Matrix Analysis and Algorithms

Transcription

1 Matrix Analysis and Algorithms Systems of Linear Equations (SLE) Least SQuares problems (LSQ) EigenValue Problems (EVP) Dr. Bjo rn Stinner, Research Fellow Mathematics Institute and Centre for Scientific Computing University of Warwick Zeeman Building, Room C2.30

2 Matrix Analysis and Algorithms Content Aim: Describe and analyse algorithms for solving some central problems which often emerge in complicated applications: 1. Systems of Linear Equations (SLE): Given A Cn n, b Cn n find x C such that Ax = b. 2. Least SQuares problems (LSQ): Given A Rm n, b Rm (typically m n) n find x R such that kax bk2 is minimal. 3. EigenValue Problems (EVP): Given A Cn n n find (x, λ) (C \{0}) C such that Ax = λx. Dr Bjo rn Stinner Term 1, 2009/2010

3 Objectives Understanding the mathematical principles underlying the design and the analysis process to solve large scale problems in numerical linear algebra. At the end of the module you will familiar with concepts and ideas related to: diverse matrix factorisations to obtain analytical results, as the basis for various algorithms, assessing algorithms with respect to computational cost efficiency, error analysis, conditioning of problems, stablity of algorithms, direct versus iterative methods. No coding, e.g. module MA4G7 Computational Linear Algebra and Optimization (Term 2).

4 Example, SLE Classical problems in linear elasticity: timber framework constructions truss brigde constructions

5 Single Bar Assumptions: bars can sustain load only in lateral direction, no shear forces (bar then rotates), deformation (compression, stretching) proportional to force: stress = elasticity modulus strain x1 Bar L of length l, both end points change the position under load: θ L x 1 = (x 11, x 12 ) R 2, x 2 = (x 21, x 22 ) R 2. For small relative deformations ( x 1 2 /l and x 2 2 /l are small), elongation to first order (linearisation): x 2 e = cos(θ)(x 11 x 21 ) + sin(θ)(x 21 x 22 ).

6 Strain m nodes, x R 2m displacement (displacement of node i: (x 2i 1, x 2i )). n bars, e R n strain. Linear relation between displacements and elongations: e = Bx, B R n 2m. Example: (0,1) m = 1 node, n = 2 bars. x = (x 1, x 2 ) R 2 displacement of the node: 1 2 (0,0) (1,1) Nodes (0, 1) and (0, 0) remain fixed, node at (1, 1) may change position under load. e 1 = 2 2 x x 2, e 2 = x 1. Hence e = Bx with «2/2 2/2 B = 1 0

7 Stress n bars, y R n stress. Recalling that stress = elasticity modulus strain: y = Ce, C R n n C contains material parameters, e.g. diagonal matrix with elasticity moduli in the diagonal. Example: (0,1) 1 Assume that «E1 0 C = 0 E 2 = « (0,0) (1,1) Then y = Ce = CBx.

8 Load m nodes, f R 2m load (force in each node). Goal: Compute deformation x for a given load such that the framework is in equilibrium, i.e. in each node, the sum of the forces is zero. Using the linearisation again, the forces on the nodes by the stresses is given by B T y. Hence, in equilibrium f = B T y = B T CBx, where B T CB R 2m 2m. Example: 1 2 f Load: f = (0, 1). System to be solved: «0 1 ««««2/2 1 = /2 2/2 x1 2/ x 2 «« x1 = Solution: x = ( 1, 5). x 2

9 Big Application Problems Optimisation problem: Ensure that stress not too large, deformations not to strong. Control by material, bar thickness etc. Solving the linear system several times required. May not want to do computations by hand.

10 Literature AM Stuart and J Voss, Matrix Analysis and Algorithms, script. LN Trefethen and D Bau, Numerical Linear Algebra, SIAM NJ Higham, Accuracy and Stability of Numerical Algorithms, SIAM G Golub and C van Loan, Matrix Computations, 3. ed., Johns Hopkins University Press D Kincaid and W Cheney, Numerical Analysis, 3. ed., AMS RA Horn and CR Johnson, Matrix Analysis, Cambridge University Press JW Demmel, Applied Numerical Linear Algebra, SIAM 1997.

11 Relation to Other Modules MA4G7 Computational Linear Algebra and Optimization (D. White) implementation, coding, software, but also towards optimisation problems. MA3H0 Numerical Analysis and PDEs (C. Elliott) discretisation of PDEs involves large systems of equations to be solved. MA3G7 Functional Analysis I (V. Gelfreich) a few ideas notions with respect to norm for the analysis are shared. MA390 Topics in Mathematical Biology (M. Kirkilionis) problems like pattern formation involve finding eigenvalues of operators. MA228 Numerical Analysis (X. He) stability and convergence, interpolation, quadrature, ODEs.

12 Other Stuff 28 lectures : No lecture on Monday 12 October. Assessment: Final exam (100%). But: Two assessed exercise sheets (not relevant for final mark). Weekly exercise sheets with solution on the web-page (one week later). Coding exercises in lecture notes (not assessed). Office hours: Wed 10-11, Mon 4-5 (not 12 October). Web-page: