LinSolve

ef6749ae-48a2-4750-b95c-6abe232af204 60 4 5 decimal

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

Implicit roots ...

guess 0021 mat 01 - 21 mat 21 mat :

x# F x# 1 el (2^x# 2 el (2^+ 5 - 5 x# 1 el * 2 x# 2 el * - 1 - 21 mat :

System of implicit fncts x y f1 x 2^y 2^+ 5 - : x y f2 5 x * 2 y * - 1 - : 21 line

x# x# 1 el x# 2 el 21 mat :

ImplicitRoots x# F x# guess 1 el roots x# F x# guess 2 el roots 21 mat :

f1 f2 2 1 sys

ImplicitRoots 1221 mat 0.6552 - 2.1379 - 21 mat 21 mat

x Clear 1

The advanced undocumented root solver

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

x μ colony f w - x solve maple μ 3 el μ 4 el * μ 2 el colony w - μ 1 el + (* colony 1 - μ 2 el + (* w 1 μ 2 el - (* + / ln - μ 3 el /

plot inverse function w q μ 3 el μ 4 el * μ 2 el colony w - μ 1 el + (* colony 1 - μ 2 el + (* w 1 μ 2 el - (* + / ln - μ 3 el / :

Maple code, active in native Smath but not documented. # # root

w sol t μ colony f w - t root : 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

simplifiy/recast
for the solve bloc 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 + /

colony 500 : μ 11000 0.76 1.35 5.037 14 mat transpose : 21 mat

solving for 't' the Cartesian X from given 'w' the Cartesian ordinate Y 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 :

Y 8000 :

given the ordinate Y

Project Notes ... 1. The original data set is not given Verhulst Logistic is so perfect. 2. In short: given f(x,μ,colony), "root" dresses the solver for solving the Cartesian X from given ordinate Y

solve variate X Y t Sol (: 4.7

track 0 Y X Y X 032 mat :

10 μ colony f 11496

x μ colony f track 2 1 sys

Inverse Verhulst from maple ...

x q 11500 10 11500 2 - 2 2 mat colony 4 colony 2 - 2 2 mat 3 1 sys

AlgLib Solver for a system of nonlinear equations(Viacheslav) result k 1 x f rows range u k el x k el : for x Jac u f u Jacobian : StepMax 0 : Eps 106 -^: u X0 StepMax Eps x f x Jac al_nleqsolve : 51 line :

Approximation by random numbers X0 u 1 el 0.5 - : i 2 x f rows range u i el u i 1 - el 31 Random * 1.5 - (1 el 12 round + : for u 31 line :

Demo ... System of 3 equations

x f x 1 el 2^x 2 el 2^+ 5 x 1 el x 2 el + (* sin (2^- 4 x 1 el x 2 el - (* cos (2^- 021 mat :

X0 0.5 - 0.0014 21 mat

result 0.3163 - 1.8383 - 21 mat

result f 0.000000000007 - 0.000000000000 21 mat

x y f x (2^y (2^+ 5 x y + (* sin (2^- 4 x y - (* cos (2^- :

x y axes x y / - :

f result transpose axes 3 1 sys

LM ... solving Tesla

m 0.001 :

vector of guesses X0 5001007008002116 mat transpose :

ε 109 -^:

t0 1 time :

Ua b s ia 2 s 3 el / 1 s 2 el / Ua * 1 s 2 el 1 s 1 el / s 6 el b + s 4 el Ua 2^+ sqrt / + (* exp + ln * (s 5 el^* :

s Φ 400 s ia 1.044 m * - 1200 s ia 2.679 m * - 2100 s ia 4.749 m * - 20 0.5 - s ia 0.18 m * - 120 0.5 - s ia 1.66 m * - 240 0.5 - s ia 4.3 m * - 61 mat :

λ s Φ X0 ε LevenbergMarquardt :

λ 501.398 140.017 704.545 800.189 1.282 0.388 61 mat

λ 501.398 140.017 704.545 800.189 1.282 0.388 61 mat :

1 time t0 - min 2.5

Solver function x λ L x 0 λ ia x 0.5 - λ ia x 1 - λ ia x 1.5 - λ ia x 2 - λ ia x 2.5 - λ ia x 3 - λ ia x 3.5 - λ ia x 4 - λ ia 91 mat :

Tesla Triode current [A] for the 9 grid bias ... [0 ... 4 V]

x λ L 1 el x λ L 2 el x λ L 3 el x λ L 4 el x λ L 5 el x λ L 6 el x λ L 7 el x λ L 8 el x λ L 9 el 9 1 sys

PseudoInverse Example 2 ...

