# An introduction to bundles, connections, metrics and by Taubes C.H.

By Taubes C.H.

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Apostolatos and U. Kulisch, Approximation der erweiterten Intervallarithmetik durch die einfache Maschinenintervallarithmetik. Computing 2, 181-194 (1967). N. Apostolatos et al, The algorithmic language Triplex-Algol 60. Numer. Math. 11, 175-180 (1968). R. Boche, Some observations on the economics of interval arithmetic. Comm. ACMS, 649 (1965). H. Christ, Realisierung einer Maschinenintervallarithmetik mit beliebigen ALGOL-60 Compilern. Elektron. Rech. 10, 217-222 (1968). J. K. S. Dewar, Procedures for interval arithmetic.

Let xia n P(x) = X i i= 0 be given as above. Then calculate the Horner scheme Pn: for i = n(- = an 1)1: Pi-1 ' = Pty + fli-i Chapter 3 34 and we have p0 = p(y). F r o m the definition cn-i=an Cn-2 c ( = Pn), = + 0 = cly l), + a (=Pi). 1 Therefore Cj_ x= ph 1 ^ / ^ n. Examples'. The examples (a) —(e) are taken from Alefeld [ 1 ] . 5], 7 = 2. 5]. coincides with the estimate given in Hansen [8] for this example. 3 (b) p(x) = x + 4x - 16, JT=[-1,3], >>=1. One gets / 7 2 = 7 3 1= = ^ = [1,17], / / ( X ) = ( / / ( X ) ) H= [ - 5 , 3 1 ] , which again coincides with the calculation in Hansen [ 8 ] .

Karlsruhe (1980). Signum Newsletter, Special Issue, October (1979). W. Wallisch and J. Grutzmann, Intervallanalytische Fehleranalyse. Beitr. Numer. Math. 3, 163-171 (1975). J. H. Wilkinson, "Rounding Errors in Algebraic Processes," H. M. Stationery Office, London, 1968. H. W. Wippermann, Realisierung einer Intervallarithmetik in einem ALGOL-60 System. Elektron. Rech. 9, 224-233 (1967). H. W. Wippermann, Definition von Schrankenzahlen in TRIPLEX-ALGOL 60. Computing 3, 99-109 (1968). Chapter 5 / COMPLEX INTERVAL ARITHMETIC We now wish to define and use a so-called complex interval arithmetic.