- #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: