The development of the internationally standardized language ALGOL has made it possible to prepare procedures which can be used without modification whenever a computer with an ALGOL translator is available. Volume Ia in this series gave details of the restricted version of ALGOL which is to be employed throughout the Handbook, and volume Ib described its implementation on a computer. Each of the subsequent volumes will be devoted to a presentation of the basic algorithms in some specific areas of numerical analysis. This is the first such volume and it was feIt that the topic Linear Algebra was a natural choice, since the relevant algorithms are perhaps the most widely used in numerical analysis and have the advantage of forming a weil defined dass. The algorithms described here fall into two main categories, associated with the solution of linear systems and the algebraic eigenvalue problem respectively and each set is preceded by an introductory chapter giving a comparative assessment.

Inhalt

I Linear Systems, Least Squares and Linear Programming.- to Part I (J. H. Wilkinson) ..- 1. Introduction.- 2. List of Procedures.- 3. Positive Definite Symmetric Matrices.- 4. Non-Positive Definite Symmetric Matrices.- 5. Non-Hermitian Matrices.- 6. Least Squares and Related Problems.- 7. The Linear Programming Problem.- Contribution I/1: Symmetric Decomposition of a Positive Definite Matrix.- Contribution I/2: Iterative Refinement of the Solution of a Positive Definite System of Equations.- Contribution I/3: Inversion of Positive Definite Matrices by the Gauss-Jordan Method.- Contribution I/4: Symmetric Decomposition of Positive Definite Band Matrices.- Contribution I/5: The Conjugate Gradient Method.- Contribution 1/6: Solution of Symmetric and Unsymmetric Band Equations and the Calculation of Eigenvectors of Band Matrices.- Contribution I/7: Solution of Real and Complex Systems of Linear Equations.- Contribution I/8: Linear Least Squares Solutions by Householder Transformations.- Contribution I/9: Elimination with Weighted Row Combinations for Solving Linear Equations and Least Squares Problems.- Contribution I/l0: Singular Value Decomposition and Least Squares Solutions.- Contribution I/l l: A Realization of the Simplex Method based on Triangular Decompositions.- II The Algebraic Eigenvalue Problem.- to Part II (J. H. Wilkinson).- 1. Introduction.- 2. List of Procedures.- 3. Real, Dense, Symmetric Matrices.- 4. Symmetric Band Matrices.- 5. Simultaneous Determination of Dominant Eigenvalues and Eigenvectors of a Symmetric Sparse Matrx.- 6. The Generalized Symetric Eigenvalue Problems Ax=?Bx and ABx = ?.- 7. Hermitian Matrices.- 8. Real Dense Unsymmetric Matrices.- 9. Unsymmetric Band Matrices.- 10. Dense Unsymmetric Matrices with Complex Elements.- Contribution II/l: The Jacobi Method for Real Symmetric Matrices.- Contribution II/2: Householder's Tridiagonalization of a Symmetric Matrix.- Contribution II/3: The QR and QL Algorithms for Symmetric Matrices.- Contribution II/4: The Implicit QL Algorithm.- Contribution II/5: Calculation of the Eigenvalues of a Symmetric Tridiagonal Matrix by the Method of Bisection.- Contribution II/6: Rational Q R Transformation with Newton Shift for Symmetric TridiagonalMatrices.- Contribution II/7: The QR Algorithm for Band Symmetric Matrices.- Contribution II/8: Tridiagonalization of a Symmetric Band Matrix.- Contribution II/9: Simultaneous Iteration Method for Symmetric Matrices.- Contribution II/l0: Reduction of the Symmetric Eigenproblem A x =?Bx and Related Problems to Standard Form.- Contribution II/11: Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors.- Contribution II/12: Solution to the Eigenproblem by a Norm Reducing Jacobi Type Method.- Contribution II/13: Similarity Reduction of a General Matrix to Hessenberg Form.- Contribution II/14: The QR Algorithm for Real Hessenberg Matrices.- Contribution II/15: Eigenvectors of Real and Complex Matrices by L R and Q R triangulari.- Contribution II/16: The Modified L R Algorithm for Complex Hessenberg Matrices.- Contribution II/l 7: Solution to the Complex Eigenproblem by a Norm Reducing Jacobi Type Method.- Contribution II/l 8: The Calculation of Specified Eigenvectors by Inverse Iteration.