ESTUDYR
AMS FOUNDATION – COMPUTATIONAL & APPLIED MATHEMATICS
(CAM) | COMPLETE EXAM
1. What does it mean for a numerical method to be A-stable?
A. Errors decay for small step sizes only
B. The method is stable for all step sizes when applied to linear test equations
C. The method converges exponentially
D. The truncation error is zero
Rationale: A-stability ensures unconditional stability for stiff problems.
2. What does L-stable mean for a numerical method?
A. It is stable only for linear systems
B. It is A-stable and strongly damps stiff modes as step size increases
C. It minimizes truncation error
D. It is explicit
Rationale: L-stability strengthens A-stability by forcing stiff components to decay to zero.
3. What does consistency of a numerical method imply?
A. Stability for all step sizes
B. The local truncation error goes to zero as the step size goes to zero
C. The solution is exact
D. The method is implicit
Rationale: Consistency ensures the numerical method approximates the differential equation
correctly.
4. What does convergence of a numerical method mean?
A. Stability only
B. Consistency only
C. The numerical solution approaches the exact solution as step size decreases
D. The truncation error is bounded
,ESTUDYR
Rationale: Convergence measures closeness to the exact solution.
5. What does zero-stability ensure for multistep methods?
A. Accuracy
B. Errors do not grow uncontrollably from initial perturbations
C. Explicitness
D. A-stability
Rationale: Zero-stability controls error propagation.
6. What is the Dahlquist First Barrier?
A. No implicit method is convergent
B. No explicit linear multistep method can be A-stable
C. All A-stable methods are implicit
D. Zero-stability implies convergence
Rationale: This is a foundational limitation of explicit LMMs.
7. What does the Lax Equivalence Theorem state?
A. Stability implies consistency
B. Consistency plus stability implies convergence
C. Convergence implies stability
D. Explicit methods are unstable
Rationale: This theorem connects the core properties of numerical methods.
INTERPOLATION & APPROXIMATION
8. What is the Runge Phenomenon?
A. Loss of numerical stability
B. Large oscillations near endpoints with high-degree polynomial interpolation
C. Failure of convergence
D. Round-off error accumulation
, ESTUDYR
Rationale: Occurs with equally spaced interpolation nodes.
9. Why are Chebyshev nodes used in interpolation?
A. They maximize oscillations
B. They simplify computation
C. They minimize interpolation error growth near endpoints
D. They guarantee exact solutions
Rationale: Node clustering reduces Runge oscillations.
10. An iterative method converges if which property of its iteration matrix holds?
A. Determinant is nonzero
B. Norm is less than 1
C. Spectral radius is less than 1
D. Trace is positive
Rationale: Spectral radius governs convergence behavior.
DIFFERENTIAL EQUATIONS & LINEAR THEORY
11. What is a Sturm–Liouville problem?
A. A nonlinear PDE
B. A second-order linear self-adjoint ODE with boundary conditions
C. An IVP with singularities
D. A numerical scheme
Rationale: Sturm–Liouville theory underpins eigenvalue problems.
12. What does self-adjoint mean?
A. Linear but unstable
B. The differential operator equals its adjoint
C. Nonlinear symmetry
D. Constant coefficients
AMS FOUNDATION – COMPUTATIONAL & APPLIED MATHEMATICS
(CAM) | COMPLETE EXAM
1. What does it mean for a numerical method to be A-stable?
A. Errors decay for small step sizes only
B. The method is stable for all step sizes when applied to linear test equations
C. The method converges exponentially
D. The truncation error is zero
Rationale: A-stability ensures unconditional stability for stiff problems.
2. What does L-stable mean for a numerical method?
A. It is stable only for linear systems
B. It is A-stable and strongly damps stiff modes as step size increases
C. It minimizes truncation error
D. It is explicit
Rationale: L-stability strengthens A-stability by forcing stiff components to decay to zero.
3. What does consistency of a numerical method imply?
A. Stability for all step sizes
B. The local truncation error goes to zero as the step size goes to zero
C. The solution is exact
D. The method is implicit
Rationale: Consistency ensures the numerical method approximates the differential equation
correctly.
4. What does convergence of a numerical method mean?
A. Stability only
B. Consistency only
C. The numerical solution approaches the exact solution as step size decreases
D. The truncation error is bounded
,ESTUDYR
Rationale: Convergence measures closeness to the exact solution.
5. What does zero-stability ensure for multistep methods?
A. Accuracy
B. Errors do not grow uncontrollably from initial perturbations
C. Explicitness
D. A-stability
Rationale: Zero-stability controls error propagation.
6. What is the Dahlquist First Barrier?
A. No implicit method is convergent
B. No explicit linear multistep method can be A-stable
C. All A-stable methods are implicit
D. Zero-stability implies convergence
Rationale: This is a foundational limitation of explicit LMMs.
7. What does the Lax Equivalence Theorem state?
A. Stability implies consistency
B. Consistency plus stability implies convergence
C. Convergence implies stability
D. Explicit methods are unstable
Rationale: This theorem connects the core properties of numerical methods.
INTERPOLATION & APPROXIMATION
8. What is the Runge Phenomenon?
A. Loss of numerical stability
B. Large oscillations near endpoints with high-degree polynomial interpolation
C. Failure of convergence
D. Round-off error accumulation
, ESTUDYR
Rationale: Occurs with equally spaced interpolation nodes.
9. Why are Chebyshev nodes used in interpolation?
A. They maximize oscillations
B. They simplify computation
C. They minimize interpolation error growth near endpoints
D. They guarantee exact solutions
Rationale: Node clustering reduces Runge oscillations.
10. An iterative method converges if which property of its iteration matrix holds?
A. Determinant is nonzero
B. Norm is less than 1
C. Spectral radius is less than 1
D. Trace is positive
Rationale: Spectral radius governs convergence behavior.
DIFFERENTIAL EQUATIONS & LINEAR THEORY
11. What is a Sturm–Liouville problem?
A. A nonlinear PDE
B. A second-order linear self-adjoint ODE with boundary conditions
C. An IVP with singularities
D. A numerical scheme
Rationale: Sturm–Liouville theory underpins eigenvalue problems.
12. What does self-adjoint mean?
A. Linear but unstable
B. The differential operator equals its adjoint
C. Nonlinear symmetry
D. Constant coefficients
