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Textbook in PDF format Why Multiplication with Unitary Matrices is a Good Thing Balancing a Matrix Wrapup Additional exercises Wrapup Additional exercises Summary Notes on the Stability of an Algorithm Launch Outline What you will learn Motivation Floating Point Numbers Notation Floating Point Computation Model of floating point computation Stability of a numerical algorithm Absolute value of vectors and matrices Stability of the Dot Product Operation An algorithm for computing Dot A simple start Preparation Target result A proof in traditional format A weapon of math induction for the war on (backward) error (optional) Results Stability of a Matrix-Vector Multiplication Algorithm An algorithm for computing Gemv Analysis Stability of a Matrix-Matrix Multiplication Algorithm An algorithm for computing Gemm Analysis An application Wrapup Additional exercises Summary Notes on Performance Notes on Gaussian Elimination and LU Factorization Opening Remarks Launch Outline What you will learn Definition and Existence LU Factorization First derivation Gauss transforms Cost of LU factorization LU Factorization with Partial Pivoting Permutation matrices The algorithm Proof of Theorem 12.3 LU with Complete Pivoting Solving A x = y Via the LU Factorization with Pivoting Solving Triangular Systems of Equations L z = y U x = z Other LU Factorization Algorithms Variant 1: Bordered algorithm Variant 2: Left-looking algorithm Variant 3: Up-looking variant Variant 4: Crout variant Variant 5: Classical LU factorization All algorithms Formal derivation of algorithms Numerical Stability Results Is LU with Partial Pivoting Stable? Blocked Algorithms Blocked classical LU factorization (Variant 5) Blocked classical LU factorization with pivoting (Variant 5) Variations on a Triple-Nested Loop Inverting a Matrix Basic observations Via the LU factorization with pivoting Gauss-Jordan inversion (Almost) never, ever invert a matrix Efficient Condition Number Estimation The problem Insights A simple approach Discussion Wrapup Additional exercises Summary Notes on Cholesky Factorization Opening Remarks Launch Outline What you will learn Definition and Existence Application An Algorithm Proof of the Cholesky Factorization Theorem Blocked Algorithm Alternative Representation Cost Solving the Linear Least-Squares Problem via the Cholesky Factorization Other Cholesky Factorization Algorithms Implementing the Cholesky Factorization with the (Traditional) BLAS What are the BLAS? A simple implementation in Fortran Implemention with calls to level-1 BLAS Matrix-vector operations (level-2 BLAS) Matrix-matrix operations (level-3 BLAS) Impact on performance Alternatives to the BLAS The FLAME/C API BLIS Wrapup Additional exercises Summary Notes on Eigenvalues and Eigenvectors Video Outline Definition The Schur and Spectral Factorizations Relation Between the SVD and the Spectral Decomposition Notes on the Power Method and Related Methods Video Outline The Power Method First attempt Second attempt Convergence Practical Power Method The Rayleigh quotient What if "026A30C 0"026A30C "026A30C 1 "026A30C? The Inverse Power Method Rayleigh-quotient Iteration Notes on the QR Algorithm and other Dense Eigensolvers Video Outline Preliminaries Subspace Iteration The QR Algorithm A basic (unshifted) QR algorithm A basic shifted QR algorithm Reduction to Tridiagonal Form Householder transformations (reflectors) Algorithm The QR algorithm with a Tridiagonal Matrix Givens' rotations QR Factorization of a Tridiagonal Matrix The Implicitly Shifted QR Algorithm Upper Hessenberg and tridiagonal matrices The Implicit Q Theorem The Francis QR Step A complete algorithm Further Reading More on reduction to tridiagonal form Optimizing the tridiagonal QR algorithm Other Algorithms Jacobi's method for the symmetric eigenvalue problem Cuppen's Algorithm The Method of Multiple Relatively Robust Representations (MRRR) The Nonsymmetric QR Algorithm A variant of the Schur decomposition Reduction to upperHessenberg form The implicitly double-shifted QR algorithm Notes on the Method of Relatively Robust Representations (MRRR) Outline MRRR, from 35,000 Feet Cholesky Factorization, Again The L D LT Factorization The U D UT Factorization The U D UT Factorization The Twisted Factorization Computing an Eigenvector from the Twisted Factorization Notes on Computing the SVD of a Matrix Outline Background Reduction to Bidiagonal Form The QR Algorithm with a Bidiagonal Matrix Putting it all together Answers Notes on Simple Vector and Matrix Operations Notes on Vector and Matrix Norms Notes on Orthogonality and the SVD Notes on Gram-Schmidt QR Factorization Notes on Householder QR Factorization Notes on Solving Linear Least-squares Problems (Answers) Notes on the Condition of a Problem Notes on the Stability of an Algorithm Notes on Peformance Noteson Gaussian Elimination and LU Factorization Notes on Cholesky Factorization Notes on Eigenvalues and Eigenvectors Notes on the Power Method and Related Methods Notes on the Symmetric QR Algorithm Notes on the Method of Relatively Robust Representations Notes on Computing the SVD How to Download LAFF Routines (FLAME@lab)