- #1
Dilon
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(I hope this is not a double posting) I want to solve this system of equations, containing logarithmic terms:
##7\ln(a/b)+A = 7\ln(d/e)+D = 7\ln(g/h)+G##
##7\ln(a/c)+B = 7\ln(d/f)+E = 7\ln(g/i)+H##
##7\ln(b/c)+C = 7\ln(e/f)+F = 7\ln(h/i)+I##
##a\phi_1+d\phi_2+g\phi_3=X##
##b\phi_1+e\phi_2+h\phi_3=Y##
##c\phi_1+f\phi_2+i\phi_3=Z##
[itex]a+b+c=1[/itex]
[itex]d+e+f=1[/itex]
[itex]g+h+i=1[/itex]
where the uppercase and [itex]\phi[/itex] coefficients are known and [itex]a,b,c...i[/itex] are the unknown coefficients.
The following are also true, but might not be important since the values are known:
[itex]\phi_1+\phi_2+\phi_3=1[/itex]
[itex]X+Y+Z=1[/itex]
My strategy so far:
I introduce the unknowns [itex]n_1,n_2, n_3[/itex] to link the equations as:
##7\ln(a)-7\ln(b)-n_1 = -A##
##7\ln(a)-7\ln(c)-n_2 = -B##
##7\ln(b)-7\ln(c)-n_3 = -C##
##7\ln(d)-7\ln(e)-n_1=-D##
##7\ln(d)-7\ln(f)-n_2=-E##
##7\ln(e)-7\ln(f)-n_3=-F##
##7\ln(g)-7\ln(h)-n_1=-G##
##7\ln(g)-7\ln(i)-n_2=-H##
##7\ln(h)-7\ln(i)-n_3=-I##
The corresponding system of equations maybe looks like this:
[tex]
\begin{bmatrix}
7L & -7L & & & & & & & & -1 & & \\
7L & & -7L & & & & & & & & -1 & \\
& 7L & -7L & & & & & & & & & -1\\
& & & 7L & -7L & & & & & -1 & & \\
& & & 7L & & -7L & & & & & -1 & \\
& & & & 7L & -7L & & & & & & -1\\
& & & & & & 7L & -7L & & -1 & & \\
& & & & & & 7L & & -7L & & -1 & \\
& & & & & & & 7L & -7L & & & -1\\
\phi_1 & & & \phi_2 & & & \phi_3 & & & & & \\
& \phi_1 & & & \phi_2 & & & \phi_3 & & & & \\
& & \phi_1 & & & \phi_2 & & & \phi_3 & & & \\
1 & 1 & 1 & & & & & & & & & \\
& & & 1 & 1 & 1 & & & & & & \\
& & & & & & 1 & 1 & 1 & & & \\
\end{bmatrix}
\begin{bmatrix}
a\\
b\\
c\\
d\\
e\\
f\\
g\\
h\\
i\\
n_1\\
n_2\\
n_3
\end{bmatrix}
=
\begin{bmatrix}
-A\\
-B\\
-C\\
-D\\
-E\\
-F\\
-G\\
-H\\
-I\\
X\\
Y\\
Z\\
1\\
1\\
1
\end{bmatrix}
[/tex]
where the "L" terms indicate that there is actually a natural log of the variable.
My main problem is: What do I do with the logarithmic terms (the 7L terms indicate 7*ln(unknown))? I want something that can use computer solvers so I need to build a system kind of like I've done, but I'm not sure how to do it even if I am on the right track. Do I need to decompose the system into a logarithmic part and a linear part first? or what?
##7\ln(a/b)+A = 7\ln(d/e)+D = 7\ln(g/h)+G##
##7\ln(a/c)+B = 7\ln(d/f)+E = 7\ln(g/i)+H##
##7\ln(b/c)+C = 7\ln(e/f)+F = 7\ln(h/i)+I##
##a\phi_1+d\phi_2+g\phi_3=X##
##b\phi_1+e\phi_2+h\phi_3=Y##
##c\phi_1+f\phi_2+i\phi_3=Z##
[itex]a+b+c=1[/itex]
[itex]d+e+f=1[/itex]
[itex]g+h+i=1[/itex]
where the uppercase and [itex]\phi[/itex] coefficients are known and [itex]a,b,c...i[/itex] are the unknown coefficients.
The following are also true, but might not be important since the values are known:
[itex]\phi_1+\phi_2+\phi_3=1[/itex]
[itex]X+Y+Z=1[/itex]
My strategy so far:
I introduce the unknowns [itex]n_1,n_2, n_3[/itex] to link the equations as:
##7\ln(a)-7\ln(b)-n_1 = -A##
##7\ln(a)-7\ln(c)-n_2 = -B##
##7\ln(b)-7\ln(c)-n_3 = -C##
##7\ln(d)-7\ln(e)-n_1=-D##
##7\ln(d)-7\ln(f)-n_2=-E##
##7\ln(e)-7\ln(f)-n_3=-F##
##7\ln(g)-7\ln(h)-n_1=-G##
##7\ln(g)-7\ln(i)-n_2=-H##
##7\ln(h)-7\ln(i)-n_3=-I##
The corresponding system of equations maybe looks like this:
[tex]
\begin{bmatrix}
7L & -7L & & & & & & & & -1 & & \\
7L & & -7L & & & & & & & & -1 & \\
& 7L & -7L & & & & & & & & & -1\\
& & & 7L & -7L & & & & & -1 & & \\
& & & 7L & & -7L & & & & & -1 & \\
& & & & 7L & -7L & & & & & & -1\\
& & & & & & 7L & -7L & & -1 & & \\
& & & & & & 7L & & -7L & & -1 & \\
& & & & & & & 7L & -7L & & & -1\\
\phi_1 & & & \phi_2 & & & \phi_3 & & & & & \\
& \phi_1 & & & \phi_2 & & & \phi_3 & & & & \\
& & \phi_1 & & & \phi_2 & & & \phi_3 & & & \\
1 & 1 & 1 & & & & & & & & & \\
& & & 1 & 1 & 1 & & & & & & \\
& & & & & & 1 & 1 & 1 & & & \\
\end{bmatrix}
\begin{bmatrix}
a\\
b\\
c\\
d\\
e\\
f\\
g\\
h\\
i\\
n_1\\
n_2\\
n_3
\end{bmatrix}
=
\begin{bmatrix}
-A\\
-B\\
-C\\
-D\\
-E\\
-F\\
-G\\
-H\\
-I\\
X\\
Y\\
Z\\
1\\
1\\
1
\end{bmatrix}
[/tex]
where the "L" terms indicate that there is actually a natural log of the variable.
My main problem is: What do I do with the logarithmic terms (the 7L terms indicate 7*ln(unknown))? I want something that can use computer solvers so I need to build a system kind of like I've done, but I'm not sure how to do it even if I am on the right track. Do I need to decompose the system into a logarithmic part and a linear part first? or what?
Last edited: