Expert: George Moustris - 11/16/2006

Question
I have five linear simultaneous equations L1=0,L2=0,L3=0,L4=0, L5=0; having five variables A11, A12,A21, A22,Z. I have to solve for these variables.
The equations are as under:
L1=
1/40320*(1764*A11+7840*A22+1764*A11*sin(3/10*pi)+4536*A11*pi^2+2016*sin(1/10*pi)*A21*cos(3/10*pi)*sin(3/10*pi)+2016*A12*sin(1/10*pi)*cos(3/10*pi)*sin(3/10*pi)-3024*A22*cos(2/5*pi)*sin(2/5*pi)*cos(3/10*pi)*sin(3/10*pi)+4536/5*A22*cos(2/5*pi)*sin(2/5*pi)*pi-9072/5*A22*pi*cos(3/10*pi)*sin(3/10*pi)+58968/25*A22*pi^2-3024/5*A12*pi+2352*A22*sin(1/10*pi)-3528*A21*cos(3/10*pi)-3136*A22*sin(3/10*pi)-378*sin(1/5*pi)*A11*pi-6048*A12*cos(3/10*pi)*sin(3/10*pi)^2-6048*sin(3/10*pi)^2*A21*cos(3/10*pi)+1008*sin(1/5*pi)*A22*pi+12096*pi*A21*sin(3/10*pi)-3024*A11*pi*sin(3/10*pi)^3*cos(3/10*pi)-5040*sin(1/10*pi)*A21*pi-40824/5*A11*pi*cos(3/10*pi)*sin(3/10*pi)-1512*sin(2/5*pi)*A22*pi-5040*sin(1/10*pi)*A12*pi+12096*A12*sin(3/10*pi)*pi-3528*A21*cos(1/10*pi)-2352*A12*sin(1/5*pi)-2352*A22*cos(1/5*pi)+9408*A22*cos(2/5*pi)+1176*A21*sin(1/5*pi)-36288/5*sin(2/5*pi)*A11*pi-3024/5*A21*pi-3528*A12*sin(2/5*pi)+6048*A11*cos(3/10*pi)^2*sin(3/10*pi)^2)/pi^2-391/11520*w^2*(-144*A21*cos(1/10*pi)-96*A12*sin(1/5*pi)-96*A22*cos(1/5*pi)+384*A22*cos(2/5*pi)+72*A11+320*A22+48*A21*sin(1/5*pi)+72*A11*sin(3/10*pi)-864/5*sin(2/5*pi)*A11*pi-144*A12*sin(2/5*pi)+1296/25*A11*pi^2+864/5*pi*A21*sin(3/10*pi)-288/5*sin(1/10*pi)*A21*pi-288/5*sin(1/10*pi)*A12*pi+864/5*A12*sin(3/10*pi)*pi+96*A22*sin(1/10*pi)-144*A21*cos(3/10*pi)-128*A22*sin(3/10*pi))/pi^2-3519/10000*w^2*(A11*sin(3/20*pi)^2+A12*sin(3/20*pi)*sin(3/10*pi)+A21*sin(3/10*pi)*sin(3/20*pi)+A22*sin(3/10*pi)^2+Z)*sin(3/20*pi)^2;

