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numerical method solutions

This document contains examples of applying the bisection method to find roots of functions. It shows the steps taken, the interval bounds being searched over at each step, the midpoint of the interval being tested, the function value at that midpoint, and whether the desired tolerance was reached. Several examples are provided of applying this method to different functions and intervals to find roots or determine that a root was not found within the tolerance.

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Laurence V. Fausett, Applied Numerical Analysis, Using MATLAB, Pearson, BISECTION METHOD p.2.1 [1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 0.2500 0.5000 2.0000 1.0000 1.5000 1.2500 -0.4375 0.2500 3.0000 1.2500 1.5000 1.3750 -0.1094 0.1250 4.0000 1.3750 1.5000 1.4375 0.0664 0.0625 5.0000 1.3750 1.4375 1.4063 -0.0225 0.0313 6.0000 1.4063 1.4375 1.4219 0.0217 0.0156 7.0000 1.4063 1.4219 1.4141 -0.0004 0.0078 zero not foundto desired tolerance p.2.2 [ 2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 1.2500 0.5000 2.0000 2.0000 2.5000 2.2500 0.0625 0.2500 3.0000 2.0000 2.2500 2.1250 -0.4844 0.1250 4.0000 2.1250 2.2500 2.1875 -0.2148 0.0625 5.0000 2.1875 2.2500 2.2188 -0.0771 0.0313 6.0000 2.2188 2.2500 2.2344 -0.0076 0.0156 7.0000 2.2344 2.2500 2.2422 0.0274 0.0078 8.0000 2.2344 2.2422 2.2383 0.0099 0.0039 zero not foundto desiredtolerence p.2.3 [ 2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 -0.7500 0.5000 2.0000 2.5000 3.0000 2.7500 0.5625 0.2500 3.0000 2.5000 2.7500 2.6250 -0.1094 0.1250 4.0000 2.6250 2.7500 2.6875 0.2227 0.0625 5.0000 2.6250 2.6875 2.6563 0.0557 0.0313 6.0000 2.6250 2.6563 2.6406 -0.0271 0.0156 zero not foundto desired tolerance
p.2.4 [1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 0.3750 0.5000 2.0000 1.0000 1.5000 1.2500 -1.0469 0.2500 3.0000 1.2500 1.5000 1.3750 -0.4004 0.1250 4.0000 1.3750 1.5000 1.4375 -0.0295 0.0625 5.0000 1.4375 1.5000 1.4688 0.1684 0.0313 6.0000 1.4375 1.4688 1.4531 0.0684 0.0156 7.0000 1.4375 1.4531 1.4453 0.0192 0.0078 8.0000 1.4375 1.4453 1.4414 -0.0053 0.0039 9.0000 1.4414 1.4453 1.4434 0.0069 0.0020 10.0000 1.4414 1.4434 1.4424 0.0008 0.0010 zero not foundto desired tolerance p.2.5 [1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 -0.6250 0.5000 2.0000 1.5000 2.0000 1.7500 1.3594 0.2500 3.0000 1.5000 1.7500 1.6250 0.2910 0.1250 4.0000 1.5000 1.6250 1.5625 -0.1853 0.0625 5.0000 1.5625 1.6250 1.5938 0.0482 0.0313 6.0000 1.5625 1.5938 1.5781 -0.0697 0.0156 7.0000 1.5781 1.5938 1.5859 -0.0111 0.0078 zero not foundto desiredtolerance p.2.6 [1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 -2.6250 0.5000 2.0000 1.5000 2.0000 1.7500 -0.6406 0.2500 3.0000 1.7500 2.0000 1.8750 0.5918 0.1250 4.0000 1.7500 1.8750 1.8125 -0.0457 0.0625 5.0000 1.8125 1.8750 1.8438 0.2677 0.0313 6.0000 1.8125 1.8438 1.8281 0.1097 0.0156 7.0000 1.8125 1.8281 1.8203 0.0317 0.0078 zero not foundto desiredtolerance
