Numerical Method · Unit 4 · 8 hrs
Solving System of Linear Equations
Exam-focused notes for Solving System of Linear Equations (Numerical Method, CSC212): what the TU syllabus asks and how it has actually been tested, with 12 solved past questions from this unit.
What this unit covers
- Review of the existence of solutions and properties of matrices
- Gaussian elimination method
- pivoting
- Gauss-Jordan method
- Inverse of matrix using Gauss-Jordan method
- Matrix factorization and Solving System of Linear Equations by using Dolittle and Cholesky's algorithm
- Iterative Solutions of System of Linear Equations
- Jacobi Iteration Method
- Gauss-Seidal Method
- Eigen values and eigen vectors problems
- Solving eigen value problems using power method
Jacobi Iteration Method
What are the limitations of direct methods for solving a system of linear equations? How does Gauss Seidel method differ from Jacobi iteration? Solve the following system of linear equations using Jacobi iteration method:
$$ \begin{aligned} 2x - 7y - 10z &= -17 \ 5x + y + 3z &= 14 \ x + 10y + 9z &= 7 \end{aligned} $$
[10]
--- Direct methods (Gauss Elimination, LU Decomposition, Cramer's Rule) obtain the solution in a finite number of steps, but have limitations: 1. High computational cost: Gauss Elimination needs about $O(n^3)$ operations; expensive for large $n$. 2. Round-o...
Full solved answer →Gauss-Jordan method
Solve the following system of linear equations using Gauss-Jordan elimination method:
$$ \begin{aligned} x + 2y - 3z &= 4 \ 2x + 4y - 6z &= 8 \ x - 2y + 5z &= 4 \end{aligned} $$
[5]
System of equations: $$x + 2y - 3z = 4 \quad \cdots (1)$$ $$2x + 4y - 6z = 8 \quad \cdots (2)$$ $$x - 2y + 5z = 4 \quad \cdots (3)$$ $$[Ab] = \begin{bmatrix} 1 & 2 & -3 & & 4 \\ 2 & 4 & -6 & & 8 \\ 1 & -2 & 5 & & 4 \end{bmatrix}$$ $R2 \to R2 - 2R1$: $[0,\ 0...
Full solved answer →How Gauss Jordan method differs from Gauss Elimination method? Solve the following system of equations using Gauss Jordan method. How can we use Gauss Jordan method to find the inverse of a matrix? Discuss.
$$ \begin{aligned} 2x - y + 4z &= 15 \ 2x + 3y - 2z &= 4 \ 3x + 2y - 4z &= -4 \end{aligned} $$
[10]
System of equations: $$2x - y + 4z = 15 \quad (1)$$ $$2x + 3y - 2z = 4 \quad (2)$$ $$3x + 2y - 4z = -4 \quad (3)$$ All coefficients and constants are present. Solvable in full. --- Feature Gauss Elimination Gauss Jordan --------- Final form Upper triangular...
Full solved answer →Solve the following set of linear equations using the Gauss-Jordan method.
$$x_2 + 2x_3 + 3x_4 = 9$$ $$7x_1 + 6x_2 + 5x_3 + 4x_4 = 33$$ $$8x_1 + 9x_2 + x_4 = 27$$ $$2x_1 + 5x_2 + 4x_3 + 3x_4 = 23$$
[5]
$$ \begin{aligned} x2 + 2x3 + 3x4 &= 9 \quad (1)\\ 7x1 + 6x2 + 5x3 + 4x4 &= 33 \quad (2)\\ 8x1 + 9x2 + 0x3 + x4 &= 27 \quad (3)\\ 2x1 + 5x2 + 4x3 + 3x4 &= 23 \quad (4) \end{aligned} $$ $$ \left[\begin{array}{ccccc} 0 & 1 & 2 & 3 & 9 \\ 7 & 6 & 5 & 4 & 33 \\...
