Solution of Simultaneous Linear Algebraic Equations

Transcription

1 Outlines Solution of Simultaneous Linear Algebraic Equations February 18, 2008

2 Outlines Part I: Review of Previous Lecture Part II: Review of Previous Lecture

3 Outlines Part I: Review of Previous Lecture Part II: Graphical Interpretation

4 Part I Review of Previous Lecture

5 Review of Previous Lecture Introduction Basic form of the equations Multiplication in matrix algebra Solution types Applications and problem setup: scaffolding Basic concepts of solution Determinants and rank Rank and determinant example

6 Part II Graphical Interpretation, Solvable and Unsolvable Problems, Linear Dependence,

7 Graphical Interpretation A system of equations with two unknowns x 1 and x 2 can be represented as a two-dimensional graph. Since each equation in the system is linear, they can be represented as straight lines on the graph. Locations where equation lines intersect represent values of (x 1, x 2 ) that satisfy those equations.

8 Graphical Interpretation Example

9 Recall from the previous lecture the conditions a non-homogeneous system of equations must satisfy in order to have a solution A non-homogeneous system of equations is one where {b} {0}. It has a proper solution if and only if rank(a) = rank([a b]) = n, where n is the number of unknowns in the system.

10 Normal Case: 2 Equations, 2 Unknowns, 1 Solution [ ] { x1 x 2 } = { 5 3 } Since n = rank(a) = rank([a unique solution. n = 2 ([ ]) 3 1 det = ([ ]) 3 1 rank = ([ ]) rank = b]), this system of equations has a

11 Abnormal Case: 2 Equations, 2 Unknowns, Infinite Solutions [ ] { x1 x 2 } = { 5 10 } n = 2 ([ ]) 3 1 det = ([ ]) 3 1 rank = ([ ]) rank = Since rank(a) = rank([a b]), this system of equations has a solution. But since n rank(a), no unique solution exists.

12 Abnormal Case: 2 Equations, 2 Unknowns, 0 Solutions [ ] { x1 x 2 } = { 5 15 } Since rank(a) rank([a solution. n = 2 ([ ]) 3 1 det = ([ ]) 3 1 rank = ([ ]) rank = b]), this system of equations has no

13 Abnormal Case: 3 Equations, 2 Unknowns, 0 Solutions { x1 x 2 } = n = det 1 2 = N/A rank 1 2 = rank = Since rank(a) rank([a solution. b]), this system of equations has no

14 Abnormal Case: 3 Equations, 2 Unknowns, 1 Solution { x1 x 2 } = n = det 1 1 = N/A rank 1 1 = rank = Since rank(a) = rank([a b]) = n, this system of equations has 1 solution.

15 Summary # Eq. # Unk. det(a) rank(a) rank([a b]) Result Solution Solutions No Solutions 3 2 N/A 2 3 No Solutions 3 2 N/A Solution

16 Each equation in a system of equations should be independent of all other equations in the system. Linearly dependent equations: x 1 x 2 + x 3 = 3 (1) 2x 1 + x 2 x 3 = 0 (2) 8x 1 + x 2 x 3 = 6 (3) If you multiply Equation 2 by 3, multiply Equation 1 by 2, then add up the results, you get Equation 3. Any values of x 1, x 2, and x 3 that satisfy the first two equations will automatically satisfy the third one.

17 Linear Dependence Graph The graphical interpretation of a system of equations that are not linearly independent is that one line lies directly on top of another one. This prevents you from finding a unique solution to the system of equations.

18 Linear Dependence Test Check the determinant of the coefficient matrix to see if the system of equations is linearly independent: = 0 If the determinant of the coefficient matrix is 0, then the system of equations is linearly dependent. If the determinant is very small, roundoff errors during computations may push the system into linear dependence, preventing an accurate solution.

19 Ill-Conditioned Equations A potential source of problems often occurring alongside linearly-dependent equations are ill-conditioned equations. A system of equations is ill-conditioned if small changes in the coefficients on either side of the equation create large variations in the solution. For example: x 1 x 2 = 5 kx 1 x 2 = 4 will have huge variations in the (x 1, x 2 ) values that solve it whenever k 1.

20 Example of x 1 x 2 = x 1 x 2 = 4 is solved with (x 1, x 2 ) = (100, 95). x 1 x 2 = x 1 x 2 = 4 is solved with (x 1, x 2 ) = ( 100, 105).

21 Identifying Ill-Conditioned Systems of Equations Ill-conditioned equations have: Determinants nearly equal to 0 Large conditioning numbers, found with the cond function in MATLAB. In general, if the condition number found with cond(a) is around 10 d, you can expect to lose approximately d digits of precision during the computational process. MATLAB will warn you when a poorly-conditioned matrix is used in a calculation: Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = e-017.