Fundamentals of matrix computations
The use of numerical methods continues to expand rapidly. At their heart lie matrix computations. Written in a clear, expository style, it allows students and professionals to build confidence in themselves by putting the theory behind matrix computations into practice instantly. Algorithms that allow students to work examples and write programs introduce each chapter. The book then moves on to discuss more complicated theoretical material. Using a step-by-step approach, it introduces mathematical material only as it is needed. Exercises range from routine computations and verifications to extensive programming projects and challenging proofs.
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Gaussian Elimination and its Variants
Sensitivity of Linear Systems Effects
Orthogonal Matrices and the LeastSquares
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approximation arithmetic back substitution bidiagonal block calculate Cholesky decomposition Cholesky factor Cholesky's method coefficient matrix columns of Q complex computation condition number convergence cost defined denote determined diagonal matrix eigenvalue problem equation example Exercise flop count forward substitution full rank Gaussian elimination Gram-Schmidt Hermitian Hessenberg matrix ill conditioned inner product invariant subspace inverse iteration Jacobi method least-squares problem linearly independent lower triangular main-diagonal entries matrix norm multiple nonsingular obtained orthogonal orthogonal matrix orthonormal pair perform perturbation polynomial positive definite Prove QR algorithm QR decomposition QR step R"Xn Rayleigh quotient iteration reduced reflectors result rotator roundoff errors row interchanges row operations rows and columns satisfy Schur's theorem Section sequence shift Show similarity transformation singular values solution solve stored submatrix subroutine Suppose symmetric matrix system Ax tridiagonal unique unitary upper Hessenberg form upper Hessenberg matrix upper triangular upper-triangular form zero