Strict error bounds in Romberg quadrature

Romberg quadrature using the Bulirsch sequence

“…Inclusions, stopping rules and monotonicity properties for the Romberg sequence are to be found in Ström [18], Albrecht [2] and Förster [12].…”

Error derivation in Romberg integration

“…From Theorem 3 we may draw several interesting conclusions. See also Strom [8]. (2), used in the extrapolation scheme.…”

“…The basic properties of the error term Z (k) in Eq (8). are described in n~ m the following theorem.…”

Monotonicity in Romberg quadrature

“…Hävie [2] suggested the approximation of j'0 f(x) dx by K^T' + U^~l)) with the error estimate } -U<km'"\. Ström [5] proved this inclusion to be strict if/<2m+2'(x) is of definite sign in (0, 1) and extended this restricted result to "majorizable" functions. In essence, / is "majorizable" if F is known such that (F ± /)<2m+2>(x) both are of (the same) definite sign in (0, 1) (Ström [6]).…”

“…Now Before we outline a proof we note that if all the even derivatives of the integrand are nonnegative (e.g., the integrand is an absolutely or completely monotonic function) then there is monotonicity column-wise ((i) and (ii)) and row-wise ((iii) and (iv)) in the scheme. Hence the best error approximation with absolutely monotonic majorants (Ström [5], [6]) appears from a "complete" extrapolation. Also the results (i)-(v) of Theorem 1, being "asymptotically true" in very general cases, in practice often hold for very moderate values of k.…”