Gaussian Elimination Method
Download as PPTX, PDF9 likes14,139 views
Gaussian elimination is a method for solving systems of linear equations. It involves converting the augmented matrix into an upper triangular matrix using elementary row operations. There are three types of Gaussian elimination: simple elimination without pivoting, partial pivoting, and total pivoting. Partial pivoting interchanges rows to choose larger pivots, while total pivoting searches the whole matrix for the largest number to use as the pivot. Pivoting strategies help prevent zero pivots and reduce round-off errors.
Gaussian Elimination Method
- 1. GAUSSIAN ELIMINATION METHOD Presented By Andi Firdaus Sudarma Student ID No. 432107963 M-505 Linear Algebra DR. Rizwan Butt
- 2. Gaussian Elimination Method • Simple elimination without pivoting • Partial pivoting • Total pivoting
- 3. I. Simple Elimination Without Pivoting Let say we have a system (size 3x3)with augmented matrix form as: a11 a12 a13 b1 A a21 a22 a23 b2 a31 a32 a33 b3
- 4. Procedure to get the solution: 1. The basic idea is to convert the system of A into upper-triangular matrix (U) form. a11 a12 a13 U 0 a22 a23 0 0 a33
- 5. 2. Eliminate the element a21 and a31 using multiple factor of a21 and a31 m 21 m 31 a11 a11 Hence, our matrix should be like this a11 a12 a13 b1 A 0 a22 a23 b2 0 a32 a33 b3
- 6. 3. Eliminate the element a32 using multiple factor of a32 to convert A into U m 32 a22 4. The possible diagonal element (pivot element) should be non-zero. If it become zero at any stage, then interchange that row with any below row with non-zero element at that pivoting position.
- 7. 5. After getting upper triangular matrix form, then use backward substitution to get the solution of the given linear system. a11x 1 a12 x 2 a13x 3 b1 a22 x 2 a23x 3 b2 a33x 3 b3
- 8. II. Pivoting Strategies The basic idea of pivoting strategies is; • To prevent diagonal (pivoting) element from becoming zero • To make diagonal element larger in magnitude than any other coefficient below it, that is, to decrease round-off errors. • After interchanging the system, use the same procedure that we discussed to get the solution.
- 9. • Partial Pivoting Interchange the largest absolute coefficient of variable X1 Example: Before After 2 2 2 4 2 2 4 2 2 2 2 2 2 3 9 2 3 9
- 10. • Total Pivoting Search the largest number in absolute, then interchange this as the pivot Example: Before After 2 2 2 9 3 2 4 2 2 2 2 4 2 3 9 2 2 2
- 11. Thank You