The Risteski [UO] reaction matrix A 7000001 - 0018240010 - 6 - 001824008 - 6 - 01 - 011100 3.99 - 0001100001 - 00202300 2.97 - 0000012 - 00000011424 - 01 - 00000710 - 002 - 99 mat :

Program PseudoInverse  Mass balance
Courtesy: Viktor Korobov [Mathcad 11] x r A rank : I A rows identity : a 1 el A tr - : B 1 el A a 1 el I * + : n 2 r range B n el A B n 1 - el * A B n 1 - el * tr - n / I * + : a n el A B n 1 - el * tr - n / : n n 1 + : 31 line for Ω A B r 1 - el (2^* a r el (2^/ : i 1 A rows range u i el 1 : for I Ω A * - (u * 81 line :

Coeffs 209279 x * :

Fractional mass ▼ balance

Integer mass ▼ balance

Components

x 0.189 2.431 1.378 1.476 1.167 0.583 1.325 3.809 1.167 91 mat

Yb C N Cs Ru F H Pt O 91 mat

Coeffs 3960050873528846530882824418612209327720079720024418691 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 1

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 66.5 26.16 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 3.1 200.77 194.67 3 3.2 190.05 179.43 4 3.3 180.2 165.73 5 3.4 171.15 153.33 6 3.5 162.83 142.06 7 3.6 155.15 131.79 8 3.7 148.05 122.42 9 3.8 141.47 113.84 10 3.9 135.37 105.96 114 mat

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

1. Solving "polyroots"

x f 42 - 27 x * - 13 x 2^* + 5 x 3^* - x 4^+ :

Undocumented x expr x f op maple :

x 1 :

coeffs x expr :

coeffs transpose 42 - 27 - 135 - 115 mat

r coeffs polyroots : 0.942929 - 0.940248 3.17495 i * + 0.940248 3.17495 i * - 4.062433 41 mat

x f

2. Solving an nth order linear system

A 51 - 51 - 7 - 923 - 333 mat :

b 7 - 61931 mat :

1. matrix must be invertible |det|≠0 2. even so: may result "no real roots"

x y z f A x y z 31 mat * b - :

xyz x y z f x y z 31 mat roots : 4.042553 11.351064 - 7.712766 - 31 mat

x y z a b c Clear 1

Utilities module adapted from Davide Carpi ... LGS system. LGS a b c solve Pre-defined 3 x 3 LGS system A 33 matrix : b 31 matrix : i 13 range x y z I LGS i el : b i el 000 I : j 13 range A i j el j 1 ≡ j 2 ≡ j 3 ≡ I b i el - : for 31 line for 31 line A b 12 mat 31 line :

LGS x 2 y * - z + a + 2 x * - 4 y * + z - b + y z - c + 31 sys :

a 5 - : b 6 : c 9 - : 13 mat sol LGS a b c solve : 21 sys

... indexed to explode the system A sol 1 col 1 el : b sol 2 col 1 el : 12 mat

A 12 - 12 - 41 - 011 - 33 mat

b 5 - 69 - 31 mat

solved system x y z 31 mat A 1 -^b * : 27 - 13 - 4 - 31 mat

sanity check x 2 y * - z + 2 x * - 4 y * + z - y z - 31 mat 5 - 69 - 31 mat

The matrix solver companion

A 51 - 211 - 7 - 3 - 25930103444 mat :

b 7 - 619241 mat :

51 - 511 - 7 - 9023 - 33120444 mat

21967 - 41 mat

w x y z f A w x y z 41 mat * b - :

wxyz w x y z f w x y z 41 mat roots : 1.2855 3.9673 7.7109 - 5.9618 41 mat

sanity check A wxyz * 7 - 619241 mat

Associated square matrix utilities

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

M c DelColRow N 0 : ct 1 : n 1 M cols range n c ≠ N 1 ct el M n col : ct ct 1 + : 21 line Unest 'N' if for v N 1 el : j 2 N cols range v v N j el augment : for v 61 line :

N M 3 DelColRow (: 05458012552080125180204512518024545801802453208012524532040512518032040550065 mat

N transpose 2 DelColRow transpose 054580125204512518024545801802453208012524532040512518032040550055 mat

A B r c replace U A : i 1 B rows range j 1 B cols range U r i + c j + el B i j el : for for U 31 line :

Mat Frame i 1 Mat rows 2 + range j 1 Mat cols 2 + range frame i j el : for for frame 21 line :

Frame M Frame :

FRAME Frame M 11 replace :

FRAME 05204580125 5204580125180 204580125180245 4580125180245320 80125180245320405 125180245320405500 88 mat

M 123424683691248121644 mat :

Insert col/row @ location

M B c InsertCol a M 1 M rows 1 c submatrix : c M 1 M rows c 1 + M cols submatrix : a B c augment 31 line :

row 15 mat :

col 41 mat :

M B r InsertRow a M 1 r 1 M cols submatrix : c M r 1 + M rows 1 M cols submatrix : a B c stack 31 line :

N M col 3 InsertCol : 123 4246 8369 124812 1645 mat

N row 3 InsertRow 123 4246 8369 12 4812 1655 mat

Frame M Frame :

FRAME Frame M 11 replace :

FRAME 1234 2468 36912 481216 66 mat

M 1232 sqrt 42463 sqrt 83695 sqrt 129637 sqrt 154812 π 1655 mat :

b 7 - 6192151 mat :

FRAME Frame M 22 replace :

FRAME 0 0 123 1.4142 4 246 1.7321 8 369 2.2361 12 963 2.6458 15004812 3.1416 1677 mat

M 1232424638369512963415481231655 mat :

b 7 - 61925 - 51 mat :

Determinant  '0'...
'M' NON invertible M det 0

M 0232 sqrt 42063 sqrt 83605 sqrt 129630154812 π 055 mat :

b 7 - 6192151 mat :

Determinant M det 3907.7814

u w x y z f M u w x y z 51 mat * b - :

sol u w x y z f u w x y z 51 mat roots : 11.6247 6.3572 - 6.4013 - 26.157 3.0183 - 51 mat

M 023456206810123601215184812020245101520030612182430066 mat :

b 12345661 mat :

t u w x y z f M t u w x y z 61 mat * b - :

t
u
v
w
x
y
z sol t u w x y z f t u w x y z 61 mat roots : 0.2 0.1 0.0667 0.05 0.04 0.0333 61 mat

M 0.1 545801252045125180245458002453208012524532040512518032040550055 mat :

b 7 - 61925 - 51 mat :

Determinant M det 3.24 105^*

u w x y z f M u w x y z 51 mat * b - :

NO real roots  ... "roots" navigates 
in NON-differential region 'diff( ▪,▪)' sol u w x y z f u w x y z 51 mat roots : #

M 1 -^b * 36.6667 - 59.9123 0.0741 - 58.6346 - 35.1296 51 mat

The two matrix solve companions

1. implicit solve

2. explicit solve

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M 0.1 545801252045125180245458002453208012524532040512518032040550055 mat :

b 7 - 61925 - 51 mat :

1. implicit matrix solve M 1 -^b * 36.6667 - 59.9123 0.0741 - 58.6346 - 35.1296 51 mat

LinSolve code solver . . .

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Clear as required w x y z Clear 1

LinSolve LinSolve Courtesy NASCH 20180117

Löst ein lineares Gleichungssystem, gegeben als Vektor von Gleichungen, nach allen
unbekannten Variablen auf. Für symbolische Ergebnisse definieren Sie bekannte Größen
mit eigenen Maßeinheiten. Solves a linear system, given as a vector of equations, for any unknown variable. 
Use custom units for parametric (symbolic) solutions. sys# LinSolve subtract right hand sides of equations n# sys# length : i# 1 n# range Nsys# i# el sys# i# el num2str ≡ -(strrep) concat str2num : sys# i# el num2str trace 21 line for Compute Jacobi matrix and it's determinant u# Nsys# Unknowns : J# Nsys# u# Jacob : d# J# det : compute right hand side of linear system i# 1 n# range RHS# i# el Nsys# i# el - J# i# j# el u# j# el * (j# 1 n# sum + : for multiply inverse jacobi matrix by rhs i# 1 n# range r# i# el J# j# i# alg d# / RHS# j# el * (j# 1 n# sum : for r# 121 line :

RHS J u * G - :

J 1 -^RHS *

Residual(sys): Berechnet den Residuenvektor des Gleichungssystems sys. Die Lösung muss den 
Unbekannten schon zugewiesen sein. Geeignet zur Prüfung von Lösungsverfahren mit eingebetteter
Ergebniszuweisung. Residual(sys): Computes the residual vector of system of equations sys. All variables must
be assigned. Suitable to check solutions with embedded assignment. sys# Residual subtract right hand sides of equations Nsys# sys# num2str ≡ - strrep str2num : 21 line :

Residual(sys,L): Berechnet den Residuenvektor des Gleichungssystems sys für die Lösung L, die
den Unbekannten nicht vorab zugewiesen wurde. Dies erfolgt innerhalb der Funktion. Geeignet zur 
Prüfung von Lösungsverfahren ohne eingebettete Ergebniszuweisung. Residual(sys,L): Computes the residual vector of system of equations sys for a given solution L. 
The solution is assigned to the unknowns and the residual is evaluated. Suitable to check 
solutions without embedded assignment. sys# L# Residual subtract right hand sides of equations n# sys# length : u# sys# Unknowns : Assign solution to unknowns i# 1 n# range u# i# el num2str : L# i# el num2str concat str2num 11 line for Nsys# sys# num2str ≡ - strrep str2num : 61 line :

xyz x y + 138 ≡ 5 x * 3 y * + 540 ≡ 5 x * 3 y * + z - 300 ≡ 31 mat :

xy x y + 138 ≡ 5 x * 3 y * + 540 ≡ 21 mat :

================= Solving as given =======================

x
y
z xyz LinSolve 637524031 mat

x
y xy LinSolve 637521 mat

================= Solving as given =======================

WXYZ 5 w * 10 x * + 32.5 y * + 122.5 z * + 0.45 ≡ 10 w * 32.5 x * + 122.5 y * + 497.125 z * + 4.555 ≡ 32.5 w * 122.5 x * + 497.125 y * + 2096.875 z * + 14.4625 ≡ 122.5 w * 497.125 x * + 2096.875 y * + 9045.15625 z * + 66.89875 ≡ 41 mat :

p WXYZ LinSolve : 2.24644709180097 - 7.06741882737669 3.91119676088135 - 0.55609748501128 41 mat

solver is reactive
assign to isolate p 2.24644709180097 - 7.06741882737669 3.91119676088135 - 0.55609748501128 41 mat :

t Poly p i el t i 1 -^* i 1 p length sum 11 line :

This is the resumed adventure of "WXYZ'

x Poly

1. Jacobi LinSolve applies to physical maths. 2. Cholesky Polyfit applies to physical data collected from experiments. SourceBook for Programmable Calculator TI 3_32

System of linear equations:

A 0 0.3251 - 0000100000 0.3251 01 - 00 0.3251 0000010000 0.3251 - 001 - 1 0.9457 - 0000000001 0.9457 - 0000 0.9457 1 - 000000000 0.9457 1 - 001000000098 mat :

B 00000000500 - 91 mat :

dot_1 dot_2 dot_3 31 mat A B dn_LinAlgLLS_COF A B dn_LinAlgLLS_QRorLQ A B dn_LinAlgLLS_SVD 31 mat :

dot_1 dot_2 dot_3 13 mat 500 - 528.7089 - 500 - 500 - 528.7089 - 500 - 171.8833 - 171.8833 81 mat 500 - 528.7089 - 500 - 500 - 528.7089 - 500 - 171.8833 - 171.8833 81 mat 500 - 528.7089 - 500 - 500 - 528.7089 - 500 - 171.8833 - 171.8833 81 mat 13 mat

A dot_1 * 00000000500 - 91 mat