Cubic(a)... algo style

84fd168c-9308-48be-bf8d-ebc2ff3e81c5 15 4 5 decimal

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

Cubic Roots Explorer ... all Numeric

v Cubic a v : S1 36 a 2 el * a 3 el * 108 a 1 el * - 8 a 3 el (3^* - 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^* + (12 /^* + (13 /^: q 13 / a 2 el * 19 / a 3 el (2^* - : Z 16 / S1 * 6 q S1 / * - 13 / a 3 el * - 1 - 12 / S1 * 3 q S1 / * + 13 / a 3 el * - i 3 sqrt 2 / * 16 / S1 * 6 q S1 / * + (* + 1 - 12 / S1 * 3 q S1 / * + 13 / a 3 el * - i 3 sqrt 2 / * 16 / S1 * 6 q S1 / * + (* - 31 mat : 31 line Z 31 line :

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 :

pre-set the vectors
for the plot explorer a 204 - 6114 mat transpose 205 - 6114 mat transpose 203 - 6114 mat transpose 204 - 5114 mat transpose 205 - 4114 mat transpose 30107114 mat transpose 61 mat :

cubic all at once very fast
...
reverse for complete real plot r a Cubic vectorize eval : r r reverse vectorize : 21 sys

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

r 0.4915 1.6194 i * - 0.4915 1.6194 i * + 6.983 - 31 mat 0.5504 1.5854 i * - 0.5504 1.5854 i * + 7.1008 - 31 mat 0.431 1.6519 i * - 0.431 1.6519 i * + 6.8619 - 31 mat 0.5864 1.7018 i * - 0.5864 1.7018 i * + 6.1729 - 31 mat 0.7751 1.7329 i * - 0.7751 1.7329 i * + 5.5501 - 31 mat 0.4164 - 2.1659 i * - 0.4164 - 2.1659 i * + 6.1673 - 31 mat 61 mat

Lagrange complex cubic plot

x 1 Lagrange x 2 Lagrange x 3 Lagrange x 4 Lagrange x 5 Lagrange x 6 Lagrange 6 1 sys

a 6 el polyroots 0.4164 - 2.1659 i * + 0.4164 - 2.1659 i * - 6.1673 - 31 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 - (* :

x Lagrange parfrac x convert maple x 3^70001 x 2^* 10000 / + 1000063921 x * 100000000 / + 300009034391211000000000000 / +

Solving a cubic polynomial

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

XY 5 - 303 - 361 - 2628642 mat :

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 30107141 mat

data XY . 14 black plot :

data x β f 2 1 sys

v 30107241 mat :

30107141 mat 0.4164 - 2.1659 i * + 0.4164 - 2.1659 i * - 6.1673 - 31 mat 12 mat

v polyroots 0.0772 - 2.116 i * + 0.0772 - 2.116 i * - 3.3456 - 31 mat