Prove Kahans Theorem 7.24. [Hint: If 1, ... , n are eigenvalues of T, then det T = -ni=1

Find the first two iterations of the SOR method with = 1.1 for the following linear systems, using x(0) = 0: a. 3x1 x2 + x3 = 1, 3x1 + 6x2 + 2x3 = 0, 3x1 + 3x2 + 7x3 = 4. b. 10x1 x2 = 9, x1 + 10x2 2x3 = 7, 2x2 + 10x3 = 6. c. 10x1 + 5x2 = 6, 5x1 + 10x2 4x3 = 25, 4x2 + 8x3 x4 = 11, x3 + 5x4 = 11. d. 4x1 + x2 + x3 + x5 = 6, x1 3x2 + x3 + x4 = 6, 2x1 + x2 + 5x3 x4 x5 = 6, x1 x2 x3 + 4x4 = 6, 2x2 x3 + x4 + 4x5 = 6.

Find the first two iterations of the SOR method with = 1.1 for the following linear systems, using x(0) = 0: a. 4x1 + x2 x3 = 5, x1 + 3x2 + x3 = 4, 2x1 + 2x2 + 5x3 = 1. b. 2x1+ x2 + 1 2 x3 = 4, x12x2 1 2 x3 = 4, x2 + 2x3 = 0. c. 4x1 + x2 x3 + x4 = 2, x1 + 4x2 x3 x4 = 1, x1 x2 + 5x3 + x4 = 0, x1 x2 + x3 + 3x4 = 1. d. 4x1 x2 = 0, x1 + 4x2 x3 = 5, x2 + 4x3 = 0, + 4x4 x5 = 6, x4 + 4x5 x6 = 2, x5 + 4x6 = 6.

Repeat Exercise 1 using = 1.3.

Repeat Exercise 2 using = 1.3.

Use the SOR method with = 1.2 to solve the linear systems in Exercise 1 with a tolerance TOL = 103 in the l norm.

Use the SOR method with = 1.2 to solve the linear systems in Exercise 2 with a tolerance TOL = 103 in the l norm.

Determine which matrices in Exercise 1 are tridiagonal and positive definite. Repeat Exercise 1 for these matrices using the optimal choice of .

Determine which matrices in Exercise 2 are tridiagonal and positive definite. Repeat Exercise 2 for these matrices using the optimal choice of .

Prove Kahans Theorem 7.24. [Hint: If 1, ... , n are eigenvalues of T, then det T = -ni=1 i. Since det D1 = det(D L)1 and the determinant of a product of matrices is the product of the determinants of the factors, the result follows from Eq. (7.18).]

The forces on the bridge truss described in the opening to this chapter satisfy the equations in the following table: Joint Horizontal Component Vertical Component F1 + 2 2 f1 + f2 = 0 2 2 f1 F2 = 0 2 2 f1 + 3 2 f4 = 0 2 2 f1 f3 1 2 f4 = 0 f2 + f5 = 0 f3 10,000 = 0 3 2 f4 f5 = 0 1 2 f4 F3 = 0 This linear system can be placed in the matrix form 10 0 2 2 10 0 0 0 1 0 2 2 00 0 0 0 0 10 00 1 2 0 000 2 2 0 1 1 2 0 000 0 10 0 1 000 0 01 0 0 000 2 2 0 0 3 2 0 000 0 00 3 2 1 F1 F2 F3 f1 f2 f3 f4 f5 = 0 0 0 0 0 10,000 0 0 . a. Explain why the system of equations was reordered. b. Approximate the solution of the resulting linear system to within 102 in the l norm using as initial approximation the vector all of whose entries are 1s and the SOR method with = 1.25.

Use the SOR method to solve the linear system Ax = b to within 105 in the l norm, where the entries of A are ai, j = 2i, when j = i and i = 1, 2, ... , 80, 0.5i, when j = i + 2 and i = 1, 2, ... , 78, j = i 2 and i = 3, 4, ... , 80, 0.25i, when j = i + 4 and i = 1, 2, ... , 76, j = i 4 and i = 5, 6, ... , 80, 0, otherwise, and those of b are bi = , for each i = 1, 2, ... , 80.

In Exercise 17 of Section 7.3 a technique was outlined to prove that the Gauss-Seidel method converges when A is a positive definite matrix. Extend this method of proof to show that in this case there is also convergence for the SOR method with 0 << 2.