## Solving a system of linear equations with a few known variables

Asked by Deepa Maheshvare

### Deepa Maheshvare (view profile)

on 11 Dec 2018
Latest activity Edited by Bruno Luong

### Bruno Luong (view profile)

on 11 Dec 2018
Accepted Answer by Bruno Luong

### Bruno Luong (view profile)

I'm solving the following system of linear equations,
Ax = b, some of the x's are knowns.
For example,
A=
-12 12 0 0 0
0 -1 1 0 0
0 0 -0.5 0.5 0
0 0 0 -17 17
x = [x1 x2 x3 x4 x5]
b = [b1 0 0 0 b5]
When some of the variables are known, say x1 and x5 are known, the system can be reduced in terms of the known variables. However, when there are around 50 variables and 5 are known re-writing the matrix in terms of the known variables is difficult.
I would like ask for suggestions on alternate ways of solving these kind of linear systems in which the values of a few variables are known.

KSSV

### KSSV (view profile)

on 11 Dec 2018
To solve b should be having a length equal to rows of A. Read about mldivide i,e \
Deepa Maheshvare

### Deepa Maheshvare (view profile)

on 11 Dec 2018
Thanks for the reply. Yes, I am considering a square matrix to solve for b.

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Answer by Bruno Luong

### Bruno Luong (view profile)

on 11 Dec 2018
Edited by Bruno Luong

### Bruno Luong (view profile)

on 11 Dec 2018

Assuming known is the logical index == TRUE for indexes of x that are known
% known = ismember(1:5,[1 5]) % in your example
x(known) = XValueYouKnow;
x(~known) = A(:,~known) \ (b-A(:,known)*x(known))

### madhan ravi (view profile)

on 11 Dec 2018
Edited by madhan ravi

### madhan ravi (view profile)

on 11 Dec 2018

One way using solve():
syms x1 x2 x3 x4 x5 b1 b5
eqn=[ -12*x1+12*x2==b1;
-x2+x3==0;
-0.5*x3+0.5*x4==0;
-17*x4+17*x5==b5];
[x1,x2,x3,x4]=solve(eqn)
Second way using linsolve():
syms b1 b5
A=[ -12 12 0 0 0
0 -1 1 0 0
0 0 -0.5 0.5 0
0 0 0 -17 17];
b = [b1;0;0;b5];
[x,R]=linsolve(A,b)
Third way using mldivide():
syms b1 b5
A=[ -12 12 0 0 0
0 -1 1 0 0
0 0 -0.5 0.5 0
0 0 0 -17 17];
b = [b1;0;0;b5];
A\b % x5 has infinity number of solutions I guess

### madhan ravi (view profile)

on 11 Dec 2018
ok now what's your query? Alternatively you can use lsqminnorm() as mentioned https://in.mathworks.com/help/matlab/ref/lsqminnorm.html#d120e717236
lsqminnorm(double(A),double(b))
Deepa Maheshvare

### Deepa Maheshvare (view profile)

on 11 Dec 2018
My query is,
eqn=[ 190*x1-190*x2==b1;
-190*x1+381*x2-190*x3==0;
-190*x2+381*x3-190*x4==0;
-190*x3+381*x4-190*x5==0;
-190*x4+190*x5==b5];
can be easily written for a small set of equations. When the size of matrix A is 50 x 50 , and 50 variables are present,it will be difficult to manually type all 50 equations in eqn=[]and use solve to reduce A matrix in terms of the unknowns.