p.2.7 [ 0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.3875 0.5000 2.0000 0.5000 1.0000 0.7500 -0.1336 0.2500 3.0000 0.7500 1.0000 0.8750 0.1362 0.1250 4.0000 0.7500 0.8750 0.8125 -0.0142 0.0625 5.0000 0.8125 0.8750 0.8438 0.0568 0.0313 6.0000 0.8125 0.8438 0.8281 0.0203 0.0156 7.0000 0.8125 0.8281 0.8203 0.0028 0.0078 zero not foundto desiredtolerance p.2.8 [ 0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.5875 0.5000 2.0000 0.5000 1.0000 0.7500 -0.3336 0.2500 3.0000 0.7500 1.0000 0.8750 -0.0638 0.1250 4.0000 0.8750 1.0000 0.9375 0.1225 0.0625 5.0000 0.8750 0.9375 0.9063 0.0245 0.0313 6.0000 0.8750 0.9063 0.8906 -0.0208 0.0156 7.0000 0.8906 0.9063 0.8984 0.0016 0.0078 zero not foundto desired tolerance p.2.9 [ 0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 0.0025 0.5000 2.0000 0 0.5000 0.2500 -0.0561 0.2500 3.0000 0.2500 0.5000 0.3750 -0.0402 0.1250 4.0000 0.3750 0.5000 0.4375 -0.0234 0.0625 5.0000 0.4375 0.5000 0.4688 -0.0117 0.0313 6.0000 0.4688 0.5000 0.4844 -0.0050 0.0156 7.0000 0.4844 0.5000 0.4922 -0.0013 0.0078 zero not foundto desiredtolerence p.2.10
[ 0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.1875 0.5000 2.0000 0.5000 1.0000 0.7500 0.0664 0.2500 3.0000 0.5000 0.7500 0.6250 -0.0974 0.1250 4.0000 0.6250 0.7500 0.6875 -0.0266 0.0625 5.0000 0.6875 0.7500 0.7188 0.0169 0.0313 6.0000 0.6875 0.7188 0.7031 -0.0056 0.0156 7.0000 0.7031 0.7188 0.7109 0.0055 0.0078 zero not foundto desired tolerance p.2.11 (a) [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -2.3750 0.5000 2.0000 0 0.5000 0.2500 -0.2344 0.2500 3.0000 0 0.2500 0.1250 0.8770 0.1250 4.0000 0.1250 0.2500 0.1875 0.3191 0.0625 5.0000 0.1875 0.2500 0.2188 0.0417 0.0313 6.0000 0.2188 0.2500 0.2344 -0.0965 0.0156 7.0000 0.2188 0.2344 0.2266 -0.0274 0.0078 8.0000 0.2188 0.2266 0.2227 0.0071 0.0039 9.0000 0.2227 0.2266 0.2246 -0.0102 0.0020 10.0000 0.2227 0.2246 0.2236 -0.0015 0.0010 zero not foundto desired tolerance (b) [2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 -4.8750 0.5000 2.0000 2.5000 3.0000 2.7500 -1.9531 0.2500 3.0000 2.7500 3.0000 2.8750 -0.1113 0.1250 4.0000 2.8750 3.0000 2.9375 0.9099 0.0625 5.0000 2.8750 2.9375 2.9063 0.3908 0.0313 6.0000 2.8750 2.9063 2.8906 0.1376 0.0156 7.0000 2.8750 2.8906 2.8828 0.0126 0.0078 8.0000 2.8750 2.8828 2.8789 -0.0495 0.0039 9.0000 2.8789 2.8828 2.8809 -0.0185 0.0020 10.0000 2.8809 2.8828 2.8818 -0.0029 0.0010 zero not foundto desired tolerance (c) [-3 -4] step a b m ym bound 1.0000 -3.0000 -4.0000 -3.5000 -9.3750 -0.5000 2.0000 -3.0000 -3.5000 -3.2500 -3.0781 -0.2500 3.0000 -3.0000 -3.2500 -3.1250 -0.3926 -0.1250
