Simultaneous Linear Equations PHYSICAL PROBLEMS Truss Problem Pressure vessel problem a a b c b
Polynomial Regression We are to fit the data to the polynomial regression model Simultaneous Linear Equations Naive Gaussian Elimination (Naïve and the Not That So Innocent Also) Naïve Gaussian Elimination A method to solve simultaneous linear equations of the form [A][X]=[C] The goal of forward elimination is to transform the coefficient matrix into an upper triangular matrix Two steps 1. 2.
Exercise: Show the steps for this slide (10 minutes). Solve each equation starting from the last equation Example of a system of 3 equations A set of n equations and n unknowns Step 1 For Equation 2, divide Equation 1 by multiply by. and (n-1) steps of forward elimination
Subtract the result from Equation 2. Repeat this procedure for the remaining equations to reduce the set of equations as... or End of Step 1 Step 2 Repeat the same procedure for the 3 rd term of Equation 3. At the end of (n-1) steps, the system of equations will look like End of Step 2 End of Step (n-1)
Matrix Form at End of Forward Elimination Solve each equation starting from the last equation Example of a system of 3 equations Starting Eqns Start with the last equation because it has only one unknown
Naïve Gauss Elimination Example http://numericalmethods.eng.usf.edu Example 1 The upward velocity of a rocket is given at three different times Table 1 Velocity vs. time data. Time, Velocity, 5 106.8 8 177.2 12 279.2 Assume Example 1 Cont. Results in a matrix template of the form: Using data from Table 1, the matrix becomes: The velocity data is approximated by a polynomial as: Find the velocity at t=6 seconds.
Example 1 Cont. 1. 2. Number of Steps of Number of steps of forward elimination is (n 1) (3 1) 2 : Step 1 Divide Equation 1 by 25 and multiply it by 64,.. Subtract the result from Equation 2 Substitute new equation for Equation 2
: Step 1 (cont.) Divide Equation 1 by 25 and multiply it by 144,. : Step 2 Divide Equation 2 by 4.8 and multiply it by 16.8,. Subtract the result from Equation 3. Subtract the result from Equation 3 Substitute new equation for Equation 3 Substitute new equation for Equation 3 Solving for a 3
(cont.) (cont.) Solving for a 2 Solving for a 1 Naïve Gaussian Elimination Solution Example 1 Cont. Solution The solution vector is The polynomial that passes through the three data points is then:
Determinant of a Square Matrix Using Naïve Gauss Elimination Example Theorem of Determinants If a multiple of one row of [A] nxn is added or subtracted to another row of [A] nxn to result in [B] nxn then det(a)=det(b) http://nm.mathforcollege.com Theorem of Determinants The determinant of an upper triangular, lower triangular or diagonal matrix [A] nxn is given by of a Square Matrix Using forward elimination to transform [A] nxn to an upper triangular matrix, [U] nxn.
Example Using Naive Gaussian Elimination method, find the determinant of the following square matrix. Finding the Determinant After forward elimination steps.