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.