c6751d89-b4a8-4f62-84b1-c81c9d0f60f2 29 4 5 decimal

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Legendre coefficients & roots

Legendre last 'n' polynomial n x L 12 n^/ 1 - (k^2 n * 2 k * - (! (* k! (n k - (! (* n 2 k * - (! (* / x n 2 k * - (^* k 0 n 2 / Floor sum * :

n 8 :

last 'n' polynomial => copy/paste in Ω1 n x L 353 x 2^* 420 - 11 x 2^* 21013 x 2^* 28 - 15 x 2^* + (* + (* + (* + 128 /

Does not fetch from upstream only as given existing argument. polyroots takes an isolated vector.

Ω 1 el 353 x 2^* 420 - 11 x 2^* 21013 x 2^* 28 - 15 x 2^* + (* + (* + (* + 128 / :

μ 1 n range Ω μ 1 + el Ω μ el (x diff μ / : for x d Ω : r 0 d polyroots : 31 line

◄ from Smath Handbook

0 d 0.2734 0 9.8438 - 0 54.1406 0 93.8438 - 0 50.2734 91 mat

R i 1 r rows range p i el . : for i 1 r rows range zero i el 0 : for r zero p augment 31 line :

R n x L 2 1 sys

R 0.1834 - 0 . 0.1834 0 . 0.5255 - 0 . 0.5255 0 . 0.7967 - 0 . 0.7967 0 . 0.9603 - 0 . 0.9603 0.83 mat

x y XY x y / - :

plot 0 x L 1 x L 2 x L 3 x L 4 x L 5 x L 61 sys :

plot XY 2 1 sys

x f 353 x 2^* 420 - 11 x 2^* 21013 x 2^* 28 - 15 x 2^* + (* + (* + (* + 128 / :

n 08 range :

x D x f x n diff vectorize :

Transit ► 0 D 0.2734375 0 19.6875 - 0 1299.375 0 67567.5 - 0 2.027025 106^* 91 mat

D 0.2734375 0 19.6875 - 0 1299.375 0 67567.5 - 0 2.027025 106^* 91 mat :

Create the 
polyroots vector v D 1 el 1! / D 2 el D 3 el 2! / D 4 el D 5 el 4! / D 6 el D 7 el 6! / D 8 el D 9 el 8! / 91 mat : 0.2734 0 9.8438 - 0 54.1406 0 93.8438 - 0 50.2734 91 mat

r v polyroots :

r 0.1834 - 0.1834 0.5255 - 0.5255 0.7967 - 0.7967 0.9603 - 0.9603 81 mat

The companion matrix =>

companion matrix dn_LinAlgEigenvalues

This work sheet will help construct the vector of coefficients required by "polyroots". The "companion matrix" provides the argument to dn_LinAlgEigenvalues(companion matrix). You may need the "companion matrix" or "coeffs" for other maths, for instance spooling little piece of information, transmit to community members for recovery at home.

x P 42 - 27 x * - 13 x 2^* + 5 x 3^* - x 4^+ : Lagrange canonical ≡

The Smath 6179 Maple wraper works a bit different than the Mathcad 11. You feed the LHS of P(x) as the argument to maple(op( ■ ))

x P

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

Undocumented

x expr 42 - 27 x * - 13 x 2^* + 5 x 3^* - x 4^+ op maple :

x 1 :

coeffs x expr :

The companion matrix, argument to dn_LinAlgEigenvalues(E) 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 :

coeffs transpose 42 - 27 - 135 - 115 mat

E coeffs companion :

E 000421002701013 - 001544 mat

polyroots E dn_LinAlgEigenvalues ≡

r coeffs polyroots :

E dn_LinAlgEigenvalues 0.943 - 0.94 3.175 i * + 0.94 3.175 i * - 4.062 41 mat

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

Lagrange complete plot, clean of residual imaginary

x Lagrange 0 r 1 el 0 . 14 black 1 5 mat r 4 el 0 . 14 black 1 5 mat 4 1 sys

r 0.943 - 0.94 3.175 i * + 0.94 3.175 i * - 4.062 41 mat