Numerical linear algebra. A concise introduction with MATLAB and Julia (Q5891029)

The book gives an introduction into the ideas of numerical mathematics. These ideas are explained by problems of linear algebra. Efficient solvers for such problems are main ingredients in computer simulations of complex problems. In Chapter 1, the basic operations with matrices and several types of matrices are introduced. The topic of Chapter 2 are matrix factorizations, e.g. LU factorization, Cholesky factorization, and QR factorization. The corresponding algorithms are explained, an implementation in MATLAB is given, and the amount of arithmetical operations and the number of floating point operations per memory access are analysed. Chapter 3 is devoted to the error analysis. Error measures, the condition of a problem, the computer representation of numbers, and the stability of algorithms are discussed and demonstrated by several examples. In Chapter 4, least squares problems are considered. In the last chapter, solvers for eigenvalue problems (vector iteration, QR iteration) are presented and analysed. Appendix A contains a short introduction into MATLAB and Appendix B an introduction into the open source program Julia which is an alternative to MATLAB. In Appendix C, vector and matrix norms as well as important inequalities are summarized. In Appendix D, the Householder method for computing a QR factorization of a matrix is explained, and in Appendix E some special questions are discussed, e.g. the global convergence of the QR iteration without shift and the local convergence of the QR iteration with shift. Each chapter contains exercises. Additional exercises are given in Appendix F.