In this post, we will see the book Computational Methods Of Linear Algebra by V. N. Faddeeva.

About the book

English-speaking physicists, mathematicians, and engineers will welcome this first English translation of a unique and valuable Russian work. Translated especially for this edition by Curtis D. Benster, it is a basic work in English that presents a systematic exposition of computational methods of linear algebra— the classical ones, as well as those developed quite recently in Russia and elsewhere, by A. N. Krylov, A. M. Danilevsky, D. K. Faddeev, and others.

This unusual computer’s guide shows in detail how to derive numerical solutions of problems in mathematical physics which are frequently connected with the numerical solution of basic problems of linear algebra. Theory as well as individual practices are given.

The book is divided into three long chapters, with numerous sub-chapters. The first chapter provides the basic material from linear algebra (matrices, linear transformations, the Jordan canonical form, etc.) that is indispensable to what follows. The second chapter describes methods of numerical solution of systems of linear equations. The third chapter provides methods of computing the proper numbers and proper vectors of a matrix.

One of the outstanding and valuable features of this work is the care which has been taken in the preparation of the twenty-three tables which accompany chapters II and III. These tables have been specially rechecked and corrected by the translator, and carefully set (with uniform double-spacing) so as to allow the user to follow the computations throughout with case.

The book was translated from Russian by Curtis D. Benster and was published in 1959.

Credits to original uploader.

You can get the book here.

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Contents

Chapter 1. Basic material from linear algebra 1

§1. Matrices 1
§2. n-Dimensional vector space 23
§3. Linear transformations 33
§4. The Jordan canonical form 49
§5. The concept of limit for vectors and matrices 54

Chapter 2. Systems of linear equations 63

§6. Gauss’s method 65
§7. The evaluation of determinants 72
§8. Compact arrangements for the solution of sacteenogeneyaes linear systems 75
§9. The connection of Gauss’s method with the decomposition of a matrix into factors 79
§10. The square-root method 81
§11. The inversion of a matrix 85
§12. The problem of elimination 90
§13. Correction of the elements of the inverse matrix 99
§14. The inversion of a matrix by partitioning 102
§15. The bordering method 105
§16. The escalator method 111
§17. The method of iteration 117
§18. The preparatory conversion of a system of linear equations into form suitable for the method of iteration 127
§19. Scidel’s method 131
§20. Comparison of the methods 142

Chapter 3. The proper numbers and proper vectors of a matrix 147

§2l. The method of A. N. Krylov 149
§22. The determination of proper vectors by the method of A. N. Krylov 159
§23. Samuelson’s method 161
§24. The method of A. M. Danthcsky 166
§25. Leverrier’s method in D. K. Faddeev’s modification 177
§26. ‘The escalator method 183
§27. ‘The method of interpolation 192
§28. Comparison of the methods 201
§29. Determination of the first proper number of a matrix, First case 202
§30. Improving the convergence of the iterative process 211
§31. Finding the proper numbers next in line 219
§32. Determination of the proper numbers next in line and their proper vectors as well 222
§ 33. Determination of the first proper number 234
§ 34. The case of a matrix with nonlinear elementary divisors 235
§ 35. Improving the convergence of the iterative process for solving
systems of linear equations 239

Bibliography 243
Index 247