2006-07 II Sem ESO218 Computational Methods in Engineering Max Marks: 100
1. Pick the best option (No negative marks) (15 Marks)
(i) In the bisection method, if the true error at any iteration is Et and the approximate error is Ea, then (a) Et>Ea (b) Et=Ea (c) Et<Ea (d) No definite relation exists
(ii) In the fixed-point iteration method written as x=g(x), as we approach the root, ξ, the true error at two successive iterations are related by Et,i+1/Et,i= (a) g(ξ) (b) g(ξ/2) (c) g′
( )
ξ (d) None of these(iii) The term quadratic convergence for the Newton-Raphson method indicates that (a) the function is replaced by a quadratic polynomial near the root (b) the error behaves as a quadratic function of the iteration number (c) the error at any iteration is proportional to the square of that at the previous iteration (d) none of these.
(iv) The Newton method is applied to find a zero of a polynomial with real coefficients. Would it be able to reach a complex root? (a) Always (b) Never (c) Only if the starting guess is complex (d) sometimes, even with real starting guess
(v)Gauss-Seidel method is applied to solve a system [A]{x}={b}. What kind of matrix should [A] be such that convergence is achieved in one iteration? (a) Diagonally dominant (b) positive definite (c) diagonal (d) tri- diagonal
(vi) Cholesky method is applicable only to matrices which are (a) positive definite (b) symmetric (c) upper triangular (d) symmetric positive definite
(vii) The L2 norm of a vector represents its (a) Euclidean length (b) maximum component (c) sum of all components (d) sum of absolute value of all components
(viii) The Fourier series is used to approximate a function which is (a) monotonic (b) periodic (c) non-negative (d) exponentially increasing
(ix) In the Bairstow method of finding roots of a polynomial, the factor is taken as
(
x2−α1x−α0)
and not (x-α).Why? (a) To find multiple roots (b) To enable us to obtain complex roots (c) To reduce the computations (x) A stiff system of ODE’s is one with (a) no solution (b) infinite solutions (c) solutions with almost similar
magnitude of “decay rates” (d) solutions with very different magnitude of “decay rates”
(xi) The power method for finding the largest Eigenvalue of a matrix is suitable for matrices which have (a) all Eigenvalues nearly equal (b) all Eigenvalues close to zero (c) One Eigenvalue much larger than others in magnitude (d) One Eigenvalue much smaller than others in magnitude
(xii) Improper integrals are integrals having (a) one limit of integral at infinity (b) both limits of integral at ±infinity (c) infinite integrand somewhere in the range of integral (d) any of the above
(xiii) A numerical method for solving an ODE is unstable if (a) the solution oscillates (b) the solution becomes zero (c) the solution grows unbounded (d) the solution is negative
(xiv) The shooting method is used to solve (a) Initial value problems (b) Eigenvalue problems (c) Boundary value problems (d) Spline interpolation problems
(xv) Out of the following methods of solving PDE’s which one is likely to have the highest accuracy (a) explicit method (b) fully-implicit method (c) Crank-Nicolson method
2. Solve the following set of equations by Gauss elimination using FIVE significant digits (with round-off) at each step. First use no pivoting, then partial pivoting, and finally complete pivoting if the pivot is small.
1 2
1 2
0.002321 0.090244 0.047443 0.3043 11.556 6.0823
x x
x x
+ =
+ =
Knowing that the true solution is x1=1 and x2=0.5, comment on all three numerical solutions. (10) 3. It is known that an angle between π and 3π/2 has its tangent equal to the angle. To obtain this value, we may
write f x
( )
= −xtan
x=0
and use the Newton method. During the iterations, it was observed that very close to the root, two successive iterations had true errors of 0.07752 and 0.02781. Estimate the error at the next iteration. The same problem could be solved using the fixed point iteration schemes x=tanx or x=tan−1x. Which of these you would prefer and why? Obtain the root using this scheme with a starting guess of x = π andwith an approximate error less than 0.01%. (10)
4. Obtain the characteristic equation and inverse of the following matrix using Fadeev-LeVerrier method:
2 1 0 1 2 1 0 1 2
Also obtain the largest Eigenvalue of the matrix with an approximate error less than 0.1% using the power
method starting with an initial vector
{
1 1 1}
T. (15)5. The velocity of an object starting at x=0 and moving along the x-axis was measured at various times as follows:
t (s) 0 10 20 30 40
v (m/s) 1.00 2.05 4.95 10.00 16.90
Estimate the acceleration at 20 s using numerical differentiation of the highest possible order. Also estimate the distance travelled in 40 s to an accuracy O(h6) applying the Romberg integration algorithm to three estimates
obtained by the trapezoidal rule. (15)
6. What is meant by local and global truncation error in the solution of an IVP? What is the difference between the Adams-Bashforth and Adams-Moulton formulae? Using Adams-Bashforth and Adams-Moulton formulae of order 2, obtain the value of y at x = 2 and 3 given dy/dx = 4 e0.8 x − 0.5 y with y(0) = 2 and y(1) = 6.1946. (10) 7. Approximate the function x3 over the range [0,1] by a straight line using the Legendre polynomials. If we want
an approximation without transformation of variable, we define functions which are orthogonal over [0,1] as opposed to Legendre polynomials which are orthogonal over [–1,1]. Taking the 0th degree polynomial as 1, derive the 1st degree orthogonal polynomial over [0,1] and obtain the best-fit straight line approximation to x3 over the range [0,1]. Compare with the previously obtained result. (15) 8. A square plate 3 cm x 3 cm is initially at a temperature of 00. Suddenly, the temperature along all the edges is
raised to 1000. The heat-conduction equation
2 2
2 2
T T T
k k
t x y
∂ = ∂ + ∂
∂ ∂ ∂
is to be solved using the Alternating Direction Implicit scheme at half-time-step and full step. Obtain the temperature at the four internal nodes at
t=0.2 s using ∆x=∆y=1 cm, k=1 cm2/s and ∆t=0.2 s. (10)
