SICP Exercise 2.16



Exercise 2.16.  Explain, in general, why equivalent algebraic expressions may lead to different answers. Can you devise an interval-arithmetic package that does not have this shortcoming, or is this task impossible? (Warning: This problem is very difficult.)

EXPLANATION

 Equivalent arithmetic expressions can lead to different answers. This is because each primitive operation with intervals i.e. addition, subtraction, multiplication, division and reciprocal introduces its own error bounds into the calculation. The center and width of the final computed interval depends upon the way in which the component intervals are combined.

 Can we devise an interval-arithmetic package that does not have this shortcoming? First of all, we need to be clear about the algebraic expressions that the package will support. For each of these expressions, we will first need to determine the algebraic form that minimizes the repetitions of intervals. Then we will need to implement this algebraic form in the package. So, yes is a difficult problem.

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