Rounding Errors in Algebraic Processes
Elementary 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.
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THE FUND AMENTAL ARITH METI c o PERAT Ions
compUTATIONS INvolving PolyNoMIALs
Accuracy of computed solution
Inversion of a general matrix
Triangular decomposition with partial pivoting
Iterative refinement of the solution
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accurate approximation assume binary digits block-floating vector coefficients column component computed inverse computed solution computed sum computed value computed zero condition number consider convergence correct correctly rounded corresponding decimal defined deflated polynomial double-precision effect eigensystem eigenvectors elements error analysis error bounds exact solution exactly example fixed-point arithmetic fixed-point computation floating-point accumulation floating-point arithmetic floating-point computation floating-point numbers 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 number of digits obtained order of magnitude original polynomial orthogonal partial pivoting perturbations practice precision relation reorthogonalization 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 theory true upper bound usually vector norm