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Matrix Multiplication Problems 1 1

1 Basic Algorithms and Notation 2 1

2 Exploiting Structure 16 1

3 Block Matrices and Algorithms 24 1

4 Vectorization and Re-Use Issues 34 1

Matrix Analysis 48 2

1 Basic Ideas from Linear Algebra 48 2

2 Vector Norms 52 2

3 Matrix Norms 54 2

4 Finite Precision Matrix Computations 59 2

5 Orthogonality and the SVD 69 2

6 Projections and the CS Decomposition 75 2

7 The Sensitivity of Square Linear Systems 80 2

General Linear Systems 87 3

1 Triangular Systems 88 3

2 The LU Factorization 94 3

3 Roundoff Analysis of Gaussian Elimination 104 3

4 Pivoting 109 3

5 Improving and Estimating Accuracy 123 3

Special Linear Systems 133 4

1 The LDMT and LDLT Factorizations 135 4

2 Positive Definite Systems 140 4

3 Banded Systems 152 4

4 Symmetric Indefinite Systems 161 4

5 Block Systems 174 4

6 Vandermonde Systems and the FFT 183 4

7 Toeplitz and Related Systems 193 4

Orthogonalization and Least Squares 206 5

1 Householder and Givens Matrices 208 5

2 The QR Factorization 223 5

3 The Full Rank LS Problem 236 5

4 Other Orthogonal Factorizations 248 5

5 The Rank Deficient LS Problem 256 5

6 Weighting and Iterative Improvement 264 5

7 Square and Underdetermined Systems 270 5

Parallel Matrix Computations 275 6

1 Basic Concepts 276 6

2 Matrix Multiplication 292 6

3 Factorizations 300 6

The Unsymmetric Eigenvalue Problem 308 7

1 Properties and Decompositions 310 7

2 Perturbation Theory 320 7

3 Power Iterations 330 7

4 The Hessenberg and Real Schur Forms 341 7

5 The Practical QR Algorithm 352 7

6 Invariant Subspace Computations 362 7

7 The QZ Method for Ax=λBx 375 7

The Symmetric Eigenvalue Problem 391 8

1 Properties and Decompositions 393 8

2 Power Iterations 405 8

3 The Symmetric QR Algorithm 414 8

4 Jacobi Methods 426 8

5 Tridiagonal Methods 439 8

6 Computing the SVD 448 8

7 Some Generalized Eigenvalue Problems 461 8

Lanczos Methods 470 9

1 Derivation and Convergence Properties 471 9

2 Practical Lanczos Procedures 479 9

3 Applications to Ax=b and Least Squares 490 9

4 Arnoldi and Unsymmetric Lanczos 499 9

Iterative Methods for Linear Systems 508 10

1 The Standard Iterations 509 10

2 The Conjugate Gradient Method 520 10

3 Preconditioned Conjugate Gradients 532 10

4 Other Krylov Subspace Methods 544 10

Functions of Matrices 555 11

1 Eigenvalue Methods 556 11

2 Approximation Methods 562 11

3 The Matrix Exponential 572 11

Special Topics 579 12

1 Constrained Least Squares 580 12

2 Subset Selection Using the SVD 590 12

3 Total Least Squares 595 12

4 Computing Subspaces with the SVD 601 12

5 Updating Matrix Factorizations 606 12

6 Modified/Structured Eigenproblems 621 12