By Vladimir Grigor'evich Boltyanskii, Izrail' Tsudikovich Gohberg, Henry Christoffers, Thomas P. Branson
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An algorithm that satisfies this property is called stable; otherwise it is unstable. Some algorithms are stable only for certain choices of initial data. These are called conditionally stable. We will characterize the stability properties of algorithms whenever possible. To further consider the subject of round-off error growth and its connection to algorithm stability, suppose an error with magnitude E 0 > 0 is introduced at some stage in the calculations and that the magnitude of the error after n subsequent operations is denoted by E n .
15 K = 17◦ C. 08206) In the laboratory, it was found that T was 15◦ C under these conditions, and when the pressure was doubled and the volume halved, T was 19◦ C. Assume that the data are rounded values accurate to the places given, and show that both laboratory figures are within the bounds of accuracy for the ideal gas law. 3 Algorithms and Convergence Throughout the text we will examine approximation procedures, called algorithms, involving sequences of calculations. An algorithm is a procedure that describes, in an unambiguous manner, a finite sequence of steps to be performed in a specified order.
D1 d2 . . dk × 10n , 1 ≤ d1 ≤ 9, and 0 ≤ di ≤ 9, for each i = 2, . . , k. Numbers of this form are called k-digit decimal machine numbers. d1 d2 . . dk dk+1 dk+2 . . × 10n . The error that results from replacing a number with its floating-point form is called round-off error, regardless of whether the rounding or chopping method is used. The floating-point form of y, denoted f l(y), is obtained by terminating the mantissa of y at k decimal digits. There are two ways of performing this termination.