Draghilev method, n=2

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Draghilev  symbolic method, n=2 2 f\032E\ ic\032E\ u\032E\ DM.2 v\0324\ us\0324\ uo\0324\ D\0324\ Clear v\0324\ N v\0324\ 1 el v\0324\ 2 el f\032E\ v\0324\ 2 el diff - v\0324\ 1 el v\0324\ 2 el f\032E\ v\0324\ 1 el diff 21 mat : k 12 range us\0324\ k el u\032E\ k el num2str (t) concat str2num : uo\0324\ k el u\032E\ k el num2str (0) concat str2num : 21 line for k 12 range D\0324\ k el us\0324\ k el t diff us\0324\ N k el ≡ (: D\0324\ k 2 + el uo\0324\ k el ic\032E\ k el ≡ (: 21 line for D\0324\ us\0324\ 12 mat 51 line :

2 f\032E\ uo\032E\ ho\032E\ N\032E\ DM.2 x# y# f\032E\ uo\032E\ ho\032E\ N\032E\ 0 DM.2 11 line :

Draghilev method, n=2 2 f\032E\ uo\032E\ ho\032E\ N\032E\ K\032E\ DM.2 u y Clear u\0324\ 1 el uo\032E\ 1 el : u\0324\ 2 el u\0324\ 1 el y f\032E\ y uo\032E\ 2 el roots : 13 mat U\0324\ u\0324\ : v\0324\ 12 range : h\0324\ ho\032E\ : b\0324\ 2 K\032E\ 0 ≠ (* 1 + : ε 103 -^: 15 mat u Δ h\0324\ b\0324\ / u 1 el u 2 el f\032E\ u 2 el diff - u 1 el u 2 el f\032E\ u 1 el diff 21 mat * eval : K\032E\ 0 ≡ n\0324\ 2 : n\0324\ N\032E\ ≤ n\0324\ n\0324\ 1 + : k\0324\.1 u\0324\ Δ : k\0324\.2 u\0324\ k\0324\.1 + Δ : k\0324\.3 u\0324\ 0.25 k\0324\.1 * + 0.25 k\0324\.2 * + Δ : 13 mat u\0324\ u\0324\ k\0324\.1 4 k\0324\.3 * + k\0324\.2 + 6 / + eval : U\0324\ v\0324\ n\0324\ el u\0324\ v\0324\ el : 12 mat 21 line for n\0324\ 2 : n\0324\ N\032E\ ≤ n\0324\ n\0324\ 1 + : k\0324\.1 u\0324\ Δ : k\0324\.2 u\0324\ k\0324\.1 + Δ : k\0324\.3 u\0324\ k\0324\.1 k\0324\.2 + 2 / + Δ : 13 mat k\0324\.4 u\0324\ 3 k\0324\.1 * 9 k\0324\.3 * + 8 / + Δ : k\0324\.5 u\0324\ 3 k\0324\.1 * 9 k\0324\.3 * - 2 / + 6 k\0324\.4 * + Δ : 12 mat k\0324\.1 92 / k\0324\.3 * - 4 k\0324\.4 * + 12 / k\0324\.5 * - max abs 5 ε * ≤ u\0324\ u\0324\ k\0324\.1 4 k\0324\.4 * + k\0324\.5 + 2 / + eval : U\0324\ v\0324\ n\0324\ el u\0324\ v\0324\ el : h\0324\ K\032E\ h\0324\ * ho\032E\ 12 mat min eval : 13 mat h\0324\ h\0324\ K\032E\ / : n\0324\ U\0324\ cols : 12 mat if 31 line for if U\0324\ transpose eval 51 line :

eq v DSol eq set convert v set convert dsolve maple maple transpose 11 line :

Example: Ellipse

Parametrization of implicit functions by Draghilev method

x y f 4 x 2^* 9 y 2^* + 36 - :

ic 0212 mat :

u x y 12 mat :

T 01 0.05 range :

U x# y# f ic 0.01 54 DM.2 :

Numerical Draghilev method

ode var 12 mat x y f ic u DM.2 :

Symbolic Draghilev method

In case you don't have maple, this is the ode solution

t x.s t y.s 12 mat ode var DSol :

Ode solver(here we use maple)

t x.s t y.s 12 mat 312 t * sin * - 212 t * cos * 12 mat :

U ic x 12 red augment T x.s vectorize T y.s vectorize o augment 3 1 sys

Ode for solve

ode t x t diff 18 t y * - ≡ t y t diff 8 t x * ≡ 0 x 0 ≡ 0 y 2 ≡ 41 mat

x' f y diff - ≡ y' f x diff ≡ 21 mat

Simbolic solution

t x.s t y.s 21 sys 312 t * sin * - 212 t * cos * 21 sys

Example: Conic

Parametrization of implicit functions by Draghilev method

x y f 4 x 2^* 9 y 2^* + 7 x * y * + 4 x * - 6 y * + :

ic 0012 mat :

u x y 12 mat :

U x# y# f ic 0.01 66 DM.2 :

Numerical Draghilev method

In case you don't have maple, this is the ode solution

ode var 12 mat x# y# f ic u DM.2 :

Symbolic Draghilev method

t x.s t y.s 12 mat ode var DSol :

Ode solver(here we use maple)

t x.s t y.s 12 mat 695 sqrt 1 - t 95 sqrt * cos + (* 5 t 95 sqrt * sin * + (* 595 sqrt * / - 45 t 95 sqrt * sin * 95 sqrt 1 t 95 sqrt * cos - (* + (* 595 sqrt * / - 12 mat :

End point:

t.2 t y.s t 0.7 roots : 0.6446

T 0 t.2 0.05 range :

t.2 x.s t.2 y.s 21 sys 1.5161 1015 -^* - 1.0108 1015 -^* - 21 sys

U ic x 12 red augment T x.s vectorize T y.s vectorize o augment 3 1 sys

Ode for solve

ode t x t diff 7 t x * 613 t y * + (* + (- ≡ t y t diff 41 - t x + t x + (* 7 t y * + ≡ 0 x 0 ≡ 0 y 0 ≡ 41 mat

The symbolic solution for the ode is the parametric equation for the function

t x.s t y.s 21 sys 695 sqrt 1 - t 95 sqrt * cos + (* 5 t 95 sqrt * sin * + (* 595 sqrt * / - 45 t 95 sqrt * sin * 95 sqrt 1 t 95 sqrt * cos - (* + (* 595 sqrt * / - 21 sys

Draghilev application: Curve length and area

A t x.s t y.s t diff (* t 0 t.2 int : 3.0943

http://www.wolframalpha.com/input/?i=conic+4*x%5E2%2B9*y%5E2%2B7*x*y-4*x%2B6*y%3D0

L t x.s t diff (2^t y.s t diff (2^+ sqrt t 0 t.2 int : 6.979

Alvaro