Numerical Methods · Floating-point arithmetic, conditioning, and stability
In four-digit decimal floating-point arithmetic with rounding to nearest, compute…
Problem
In four-digit decimal floating-point arithmetic with rounding to nearest, compute \(\mathrm{fl}(a+b)\) and \(\mathrm{fl}(a \times b)\) for \(a = 0.2345 \times 10^{4}\) and \(b = 0.7654 \times 10^{0}\). Then compute the absolute and relative errors of each result relative to the exact \(a+b\) and \(a \times b\). Finally, determine whether \(\mathrm{fl}(a+b)\) equals the exact sum rounded to four significant digits, and explain any discrepancy using the definition of a floating-point significand of length 4.
Hint
Convert to ordinary decimals, form the exact sum and product, then round each to four significant digits.
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