Rounding Errors in Algebraic ProcessesElementary introduction to problem of cumulative effect of rounding errors in a very large number of arithmetical calculations—particularly applicable to computer operations. Simple representative analyses illustrate techniques. Topics include fundamental arithmetic operations, computations involving polynomials and matrix computations. Results deal exclusively with digital computers but are equally applicable to desk calculators. Bibliography. |
Contents
THE FUNDAMENTAL ARITHMETIC OPERATIONS | 1 |
Blockfloating vectors and matrices | 26 |
Additional comments | 33 |
MATRIX COMPUTATIONS | 79 |
Accuracy of computed solution | 102 |
Inversion of a general matrix | 109 |
Triangular decomposition with partial pivoting | 115 |
Iterative refinement of the solution | 121 |
Other editions - View all
Common terms and phrases
a₁ accurate approximation assume b₁ b₂ block-floating vector calculation cancellation coefficients column component computed inverse computed solution computed sum computed value computed zero condition number consider convergence correctly rounded corresponding decimal defined deflated polynomial diag double-precision E₁ effect eigensystem eigenvalues eigenvectors elements error analysis error bound Euclidean norm exact solution exactly example fixed-point arithmetic fixed-point computation floating-point accumulation floating-point arithmetic floating-point computation Gaussian elimination give given Hence ill-conditioned inner-products interval iterative methods Laguerre's method limiting accuracy low relative error mantissa mathematical matrix norm maximum modulus multiple obtained order of magnitude original polynomial orthogonal partial pivoting perturbations practice precision problem relation residual vector result right-hand side root-squaring rounding errors satisfies sensitivity set of equations significant figure single-precision smaller squared polynomial stage standard floating-point subroutines symmetric matrix t-digit technique term tri-diagonal true u₁ upper bound usually vector norm x₁ δα λ₁