MA2501 Numerical methods
spring 2005
Examination, course outline and aim with the
course
- Exam: 30. may.
- Allowed aids for the exam:
- W. Cheney and D. Kincaid:
Numerical Mathematics and Computing, 5th edition or 4th
edition. The book should be free from own notations.
- Approved calculator.
- Course outline:
- Chap. 3.1, 3.2 and lecture notes about
fix point iterations.
- Chap. 4.1, only p.140-144 (until Newton Form) and chap. 4.2, p.169-174,
(until Theorem 3).
- Chap. 4.3 except Richardson-extrapolation
(s.182-186).
- Chap. 5 and 6.1 except multidimensional integration
(p.215-216). 5.1 should be known from earlier courses.
- Chap. 7.1,7.2, 8.1 (except p.323-331)), 8.2 and 8.4.
- Chap. 9.1 and 9.2.
- Chap. 10.0-10.3, 11.1-11.3. Lecture notes on convergence.
- Chap. 14.
- Chap. 15.0-15.3, except p.647-649 (Finite Element
Methods)
- Exercise with solutions.
Aim with the course:
- Nonlinear equations.
- Outline: Chap. 3.1, 3.2 and lecture notes on
fix point iterations.
- Exercise: nr. 2 and 3.
- You should know:
- how to derive and apply the bisection method, Newton's method
and fix point iterations for scalar equations.
- how to perform convergence- and error estimation for different methods for scalar problems.
- Newton's method for systems of equations.
- how to implement Newton's method and fix point iterations in Matlab.
- Polynomial interpolation.
- Outline: Chap. 4.1, p.140-144 until Newton Form and chap. 4.2, p.169-174,
until Theorem 3.
- Exercise nr. 4.
- You should:
- be able to calculate a interpolation polynomial
by means of Lagrange-interpolation.
- know how to use the error formulae.
- know what makes the Chebyshew nodes attractive,
and how to find them on an arbitrary interval.
- be able to use the Matlab-commands Polyval and
polyfit .
- Spline interpolation
- Outline: Chap. 9.1 and 9.2.
- Exercise nr. 5.
- You should:
- know what a spline of degree k is, and what it is useful for.
- know what a natural cubic spline is, and what makes them attractive.
- know how to find a spline from given interpolation point and other additional conditions.
- be able to use the Matlab-command spline .
- Numerical differentiation
- Outline: Chap. 4.3 minus Richardson-extrapolation (p.182-186).
- Exercise nr. 6.
- You should:
- know the approximation formulae for f'(x) and
f''(x) by heart.
- be able to develop formulae for numerical differentiation by means of the method of unknown coefficients.
- be able to determine the error in the differentiation formulae,
by means of Taylor expansion or differentiation of
interpolation formulae.
- understand why rounding error can be devastating for the approximations (see. chap. 2.2)
- Numerical integration
- Outline: Chap. 5 and 6.1 except multidimensional integration
(p.215-216). 5.1 are supposed to be known from earlier courses.
- Exercise nr. 7.
- You should:
- be able to use the composite trapezoid rule, the Simpson's rule and the Romberg algorithm.
- Know how to use the error formulae for the trapezoid- and Simpson's rule.
- derive the error formulae for trapezoid rule (og tilsvarende).
- understand how the adaptive Simpson algorithm works, and the meaning of adaptive algorithm.
- be able to use Matlab's routines for integration.
- Numerical linear algebra
- Outline: Chap. 7.1,7.2, 8.1 (except p.323-331)), 8.2 and 8.4.
- Exercise nr. 8 and 9.
- You should:
- know how to perform a Gauss elimination with and without scalar
pivoting, and calculate the LU factorizing of a matrix (if possible).
- be able to solve linear system of equations by means of Jacobi,
Gauss-Seidel and SOR iterations.
- know how to find eigenvalues and the their corresponding eigenvectors via the power method
- be familiar with the notions of condition number, norm, spectral radius,
diagonal dominance and symmetric positive definite.
- be able to determine if an iterative method is converging.
- Ordinary differential equations (ODE)
- Outline: Chap. 10.0-10.3, 11.1-11.3. Lecture notes on convergence.
- Exercise nr. 10 and 11.
- You should:
- know how to perform one step for a given Runge-Kutta method
or a multi step method an a system of ODE.
- be able to rewrite a system of higher order equations to a system of first order equations.
In addition you should be able to rewrite a system of non-autonomous equations to an autonomous system.
- understand how error estimation and step length control works for different Runge-Kutta methods.
- know how to develop simple methods by your self.
- know how to use Matlab's ODE-solvers, eg. ode23 and ode45.
- Boundary value problems
- Outline: Chap. 14.
- You should:
- know how to perform the shooting method for both linear and nonlinear equations.
- know how to solve a linear problem by means of a difference method.
- Partial differential equations
- Outline: Chap. 15.0-15.3, except p.647-649 (Finite Element
Methods)
- Exercise nr. 11
- You should:
- know the difference between hyperbolic, parabolic and elliptic equations.
- be able to construct a finite difference scheme for a given equation, for some boundary conditions.
- be able to solve the latter problem, by hand (for a few number of discretization points) or in Matlab.
- understand why explicit methods may cause stability problem for parabolic equations.