4.4 Summing Addends with Different Signs
4.4 Summing Addends with Different Signs The previous example involved addends that were all of the same sign. What happens when we sum both positive and negative numbers? Our next two programs experiment with the following series: which has alternating signs. [2]
Program 4-7 performs a simple straight summation of the series. See Listing 4-7. The summation stops as soon as the fractions are too small to change the value of the running sum. Listing 4-7 Summing numbers with mixed signs. package numbercruncher.program4_7; import numbercruncher.mathutils.AlignRight; /** * Program 4-7: Sum Numbers with Mixed Signs * * Sum a sequence numbers with alternating signs. The sum should be 0. */ public class SumMixedSigns { private static AlignRight ar = new AlignRight(); public static void main(String args[]) { int k = 0; int odd = 1; // odd number float kFactorial = 1; // k! boolean subtract = false; // true if subtract, false if add float sum = 0; // running sum float prevSum = -1; // previous value of running sum ar.print("k", 2); ar.print("Numerator", 11); ar.print("Factorial", 16); ar.print("Fraction", 16); ar.print("Running sum", 16); ar.underline(); // Loop until the running sum stops changing. do { float numerator = odd*odd; numerator += odd*numerator; float fraction = numerator/kFactorial; prevSum = sum; // Add or subtract the next fraction. if (subtract) fraction = -fraction; sum += fraction; ar.print(k, 2); ar.print(numerator, 11); ar.print(kFactorial, 16); ar.print(fraction, 16); ar.print(sum, 16); ar.println(); ++k; kFactorial *= k; odd += 2; subtract = !subtract; } while (sum != prevSum); } } Output: k Numerator Factorial Fraction Running sum ------------------------------------------------------------- 0 2.0 1.0 2.0 2.0 1 36.0 1.0 -36.0 -34.0 2 150.0 2.0 75.0 41.0 3 392.0 6.0 -65.333336 -24.333336 4 810.0 24.0 33.75 9.416664 5 1452.0 120.0 -12.1 -2.6833363 6 2366.0 720.0 3.286111 0.60277486 7 3600.0 5040.0 -0.71428573 -0.11151087 8 5202.0 40320.0 0.12901786 0.017506987 9 7220.0 362880.0 -0.019896384 -0.0023893975 10 9702.0 3628800.0 0.002673611 2.8421357E-4 11 12696.0 3.99168E7 -3.1806156E-4 -3.3847988E-5 12 16250.0 4.790016E8 3.392473E-5 7.674316E-8 13 20412.0 6.2270208E9 -3.277972E-6 -3.2012288E-6 14 25230.0 8.7178289E10 2.8940693E-7 -2.9118219E-6 15 30752.0 1.30767428E12 -2.351656E-8 -2.9353384E-6 16 37026.0 2.09227885E13 1.7696494E-9 -2.9335688E-6 17 44100.0 3.55687415E14 -1.2398527E-10 -2.9336927E-6 18 52022.0 6.4023735E15 8.125424E-12 -2.9336845E-6 19 60840.0 1.21645096E17 -5.001435E-13 -2.933685E-6 20 70602.0 2.43290202E18 2.9019664E-14 -2.933685E-6 The values of the running sum are supposed to oscillate about 0, alternating between positive and negative in step with the signs of the fractions. After the first couple of values, the running sum values should steadily decrease in magnitude. However, the pattern breaks when k = 12. The running sum's value seems smaller than expected, the next value has a greater magnitude, and all subsequent values are negative. The pattern breaks when the exponents of the fraction and of the running sum are the same. Because the fraction and the running sum have opposite signs, cancellation error is the culprit. When k = 12, we subtract Losing the leading significant digits throws off the remaining values of the running sum. It is also at this point that the factorial values (and hence the fraction values) begin to lose accuracy due to insufficient digits of precision. Program 4-8 uses double arithmetic to show one solution to the cancellation problem. See Listing 4-8. It computes separate subtotals for the positive fractions and for the negative fractions, thus removing the cancellation problem. Then, it computes the correct final sum by adding together the two subtotals. So we see that, when summing numbers with mixed signs, it is better to sum the positive numbers and the negative numbers separately. Listing 4-8 Summing numbers with mixed signs using separate positive and negative subtotals. package numbercruncher.program4_8; import numbercruncher.mathutils.AlignRight; /** * Program 4-8: Sum Numbers with Mixed Signs, * Positive and Negative Subtotals * * Sum a sequence of double numbers with alternating signs. * Compute separate positive and negative subtotals. The final sum * should be 0. */ public class SumMixedSignsPosNeg { public static void main(String args[]) { AlignRight ar = new AlignRight(); int k = 0; int odd = 1; // odd number double kFactorial = 1; // k! double posSum = 0; // running subtotal of pos fractions double negSum = 0; // running subtotal of neg fractions double prevPosSum = -1; // previous value of positive sum double prevNegSum = 1; // previous value of negative sum ar.print("k", 2); ar.print("Fraction", 25); ar.print("Positive subtotal", 20); ar.print("Negative subtotal", 21); ar.underline(); // Loop until the positive and negative subtotals // no longer change. do { // Positive fraction and subtotal. double posNumerator = odd*odd; posNumerator += odd*posNumerator; double posFraction = posNumerator/kFactorial; prevPosSum = posSum; posSum += posFraction; ar.print(k, 2); ar.print(posFraction, 25); ar.print(posSum, 20); ar.println(); ++k; kFactorial *= k; odd += 2; // Negative fraction and subtotal. double negNumerator = odd*odd; negNumerator += odd*negNumerator; double negFraction = -negNumerator/kFactorial; prevNegSum = negSum; negSum += negFraction; ar.print(k, 2); ar.print(negFraction, 25); ar.print(" ", 20); ar.print(negSum, 21); ar.println(); ++k; kFactorial *= k; odd += 2; } while ((posSum != prevPosSum) (negSum != prevNegSum)); System.out.println("\nFinal sum = " + (posSum + negSum)); } } Output: k Fraction Positive subtotal Negative subtotal -------------------------------------------------------------------- 0 2.0 2.0 1 -36.0 -36.0 2 75.0 77.0 3 -65.33333333333333 -101.33333333333333 4 33.75 110.75 5 -12.1 -113.43333333333332 6 3.286111111111111 114.03611111111111 7 -0.7142857142857143 -114.14761904761903 8 0.12901785714285716 114.16512896825397 9 -0.019896384479717814 -114.16751543209875 10 0.002673611111111111 114.16780257936507 11 -3.1806156806156807E-4 -114.16783349366682 12 3.392473010528566E-5 114.16783650409518 13 -3.277972027972028E-6 -114.16783677163885 14 2.89406911430721E-7 114.16783679350209 15 -2.35165579080923E-8 -114.16783679515541 16 1.7696492770897533E-9 114.16783679527174 17 -1.2398526491663746E-10 -114.1678367952794 18 8.125423849197927E-12 114.16783679527987 19 -5.001434484046243E-13 -114.1678367952799 20 2.901966448410855E-14 114.1678367952799 21 -1.5923761931532594E-15 -114.1678367952799 22 8.287361182067708E-17 114.1678367952799 23 -4.101498195323127E-18 -114.1678367952799 Final sum = 0.0 |
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