Q1. Numerical Differentiation: Exercise Set 4.1: 1a, 20.
Q2. (Excise Set 4.3: 2a, 6a,10a) Elements of Numerical Integration: Approximate the following integrals using the Trapezoidal, Simpsonís and Midpoint rules.
Q3 (Exercise Set 4.4: 2a, 4a, 6a)† Composite Numerical Integration: Use Composite Trapezoidal, Simpsonís and Midpoint rules to approximate the integral
Q4. (Exercise Set 4.7: 1a) Gaussian Quadrature
Approximate the following integral using Gaussian quadrature with n=3.
Q1. (Exercise Set 5.2: 2c and 4c) Eulerís Method: Given the initial-value problem
†††††† with the exact solution .
†††††† Use Eulerís method with h=0.25 to approximate the solution, and compare your computed solution with the exact solution at each step.
Q1: (Exercise Set 6.1: 6d)† Use Gaussian Elimination with Backward Substitution method (Algorithm 6.1) to solve the following linear system, if possible, and determine whether row interchanges are necessary.
†x1†††††† +x2††††††††††††††† +x4 = 2,
2x1††††† +x2††††† -x3†††† +x4 = 1,
-x1†††† +2x2†† +3x3††††† -x4 = 4,
3x1†††††† -x2††††† -x3††† +2x4 =-3.
Q2: (Exercise Set 6.2: 2c,4c)† Find the row interchanges that are required to solve the following linear system using (a) Algorithm 61: Gaussian Elimination with Backward Substitution; (b) Algorithm 6.2: Gaussian Elimination with Partial Pivoting.
5x1†††††† +x2†††† -6x3††† = 7,
2x1†††††† +x2††††† -x3†††† = 8,
6x1†† +12x2†††† +x3†††† = 9.
Q3: (Exercise Set 6.5: 2a) Solve the following linear system using matrix factorization technique.
Q4: (Exercise Set 6.5: 6a): Factor the matrices into LU decomposition using the LU Factorization Algorithm with lii=1 for all i.
Q1: (Exercise Set 7.3: 2c,4c) Find the first two iterations using (a) Jacobi method and (b) Gauss-Seidel method for the following linear system, using x(0)=0
4x1†††††† +x2†††† -x3†††† +x4 = -2,
†x1††††† +4x2††††† -x3††† -x4 = -1,
-x1 ††††††-x2†††† +5x3††† +x4 = 0,
†x1†††††† -x2††††† +x3††† +3x4 =-1.
Q2: (Exercise Set 7.3: 9a) Find the first two iterations of the SOR method with †for the linear system, using x(0)=0
3x1†††††† -x2†††† +x3††† = 1,
3x1†† +6x2†††† +2x3†† = 0,
3x1† +3x2†††† +7x3†† = 0.
Last Revised: 30/8/2004