Visvesvaraya National Institute of Technology, Nagpur
Department of Mathematics
Mathematics (MAL-205)
Assignment - 2
1. Determine p, q and r so that order of the iterative method
xn+1 = pxn + qa/x2n + ra2 /x5n
for a1/3 becomes as high as possible. For this choice of p, q and r, indicate how the error in
xn+1 depends on the error in xn .
2. A sequence {xn }∞
1 is defined by
x0 = 5
1 4 1 3
xn+1 = x − x + 8xn − 12.
16 n 2 n
Show that it gives cubic convergence to α = 4.
3. The system of equations x2 y + y 3 = 10, xy 2 − x2 = 3 has a solution near x = 0.8, y = 2.2.
Perform two iterations of Newton’s method to obtain this root.
√
4. The system of equations loge (x2 + y) − 1 + y = 0, x + xy = 0 has one approximate solution
(x0 , y0 ) = (2.4, −0.6). Improve this solution and estimate the accuracy of the result.
5. The system of equations y cos(xy) + 1 = 0, sin(xy) + x − y = 0 has one solution close to
(x, y) = (1, 2). Calculate this solution correct to four decimal places.
6. Calculate the solution of the system x2 + y 2 = 1.12, xy = 0.23 correct up to three decimal
place(take (x0 = y0 = 1)).
7. Calculate the solution of the system of equations x3 + y 3 = 53, 2y 3 + z 4 = 69, 3x5 + 10z 2 =
770, which is close to (x, y, z) = (3, 3, 2).
8. Solve the system using Gauss Elimination method (Check the result by back substitution)
(i) 8x2 + 2x3 = −7 (ii) 6x2 + 13x3 = 61 (iii) 10x1 − x2 + 2x3 = 4
3x1 + 5x2 + 2x3 = 8 6x1 − 8x3 = −38 x1 + 10x2 − x3 = 8
6x1 + 2x2 + 8x3 = 26, 13x1 − 8x2 = 79, 2x1 + 3x2 + 20x3 = 7.
9. Solve the system of equations by LU decomposition (Doolittle’s method)
(i) 5x1 + 4x2 + x3 = 3.4 (ii) x1 + x2 + x3 = 1
10x1 + 9x2 + 4x3 = 8.8 4x1 + 3x2 − x3 = 6
10x1 + 13x2 + 15x3 = 19.2, 3x1 + 5x2 + 3x3 = 4.
10. Solve the system of equations by LU decomposition (Crout’s method)
(i) x1 − 4x2 + 2x3 = 81 (ii) x1 + x2 + x3 = 1
−4x1 + 25x2 + 4x3 = −153 4x1 + 3x2 − x3 = 6
2x1 + 4x2 + 15x3 = 324, 3x1 + 5x2 + 3x3 = 4.
11. Show that the LU decomposition method fails to solve the system of equations
x1 + x2 − x3 = 2
2x1 + 2x2 + 5x3 = −3
3x1 + 2x2 − 3x3 = 6.
1 −k
12. Let Ax = b (for arbitrary b). If A = , k ∈ R, then determine k such that Gauss
−k 1
Seidel method converges.
13. Find the sufficient condition on k so that the Gauss-seidel
iterative
method converges for solv-
1 0 k
ing the system of equations Ax = b, where A = 2 1 3 and b is arbitrary.
k 0 1
14. Discuss the convergence of the Gauss-Seidel iterative method
for
solving thesystem
of equa-
1 2 −2 1
tions Ax = b, and hence solve the system, where A = 1 1 1 and b = 2.
2 2 1 3
2x − y = 1
15. Solve the system of equations −x + 2y − z = 0 using Gauss-Seidel iterative method taking
−y + 2z − w = 0
. − z + 2w = 1
(0) T
initial guess x = (0, 0, 0, 0) .(perform three iterations)
16. Using
Jacobimethod
find √ all eigenvaluesand the corresponding
eigen vectors of the matrices
3 2 2 √2 2 √4 3 2 1
(i) 2 5 2 (ii) 2 √6 2 (iii) 2 3 2(In(i) for first rotation use a13 as a largest
2 2 3 4 2 2 1 2 3
off diagonal element, for (iii) Iterate till the off-diagonal elements in magnitude are less than
0.005).
