5f8ff4f5-7091-48c1-a525-d402a4d4e555 35 6 9 decimal

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

1. Quadratic roots

a b c f a 2^2 b * - 2 c * / 2 - :

b 1 : c 2 : 21 line

s x b c f x 5 - 5 solve : 3.162278 - 3.162278 21 mat

xy s 1 el 0.15 red s 2 el 0.15 red 25 mat :

x b c f xy 2 1 sys

2. Solving a system of implicit fncts

System of implicit fncts x y f1 x 2^y 2^+ 5 - : x y f2 5 x * 2 y * - 1 - : 21 line

f1 f2 2 1 sys

In mathematics, an "Implicit fnct" is implicit of f(x,y) = 0 i.e: plot the contour of level=0 Maxima X_Y does that for you.

1.AutoSolve w/o guesses

2.Option that may require guesses

x F x 1 el (2^x 2 el (2^+ 5 - 5 x 1 el * 2 x 2 el * - 1 - 21 mat :

r x y f1 x y f2 21 mat x y 21 mat roots : 1221 mat

x x 1 el x 2 el 21 mat :

guess 0021 mat :

ImplicitRoots x F x guess roots : 1221 mat

Sanity check r 1 el r 2 el f1 r 1 el r 2 el f2 21 mat 0021 mat

Sanity check ImplicitRoots F 0021 mat

3. Solving an nth order linear system

A 51 - 51 - 7 - 923 - 333 mat :

b 7 - 61931 mat :

x y z f A x y z 31 mat * b - :

xyz x y z f x y z 31 mat roots : 4.042553 11.351064 - 7.712766 - 31 mat

4. Solving "polyroots"

Undocumented

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

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

x 1 :

coeffs x expr :

coeffs transpose 42 - 27 - 135 - 115 mat

r coeffs polyroots : 0.942929 - 0.940248 3.17495 i * + 0.940248 3.17495 i * - 4.062433 41 mat

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

5. Solving "wisely" the classical

x f 6 x 3^* 7 x 2^* + 94 x * - 105 + :

Intuitively, roots look weird w x f x 5 - 2.5 solve : 5 - 1.49999999347833 2.3333333284785 31 mat

x f 100 2 1 sys

2.333333 73 / ≡

last two roots confirmed inexact.
You can guess, but that's no maths w 1 el f w 2 el f w 3 el f 31 mat 0 0.000000211954287 0.000000178010538 - 31 mat

x f 100 ≡ x 5 - 4 solve 4.606745 - 0.053414 3.386664 31 mat

5/a. Precise the two roots

a 1 : b 2 : xn b a + (2 / : ε 1010 -^: 41 line

Dichotomy n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

.. sanity ..

xn 1.5

xn f 0

n 0

a 2 : b 2.5 : xn b a + (2 / : ε 1012 -^: 41 line

Dichotomy n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

.. sanity ..

xn 2.33333333333334

xn f 0

n 42

========================= solve block NOT robust ============================

x f x exp x 2^+ 5 x * - :

The solve block is underdesigned: It should include the dichotomy module for each root.

x f

s x f x 02 solve : 0.280490945389913 1.73399571258953 21 mat

s 1 el f s 2 el f 21 mat 2.034 1010 -^* - 2.824 109 -^* - 21 mat

Address dichotomy to refine roots ρ n Solve a ρ n el 0.1 ρ n el * - : b ρ n el 0.1 ρ n el * + : xn b a + (2 / : ε 1015 -^: n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 61 line xn 21 line :

X s 1 Solve s 2 Solve 21 mat : 0.280490945324613 1.73399571327305 21 mat

X 1 el f X 2 el f 21 mat 0021 mat

Extend to 'n' roots

X 1 el f X 2 el f # 31 mat

X s 1 Solve s 2 Solv # 31 mat :