```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
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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