Golden ratio

350ff2b3-c995-4434-a681-9ce2ca374d96 88 4 5 decimal

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

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a 204 - 6114 mat transpose :

z a polyroots : 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat

x sanity a 1 el a 2 el x * + a 3 el x 2^* + a 4 el x 3^* + :

x L x z n el - (n 13 product :

x L 2 1 sys x sanity 1 1 sys

x Lagrange x z n el abs - (n 13 product :

0 Lagrange 20 -

datum 16 - :

x Lagrange datum - 0 ≡ x 08 solve 0.1609 4.0137 6.1931 31 mat

plot x Lagrange z 1 el abs 10 z 1 el abs 10 - 22 mat z 3 el abs 10 z 3 el abs 10 - 22 mat datum 41 sys :

plot

z 1 el abs 1.6924

z 3 el abs 6.983

x q2 x z 1 el - (x z 2 el - (* Re :

x q2 16 - 0 ≡ x 10 - 10 solve 3.166 - 4.149 21 mat

T P char size color plotG n T length : plot T 1 el P 1 el char size color augment : i 2 n range plot plot T i el P i el char size color augment stack : for plot 41 line :

z Re z Im 21 mat 0.4915 0.4915 6.983 - 31 mat 1.6194 1.6194 - 031 mat 21 mat

x sanity x 10 - 0 solve x Lagrange x 010 solve 21 mat 6.983 - 6.983 21 mat

Golden ratio

a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + :

TOL 1012 -^:

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema y1 y2 ≥ max in range ≡ y1 y2 ≤ min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 61 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

x f x q2 : Xre f 01 y1 y2 ≤ 1012 -^Extrema 1 el : 21 sys

The roots are complex conjugates, we can solve for imaginary

Xim 0.4915 2^w 2^+ sqrt z 1 el abs ≡ w 02 solve : 1.6194

r z 3 el Xre Xim 31 mat :

u 00031 mat :

roots r u o 10 black plotG :

x L x q2 roots Xre 0 + 50 red 1 5 mat Xim 0 + 50 green 1 5 mat 5 1 sys

How cubic roots appear on Cartesian 'X'

from cubic solver(s) r 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat :

Lagrange roots style x L x r n el - (n 13 product :

a 204 - 6114 mat transpose :

a polyroots 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat

Lagrange canonic x canonic a 1 el a 2 el x * + a 3 el x 2^* + a 4 el x 3^* + :

x c 16 - x ≤ (x 2 ≤ (& cases :

x L x c * x canonic 2 1 sys

a 1 el a 2 el x * + a 3 el x 2^* + x 3^+ canonic format ≡

x^3 silent Cubic Solver a 204 - 613 mat transpose :

complete for polyroots 204 - 6114 mat transpose

In Mathematics, polynomials are represented in Lagrange format, thus wrt their roots, which results in the canonic format. In the cubic example, the polynomial does cross the X only once, by inspection it will have 2 complex roots. This cubic solver is extracted from Mathcad 8/Maple, roots ordered same as polyroots.

p 36 a 2 el * a 3 el * 108 a 1 el * - 8 a 3 el (3^* - : S p 1212 a 2 el (3^* 3 a 2 el (2^* a 3 el (2^* - 54 a 1 el * a 2 el * a 3 el * - 81 a 1 el (2^* + 12 a 1 el * a 3 el (3^* + sqrt * + 3 nthroot : q 13 / a 2 el * 19 / a 3 el (2^* - : Z 1 - 12 / S * 3 q S / * + 13 / a 3 el * - i 3 sqrt 2 / * 16 / S * 6 q S / * + (* + 1 - 12 / S * 3 q S / * + 13 / a 3 el * - i 3 sqrt 2 / * 16 / S * 6 q S / * + (* - 16 / S * 6 q S / * - 13 / a 3 el * - 31 mat : 41 line

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

◄ Cubic roots invariants Cubic solver = polyroots = Mathsoft Companion Matrix

recast Lagrange r Z :

x Lagrange x r n el - (n 13 product :

a Z 12 mat 204 - 631 mat 0.491489 1.619428 i * + 0.491489 1.619428 i * - 6.982977 - 31 mat 12 mat

Lagrange complex cubic plot

x Lagrange

a 204 - 6114 mat transpose :

a polyroots 0.491489 1.619428 i * + 0.491489 1.619428 i * - 6.982977 - 31 mat

a 204 - 6114 mat : x poly a 1 el a 2 el x * + a 3 el x 2^* + a 4 el x 3^* + : datum 30 : Sol x poly datum ≡ x 10 - 2 solve : 41 line

x poly datum 2 1 sys

Sol 6.3813 - 1.0756 - 1.4569 31 mat

reverse for dotted plot r Z reverse :

x Lagrange x r n el - (n 13 product :

Lagrange complex cubic plot ... dotted style

x Lagrange

Otherwise cubic ...

