Nepal Engineering College Changunarayan VDC-9, Bhaktapur email: [email protected] Subject: Numerical Methods Teacher: [email protected] Subject Code: MTH 317.3 Tutorial No.: 4 Title: Solution of System of Linear Equations Date: June 17, 2007 Direct Method: (A) Gauss Elimination Method: 1. Solve the following set of linear simultaneous equations by Gauss Elimination. x1 + x2 + 4 x3 = 43; 2 x1 + 10 x2 + x3 = 63; 25 x1 + 2 x2 + x3 = 69 2. Solve the following system of linear equations by Gauss elimination method. 4y + 3 z = 4; 4 x + 2z = 2; 5 x + 5 y + 10 z = -3 3. Solve the following equations by using Gauss-Elimination method. 3 w - 2 x + y = 7; -2 w + 5 x – y + z = -11; x – 2 y + 2 z = -29; w - x + 11y –2 z = 28 Direct Method: (B) Cholesky's (LU Decomposition) Method: 4. Solve the following set of equations by Cholesky's and Dolittle’s method. 3x1 - 4 x2 + 7x3 = 6; - 4x1 + x2 - x3 = - 4; 7x1 - x2 - 3 x3 = 3 5. Solve the set of equations by Cholesky's or Crout’s method. a + b + c + d = 5; 4a + 3b - c + 5d = 2; 2a + 5b - 7c - 9d = 0; a + 2b + 3c + 4d =10 Direct Method: (C) Gauss Jordan Method: 6. Solve the following set of linear simultaneous equations by Gauss Jordan method. x1 + x2 + x3 = 9; 2 x1 - 3 x2 + 4 x3 = 13; 3 x1 + 4 x2 + 5 x3 = 40 7. Solve the following set of linear simultaneous equations by Gauss Jordan method. 8x1 + 4x2 + 2x3 = 24; 4x1 + 10 x2 + 5 x3 + 4 x4= 32; 2 x1 + 5 x2 + 6.5 x3 +4 x4= 26; 4 x2 + 4 x3 + 9 x4= 21 8. Solve the following set of linear simultaneous equations by Gauss Jordan method. 3x1 + x2 + 2 x3 = 3; 2 x1 - 3 x2 - x3 = - 3; x1 + 2 x2 + x3 = 4 Indirect (Iterative) Methods for solving System of Linear Equations: 9. Solve the following set of linear simultaneous equations by Gauss Jacobi iteration method. 3 x1 - 20 x2 - x3 = - 18; 2 x1 - 3 x2 + 20 x3 = 25; 20 x1 + x2 - 2 x3 = 17 10. Solve the following set of linear simultaneous equations by Gauss Seidel iteration method. 3 x1 - 20 x2 - x3 = - 18; 2 x1 - 3 x2 + 20 x3 = 25; 20 x1 + x2 - 2 x3 = 17 11. Solve the following set of linear simultaneous equations by Gauss Seidel iteration method. 10 x – 2 y –z - w = 3; -2 x + 10 y – z – w = 15; - x - y + 10 z – 2w = 27; - x – y – 2z + 10w = -9 Matrix Inversion Problem: 1 2 1 12. Find the inverse of the matrix by the Gauss-Elimination method. 2 3 1 2 1 3 13. Find the inverse of the matrix by the Gauss-Elimination 3 1 10 5 1 2 method. 9 7 39 1 2 2 Eigen Values, Eigen Vectors, and Power Method Problems 14. Find the all the eigen values of the matrix 4 1 using the Power Method. 2 6 6 4 3 15. Find the largest eigenpair of 2 3 2 using the Power Method. 1 2 4 Submission deadline: December 7, 2006 2 3 4 1 16. Find the highest eigenvalue and the corresponding eigenvector of symmetric matrix A using the Power Method. Verify the results manually using the Rayleigh quotient. 0 120 90 0 160 70 0 A 130 40 80 $ Matrix Problems 1 2 1 14. Find the inverse of the matrix by the Gauss-Jordan method. 2 3 1 2 1 3 2 1 2 15. Find the inverse of the matrix by the Cholesky method. 1 1 1 2 1 3 16. Find the inverse of the coefficient matrix of the following linear equation and solve the system of equation as X = A-1b where b is right hand side. x1 + x2 + x3 = 2. 2x1 + 3x2 – 20 x3 = 7 4x1 - x2 + 3 x3 = 10

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