Applied Numerical Methods Using MATLAB, 2nd Edition

2System of Linear Equations

Chapter Outline

• 2.1 Solution for a System of Linear Equations
  • 2.1.1 The Nonsingular Case (M = N)
  • 2.1.2 The Underdetermined Case (M < N): Minimum‐norm Solution
  • 2.1.3 The Overdetermined Case (M > N): Least‐squares Error Solution
  • 2.1.4 Recursive Least‐Squares Estimation (RLSE)
• 2.2 Solving a System of Linear Equations
  • 2.2.1 Gauss(ian) Elimination
  • 2.2.2 Partial Pivoting
  • 2.2.3 Gauss‐Jordan Elimination
• 2.3 Inverse Matrix
• 2.4 Decomposition (Factorization)
  • 2.4.1 LU Decomposition (Factorization) – Triangularization
  • 2.4.2 Other Decomposition (Factorization) – Cholesky, QR and SVD
• 2.5 Iterative Methods to Solve Equations
  • 2.5.1 Jacobi Iteration
  • 2.5.2 Gauss‐Seidel Iteration
  • 2.5.3 The Convergence of Jacobi and Gauss‐Seidel Iterations
  • Problems

In this chapter, we discuss several numerical schemes for solving a system of equations

(2.0.1a)

which can be written in a compact form by using a matrix‐vector notation as

(2.0.1b)

where

We will deal with the three cases:

1. (i) The case where the number (M) of equations and the number (N) of unknowns are equal (M = N) so that the coefficient matrix A_M×N is square.
2. (ii) The case where the number (M) of equations ...