4.0000 -3.0000 -3.1250 -3.0625 0.8396 -0.0625 5.0000 -3.0625 -3.1250 -3.0938 0.2326 -0.0313 6.0000 -3.0938 -3.1250 -3.1094 -0.0777 -0.0156 7.0000 -3.0938 -3.1094 -3.1016 0.0780 -0.0078 8.0000 -3.1016 -3.1094 -3.1055 0.0003 -0.0039 zero not foundto desired tolerance p.2.12 one real and 2 imaginary roots step a b m ym bound 1.0000 2.0000 3.0000 2.5000 -1.8750 0.5000 2.0000 2.5000 3.0000 2.7500 0.6719 0.2500 3.0000 2.5000 2.7500 2.6250 -0.6934 0.1250 4.0000 2.6250 2.7500 2.6875 -0.0344 0.0625 5.0000 2.6875 2.7500 2.7188 0.3127 0.0313 6.0000 2.6875 2.7188 2.7031 0.1377 0.0156 7.0000 2.6875 2.7031 2.6953 0.0512 0.0078 8.0000 2.6875 2.6953 2.6914 0.0083 0.0039 9.0000 2.6875 2.6914 2.6895 -0.0131 0.0020 10.0000 2.6895 2.6914 2.6904 -0.0024 0.0010 zero not foundto desiredtolerence p.2.13 (a) [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.1250 0.5000 2.0000 0.5000 1.0000 0.7500 1.1094 0.2500 3.0000 0.5000 0.7500 0.6250 0.4160 0.1250 4.0000 0.5000 0.6250 0.5625 0.1272 0.0625 5.0000 0.5000 0.5625 0.5313 -0.0034 0.0313 6.0000 0.5313 0.5625 0.5469 0.0608 0.0156 7.0000 0.5313 0.5469 0.5391 0.0284 0.0078 8.0000 0.5313 0.5391 0.5352 0.0124 0.0039 9.0000 0.5313 0.5352 0.5332 0.0045 0.0020 10.0000 0.5313 0.5332 0.5322 0.0006 0.0010 zero not foundto desiredtolerence (b) [0 -1] step a b m ym bound 1.0000 0 -1.0000 -0.5000 -0.3750 -0.5000 2.0000 -0.5000 -1.0000 -0.7500 0.2656 -0.2500 3.0000 -0.5000 -0.7500 -0.6250 -0.0723 -0.1250 4.0000 -0.6250 -0.7500 -0.6875 0.0930 -0.0625
5.0000 -0.6250 -0.6875 -0.6563 0.0094 -0.0313 6.0000 -0.6250 -0.6563 -0.6406 -0.0317 -0.0156 7.0000 -0.6406 -0.6563 -0.6484 -0.0112 -0.0078 8.0000 -0.6484 -0.6563 -0.6523 -0.0009 -0.0039 9.0000 -0.6523 -0.6563 -0.6543 0.0042 -0.0020 10.0000 -0.6523 -0.6543 -0.6533 0.0016 -0.0010 (c) [-2 -3] step a b m ym bound 1.0000 -2.0000 -3.0000 -2.5000 2.1250 -0.5000 2.0000 -2.5000 -3.0000 -2.7500 0.8906 -0.2500 3.0000 -2.7500 -3.0000 -2.8750 0.0332 -0.1250 4.0000 -2.8750 -3.0000 -2.9375 -0.4607 -0.0625 5.0000 -2.8750 -2.9375 -2.9063 -0.2082 -0.0313 6.0000 -2.8750 -2.9063 -2.8906 -0.0861 -0.0156 7.0000 -2.8750 -2.8906 -2.8828 -0.0261 -0.0078 8.0000 -2.8750 -2.8828 -2.8789 0.0036 -0.0039 9.0000 -2.8789 -2.8828 -2.8809 -0.0112 -0.0020 10.0000 -2.8789 -2.8809 -2.8799 -0.0038 -0.0010 zero not foundto desired tolerance p.2.14 (a) [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.8750 0.5000 2.0000 0 0.5000 0.2500 0.0156 0.2500 3.0000 0.2500 0.5000 0.3750 -0.4473 0.1250 4.0000 0.2500 0.3750 0.3125 -0.2195 0.0625 5.0000 0.2500 0.3125 0.2813 -0.1028 0.0313 6.0000 0.2500 0.2813 0.2656 -0.0438 0.0156 7.0000 0.2500 0.2656 0.2578 -0.0141 0.0078 8.0000 0.2500 0.2578 0.2539 0.0007 0.0039 9.0000 0.2539 0.2578 0.2559 -0.0067 0.0020 10.0000 0.2539 0.2559 0.2549 -0.0030 0.0010 zero not foundto desired tolerance (b) [1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 -1.6250 0.5000 2.0000 1.5000 2.0000 1.7500 -0.6406 0.2500 3.0000 1.7500 2.0000 1.8750 0.0918 0.1250 4.0000 1.7500 1.8750 1.8125 -0.2957 0.0625 5.0000 1.8125 1.8750 1.8438 -0.1073 0.0313 6.0000 1.8438 1.8750 1.8594 -0.0091 0.0156
