<p> eigenall </p>

b223fd34-594a-4d28-afe0-ef609fff30b1 10 4 5 decimal

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

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 p x 3^10 x * - 2 + :

Vector 'v' is constructed manually in that order. Missing term set '0'

v 210 - 0141 mat :

r v polyroots : 0.2008 3.0571 3.2579 - 31 mat

order of the roots is immaterial r r reverse : 3.2579 - 3.0571 0.2008 31 mat

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

roots r r p vectorize augment : R roots . 14 red plot : 21 sys

x f R 2 1 sys

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

Differently than Mathcad,
Smath Polyroots plugin 
does NOT suport  complex
polyroots(v). v 8 i * - 4 - 6 i * + 32 i * + 141 mat :

==================================

install  the ...
companion vector v 42 - 27 - 135 - 41 mat :

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

vx companion c vx : a 0 : leading c c rows el : i 1 c length range a i el c i el - leading / : for b a 1 c length 1 - 11 submatrix : j 1 c length 2 - (range M j 1 + j el 1 : for M b augment 71 line :

E v companion :

The companion matrix E 00 8.4 - 10 5.4 - 01 2.6 33 mat

x P 0 2 1 sys

polyroots is Smath built-in r v polyroots : 0.9551 - 1.7775 2.374 i * + 1.7775 2.374 i * - 31 mat

this Polyroots needs be scaled ! scale v 4 el :

x Φ x r n el - (n 13 product :

x c 12 - x ≤ (x 2 ≤ (& cases :

Independent complete plots

scale x Φ Re * x c * x P 2 1 sys

. . . The eigen companion . . .

]]> <p> eigenall </p>

Function for calculating the coefficients of the characteristic polynomial, eigenvalues and eigenvectors Adapted from: "Introduction to the use of SMath Studio" G. Urroz, May 2010

A eig n A rows : p n 1 matrix : I n identity : p n 1 + 1 el 1.0 - : B A : p n 1 el B tr : j n 1 - 1 n 2 - range B A B p j 1 + 1 el I * - (* : p j 1 el B tr n j - 1 + / : 21 line for p 1 - (n^(- (p * : λ p polyroots : V n 1 - 1 matrix : j 1 n range A.L A λ j el I * - : A.Lm A.L 1 n 1 - 1 n 1 - submatrix : b.L A.L 1 n 1 - n n submatrix : x.L A.Lm 1 -^b.L * : V x.L V augment : 51 line for V V 1 n 1 - 1 n submatrix : V.b 1 n matrix : j 1 n range V.b 1 j el 1 : for V V V.b stack : j 1 n range x.L V j col : x.L x.L x.L norme / : V V x.L augment : 31 line for V V 1 n n 1 + 2 n * submatrix : p λ V 31 sys 181 line :

12 - 422 - 22 - 642 - 38268444 mat eig 1.3 103^* - 530 - 93101 - 51 mat 1.95 - 6.33 7.83 - 13.45 41 mat 0.35 - 0.1 0.79 0.7 - 0.21 - 0.51 0.52 - 0.46 0.59 - 0.57 0.26 0.54 - 0.69 0.64 0.18 0.11 44 mat 31 sys

2 - i + 0002 - i - 0001 - 33 mat eig 5 - 9 - 5 - 1 - 41 mat 1 - 2 - i + 2 - i - 31 mat 00000011133 mat 31 sys

1 - 004 - 22 mat eig 2 el 1 - 4 - 21 mat

ε 106 -^:

EIGENVECTORS CALCULATION FAILS FOR 2x2 MATRICES:

1 - ε + 0 ε + 0 ε - 4 - 22 mat eig 3 el 3.33 107 -^* 11 3.35 107 -^* 22 mat

1 - 004 - 22 mat eig 1 el 45131 mat

1 - i - 001 - i + 22 mat eig 1 el 22131 mat

1 - i - 001 - i + 22 mat eig 2 el 1 - i + 1 - i - 21 mat

install the companion matrix
... in 'eigen' system ... E 00 8.4 - 10 5.4 - 01 2.6 33 mat :

E eig 8.4 - 5.4 - 2.6 1 - 41 mat 0.9551 - 1.7775 2.374 i * + 1.7775 2.374 i * - 31 mat 0.4335 0.579 i * + 0.4335 0.579 i * - 0.922 - 0.21 0.6062 i * + 0.21 0.6062 i * - 0.3727 0.2554 0.2554 0.1048 33 mat 31 sys

roots E eig 2 el : 0.955057 - 1.777529 2.373958 i * + 1.777529 2.373958 i * - 31 mat

M E eig 3 el :

eigvec M 1 col : 0.43352 0.578983 i * + 0.210031 0.606229 i * + 0.255366 31 mat

eigval M 1 M rows 2 M cols submatrix : 0.43352 0.578983 i * - 0.922019 - 0.210031 0.606229 i * - 0.372681 0.255366 0.104831 32 mat

====== This complex Polyroots needs NOT be scaled ========

a 204 - 6114 mat transpose :

x L a 1 el a 2 el x * + a 3 el x 2^* + a 4 el x 3^* + (: (Lagrange canonic format ≡

r a polyroots :

r 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat

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

x c 15 - x ≤ (x 3 ≤ (& cases :

Independent complete plots

x f Re x c * x L 2 1 sys

polyroots("vector") : Returns the roots of the polynomial whose coefficients are in "vector"