L2=
1/40320*(-3136*A21*sin(3/10*pi)+9072/5*A21*pi^2+3024/5*A22*pi+2352*A21*sin(1/10*pi)-3024/5*A11*pi+10332/5*sin(1/5*pi)*A12*pi-4032*A12*sin(1/10*pi)*sin(3/10*pi)+6048*A21*sin(3/10*pi)^2+672*A21*sin(1/10*pi)^2-3528*A11*sin(2/5*pi)-18144/5*sin(2/5*pi)*A12*pi+588*A22*sin(1/5*pi)-588*A22*sin(2/5*pi)-2352*A11*sin(1/5*pi)-2352*A21*cos(1/5*pi)+9408*A21*cos(2/5*pi)+1764*A22*cos(1/10*pi)-4608*A22*sin(1/10*pi)*pi+672*A12*sin(1/10*pi)^2-882*A12*sin(1/10*pi)+77112/25*A12*pi^2+15120*pi*A22*sin(3/10*pi)+1008*sin(1/5*pi)*A21*pi-6048*A11*cos(3/10*pi)*sin(3/10*pi)^2-4032*A21*sin(1/10*pi)*sin(3/10*pi)-1008*sin(1/10*pi)*A22*cos(2/5*pi)*sin(2/5*pi)-5040*sin(1/10*pi)*A11*pi+6048*A12*sin(3/10*pi)^2-882*A12*sin(3/10*pi)-1512*sin(2/5*pi)*A21*pi+12096*A11*sin(3/10*pi)*pi+2016*A11*sin(1/10*pi)*cos(3/10*pi)*sin(3/10*pi)+3024*A22*cos(2/5*pi)*sin(2/5*pi)*sin(3/10*pi)+7840*A21)/pi^2-391/11520*w^2*(72*A22*cos(1/10*pi)+384*A21*cos(2/5*pi)-96*A21*cos(1/5*pi)-96*A11*sin(1/5*pi)-24*A22*sin(2/5*pi)+24*A22*sin(1/5*pi)+320*A21-36*A12*sin(3/10*pi)-432/5*sin(2/5*pi)*A12*pi-144*A11*sin(2/5*pi)-36*A12*sin(1/10*pi)+864/5*pi*A22*sin(3/10*pi)-288/5*A22*sin(1/10*pi)*pi-288/5*sin(1/10*pi)*A11*pi+216/5*sin(1/5*pi)*A12*pi+96*A21*sin(1/10*pi)+1296/25*A12*pi^2+864/5*A11*sin(3/10*pi)*pi-128*A21*sin(3/10*pi))/pi^2-3519/10000*w^2*(A11*sin(3/20*pi)^2+A12*sin(3/20*pi)*sin(3/10*pi)+A21*sin(3/10*pi)*sin(3/20*pi)+A22*sin(3/10*pi)^2+Z)*sin(3/20*pi)*sin(3/10*pi);

L3=
1/40320*(77112/25*A21*pi^2+3024/5*A22*pi-3024/5*A11*pi+1008*sin(1/5*pi)*A12*pi-4032*A12*sin(1/10*pi)*sin(3/10*pi)+6048*A21*sin(3/10*pi)^2+672*A21*sin(1/10*pi)^2-1512*sin(2/5*pi)*A12*pi+1176*A11*sin(1/5*pi)-882*A21*cos(1/5*pi)-882*A21*cos(2/5*pi)+1176*A22*cos(1/10*pi)-4608*A22*sin(1/10*pi)*pi+672*A12*sin(1/10*pi)^2+2352*A12*sin(1/10*pi)+9072/5*A12*pi^2+15120*pi*A22*sin(3/10*pi)+10332/5*sin(1/5*pi)*A21*pi-6048*A11*cos(3/10*pi)*sin(3/10*pi)^2-4032*A21*sin(1/10*pi)*sin(3/10*pi)-1008*sin(1/10*pi)*A22*cos(2/5*pi)*sin(2/5*pi)-5040*sin(1/10*pi)*A11*pi+6048*A12*sin(3/10*pi)^2-3136*A12*sin(3/10*pi)-18144/5*sin(2/5*pi)*A21*pi+12096*A11*sin(3/10*pi)*pi+2016*A11*sin(1/10*pi)*cos(3/10*pi)*sin(3/10*pi)+3024*A22*cos(2/5*pi)*sin(2/5*pi)*sin(3/10*pi)+588*A22*cos(3/10*pi)-3528*A11*cos(1/10*pi)-2352*A12*cos(1/5*pi)+9408*A12*cos(2/5*pi)-3528*A11*cos(3/10*pi)+7840*A12)/pi^2-391/11520*w^2*(384*A12*cos(2/5*pi)-96*A12*cos(1/5*pi)-144*A11*cos(1/10*pi)+24*A22*cos(3/10*pi)+48*A22*cos(1/10*pi)-36*A21*cos(1/5*pi)+320*A12+48*A11*sin(1/5*pi)-432/5*sin(2/5*pi)*A21*pi+864/5*A11*sin(3/10*pi)*pi-288/5*sin(1/10*pi)*A11*pi-288/5*A22*sin(1/10*pi)*pi+216/5*sin(1/5*pi)*A21*pi+96*A12*sin(1/10*pi)+1296/25*A21*pi^2+864/5*pi*A22*sin(3/10*pi)-144*A11*cos(3/10*pi)-128*A12*sin(3/10*pi)-36*A21*cos(2/5*pi))/pi^2-3519/10000*w^2*(A11*sin(3/20*pi)^2+A12*sin(3/20*pi)*sin(3/10*pi)+A21*sin(3/10*pi)*sin(3/20*pi)+A22*sin(3/10*pi)^2+Z)*sin(3/20*pi)*sin(3/10*pi);