7.0000 1.8594 1.8750 1.8672 0.0410 0.0078 8.0000 1.8594 1.8672 1.8633 0.0158 0.0039 9.0000 1.8594 1.8633 1.8613 0.0033 0.0020 10.0000 1.8594 1.8613 1.8604 -0.0029 0.0010 zero not foundto desired tolerance (c) [-2 -3] step a b m ym bound 1.0000 -2.0000 -3.0000 -2.5000 -4.6250 -0.5000 2.0000 -2.0000 -2.5000 -2.2500 -1.3906 -0.2500 3.0000 -2.0000 -2.2500 -2.1250 -0.0957 -0.1250 4.0000 -2.0000 -2.1250 -2.0625 0.4763 -0.0625 5.0000 -2.0625 -2.1250 -2.0938 0.1964 -0.0313 6.0000 -2.0938 -2.1250 -2.1094 0.0519 -0.0156 7.0000 -2.1094 -2.1250 -2.1172 -0.0215 -0.0078 8.0000 -2.1094 -2.1172 -2.1133 0.0153 -0.0039 9.0000 -2.1133 -2.1172 -2.1152 -0.0031 -0.0020 10.0000 -2.1133 -2.1152 -2.1143 0.0061 -0.0010 zero not foundto desiredtolerence >> p.2.15 [2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 -3.6250 0.5000 2.0000 2.5000 3.0000 2.7500 -0.7656 0.2500 3.0000 2.7500 3.0000 2.8750 0.9980 0.1250 4.0000 2.7500 2.8750 2.8125 0.0872 0.0625 5.0000 2.7500 2.8125 2.7813 -0.3464 0.0313 6.0000 2.7813 2.8125 2.7969 -0.1314 0.0156 7.0000 2.7969 2.8125 2.8047 -0.0226 0.0078 8.0000 2.8047 2.8125 2.8086 0.0322 0.0039 9.0000 2.8047 2.8086 2.8066 0.0048 0.0020 10.0000 2.8047 2.8066 2.8057 -0.0089 0.0010 zero not foundto desired tolerance p.2.16 [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.8750 0.5000 2.0000 0.5000 1.0000 0.7500 0.2969 0.2500 3.0000 0.5000 0.7500 0.6250 -0.2246 0.1250 4.0000 0.6250 0.7500 0.6875 0.0515 0.0625
5.0000 0.6250 0.6875 0.6563 -0.0826 0.0313 6.0000 0.6563 0.6875 0.6719 -0.0146 0.0156 7.0000 0.6719 0.6875 0.6797 0.0187 0.0078 8.0000 0.6719 0.6797 0.6758 0.0021 0.0039 9.0000 0.6719 0.6758 0.6738 -0.0062 0.0020 10.0000 0.6738 0.6758 0.6748 -0.0020 0.0010 zero not foundto desiredtolerence p.2.17 (a) [ 1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 -1.5000 0.5000 2.0000 1.0000 1.5000 1.2500 0.7813 0.2500 3.0000 1.2500 1.5000 1.3750 -0.3867 0.1250 4.0000 1.2500 1.3750 1.3125 0.1948 0.0625 5.0000 1.3125 1.3750 1.3438 -0.0971 0.0313 6.0000 1.3125 1.3438 1.3281 0.0486 0.0156 7.0000 1.3281 1.3438 1.3359 -0.0243 0.0078 8.0000 1.3281 1.3359 1.3320 0.0122 0.0039 9.0000 1.3320 1.3359 1.3340 -0.0061 0.0020 10.0000 1.3320 1.3340 1.3330 0.0030 0.0010 zero not foundto desiredtolerence (b) [2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 0 0.5000 bisectionhas converged (c) 0 p.2.18 (a) [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -2.8750 0.5000 2.0000 0 0.5000 0.2500 1.4844 0.2500 3.0000 0.2500 0.5000 0.3750 -0.7324 0.1250 4.0000 0.2500 0.3750 0.3125 0.3689 0.0625 5.0000 0.3125 0.3750 0.3438 -0.1838 0.0313 6.0000 0.3125 0.3438 0.3281 0.0921 0.0156 7.0000 0.3281 0.3438 0.3359 -0.0460 0.0078 8.0000 0.3281 0.3359 0.3320 0.0230 0.0039 9.0000 0.3320 0.3359 0.3340 -0.0115 0.0020
10.0000 0.3320 0.3340 0.3330 0.0058 0.0010 zero not foundto desiredtolerence (b) [-2 -3] step a b m ym bound 1.0000 -2.0000 -3.0000 -2.5000 -2.1250 -0.5000 2.0000 -2.0000 -2.5000 -2.2500 7.2656 -0.2500 3.0000 -2.2500 -2.5000 -2.3750 2.9199 -0.1250 4.0000 -2.3750 -2.5000 -2.4375 0.4871 -0.0625 5.0000 -2.4375 -2.5000 -2.4688 -0.7963 -0.0313 6.0000 -2.4375 -2.4688 -2.4531 -0.1490 -0.0156 7.0000 -2.4375 -2.4531 -2.4453 0.1704 -0.0078 8.0000 -2.4453 -2.4531 -2.4492 0.0111 -0.0039 9.0000 -2.4492 -2.4531 -2.4512 -0.0689 -0.0020 10.0000 -2.4492 -2.4512 -2.4502 -0.0289 -0.0010 zero not foundto desired tolerance (c) [2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 1.6250 0.5000 2.0000 2.0000 2.5000 2.2500 -5.3906 0.2500 3.0000 2.2500 2.5000 2.3750 -2.2012 0.1250 4.0000 2.3750 2.5000 2.4375 -0.3699 0.0625 5.0000 2.4375 2.5000 2.4688 0.6068 0.0313 6.0000 2.4375 2.4688 2.4531 0.1133 0.0156 7.0000 2.4375 2.4531 2.4453 -0.1295 0.0078 8.0000 2.4453 2.4531 2.4492 -0.0084 0.0039 9.0000 2.4492 2.4531 2.4512 0.0524 0.0020 10.0000 2.4492 2.4512 2.4502 0.0220 0.0010 zero not foundto desired tolerance p.2.19 (a) [-1 -2] step a b m ym bound 1.0000 -1.0000 -2.0000 -1.5000 -1.6250 -0.5000 2.0000 -1.5000 -2.0000 -1.7500 1.5781 -0.2500 3.0000 -1.5000 -1.7500 -1.6250 0.0684 -0.1250 4.0000 -1.5000 -1.6250 -1.5625 -0.7561 -0.0625 5.0000 -1.5625 -1.6250 -1.5938 -0.3382 -0.0313 6.0000 -1.5938 -1.6250 -1.6094 -0.1335 -0.0156 7.0000 -1.6094 -1.6250 -1.6172 -0.0322 -0.0078 8.0000 -1.6172 -1.6250 -1.6211 0.0182 -0.0039 9.0000 -1.6172 -1.6211 -1.6191 -0.0070 -0.0020 10.0000 -1.6191 -1.6211 -1.6201 0.0056 -0.0010 zero not foundto desiredtolerence
(b) [5 6] step a b m ym bound 1.0000 5.0000 6.0000 5.5000 -27.8750 0.5000 2.0000 5.5000 6.0000 5.7500 -12.9531 0.2500 3.0000 5.7500 6.0000 5.8750 -4.7363 0.1250 4.0000 5.8750 6.0000 5.9375 -0.4338 0.0625 5.0000 5.9375 6.0000 5.9688 1.7666 0.0313 6.0000 5.9375 5.9688 5.9531 0.6623 0.0156 7.0000 5.9375 5.9531 5.9453 0.1132 0.0078 8.0000 5.9375 5.9453 5.9414 -0.1606 0.0039 9.0000 5.9414 5.9453 5.9434 -0.0238 0.0020 10.0000 5.9434 5.9453 5.9443 0.0447 0.0010 zero not foundto desiredtolerence (c) [-3 -4] step a b m ym bound 1.0000 -3.0000 -4.0000 -3.5000 -3.1250 -0.5000 2.0000 -3.0000 -3.5000 -3.2500 1.1094 -0.2500 3.0000 -3.2500 -3.5000 -3.3750 -0.8340 -0.1250 4.0000 -3.2500 -3.3750 -3.3125 0.1804 -0.0625 5.0000 -3.3125 -3.3750 -3.3438 -0.3160 -0.0313 6.0000 -3.3125 -3.3438 -3.3281 -0.0651 -0.0156 7.0000 -3.3125 -3.3281 -3.3203 0.0583 -0.0078 8.0000 -3.3203 -3.3281 -3.3242 -0.0032 -0.0039 9.0000 -3.3203 -3.3242 -3.3223 0.0276 -0.0020 10.0000 -3.3223 -3.3242 -3.3232 0.0122 -0.0010 zero not foundto desiredtolerence p.2.20 (a) [3 4] step a b m ym bound 1.0000 3.0000 4.0000 3.5000 -0.8750 0.5000 2.0000 3.5000 4.0000 3.7500 -0.2031 0.2500 3.0000 3.7500 4.0000 3.8750 0.3262 0.1250 4.0000 3.7500 3.8750 3.8125 0.0442 0.0625 5.0000 3.7500 3.8125 3.7813 -0.0837 0.0313 6.0000 3.7813 3.8125 3.7969 -0.0208 0.0156 7.0000 3.7969 3.8125 3.8047 0.0114 0.0078 8.0000 3.7969 3.8047 3.8008 -0.0048 0.0039 9.0000 3.8008 3.8047 3.8027 0.0033 0.0020 10.0000 3.8008 3.8027 3.8018 -0.0007 0.0010 zero not foundto desiredtolerence (b) [0 1]
step a b m ym bound 1.0000 0 1.0000 0.5000 -1.6250 0.5000 2.0000 0.5000 1.0000 0.7500 -0.0156 0.2500 3.0000 0.7500 1.0000 0.8750 0.5605 0.1250 4.0000 0.7500 0.8750 0.8125 0.2903 0.0625 5.0000 0.7500 0.8125 0.7813 0.1419 0.0313 6.0000 0.7500 0.7813 0.7656 0.0643 0.0156 7.0000 0.7500 0.7656 0.7578 0.0246 0.0078 8.0000 0.7500 0.7578 0.7539 0.0046 0.0039 9.0000 0.7500 0.7539 0.7520 -0.0055 0.0020 10.0000 0.7520 0.7539 0.7529 -0.0005 0.0010 zero not foundto desiredtolerence >> SECANT METHOD p.2.1 [-1 2] secant(inline('x^2-2'),-1,2,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1 -1 2 0 -2 -2 2 2 0 1 -1 1 3 0 1 2 2 1 4 1.0000 2.0000 1.3333 -0.2222 -0.6667 5 2.0000 1.3333 1.4000 -0.0400 0.0667 6 1.3333 1.4000 1.4146 0.0012 0.0146 secantmethod has converged ans = 1.4146 p.2.2 [ 2 3] secant(inline('x^2-5'),2,3,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1 2.0000 3.0000 2.2000 -0.1600 -0.8000 2 3.0000 2.2000 2.2308 -0.0237 0.0308 3 2.2000 2.2308 2.2361 0.0002 0.0053 secantmethod has converged ans = 2.2361
p.2.3 [ 2 3] >> secant(inline('x^2-7'),2,3,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1 2.0000 3.0000 2.6000 -0.2400 -0.4000 2 3.0000 2.6000 2.6429 -0.0153 0.0429 3 2.6000 2.6429 2.6458 0.0001 0.0029 secantmethod has converged ans = 2.6458 p.2.4 [1 2] secant(inline('x^3-6'),1,2,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1 1.0000 2.0000 1.7143 -0.9621 -0.2857 2 2.0000 1.7143 1.8071 -0.0988 0.0928 3 1.7143 1.8071 1.8177 0.0059 0.0106 4. 1.8071 1.8177 1.8171 -0.0000 -0.0006 secantmethod has converged ans = 1.8171 p.2.5 [1 2] secant(inline('x^3-4'),1,2,0.005,7) step x(k-1) x (k) x(k+1) y(k+1) dx(k+1) 1 1.0000 2.0000 1.4286 -1.0845 -0.5714 2 2.0000 1.4286 1.5505 -0.2728 0.1219 3 1.4286 1.5505 1.5914 0.0305 0.0410 4 1.5505 1.5914 1.5873 -0.0007 -0.0041 secantmethod has converged ans = 1.5873 p.2.6
[1 2] secant(inline('x^3-6'),1,2,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 1.0000 2.0000 1.7143 -0.9621 -0.2857 2.0000 2.0000 1.7143 1.8071 -0.0988 0.0928 3.0000 1.7143 1.8071 1.8177 0.0059 0.0106 4.0000 1.8071 1.8177 1.8171 -0.0000 -0.0006 secantmethod has converged ans = 1.8171 p.2.7 [ 0 1] secant(inline('x^4-0.45'),0,1,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.4500 -0.4090 -0.5500 2.0000 1.0000 0.4500 0.6846 -0.2304 0.2346 3.0000 0.4500 0.6846 0.9871 0.4995 0.3026 4.0000 0.6846 0.9871 0.7801 -0.0797 -0.2071 5.0000 0.9871 0.7801 0.8086 -0.0226 0.0285 6.0000 0.7801 0.8086 0.8198 0.0017 0.0113 secantmethod has converged ans = 0.8198 p.2.8 [ 0 1] secant(inline('x^4-0.65'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.6500 -0.4715 -0.3500 2.0000 1.0000 0.6500 0.8509 -0.1258 0.2009 3.0000 0.6500 0.8509 0.9240 0.0789 0.0731 4.0000 0.8509 0.9240 0.8958 -0.0060 -0.0282 5.0000 0.9240 0.8958 0.8978 -0.0003 0.0020 secantmethod has converged ans = 0.8978
