Matlab Tutorial 5: Linear Equations

In this Matlab tutorial we will deal with linear equations, the least square method, condition numbers over & under-determined equations.

Let’s start with an example.

Example 1

We have an equation system with three unknown variables and three equations. What will be the solution to the system below?

3w-2y+4z=8
5w+8y-6z=-5
9w-2y+7z=-17

Assume a matrix A containing the coefficients multiplied with x, y and z, and a vector with the numbers on the right-hand side of the equations. We can thus rewrite our equations as:

AX=b , where X contains the unknown (w, y and z), A and b are shown below.

b=     A=
  8    3 -2  4
 -5    5  8 -6
-17    9 -2  7

How will we find the solution?

>> X=inv(A)*b

or

>> X=A\b

both gives a correct answer. The last method produces a Gaussian elimination if A is a quadratic matrix. In our case X becomes:

X=
-36.7778
71.2778
65.2222

This is no exact solution, only an approximation. Try also a specific command:

>> lsqr(A,b)

The condition number of an equation can be examined. It means in short, that you can investigate the sensitivity of a linear equation system to disturbances in A or b. The condition number is always >1. The greater the sensitivity, the greater the number.

Calculate the sensitivity in our system.

>> cond(A)

The condition number partly indicates what kind of trust one should put in a solution given from lsqr.

>> cond(A\b) or cond(lsqr(A,b))

As you have seen there are many different commands to use for solving equation systems. Wee will look at a few. They use the same argument as lsqr. Try the following: pcg, qmr, symmlq, minres . Some of them are successful and one or two will fail. try to figure out which one of these methods produces a correct answer?

Round-off errors can be magnified and we can lose accuracy. For a badly conditioned matrix A, A*A^-1 is not equal to the identity matrix. Return to the matrix A and change the element A(3,3). See below!

A =
3.0000 -2.0000 4.0000
5.0000 8.0000 -6.0000
9.0000 -2.0000 7.5600

What will the solution become and what happens to the condition number? The right-hand side is the same as previous. The equation system above has been applied to a quadratic matrix A (nxn) , where b is a column vector with n elements.

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