L4=
1/40320*(7840*A11+441*A22+1512*A22*pi*sin(2/5*pi)^3*cos(2/5*pi)+1764*A12*cos(1/10*pi)-2352*A11*cos(1/5*pi)+9408*A11*cos(2/5*pi)-1008*sin(1/10*pi)*A12*cos(2/5*pi)*sin(2/5*pi)-1008*A21*cos(2/5*pi)*sin(2/5*pi)*sin(1/10*pi)+1512*A22*cos(2/5*pi)^2*sin(2/5*pi)^2+2352*A11*sin(1/10*pi)+3024*A12*cos(2/5*pi)*sin(2/5*pi)*sin(3/10*pi)+3024*A21*cos(2/5*pi)*sin(2/5*pi)*sin(3/10*pi)-3024*A11*cos(2/5*pi)*sin(2/5*pi)*cos(3/10*pi)*sin(3/10*pi)+4536/5*A11*cos(2/5*pi)*sin(2/5*pi)*pi-3136*A11*sin(3/10*pi)+58968/25*A11*pi^2+20412/5*A22*cos(2/5*pi)*sin(2/5*pi)*pi+4536*A22*pi^2+3024/5*A12*pi+588*A21*cos(3/10*pi)+1008*sin(1/5*pi)*A11*pi+18144/5*sin(1/5*pi)*A22*pi+15120*pi*A21*sin(3/10*pi)-4608*sin(1/10*pi)*A21*pi-9072/5*A11*pi*cos(3/10*pi)*sin(3/10*pi)+189*sin(2/5*pi)*A22*pi-4608*sin(1/10*pi)*A12*pi+15120*A12*sin(3/10*pi)*pi+1176*A21*cos(1/10*pi)+588*A12*sin(1/5*pi)-441*A22*cos(2/5*pi)-1512*sin(2/5*pi)*A11*pi+3024/5*A21*pi-588*A12*sin(2/5*pi))/pi^2-391/11520*w^2*(72*A12*cos(1/10*pi)+24*A21*cos(3/10*pi)-96*A11*cos(1/5*pi)+384*A11*cos(2/5*pi)-24*A12*sin(2/5*pi)+48*A21*cos(1/10*pi)+24*A12*sin(1/5*pi)+18*A22+320*A11-18*A22*cos(2/5*pi)-288/5*sin(1/10*pi)*A21*pi+864/5*A12*sin(3/10*pi)*pi-288/5*sin(1/10*pi)*A12*pi+432/5*sin(1/5*pi)*A22*pi+1296/25*A22*pi^2+864/5*pi*A21*sin(3/10*pi)+96*A11*sin(1/10*pi)-128*A11*sin(3/10*pi))/pi^2-3519/10000*w^2*(A11*sin(3/20*pi)^2+A12*sin(3/20*pi)*sin(3/10*pi)+A21*sin(3/10*pi)*sin(3/20*pi)+A22*sin(3/10*pi)^2+Z)*sin(3/10*pi)^2;

L5=
-3519/10000*w^2*(A11*sin(3/20*pi)^2+A12*sin(3/20*pi)*sin(3/10*pi)+A21*sin(3/10*pi)*sin(3/20*pi)+A22*sin(3/10*pi)^2+Z)+210420*Z;

Plase suggest how to form a matrix to solve it.

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The text above is a follow-up to ...

-----Question-----
In matlab , I get  five simultaneous equations in five variables( say x1 x2 x3 x4 x5) . In every equation , several times x1 x2 etc gets repeated. Is there any method so that we can rearrange the all coefficients of one variable at one place.example, we get, 2x1+2x2+5x3+6x4+7x5-9x1-2x3-66x3-6262x1+ so many terms
I want (    )x1+ (      )x2+(    )x3+(    )x4+(   )x5=0;
Alternatively can I directly convert an equation into matrix form.
-----Answer-----
Hi vinayakranjan,

i can't say i understand fully what you want to do. You say you get 5 simultaneous equations in Matlab. Are these equations linear? In other words, you want to solve a 5x5 linear system? Are the equations non linear? The equations are in what form? Symbolic math or matrices?
Please explain in detail what you want to do and what your problem is.

Best
G.

Answer
The only way to do this is by symbolic manipulation. Matrix forms cannot do what you ask. You must declare each variable as a symbolic variable e.g. A11=sym('A11');A22=sym('A2');Then, if you type L1, Matlab will automatically simplify the equations. You can even solve the system symbolically. I also noticed that you have 2 more unknowns; w and Z. You must either define them numerically (e.g. w=2) or convert them to symbolic variables as well (w=sym('w'); Z=sym('Z');). Matlab will treat them as free variables and can solve the system symbolically .If you have trouble, read the help file on the symbolic toolbox. It has many examples.

Best
G.