coeffs(M,vars)

3173ddec-40ee-4535-aa05-cccc143b9f34 51 4 5 decimal

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

LinSolve code exploder . . . Davide Carpi

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LGS x 2 y * - z + 4 - 2 x * - 4 y * + z - 2 + y z - 3 - 31 sys :

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 12 - 12 - 41 - 011 - 33 mat

b 4 - 23 - 31 mat

A 1 -^b * 16 - 9 - 6 - 31 mat

coeffs(M,vars)

M vars coeffs M.str M num2str : // accept only systems or column vectora M.str sys(findstr 1 - ≡ (M.str sys(findstr length 1 > (| M.str mat(findstr 1 ≡ (M cols 1 ≡ (& M.mat M : invalid input, system or column matrix needed error if M.mat M.str sys(mat(strrep str2num : if // preliminary checks count vars length : count M.mat rows ≠ number of unknowns doesn't match number of equations error go on if // core function vars n f // enable reusable symbolic assignment vars num2str : 1 count range n ≡ vectorize num2str concat str2num M.mat eval 31 line : // initialize the output A count 0 matrix : b vars 0 f : i 1 count range A A vars i f b - augment eval : for A b 12 mat 131 line :

A b Clear 1

LGS 22 y + (* - x + z + 21 x - 2 y * + (* z - 3 y - z + (- 31 sys

system LGS x y z 31 mat coeffs : 12 - 12 - 41 - 011 - 33 mat 4 - 23 - 31 mat 12 mat

Explode the system A system 1 col 1 el : b system 2 col 1 el : 12 mat

A 12 - 12 - 41 - 011 - 33 mat

b 4 - 23 - 31 mat

x y z 31 mat A 1 -^b * : 16 - 9 - 6 - 31 mat

12 / LGS * 4 - 23 - 31 sys

]]> coeffs(M)

M coeffs M.str M num2str : // accept only systems or column vectora M.str sys(findstr 1 - ≡ (M.str sys(findstr length 1 > (| M.str mat(findstr 1 ≡ (M cols 1 ≡ (& M.mat M : invalid input, system or column matrix needed error if M.mat M.str sys(mat(strrep str2num : if // preliminary checks vars M.mat Unknowns : count vars length : count M.mat rows ≠ number of unknowns doesn't match number of equations error go on if // core function vars n f // enable reusable symbolic assignment vars num2str : 1 count range n ≡ vectorize num2str concat str2num M.mat eval 31 line : // initialize the output A count 0 matrix : b vars 0 f : i 1 count range A A vars i f b - augment eval : for A b 12 mat 141 line :

A b x y z Clear 1

A b 12 mat LGS coeffs : 12 - 12 - 41 - 011 - 33 mat 4 - 23 - 31 mat 12 mat

A 12 - 12 - 41 - 011 - 33 mat

b 4 - 23 - 31 mat

x y z 31 mat A 1 -^b * : 16 - 9 - 6 - 31 mat

LinSolve code direct solver . . . NASCH

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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 :

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

X t Poly t 05 solve : 0.4023 2.3402 4.2908 31 mat

This is the resumed adventure of "WXYZ'

x Poly X 1 el 0 + 20 red 1 5 mat X 2 el 0 + 20 red 1 5 mat X 3 el 0 + 20 red 1 5 mat 4 1 sys

t Poly t X 1 el X 3 el int 0.0347 -

x y z Clear 1

Advanced infinite algebraic solve

User in-situ system
or imported system.
Triplet in demo. system 4 x + y + z - x z - e + 3 y * + (- x y - z + 2 sqrt + 31 mat :

system 1 el 0 ≡ x solve maple 4 y + z - (-

system 2 el 0 ≡ x solve maple z - 3 y * + 1 exp + (-

system 3 el 0 ≡ x solve maple y - z + 2 sqrt + (-

system 1 el 0 ≡ y solve maple 4 x + z - (-

system 2 el 0 ≡ y solve maple x z - 1 exp + 3 / -

system 3 el 0 ≡ y solve maple x z + 2 sqrt +

system 1 el 0 ≡ z solve maple 4 x + y +

system 2 el 0 ≡ z solve maple x 3 y * + 1 exp +

system 3 el 0 ≡ z solve maple x y - 2 sqrt + (-

Solving a Jacobian system . . .

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x y z Clear 1

x y z f1 x sin y 2^+ z ln + 7 - : x y z f2 3 x * 2 y^+ 1 + z 3^- : x y z f3 x y + z + 5 - : 31 sys

Jacobian all modules numeric

As exemplified, this system works well.

That it will solve any/all user systems ?

