Numerical Linear Algebra
This is a concise, insightful introduction to the field of numerical linear algebra. The clarity and eloquence of the presentation make it popular with teachers and students alike. The text aims to expand the reader's view of the field and to present standard material in a novel way. All of the most important topics in the field are covered with a fresh perspective, including iterative methods for systems of equations and eigenvalue problems and the underlying principles of conditioning and stability. Presentation is in the form of 40 lectures, which each focus on one or two central ideas. The unity between topics is emphasized throughout, with no risk of getting lost in details and technicalities. The book breaks with tradition by beginning with the QR factorization - an important and fresh idea for students, and the thread that connects most of the algorithms of numerical linear algebra.
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accuracy applied approximation Arnoldi iteration backward stable bidiagonalization CG iteration Cholesky factorization coefficients column spaces columns of Q condition number convergence corresponding defined denote diagonal matrix dimension eigenvalue problem eigenvectors entries example Exercise Figure floating point arithmetic flops follows formula full rank function Gaussian elimination geometric GMRES Hessenberg Householder triangularization idea ill-conditioned introduce zeros inverse iteration iterative methods Krylov subspace Lanczos iteration least squares problem linear systems machine mathematical Matlab matrix norm minimal multiplication mXn matrix nonsingular nonzero normal equations numerical analysis numerical linear algebra O(machine operation count orthogonal projector orthonormal partial pivoting perturbations polynomial preconditioner QR algorithm random matrices range(A Rayleigh quotient Rayleigh quotient iteration reduced QR factorization result Ritz values rounding errors satisfying sequence SIAM singular values singular vectors solution solve stability step Suppose symmetric system of equations Theorem tridiagonal unitary matrix upper-triangular XXXXX