TY - JOUR

T1 - Stability and length of computations in the variational-difference method

AU - Dem'yanovich, Yu K.

PY - 1985/2/1

Y1 - 1985/2/1

N2 - The article considers a stable algorithm for computing the matrix and the righthand side of a variational-difference system of equations for one-dimensional nonsingular problems with a differential operator of order 2k. Since usually such systems are O(h-2k) conditioned, an error ε in the coefficients in general leads to an asymptotic error ε O(h-2k) in the solution of the system. A matrix subspace is identified in which an error ε leads to an error Cε in the solution of the system (in the energy norm) the constant C is independent of h), and then an algorithm is proposed which leaves the matrix computation error in this subspace. An approximate solution is represented as a sequence (a "word") of elementary arithmetic operations, and a bound on the computation length is derived. A measure of the computation length is a nonnegative functional defined on the set of words of some alphabet which has certain desirable properties. Particular instances of this functional include the number of operations, the weighted average number of macroinstructions of different types, the computation time, etc.

AB - The article considers a stable algorithm for computing the matrix and the righthand side of a variational-difference system of equations for one-dimensional nonsingular problems with a differential operator of order 2k. Since usually such systems are O(h-2k) conditioned, an error ε in the coefficients in general leads to an asymptotic error ε O(h-2k) in the solution of the system. A matrix subspace is identified in which an error ε leads to an error Cε in the solution of the system (in the energy norm) the constant C is independent of h), and then an algorithm is proposed which leaves the matrix computation error in this subspace. An approximate solution is represented as a sequence (a "word") of elementary arithmetic operations, and a bound on the computation length is derived. A measure of the computation length is a nonnegative functional defined on the set of words of some alphabet which has certain desirable properties. Particular instances of this functional include the number of operations, the weighted average number of macroinstructions of different types, the computation time, etc.

UR - http://www.scopus.com/inward/record.url?scp=34250116528&partnerID=8YFLogxK

U2 - 10.1007/BF02104302

DO - 10.1007/BF02104302

M3 - Article

AN - SCOPUS:34250116528

VL - 28

SP - 275

EP - 293

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 3

ER -