Ebook: Handbook for Automatic Computation: Volume II: Linear Algebra

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.

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Content:
Front Matter....Pages I-IX
Front Matter....Pages 1-8
Symmetric Decomposition of a Positive Definite Matrix....Pages 9-30
Iterative Refinement of the Solution of a Positive Definite System of Equations....Pages 31-44
Inversion of Positive Definite Matrices by the Gauss-Jordan Method....Pages 45-49
Symmetric Decomposition of Positive Definite Band Matrices....Pages 50-56
The Conjugate Gradient Method....Pages 57-69
Solution of Symmetric and Unsymmetric Band Equations and the Calculations of Eigenvectors of Band Matrices....Pages 70-92
Solution of Real and Complex Systems of Linear Equations....Pages 93-110
Linear Least Squares Solutions by Housholder Transformations....Pages 111-118
Elimination with Weighted Row Combinations for Solving Linear Equations and Least Squares Problems....Pages 119-133
Singular Value Decomposition and Least Squares Solutions....Pages 134-151
A Realization of the Simplex Method Based on Triangular Decompositions....Pages 152-190
Front Matter....Pages 191-201
The Jacobi Method for Real Symmetric Matrices....Pages 202-211
Householder’s Tridiagonalization of a Symmetric Matrix....Pages 212-226
The QR and QL Algorithms for Symmetric Matrices....Pages 227-240
The Implicit QL Algorithm....Pages 241-248
Calculation of the Eigenvalues of a Symmetric Tridiagonal Matrix by the Method of Bisection....Pages 249-256
Rational QR Transformation with Newton Shift for Symmetric Tridiagonal Matrices....Pages 257-265
The Q R Algorithm for Band Symmetric Matrices....Pages 266-272
Tridiagonalization of a Symmetric Band Matrix....Pages 273-283
Simultaneous Iteration Method for Symmetric Matrices....Pages 284-302
Front Matter....Pages 191-201
Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors....Pages 303-314
Solution to the Eigenproblem by a Norm Reducing Jacobi Type Method....Pages 315-326
Similarity Reduction of a General Matrix to Hessenberg Form....Pages 327-338
The QR Algorithm for Real Hessenberg Matrices....Pages 339-358
Eigenvectors of Real and Complex Matrices by LR and QR triangularizations....Pages 359-371
The Modified LR Algorithm for Complex Hessenberg Matrices....Pages 372-395
Solution to the Complex Eigenproblem by a Norm Reducing Jacobi Type Method....Pages 396-403
The Calculation of Specified Eigenvectors by Inverse Iteration....Pages 404-417
Back Matter....Pages 418-439
....Pages 440-441