Consider the quantitya2 − b2 where a and b are real numbers. (a) Under what conditions should one expect an unusually large relative error in the computed value of a2 − b2 when this expression is evaluated in finite precision arithmetic? (b)cWs 4-digit (decimal) rounding arithmetic to evaluate both a2 − b2 and (a + b)(a − b) with a =995.1 and b=995.0. Calculate th relative error in each result. (c) The expression (a + b)(a − b) is algebraically equivalent to a2 − b2, but it is a more accurate way to calculate this quantity if both a and b have exact floating point representations. Why?

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Consider the quantitya2 − b2 where a and b are real numbers. (a) Under what conditions should one expect an unusually large relative error in the computed value of a2 − b2 when this expression is evaluated in finite precision arithmetic? (b)cWs 4-digit (decimal) rounding arithmetic to evaluate both a2 − b2 and (a + b)(a − b) with a =995.1 and b=995.0. Calculate th relative error in each result. (c) The expression (a + b)(a − b) is algebraically equivalent to a2 − b2, but it is a more accurate way to calculate this quantity if both a and b have exact floating point representations. Why?

Arnold Odonnell · Accepted Answer

Solution:  a.The condition is that the computed value is rounded off when this expression is evaluated in finite precision arithmetic.  b.Given: a=995.1 and b=995.0  a2 − b2=(995.1)2 − (995.0)2  =990224.01 − 990025  =199.01  4 digit rounding arithmetic we have a2 − b2=199.0.  The relative error becomes |199.01 − 199.0|199.01=0.01199.01≈0.525 × 10−4  Conclusion:  Given: a=995.1 and b=995.0  (a + b)(a − b)=(995.1 + 995.0)(995.1 − 995.0)  =(199.01)(0.1)  4 digit rounding arithmetic we have (a + b)(a − b)=199.0.  The relative error becomes |199.01 − 199.0|199.01=0.01199.01≈0.525 × 10−4  c.The expression (a + b)(a − b) is more accurate than a2 − b2 when a and b have exact floating point representations because the expression  (a + b)(a − b) involves simple addition and subtraction of decimals, then easy multiplication takes place,   however the expression a2 − b2 involves squaring of decimals resulting in more decimals , then subtraction takes place. The latter may involves round off whereas the former may not.

Consider the quantitya^{2} - b^{2} where a and b are real numbers. (a) Under what conditions should one expect an unusually large relative error in th

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