Guaranteed Accuracy In Numerical Linear Algebra Book

	Introduction
1	Singular Value Decomposition	1
1.1	Singular Value Decomposition and Singular Values of Square Matrix	3
1.2	Elementary Orthogonal Transformations	9
1.3	Singular Value Decomposition of Rectangular Matrices	24
1.4	Norm of Matrix. Singular Values and Singular Vectors	34
1.5	Some Numerical Characteristics of Matrices	49
1.6	Some Properties of Bidiagonal Square Matrices. Singular Values and Singular Vectors	58
1.7	Simplification of Matrix Form by Usage of Orthogonal Transformations	71
1.8	Simplification of Matrix Form by Deflation	86
1.9	Extension of Results for Complex Matrices	98
2	Systems of Linear Equations	109
2.1	Condition Number for Square Matrix	111
2.2	Systems of Linear Equations with Simplest Band Matrices of Coefficients	121
2.3	Generalized Normal Solutions of Systems with Arbitrary Matrices of Coefficients	136
2.4	Conditioning of Generalized Normal Solutions of Systems of Full Rank	153
2.5	Angles between Spaces and Their Conditioning	162
2.6	Conditioning of the Generalized Normal Solutions in Case of Not Full Rank	181
2.7	Generalized Normal r-solution of the Systems of Linear Equations	193
2.8	General Scheme of Finding of r-solution of Linear System	207
3	Deflation Algorithms for Band Matrices	215
3.1	Transformations of Hessenberg Matrices by Chains of Rotations	217
3.2	Deflation of Degenerate Bidiagonal Matrices	235
3.3	Singular Deflation of Non-Degenerate Bidiagonal Matrices	247
3.4	Spectral Deflation of Hessenbergian and Symmetric Tridiagonal Matrices	259
3.5	Theory of Perturbations of Singular Deflation of Non-Degenerate Bidiagonal Matrices	267
3.6	Theory of Perturbations of Singular Deflation of Degenerate Bidiagonal Matrices	290
3.7	Theory of Perturbations of Singular Deflation of Symmetric Tridiagonal Matrices	296
4	Sturm Sequences of Tridiagonal Matrices	313
4.1	Elementary Proof of Sturm Theorem	315
4.2	Algorithm of Computation of Eigenvalues of Symmetric Tridiagonal Matrix	323
4.3	Trigonometric Parametrization of Rational Relations	336
4.4	Sturm Sequences of Second Kind	349
4.5	One-Side Sturm Sequences for Tridiagonal Matrices	355
4.6	Two-Side Sturm Sequences for Tridiagonal Symmetric Matrices	368
4.7	Examples of Calculations in Problems of Finding Eigenvalues and Sturm Sequences	391
4.8	Two-Side Sturm Sequences for Bidiagonal Matrices	404
4.9	Examples of Computations of Singular Values and Two-Side Sturm Sequences of Bidiagonal Matrices	417
5	Peculiarities of Computer Computations	425
5.1	Modelling of Errors in Computer Arithmetic Operations	428
5.2	Machine Operations on Vectors and Matrices	443
5.3	Machine Realization of Reflections	452
5.4	Analysis of Errors in Reduction of Matrices into Bi- and Tridiagonal Form	462
5.5	Machine Solution of Systems of Equations with Bidiagonal Coefficient Matrices	471
5.6	Numerical Examples	478
5.7	Machine Realization of Sturm Algorithm. Estimates of Errors in Computation of Eigenvalues and Singular Values	484
5.8	Computation of Two-Side Sturm Sequence and Components of Eigenvector of Tridiagonal Symmetric Matrix	493
5.9	Machine Realization of Computations of Two-Side Sturm Sequences for Bidiagonal Matrices	502
5.10	Machine Realization of Deflation Algorithm for Bidiagonal Matrices	508
	Bibliography	523
	Index	529