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Numerical Linear Algebra

# Numerical Linear Algebra - Chapter 16 Numerical Linear...

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Chapter 16 Numerical Linear Algebra 16.1 Sets of Linear Equations MATLAB was developed to handle problems involving matrices and vectors in an efficient way. One of the most basic problems of this type involves the solution of a system of linear equations. The built-in matrix algebra functions of MATLAB facilitate the process of establishing whether a solution exists and finding it when it does. For example, the 4 × 3 system below 4 7 5 1 2 3 4 0 3 0 x y z x y z x y z x y z + + = - + + = + + = - - = is a system of equations which can be represented as Ax = b where 1 4 7 -5 1 1 1 1 , = , = 2 3 4 0 1 1 3 0 x A x y b z = - - To check if a unique solution exists, the MATLAB function ' rank ' can be used with the coefficient matrix A and the augmented matrix ( A|b ). If the ranks of both matrices are equal to the number of unknown variables, in this case 3, a unique solution exists. If the ranks are equal and less than 3, an infinite number of solutions exist, i.e. the system is underconstrained. Finally if the rank of A is one less than the rank of ( A|b ), the equations are inconsistent, i.e the system is overconstrained and there is no solution. Example 16.1.1 A=[1,4,7; 1,1,1; 2,3,4; 1 -1 -3]; b=[-5;1;0;0]; rA=rank(A) % Check rank of A rAb=rank([A b]) % Check rank of (A|b) rA = 2 rAb = 3 Since the ranks are unequal, the system of equations are inconsistent and no solution exists. Suppose the A (4,3) element were +3 instead of -3. 1

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Example 16.1.2 A=[1,4,7; 1,1,1; 2,3,4; 1 -1 3]; b=[-5;1;0;0]; rA=rank(A) % Check rank of A rAb=rank([A b]) % Check rank of (A|b) rA = 3 rAb = 3 In this case, the ranks are both equal to the number of unknowns so there is a unique solution to the system. The solution can be determined by transforming the augmented matrix into an echelon form using the ' rref ' function, which stands for row reduced echelon form, a Gauss elimination type of solution. Example 16.1.3 format rat rref([A b]) ans = 1 0 0 13/6 0 1 0 -1/3 0 0 1 -5/6 0 0 0 0 From the echelon form of ( A|b ) the solution is given by x = 13/6, y = -1/3, z =-5/6 When the coefficient matrix A is square (same number of equations as unknowns) the MATLAB function ' det ' will also reveal whether there is a unique solution. A
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