Full solved answer →Matrix factorization and Solving System of Linear Equations by using Dolittle and Cholesky's algorithm
Factorise the following matrix using Cholesky method. $$\begin{bmatrix} 2 & 1 & 1 \ 3 & 2 & 3 \ 1 & 4 & 9 \end{bmatrix}$$ [5]
$$A = \begin{bmatrix} 2 & 1 & 1 \\ 3 & 2 & 3 \\ 1 & 4 & 9 \end{bmatrix}$$ Cholesky's method requires the matrix to be symmetric (and positive definite), factorising it as: $$A = L L^T$$ where $L$ is lower triangular. The given matrix is NOT symmetric since ...
Full solved answer →Matrix Factorization and Solving Linear Systems
Matrix factorization expresses a square matrix $A$ as a product of two triangular matrices: $$A = LU$$ where $L$ is lower triangular (with unit diagonal in Doolittle's method) and $U$ is upper triangular. This is called LU decomposition. $$A = \begin{bmatri...
Full solved answer →Discuss the Doolittle LU decomposition method for matrix factorization. [5]
The coefficient matrix A of a system of linear equations can be decomposed (factorized) into two triangular matrices L (lower triangular) and U (upper triangular) such that: $$A = L \cdot U$$ $$L = \begin{bmatrix} 1 & 0 & 0 \\ l{21} & 1 & 0 \\ l{31} & l{32}...
Full solved answer →Matrix Factorization and LU Decomposition
Matrix factorization (decomposition) means expressing a matrix $A$ as a product of two or more matrices. In LU decomposition, $A = LU$, where $L$ is lower triangular and $U$ is upper triangular. The Doolittle method fixes the diagonal of $L$ to be all 1's. ...
Full solved answer →pivoting
What is pivoting? Why is it necessary? Write an algorithm and program to solve the set of n linear equations using Gaussian elimination method.[10]
Pivoting is the process of rearranging (swapping) rows (or columns) of an augmented matrix during Gaussian elimination so that the element used as the divisor (pivot element) in each forward elimination step is the largest possible value in absolute magnitu...
Full solved answer →Why partial pivoting is used with Naive Gauss Elimination method? Solve the following system of equations using Gauss Elimination method with partial pivoting? How Gauss Jordan method differs from Gauss elimination method? $2x + 2y - z = 6$, $4x + 2y + 3z = 4$, $x + y + z = 0$ [10]
System of equations: $$2x + 2y - z = 6 \quad \cdots (1)$$ $$4x + 2y + 3z = 4 \quad \cdots (2)$$ $$x + y + z = 0 \quad \cdots (3)$$ All data present. Sub-questions: (a) why partial pivoting, (b) solve by Gauss elimination with partial pivoting, (c) differenc...
Full solved answer →Gauss-Seidal Method
Solve the following set of equations using Gauss Siedal method.
$$ \begin{aligned} x + 2y + 3z &= 4 \ 6x + 4y + 5z &= 16 \ 5x + 2y + 3z &= 12 \end{aligned} $$
[5]
The system of equations: $$x + 2y + 3z = 4 \quad \text{...(1)}$$ $$6x + 4y + 5z = 16 \quad \text{...(2)}$$ $$5x + 2y + 3z = 12 \quad \text{...(3)}$$ Initial guess (standard assumption): $x^{(0)} = y^{(0)} = z^{(0)} = 0$. For Gauss-Seidel to converge reliabl...
Full solved answer →Solve the following set of equations using Gauss Seidel method.
$$ \begin{aligned} x + 2y + 3z &= 4 \ 6x - 4y + 5z &= 10 \ 5x + 2y + 2z &= 25 \end{aligned} $$
[5]
System of equations: $$x + 2y + 3z = 4 \quad \cdots (1)$$ $$6x - 4y + 5z = 10 \quad \cdots (2)$$ $$5x + 2y + 2z = 25 \quad \cdots (3)$$ Initial guess (standard): $x^{(0)} = y^{(0)} = z^{(0)} = 0$. To apply Gauss-Seidel we solve equation $(i)$ for $x$, $(j)$...
Full solved answer →Make Unit 4 stick
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