Error analysis and stability – MCQs – EE

1. Round-off error occurs due to:
(A) Truncating infinite series
(B) Limited precision of computer arithmetic
(C) Incorrect boundary conditions
(D) Poor initial guess

2. Truncation error arises from:
(A) Approximating mathematical procedures
(B) Rounding numbers
(C) Hardware malfunction
(D) Using exact formulas

3. Absolute error is:
(A) Difference between true value and approximate value
(B) Ratio of true value to approximate value
(C) Difference between successive approximations
(D) Maximum possible error

4. Relative error is:
(A) Absolute error divided by true value
(B) Absolute error multiplied by true value
(C) Square of absolute error
(D) Difference between successive approximations

5. Round-off errors are significant when:
(A) Using very small step sizes
(B) Using large numbers
(C) Using exact formulas
(D) None of the above

6. Total numerical error is the sum of:
(A) Round-off and truncation errors
(B) Only truncation errors
(C) Only round-off errors
(D) Absolute and relative errors

7. Stability in numerical methods means:
(A) Small errors do not grow significantly during computation
(B) Computation is very fast
(C) Only exact solutions are obtained
(D) Errors always cancel out

8. An unstable numerical method:
(A) Amplifies errors during computation
(B) Reduces all errors
(C) Always converges
(D) Works only for linear problems

9. Consistency of a numerical method ensures:
(A) Local truncation error tends to zero as step size decreases
(B) Round-off error is zero
(C) Absolute error is minimized
(D) Step size is constant

10. Convergence of a numerical method means:
(A) Approximate solution approaches exact solution as step size decreases
(B) Errors grow with each step
(C) Method is explicit
(D) Step size is fixed

11. Stability and convergence are related through:
(A) Lax equivalence theorem
(B) Taylor series
(C) Euler’s formula
(D) Runge-Kutta method

12. Propagation of errors in iterative methods depends on:
(A) Spectral radius of iteration matrix
(B) Step size only
(C) Derivative of function only
(D) Initial guess only

13. A method is conditionally stable if:
(A) Stability depends on step size
(B) Always stable
(C) Always unstable
(D) Independent of step size

14. Forward Euler method is:
(A) Conditionally stable
(B) Unconditionally stable
(C) Always convergent
(D) Implicit method

15. Backward Euler method is:
(A) Unconditionally stable
(B) Conditionally stable
(C) Explicit
(D) Only for linear problems

16. Round-off errors increase when:
(A) Step size is too small
(B) Step size is too large
(C) Iterations are few
(D) None of the above

17. Truncation error decreases when:
(A) Step size is reduced
(B) Step size is increased
(C) Step size remains constant
(D) Step size is random

18. In finite difference methods, stability analysis is required to:
(A) Ensure solution does not diverge
(B) Reduce truncation error only
(C) Eliminate round-off error
(D) Compute derivatives exactly

19. Numerical stability is essential for:
(A) Long-time integration
(B) Short-time integration
(C) Single-step problems
(D) Analytical solutions only

20. Absolute stability region of a method indicates:
(A) Step sizes for which the method is stable
(B) Maximum possible error
(C) Convergence rate
(D) Round-off error magnitude

21. In iterative solution of linear systems, divergence occurs if:
(A) Spectral radius ≥ 1
(B) Spectral radius < 1
(C) Matrix is positive definite
(D) Step size is small

22. Local truncation error is:
(A) Error made in a single step
(B) Error accumulated over all steps
(C) Same as round-off error
(D) Relative error

23. Global error is:
(A) Accumulated error over all steps
(B) Single step error
(C) Only round-off error
(D) Only truncation error

24. Round-off error can be reduced by:
(A) Using higher precision arithmetic
(B) Decreasing step size always
(C) Ignoring small terms
(D) Reducing iterations

25. Truncation error depends on:
(A) Step size and method order
(B) Precision of computer
(C) Number of iterations only
(D) Boundary conditions

26. A-stability is a property of:
(A) Unconditionally stable methods
(B) Conditionally stable methods
(C) Euler’s explicit method only
(D) Trapezoidal rule only

27. For stiff differential equations, preferred methods are:
(A) Implicit methods
(B) Explicit methods
(C) Euler’s forward method
(D) Runge-Kutta only

28. Forward difference approximation is:
(A) Conditionally stable
(B) Unconditionally stable
(C) Implicit
(D) Predictor-corrector method

29. Backward difference approximation is:
(A) Unconditionally stable
(B) Conditionally stable
(C) Explicit method
(D) Forward method

30. Stability analysis ensures:
(A) Errors do not grow uncontrollably
(B) Faster computation
(C) Elimination of all errors
(D) Exact solutions