820b889b-2bc7-497b-a956-55d3255c21c8 7 4 5 false false 0 0 decimal

Find the real roots of

d f 0.5 5.96 π 32 / 0.033 4^d 4^- (0.033 / * / (* 0.5 5.96 π 32 / 0.033 4^d 4^- (0.033 / * / (2^46114 / π * 0.033 2^d 2^- (* / (2^* + sqrt (* + 400106^* - :

Isolating a from

0.5 a * 0.5 a 2^4 b 2^* + sqrt * + c - 0 ≡

we get

ao b c - (b c + (* c / - :

Identifying

a 5.96 π 32 / 0.033 4^d 4^- (0.033 / * / :

b 61 π 4 / 0.033 2^d 2^- (* / :

c 400106^* :

the roots of f are in the roots of the polynom P

d P a ao - numden 1 el :

which is a polynom of degree 8 because

x P x 9 diff 0

for all x in R

Extracting the polynomial coefficients

x 0 : k 08 range C k 1 + el 1 k! / x P x k diff * : for 21 line

C 1030^/ 0.00000000552883 - 0 0.000010153958008 0 0.000019772281339 0 8.59839935422905 - 0 3947.84176043574 91 mat

We can try to solve it with

C polyroots 0.032817448845286 0.032817448316602 - 0.033073822796661 - 0.000145703373189 i * + 0.033073822796661 - 0.000145703373189 i * - 0.033073822531374 0.000145695492564 i * + 0.033073822531374 0.000145695492564 i * - 9.45632461224477 1013 -^* 0.032965102242803 i * + 9.45632461224477 1013 -^* 0.032965102242803 i * - 81 mat

but we can make a better calculus. Notice that all coefficients of odd powers are zero. Thus, this polynom can be symbolically solved as a quartic equation in d^2. Let's try to solve it numerically.

Solving for d^2

r2 C 193 range el polyroots : 0.001086869787103 0.001088249362357 0.001089578855439 0.001086698004898 - 41 mat

ro r2 sqrt vectorize : 0.032967708247663 0.032988624741826 0.033008769371775 0.032965102834634 i * 41 mat

we can get some numerical aproach to the roots

Lets to find intervals with amplitude 2ε for the roots using above results as guesses

ε 1015 -^:

d.1 d P d ro 1 el roots : 0.03296485612112

d.1 ε - f 0.0071 -

d.1 ε + f 0.0156

d.2 d P d ro 3 el roots : 0.033000246667148

d.2 ε - f 1.206

d.2 ε + f 2.0333 -

For symmetry

d.3 d.1 - :

d.3 ε - f 0.0156

d.4 d.2 - :

d.4 ε - f 2.0333 -

d.3 ε + f 0.0071 -

d.4 ε + f 1.206

That is:four real roots, very closed to the discontinuity at d = 0.033. Notice that using do(2) as guess we get it

d P d ro 2 el roots 331000 /

Alvaro