>> p.2.9 sm prob of e p.2.10 [ 0 1] secant(inline('x^4-0.25'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.2500 -0.2461 -0.7500 2.0000 1.0000 0.2500 0.4353 -0.2141 0.1853 3.0000 0.2500 0.4353 1.6751 7.6241 1.2398 4.0000 0.4353 1.6751 0.4692 -0.2016 -1.2060 5.0000 1.6751 0.4692 0.5002 -0.1874 0.0311 6.0000 0.4692 0.5002 0.9112 0.4395 0.4110 7.0000 0.5002 0.9112 0.6231 -0.0993 -0.2881 ans = 0.6231 >> p.2.11 (a) [0 1] secant(inline('x^3-9*x+2'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.2500 -0.2344 -0.7500 2.0000 1.0000 0.2500 0.2195 0.0350 -0.0305 3.0000 0.2500 0.2195 0.2235 -0.0001 0.0040 secantmethod has converged ans = 0.2235 (b) [-3 -4] secant(inline('x^3-9*x+2'),-3,-4,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -3.0000 -4.0000 -3.0714 0.6680 0.9286 2.0000 -4.0000 -3.0714 -3.0947 0.2141 -0.0233 3.0000 -3.0714 -3.0947 -3.1057 -0.0035 -0.0110 secantmethod has converged ans =
-3.1057 (c) [2 3] secant(inline('x^3-9*x+2'),2,3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.8000 -1.2480 -0.2000 2.0000 3.0000 2.8000 2.8768 -0.0821 0.0768 3.0000 2.8000 2.8768 2.8823 0.0038 0.0054 secantmethod has converged ans = 2.8823 >> p.2.12 secant(inline('x^3-2*x^2-5'),2,3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.5556 -1.3717 -0.4444 2.0000 3.0000 2.5556 2.6691 -0.2338 0.1135 3.0000 2.5556 2.6691 2.6924 0.0189 0.0233 4.0000 2.6691 2.6924 2.6906 -0.0002 -0.0017 secantmethod has converged ans = 2.6906 p.2.13 (a) [-2 -3] secant(inline('x^3+3*x^2-1'),-2,-3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -2.0000 -3.0000 -2.7500 0.8906 0.2500 2.0000 -3.0000 -2.7500 -2.8678 0.0875 -0.1178 3.0000 -2.7500 -2.8678 -2.8806 -0.0092 -0.0128 4.0000 -2.8678 -2.8806 -2.8794 0.0001 0.0012 secantmethod has converged ans = -2.8794 (b) [0 -1] secant(inline('x^3+3*x^2-1'),0,-1,0.005,8)
step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 -1.0000 -0.5000 -0.3750 0.5000 2.0000 -1.0000 -0.5000 -0.6364 -0.0428 -0.1364 3.0000 -0.5000 -0.6364 -0.6539 0.0033 -0.0176 secantmethod has converged ans = -0.6539 (c) [0 1] secant(inline('x^3+3*x^2-1'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.2500 -0.7969 -0.7500 2.0000 1.0000 0.2500 0.4074 -0.4344 0.1574 3.0000 0.2500 0.4074 0.5961 0.2777 0.1887 4.0000 0.4074 0.5961 0.5225 -0.0383 -0.0736 5.0000 0.5961 0.5225 0.5314 -0.0027 0.0089 secantmethod has converged ans = 0.5314 p.2.14 (a) [0 1] secant(inline('x^3-4*x+1'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.3333 -0.2963 -0.6667 2.0000 1.0000 0.3333 0.2174 0.1407 -0.1159 3.0000 0.3333 0.2174 0.2547 -0.0024 0.0373 secantmethod has converged ans = 0.2547 (b) [1 2] secant(inline('x^3-4*x+1'),1,2,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 1.0000 2.0000 1.6667 -1.0370 -0.3333 2.0000 2.0000 1.6667 1.8364 -0.1528 0.1697 3.0000 1.6667 1.8364 1.8657 0.0313 0.0293 4.0000 1.8364 1.8657 1.8607 -0.0007 -0.0050 secantmethod has converged ans = 1.8607