a 422713 - 541 mat :

r a polyroots : 0.9551 - 1.7775 2.374 i * + 1.7775 2.374 i * - 31 mat

x Lagrange x r n el - (n 13 product :

Lagrange complex cubic plot ... dotted style

x Lagrange

r r reverse :

x Lagrange x r n el - (n 13 product :

Lagrange complex cubic plot ... dotted style

x Lagrange

Companion Matrix E 000421002701013 - 001544 mat :

r E dn_LinAlgEigenvalues : 0.9429 - 0.9402 3.1749 i * + 0.9402 3.1749 i * - 4.0624 41 mat

x Lagrange x r j el - j 14 product Re :

Lagrange roots style x L x r n el - (n 14 product :

0.943 - 0 . 14 black 1 5 mat x L 4.062 0 . 14 black 1 5 mat 3 1 sys x Lagrange

Style 2 cubic solver ...

x^3 silent Cubic Solver a 40401113 mat transpose :

complete for polyroots b 404011114 mat transpose :

Q 13 / a 2 el * 19 / a 3 el (2^* - : R 16 / a 2 el a 3 el * 3 a 1 el * - (* 127 / a 3 el (3^* - : 12 mat u1 R Q 3^R 2^+ sqrt + 3 nthroot : u2 R Q 3^R 2^+ sqrt - 3 nthroot : 12 mat Z u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* + u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* - u1 u2 + (a 3 el 3 / - 31 mat : 31 mat

Z 4.6974 - 1.6894 i * + 4.6974 - 1.6894 i * - 1.6051 - 31 mat

r b polyroots : 1.6051 - 4.6974 - 1.6894 i * + 4.6974 - 1.6894 i * - 31 mat

x Lagrange x Z n el - (n 13 product :

x Lagrange r 1 el 0 + 40 red 1 5 mat 2 1 sys

for whatever use ? Q R u1 u2 41 mat 0.1111 - 4.037 2.0061 0.0554 41 mat

r 0.4164 - 2.1659 i * - 0.4164 - 2.1659 i * + 6.1673 - 31 mat :

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

r 1371041 mat :

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 :

x f x r n el - (n 14 product :

x f maple 1 - x + (3 - x + (* 7 - x + (* 10 - x + (*

i 1 r rows range u i el 0 : for

XY r u augment :

roots XY o 7 red plot :

Lagrange root(s) construction

x f roots 2 1 sys

jmGiraud: 2015:07:25

linfit modules

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 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 :

Applicable parts are: Optimiz "numeric" XY f φ β βlinfit X XY 1 col : Y XY 2 col : 12 mat n XY rows : k x β φ length 1 - : vx raw initial vector fit i 1 X length range vx i el X i el β f eval : for M-> Cholesky Gradient Matrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for Cholesky solver [LS] Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 121 line :

Vector gradient ... Optimiz Symbolic x β f n φ ct 1 : i 1 n range PD ct el x β f (β ct el diff : stack f(x,β) & PD V 1 el x β f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

set N = even/odd to escape divide by 0 maybe other singularities.

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

XY 5 - 303 - 3602028642 mat :

Solve/Export a cubic polynomial

x β f β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + :

Model is a linear combination ... 'β' are simply '1' β 111114 mat transpose :

last index β

<= ConjugateGradient Vector Fnct x α φ x β f 4 φ :

β XY f φ β βlinfit :

β 20 9.6667 9 1.3333 41 mat

data XY . 14 black plot :

β β eval :

in many instances, reverse(r) r β polyroots :

x Project β 4 el x r n el - (n 13 product * :

plot x β f data 31 sys :

plot

r 0.3972 - 1.5365 i * + 0.3972 - 1.5365 i * - 5.9556 - 31 mat

x Project data 2 1 sys

Revisit native Cholesky . . .

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

vx vy a b c plotG 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 41 line :

vx 0123 4.5 51 mat :

vy 2 - 0.04 2.5 1.6 - 1.1 51 mat :

x φ 1 x x 2^x 3^41 mat :

Cholesky companion solver jj 1 U φ length range ii 1 vx length range M ii jj el vx ii el φ jj el : for for Φ M transpose M * : Δ M transpose vy * : 12 mat β Φ 1 -^eval Δ * : 31 line

β β eval :

β 2.3069 - 6.4102 3.3413 - 0.4618 41 mat

recast the fit x Poly x φ β * :

Polyfit solves directly from [X, Y] WXYZ is supplementary insight.

x Poly vx vy . 14 black plotG 2 1 sys

β β eval :

The 5 points unknown data set is now reduced to 4 coefficients function ... we have reduced the XY system by another one, now: productive

r β polyroots : 0.4656 2.5313 4.2383 31 mat

x λ 1 0.5 - x ≤ (x 5 ≤ (& cases :

assign/isolate r r eval :

Polyroots representation
is factors wrt β4 x Polyroots β 4 el x r n el - (n 1 r rows product * :

x Polyroots x λ * x Poly vx vy . 14 black plotG 3 1 sys