I want to solve this equation. please help me
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3.8 = a (1 - exp(b×8.2)) + c (exp(d×8.1) - 1);
27.9 = a (1 exp(b×63)) + c (exp(d×35.667) - 1);
23.5 = a (1 - exp(b×59.5)) + c (exp(d×59.5) - 1);
13.7 = a (1 - exp(b×35.7)) + c (exp(d×35.7) - 1)};
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Answers (2)
madhan ravi
on 9 Oct 2018
syms a b c d
eq1 = 3.8 == a*(1 - exp(b*8.2)) + c*(exp(d*8.1) - 1);
eq2 = 27.9 == a*(1 - exp(b*63)) + c*(exp(d*35.667) - 1);
eq3 = 23.5 == a*(1 - exp(b*59.5)) + c*(exp(d*59.5) - 1);
eq4 = 13.7 == a*(1 - exp(b*35.7)) + c*(exp(d*35.7) - 1);
[a,b,c,d]=vpasolve(eq1,eq2,eq3,eq4,a,b,c,d)
10 Comments
Walter Roberson
on 10 Oct 2018
Looking again, there are a lot more points with very close residue than I had thought I had. Residues that are approximately 0.44011 were found for over 550 points, with a values ranging from 1337.8 to 171822.3, with b from -0.000388 to -2.965e-06, with c from 52.7 to 114.7, and d from -0.00244 to -0.00104 . a and b are practically linear with each other, and c and d are practically linear with each other, but the other pairs are only mostly linear with each other.
The residue was being calculated as
F = (a*(exp((41*b)/5) - 1) - c*(exp((81*d)/10) - 1) + 19/5)^2 + (a*(exp((119*b)/2) - 1) - c*(exp((119*d)/2) - 1) + 47/2)^2 + (a*(exp((357*b)/10) - 1) - c*(exp((357*d)/10) - 1) + 137/10)^2 + (a*(exp(63*b) - 1) - c*(exp((35667*d)/1000) - 1) + 279/10)^2
My tests did not show any sign of a "bowl" -- no minima that the values then rise again on the other side. I think doing better would require higher precision.
The marginally best combination I found was near 33762.2195025919 -1.50991408545558e-05 110.079764424126 -0.00109033033598055 but near 740.322188783004 -0.000712129376445178 37.1451062324705 -0.00368725773398294 is not far off
Alex Sha
on 19 Dec 2019
Here is the right solution, obtained from 1stOpt:
a: -2.38880243947768E-9
b: 0.357245116257183
c: -32.3471147348507
d: -0.0154282661155712
4 Comments
Walter Roberson
on 19 Dec 2019
The residue
F = (a*(exp((41*b)/5) - 1) - c*(exp((81*d)/10) - 1) + 19/5)^2 + (a*(exp((119*b)/2) - 1) - c*(exp((119*d)/2) - 1) + 47/2)^2 + (a*(exp((357*b)/10) - 1) - c*(exp((357*d)/10) - 1) + 137/10)^2 + (a*(exp(63*b) - 1) - c*(exp((35667*d)/1000) - 1) + 279/10)^2
is quite small, on the order of 1e-23 which is quite good for this situation.
Alex Sha
on 20 Dec 2019
Hi, for "(a*(exp((41*b)/5) - 1) - c*(exp((81*d)/10) - 1) + 19/5)^2 + (a*(exp((119*b)/2) - 1) - c*(exp((119*d)/2) - 1) + 47/2)^2 + (a*(exp((357*b)/10) - 1) - c*(exp((357*d)/10) - 1) + 137/10)^2 + (a*(exp(63*b) - 1) - c*(exp((35667*d)/1000) - 1) + 279/10)^2", the minmum value:
Objective Function (Min.): 1.67130234831156E-23
a: -2.38880243946672E-9
b: 0.357245116257267
c: -32.347114734847
d: -0.015428266115563
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