# Math 373 Fall 2000 Solution of exercise

Programming the bisection method in a spreadsheet leads to the following results for parts a and d of this exercise. The function whose roots we are finding is

f(x) = x4 - 2x3 - 4x2 + 4x + 4

and in both cases, a refers to a point where f(x) is negative and b refers to a point where f(x) is positive, so b is always to the left of a in the first example and always to the right in the second. In each case x is the midpoint, and a, b, and x are shown to full computer accuracy. The function values in the last column are shown to six decimal places since only the sign of these terms is used in the computation. However, it is interesting to note how this value is (more or less) approaching zero as we step through the method.
a b x f(x)
-1 -2 -1.5 +0.812500
-1 -1.5 -1.25 -0.902343
-1.25 -1.5 -1.375 -0.288818
-1.375 -1.5 -1.4375 +0.195327
-1.375 -1.4375 -1.40625 -0.062666
-1.40625 -1.4375 -1.421875 +0.062262
-1.40625 -1.421875 -1.4140625 -0.001208
-1.4140625 -1.421875 -1.41796875 +0.030274
-1.4140625 -1.41796875 -1.416015625 +0.014470
-1.4140625 -1.416015625 -1.4150390625 +0.006615
-1.4140625 -1.4150390625 -1.41455078125 +0.002699
-1.4140625 -1.41455078125 -1.414306640625 +0.000744
-1.4140625 -1.414306640625 -1.4141845703125 -0.000231
-1.4141845703125 -1.414306640625 -1.4142456054688 +0.000256
-1.4141845703125 -1.4142456054688 -1.4142150878906 +0.000012
-1.4141845703125 -1.4142150878906 -1.4141998291016 -0.000109
-1.4141998291016 -1.4142150878906 -1.4142074584961 -0.000048
-1.4142074584961 -1.4142150878906 -1.4142112731934 -0.000018
-1.4142112731934 -1.4142150878906 -1.414213180542 -0.000003
-1.414213180542 -1.4142150878906 -1.4142141342163 +0.000004
-1.414213180542 -1.4142141342163 -1.4142136573792 +0.000000
the interval [-2,-1]

a b x f(x)
-1 0 -0.5 +1.312500
-1 -0.5 -0.75 -0.089843
-0.75 -0.5 -0.625 +0.578369
-0.75 -0.625 -0.6875 +0.232681
-0.75 -0.6875 -0.71875 +0.068085
-0.75 -0.71875 -0.734375 -0.011768
-0.734375 -0.71875 -0.7265625 +0.027943
-0.734375 -0.7265625 -0.73046875 +0.008032
-0.734375 -0.73046875 -0.732421875 -0.001881
-0.732421875 -0.73046875 -0.7314453125 +0.003072
-0.732421875 -0.7314453125 -0.73193359375 +0.000594
-0.732421875 -0.73193359375 -0.732177734375 -0.000643
-0.732177734375 -0.73193359375 -0.7320556640625 -0.000024
-0.7320556640625 -0.73193359375 -0.7319946289063 +0.000284
-0.7320556640625 -0.7319946289063 -0.7320251464844 +0.000130
-0.7320556640625 -0.7320251464844 -0.7320404052734 +0.000052
-0.7320556640625 -0.7320404052734 -0.732048034668 +0.000014
-0.7320556640625 -0.732048034668 -0.7320518493652 -0.000005
-0.7320518493652 -0.732048034668 -0.7320499420166 +0.000004
-0.7320518493652 -0.7320499420166 -0.7320508956909 -0.000000
-0.7320508956909 -0.7320499420166 -0.7320504188538 +0.000001
the interval [-1,0]

Since each table starts with an interval of length 1 and has 20 more rows, the final interval has a length of 2-20 which is less than 10-6.