6. Interpolation & Numerical Methods

Try these first

1. 1.

   What is a system of linear equations?

   Hint

   Think about what characterizes a linear equation.

2. 2.

   Write the matrix form of the equations: x + 2y = 5 and 3x - y = 4.

   Hint

   Recall how to form matrices from equations.

3. 3.

   What does LU decomposition achieve?

   • It solves systems directly.
   • It produces an upper and lower triangular matrix.
   • It requires large matrix sizes.

   Hint

   Remember what LU stands for in this context.

4. 4.

   True or False: The Gauss-Jacobi method always converges for any matrix.

   • True
   • False

   Hint

   Consider the conditions for convergence we discussed.

5. 5.

   Given the matrix A = [[4, -2, 1], [1, 1, -1], [-1, 2, 5]] and vector B = [3, 1, 0], use Gaussian elimination to find the solution.

   Hint

   Keep track of your row operations carefully.

6. 6.

   Consider a large sparse matrix derived from a real-world dataset. Discuss when you would prefer to use Gauss-Seidel over direct methods and justify your choice.

   Hint

   Reflect on the characteristics of sparse matrices.