Numerical Methods · Unit 7
Systems of Linear Equations
Exam-focused notes for Systems of Linear Equations (Numerical Methods, BIT203): what the TU syllabus asks and how it has actually been tested, with 8 solved past questions from this unit.
What this unit covers
- Gaussian elimination method
- Gaussian elimination with partial pivoting
- Jacobi iteration method
- Gauss-Seidel iteration method
- Comparison of iterative methods
- LU decomposition and Doolittle method
- Cholesky decomposition method
- Applications of linear equation systems
Jacobi iteration method
List out any two applications of system of linear equation. Differentiate between Gauss-Seidel and Jacobi iteration method. Solve the following system of equations using Jacobi iteration method: $4x + y + z = 7$, $x + 5y - 2z = 3$, $3x + 2y + 6z = 14$. [2+3+5]
1. Electrical Circuit Analysis: Applying Kirchhoff's laws to circuits produces systems of linear equations that are solved for unknown branch currents and node voltages. 2. Structural/Engineering Analysis: Truss and frame analysis in civil and mechanical en...
Full solved answer →Explain the working of Jacobi Iteration method? Solve the following system of equations using the method. Assume error precision is 0.01. Compare Jacobi Iteration method with Gauss-Seidel method. $5x-2y+3z=-1$, $-3x+9y+z=2$, $2x-y-7z=-3$ [10]
System of equations: $$5x - 2y + 3z = -1 \quad (1)$$ $$-3x + 9y + z = 2 \quad (2)$$ $$2x - y - 7z = -3 \quad (3)$$ Error precision (tolerance): $\epsilon = 0.01$ Initial guess (assumed standard): $x^{(0)} = y^{(0)} = z^{(0)} = 0$ --- The Jacobi method is an...
Full solved answer →Cholesky decomposition method
Solve the following system of linear equation using Cholesky decomposition method:
$$ \begin{aligned} 4x + 6y - 8z &= -8 \ 6x + 13y - 11z &= -1 \ -8x - 11y + 29z &= 57 \end{aligned} $$
[5]
$$A = \begin{bmatrix} 4 & 6 & -8 \\ 6 & 13 & -11 \\ -8 & -11 & 29 \end{bmatrix}, \quad B = \begin{bmatrix} -8 \\ -1 \\ 57 \end{bmatrix}$$ $A$ is symmetric, so $A = LL^T$ applies. Column 1: $$l{11} = \sqrt{4} = 2$$ $$l{21} = \frac{6}{2} = 3, \quad l{31} = \f...
Full solved answer →Gauss-Seidel iteration method
Compare and contrast between Jacobi iterative methods and Gauss Seidal method? Solve the following equation using Gauss Seidal method.
$$ \begin{aligned} x + 2y + 3z &= 5 \ 2x + 8y + 22z &= 6 \ 3x + 22y + 82z &= -10 \end{aligned} $$
[10]
The equations as written in the question are garbled. Reading them carefully, the intended distinct system is: $$x + 2y + 3z = 5 \quad \cdots (1)$$ $$2x + 8y + 22z = 6 \quad \cdots (2)$$ $$3x + 22y + 82z = -10 \quad \cdots (3)$$ Coefficient matrix: $$A = \b...
Full solved answer →Solve the following system of linear equations using Gauss-Seidal method.
$$\begin{aligned} 10x + y + z &= 12 \ 2x + 10y + z &= 13 \ 2x + 2y + 10z &= 14 \end{aligned}$$
[5]
System of linear equations: $$10x + y + z = 12 \quad \cdots (1)$$ $$2x + 10y + z = 13 \quad \cdots (2)$$ $$2x + 2y + 10z = 14 \quad \cdots (3)$$ Initial approximation: $x^{(0)} = y^{(0)} = z^{(0)} = 0$ Row 1: $10 \geq 1+1$; Row 2: $10 \geq 2+1$; Row 3: $10 ...
Full solved answer →Gaussian elimination method
Solve the following system of linear equations using Gaussian elimination method.
$$\begin{aligned} 2x + 2y + z &= 12 \ 3x + 2y + 2z &= 8 \ 5x + 10y - 8z &= 10 \end{aligned}$$
[5]
System of equations: - $2x + 2y + z = 12$ - $3x + 2y + 2z = 8$ - $5x + 10y - 8z = 10$ Augmented matrix: $$[Ab] = \begin{bmatrix} 2 & 2 & 1 & & 12 \\ 3 & 2 & 2 & & 8 \\ 5 & 10 & -8 & & 10 \end{bmatrix}$$ Eliminate x from R2: $R2 \to R2 - \frac{3}{2}R1$ $$\le...
Full solved answer →Gaussian elimination with partial pivoting
Solve the following system of linear equation by Gauss Elimination with Pivoting $2x + 2y + z = 6$, $4x + 2y + 3z = 4$, $x - y + 1 = 0$. [5]
Equations: - $2x + 2y + z = 6$ - $4x + 2y + 3z = 4$ - $x - y + 1 = 0 \Rightarrow x - y = -1$ Augmented matrix: $$[Ab] = \begin{bmatrix} 2 & 2 & 1 & & 6 \\ 4 & 2 & 3 & & 4 \\ 1 & -1 & 0 & & -1 \end{bmatrix}$$ Largest magnitude in column 1: $4 2 1$. Swap $R1 ...
Full solved answer →LU decomposition and Doolittle method
Derive formula for the Doolittle LU decomposition matrix factorization method. [5]
In Doolittle's method, a square matrix A is factored as: $$A = LU$$ where: - L is a lower triangular matrix with unit diagonal (all diagonal entries = 1) - U is an upper triangular matrix --- Let A be an $n \times n$ matrix. We write: $$A = LU$$ $$\begin{bm...
Full solved answer →Make Unit 7 stick
Practice BIT203 with flashcards & quizzes