Let us see how to solve a
system of linear equations in MATLAB. Here are the various operators that we will be deploying to execute our task :
- \ operator :
A \ B is the matrix division of A into B, which is roughly the same as INV(A) * B. If A is an NXN matrix and B is a column vector with N components or a matrix with several such columns, then X = A \ B is the solution to the equation A * X = B. A warning message is printed if A is badly scaled or nearly singular. A\EYE(SIZE(A)) produces the inverse of A.
- linsolve operator :
X = LINSOLVE(A, B) solves the linear system A * X = B using LU factorization with partial pivoting when A is square, and QR factorization with column pivoting. A warning is given if A is ill conditioned for square matrices and rank deficient for rectangular matrices.
Example 1 : Non-homogeneous System Ax = b, where A is a square and is invertible. In our example we will consider the following equations :
2x + y - z = 7
x -2y + 5z = -13
3x + 5y - 4z = 18
We will convert these equations into matrices A and b :
Output :
A =
2 1 -1
1 -2 5
3 5 -4
b =
7
-13
18
Now we will create an augmented matrix Ab. We will compare the ranks of Ab and A, if the ranks are equal then a unique solution exists.
Output :
Ab =
2 1 -1 7
1 -2 5 -13
3 5 -4 18
Unique solution exists
Now we can find the solution to this system of equations by using 3 methods:
- conventional way :
inv(A) * b
- using mid-divide routine :
A \ b
- using linsolve routine :
linsolve(A, b)
Output :
x_inv =
2.0000e+00
8.8818e-16
-3.0000e+00
x_bslash =
2.0000e+00
9.6892e-16
-3.0000e+00
x_linsolve =
2.0000e+00
9.6892e-16
-3.0000e+00
We can verify the correctness of the solution by finding the error using
A * x - b. The error should be 0.
Output :
Er1 =
-8.8818e-16
-3.5527e-15
0.0000e+00
Er2 =
-1.7764e-15
-1.7764e-15
0.0000e+00
Er3 =
-1.7764e-15
-1.7764e-15
0.0000e+00
As all the errors are close to 0, we can say that the solution is correct.
Example 2 : Non-homogeneous system Ax = b, where A is a square and it is not invertible. In our example we will consider the following equations :
2x + 4y + 6z = 7
3x -2y + 1z = 2
1x + 2y + 3z = 5
Output :
A =
2 4 6
3 -2 1
1 2 3
b =
7
2
5
Ab =
2 4 6 7
3 -2 1 2
1 2 3 5
Unique solution does not exist
warning: matrix singular to machine precision
warning: called from
testing at line 17 column 7
x_inv =
Inf
Inf
Inf
warning: matrix singular to machine precision
warning: called from
testing at line 21 column 10
x_bslash =
-Inf
-Inf
Inf
Er1 =
Inf
NaN
Inf
Er2 =
NaN
NaN
NaN
Example 3 : Non-homogeneous system Ax = b where A is not a square. In our example we will consider the following equations :
2a + c - d + e = 2
a + c - d + e = 1
12a + 2b + 8c + 2e = 12
Output :
A =
2 0 1 -1 1
1 0 1 -1 1
12 2 8 0 2
b =
2
1
12
Ab =
2 0 1 -1 1 2
1 0 1 -1 1 1
12 2 8 0 2 12
Solution exists
Unique solution does not exist
Example 4 : Homogeneous system Ax = 0 where A is a square and is invertible. In our example we will consider the following equations :
6x + 2y + 3z = 0
4x - y + 2z = 0
2x + y + 5z = 0
Output :
A =
6 2 3
4 -1 2
2 1 5
b =
0
0
0
Unique solution exists
x =
0
0
0
x = [](3x0)
Example 5 : Homogeneous system Ax = 0 where A is a square and is not invertible. In our example we will consider the following equations :
1x + 2y + 3z = 0
4x + 5y + 6z = 0
7x + 8y + 9z = 0
Output :
A =
1 2 3
4 5 6
7 8 9
b =
0
0
0
Unique solution does not exist
warning: matrix singular to machine precision, rcond = 1.54198e-18
warning: called from
testing at line 13 column 3
x =
0
0
0
x =
0.40825
-0.81650
0.40825
Err =
-1.3323e-15
-4.4409e-16
4.4409e-16