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[1]

x y z J x y z f1 x diff x y z f1 y diff x y z f1 z diff x y z f2 x diff x y z f2 y diff x y z f2 z diff x y z f3 x diff x y z f3 y diff x y z f3 z diff 33 mat :

x y z J x cos 2 y * 1 z / 32 y^2 ln * 3 z 2^* - 11133 mat

x f x 1 el x 2 el x 3 el f1 x 1 el x 2 el x 3 el f2 x 1 el x 2 el x 3 el f3 31 mat :

in this order ... ! x Minimize x x Ω (1 -^x f * - : x Ω x 1 el x 2 el x 3 el J eval : 21 sys

x x 1 el 11113 mat transpose : iter 7 : 12 mat j 1 : j iter ≤ x j 1 + el x j el Minimize eval : j j 1 + : 21 line while x x iter el : 41 line :

t0 1 time :

exp'l threshold 15
sanity implicit [1] x f 0071015 -^* 31 mat

x 0.599053756640566 2.39593140237782 2.00501484098162 31 mat

1 time t0 - 0.5 s *

Jacobian to fit a parametric function . . .

]]> Pearson correlation

p# corr Pearson correlation X XY 1 col : Y XY 2 col : 12 mat χ X Mean : μ Y Mean : n XY rows : 13 mat X i el χ - (Y i el μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum Y i el μ - (2^(i 1 n sum * sqrt / X i el χ - (X i el p# f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el p# f eval μ - (2^(i 1 n sum * sqrt / / 41 line :

===== Thurber data set =====

ListCollect

data list ListCollect collects index list select list : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

XY Thurber

XY 2.981 - 84.248 2.921 - 87.264 2.912 - 87.185 2.84 - 89.076 2.797 - 89.608 2.702 - 89.868 2.699 - 90.101 2.633 - 92.405 2.481 - 95.854 2.363 - 100.696 2.322 - 101.06 1.501 - 401.672 1.46 - 390.724 1.274 - 567.534 1.212 - 635.316 1.1 - 7330.54 1.046 - 759.087 0.915 - 894.206 0.714 - 990.785 0.566 - 1090 0.545 - 1080 0.4 - 1120 0.309 - 1180 0.109 - 1260 0.103 - 1270 0.01 1290 0.119 1330 0.377 1350 0.79 1420 0.963 1430 1.006 1420 1.115 1440 1.572 1460 1.841 1470 2.047 1450 2.2 1460362 mat :

1. select the support points
2. collect for al_nleqsolve list 113243414 mat : xy XY list ListCollect : 21 sys

raw XY o 5 red plot : support xy . 16 black plot : 21 line

support raw 2 1 sys

xy 2.981 - 84.248 1.46 - 390.724 0.109 - 1260 1.841 147042 mat

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

vector of guesses ... 
from robust2crucifying ! X0 8014001014 mat transpose :

t0 1 time :

AlgLib Solver implicit equalities [Courtesy  Viacheslav] sol k 1 μ f rows range u k el μ k el : for μ Jac u f u Jacobian : StepMax 0 : ε 1012 -^: 12 mat u X0 StepMax ε μ f μ Jac al_nleqsolve : 41 line :

SGompertz . . .

is extremely reflexive, declared unfittable

in Smath [L-M, CG]. Possible attempt

OriginLab, Mathcad 11. As offered, and

fitted by al_nleqsolve, at this point, you

may want to round/refactor the coefficients.

The fillet region is a bit sparse !

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SGompertz Thurber model fnct x m Φ m 1 el m 2 el m 3 el x m 4 el - (* - exp - exp * + :

Implicit equalities = 0 μ f x 1 el μ Φ y 1 el - x 2 el μ Φ y 2 el - x 3 el μ Φ y 3 el - x 4 el μ Φ y 4 el - 41 mat :

μ sol : 84.25 1396.05 1.61 1.2 - 41 mat

Notes . . . Smath 6179

1. solves a system of [n equalities = n parameters].

2. The order of equalities is immaterial.

3. Guess wisely the best support equalities !

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1 time t0 - s 1

The support points X XY 1 col : Y XY 2 col : 12 mat data XY o 5 black plot : 21 sys

μ f 0.000000000000 0.000000000002 - 0.000000000002 - 0.000000000001 41 mat

x c 14 - x ≤ (x 3 ≤ (& cases :

native al_nleqsolve from

113243414 mat

μ 84.25 1396 1.61 1.2 - 41 mat 87.5 1375 1.6 1.25 - 41 mat 901375 1.6 1.25 - 41 mat 801400 1.5 1.25 - 41 mat 901370 1.6 1.25 - 41 mat 15 mat :

observe fillet region j 3 :

data x μ 1 j el Φ x c * support 3 1 sys

x μ f x μ 1 j el Φ :

μ corr 0.4505

ListCollect

data list ListCollect collects index list select list : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

XY 304 0.1653 - 5 0.1313 6 0.219 7 0.8097 8 1.2045 9 1.302 10 2.1009 11 2.6004 12 3.7002 13 4.3001 112 mat :

1. select the support points
2. collect for al_nleqsolve list 13791115 mat : xy XY list ListCollect : 21 sys

raw XY o 5 red plot : support xy . 14 black plot : 21 line

raw support 2 1 sys

xy 305 0.131 9 1.302 11 2.6 13 4.3 52 mat

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

vector of guesses ... NOT easy ! X0 0.01 1 1.5 0515 mat transpose :

AlgLib Solver for a system of nonlinear equations(Uni) sol k 1 α f rows range u k el α k el : for α Jac u f u Jacobian : StepMax 0 : ε 109 -^: 12 mat u X0 StepMax ε α f α Jac al_nleqsolve : 41 line :

Model function.
Specific for the
flat low region. x α Φ α 1 el x * 1 x α 5 el - α 3 el / exp + ln (α 2 el^(* α 4 el + :

0.005 0.007 - 0.006 0.005 - 0.001 51 mat

Implicit equalities = 0 α f x 1 el α Φ y 1 el - x 2 el α Φ y 2 el - x 3 el α Φ y 3 el - x 4 el α Φ y 4 el - x 5 el α Φ y 5 el - 51 mat :

α sol : 0.10919 0.85833 1.61119 0.03206 - 7.12811 51 mat

Notes . . . Smath 6179

1. solves a system of [n equalities = n parameters.

2. The order of equalities is immaterial.

3. Guess wisely the best support equalities !

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The support points X XY 1 col : Y XY 2 col : 12 mat data XY o 5 black plot : 21 sys

sanity 0 α Φ 1 α Φ 2 α Φ 3 α Φ 41 mat : 0.032 - 0.028 - 0.018 - 0.003 41 mat

α f 0.003 0.005 - 0.005 0.005 - 0.001 51 mat

x c 10 x ≤ (x 15 ≤ (& cases :

user adjust '0' flat region bias 0.0320615 - :

data x α Φ x c * bias - support 3 1 sys

Support points xy 305 0.131 9 1.302 11 2.6 13 4.3 52 mat

Conclusion . . .

The experimental model function is very

reactive. Hyper difficult Genfit.

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0 α Φ bias - 2.096 107 -^* -

sanity flat '0' 0 α Φ bias - 0.0000002 -

Altitude m . . . vs kPa

]]> ListCollect ... Jacobian

data list ListCollect collects index list select list : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

sol(init,Tol) ... AlgLib Solver for a system of nonlinear equations
adapted from Viacheslav original init sol k 1 ζ f rows range u k el ζ k el : for ζ Jac u f u Jacobian : StepMax 0 : ε 1015 -^: 12 mat u init StepMax ε ζ f ζ Jac al_nleqsolve : 41 line :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

XY 060000 0.5 35462130747 2.1 26217 3.6 22781 5.6 19921 9.7 16453 14.7 13766 22.3 11115 32.4 8654 50.2 5563 75.5 2423 101.3 0132 mat :

1. selct the support points
2. collect for al_nleqsolve
3. collect variates [x,y] list 91011121315 mat : xy XY list ListCollect : X XY 1 col : Y XY 2 col : 12 mat 31 sys

raw XY o 5 red plot : support xy + 20 blue plot : 21 line

raw support 2 1 sys

xy 22.3 11115 32.4 8654 50.2 5563 75.5 2423 101.3 052 mat

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

al_nleqsolve COMPACTUM
... NUMERIC/SYMBOLIC x ζ Φ ζ 1 el ζ 2 el * ζ 3 el x ζ 5 el^* + ζ 2 el x ζ 4 el^+ / 0 + : init 60100361000 - 0.2 0.25 15 mat transpose : ζ f ct 1 : j 1 x rows range Q ct el x j el ζ Φ y j el - : ct ct 1 + : 21 line for Q 31 line : 31 line

t0 1 time :

ζ init sol : 59814.15162 2.56094 60887.56868 - 0.01159 0.19977 51 mat

1 time t0 - s 3

datum [0] -2.3 adjusted by user x ζ Φ ζ 1 el ζ 2 el * ζ 3 el x ζ 5 el^* + ζ 2 el x ζ 4 el^+ / 2.3 - :

x c 1 9.7 x ≤ (x 101.325 ≤ (& cases :

[-2.3] adjust local datum [0]

Absolute  residuals δ j 2 XY rows range Δ j el Y j el X j el ζ Φ - eval : for X Y Δ 1 round vectorize augment 21 line :

δ δ : Δ δ 813 range 13 range el : header kPa alt[m] Δm 13 mat : Δ header Δ stack : data XY o 5 black plot : 51 sys

14.7 25000 14.7 10000 - 2 2 mat x ζ Φ x c * support data 4 1 sys

EXEPTIONAL transmute Φ(x,ζ)

Adjusted to local landing datum [0]

0 . . . 14 km

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Δ kPa alt[m] Δm 14.7 13766 124.6 22.3 11115 6.1 32.4 8654 5.9 - 50.2 5563 6.6 75.5 2423 4.7 101.3 0073 mat

Next solving session is specific to fitting model(s)

to data sets . . . an immense Treasury !

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