19.602137

Looking at the complex cubic does not induce understanding. It is the core component in modeling the PID control algorithm. PID = Proportional + Integral + Derivative. There remains ambiguaty: While in the process of modeling a closed control loop, the parametric model ends with only 3 combined parameters. Thus, we use the "Cubic solver" in the form of a1,a2,a3 to accomodate the Laplace suite.

a 204 - 613 mat transpose :

The "Cubic solver" and polyroots have same set of complex roots. In the modeling procedure, we could add the silent x^3 but that makes no sense. So, we confine the modeling wrt the "Cubic solver"... =>

Cubic roots invariants...=>

... => once the cubic roots invariants in hand, they enter in subconstruct of each model P, I, D and combined models P+I, P+D, P+I+D. They enter in Laplace algebra in simple form. The Laplace control modeling was wisely constructed around the "complex cubic invariants". Thanks to Maple & Mathematica, and to Smath genius advanced math tool.

Z 1 el Z 2 el * Z 3 el * abs Z 1 el Z 2 el * Z 1 el Z 3 el * + Z 2 el Z 3 el * + Z 1 el Z 2 el + Z 3 el + (- 31 mat 204 - 631 mat

a 204 - 631 mat

Solving for a bivalued function

Just different but not more difficult

β 0.7819 1.0676 1.2636 4.8488 41 mat :

Model it to unknown Pharmacokinetics sampled data x β f β 2 el β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ / + β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ (2^/ x β 3 el / (β 4 el^* β 4 el * - :

x β f x diff 0 ≡ x 12 solve 1.3775

λ 1.3775 2 1.3775 2 - 22 mat :

asymptote 5 β f : 1.07

Solving for either region

x Dose 1.125 :

X x β f x Dose - x 1 roots : 1.02

Enable/diable to view either  region X x β f x Dose - x 1.5 roots : 2.26

If Dose(x) is below the asymptote, the 2nd root will turn red. This is normal whereas there is no solution in that region below the asymptote ~ 1.07 ... pratice to become fluent at it !

Solving for bivalued function ... two sol => select wisely

x β f 0.78 X x Dose + 20 red 1 5 mat λ 1.07 5 1 sys

Search/solve region

x β f x diff λ 2 1 sys

Unknown in Pharmacokinetics