(c) [-2 -3] secant(inline('x^3-4*x+1'),-2,-3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -2.0000 -3.0000 -2.0667 0.4397 0.9333 2.0000 -3.0000 -2.0667 -2.0951 0.1842 -0.0284 3.0000 -2.0667 -2.0951 -2.1156 -0.0063 -0.0205 4.0000 -2.0951 -2.1156 -2.1149 0.0001 0.0007 secantmethod has converged ans = -2.1149 p.2.15 [2 3] secant(inline('x^3-x^2-4*x-3'),2,3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.7000 -1.4070 -0.3000 2.0000 3.0000 2.7000 2.7958 -0.1466 0.0958 3.0000 2.7000 2.7958 2.8069 0.0087 0.0111 4.0000 2.7958 2.8069 2.8063 -0.0000 -0.0006 secantmethod has converged ans = 2.8063 p.2.16 [0 1] secant(inline('x^3-6*x^2+11*x-5'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.8333 0.5787 -0.1667 2.0000 1.0000 0.8333 0.6044 -0.3226 -0.2289 3.0000 0.8333 0.6044 0.6863 0.0467 0.0819 4.0000 0.6044 0.6863 0.6760 0.0030 -0.0104 secantmethod has converged ans = 0.6760
p.2.17 (a) [ 1 2] secant(inline('6*x^3-23*x^2+20*x'),1,2,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 1.0000 2.0000 1.4286 -0.8746 -0.5714 2.0000 2.0000 1.4286 1.2687 0.6062 -0.1599 3.0000 1.4286 1.2687 1.3341 -0.0073 0.0655 4.0000 1.2687 1.3341 1.3333 -0.0000 -0.0008 secantmethod has converged ans = 1.3333 (b) [2 3] Secant(inline('6*x^3-23*x^2+20*x'),2,3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.2105 -3.3678 -0.7895 2.0000 3.0000 2.2105 2.3553 -2.0900 0.1448 3.0000 2.2105 2.3553 2.5920 1.8018 0.2368 4.0000 2.3553 2.5920 2.4824 -0.3007 -0.1096 5.0000 2.5920 2.4824 2.4981 -0.0330 0.0157 6.0000 2.4824 2.4981 2.5000 0.0007 0.0019 secantmethod has converged ans = 2.5000 p.2.18 (a) [2 3] secant(inline('3*x^3-x^2-18*x+6'),2,3,0.005,8) step x(k-1) x(k) x (k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.2941 -4.3354 -0.7059 2.0000 3.0000 2.2941 2.4021 -1.4263 0.1080 3.0000 2.2941 2.4021 2.4551 0.1743 0.0530 4.0000 2.4021 2.4551 2.4493 -0.0057 -0.0058 5.0000 2.4551 2.4493 2.4495 -0.0000 0.0002 secantmethod has converged ans = 2.4495
(b) [-2 -3] secant(inline('3*x^3-x^2-18*x+6'),-2,-3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -2.0000 -3.0000 -2.3182 4.9798 0.6818 2.0000 -3.0000 -2.3182 -2.4152 1.3736 -0.0971 3.0000 -2.3182 -2.4152 -2.4522 -0.1118 -0.0370 4.0000 -2.4152 -2.4522 -2.4494 0.0022 0.0028 secantmethod has converged ans = -2.4494 (c) [0 1] secant(inline('3*x^3-x^2-18*x+6'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.3750 -0.7324 -0.6250 2.0000 1.0000 0.3750 0.3256 0.1366 -0.0494 3.0000 0.3750 0.3256 0.3334 -0.0007 0.0078 secantmethod has converged ans = 0.3334 p.2.19 (a) [-1 -2] secant(inline('x^3-x^2-24*x-32'),-1,-2,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -1.0000 -2.0000 -1.7143 1.1662 0.2857 2.0000 -2.0000 -1.7143 -1.5967 -0.2993 0.1176 3.0000 -1.7143 -1.5967 -1.6207 0.0133 -0.0240 4.0000 -1.5967 -1.6207 -1.6197 0.0001 0.0010 secantmethod has converged ans = -1.6197 (b) [5 6] secant(inline('x^3-x^2-24*x-32'),5,6,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 5.0000 6.0000 5.9286 -1.0565 -0.0714 2.0000 6.0000 5.9286 5.9435 -0.0142 0.0149 3.0000 5.9286 5.9435 5.9437 0.0001 0.0002 secantmethod has converged ans =
5.9437 (c) [-3 -4] secant(inline('x^3-x^2-24*x-32'),-3,-4,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -3.0000 -4.0000 -3.2000 1.7920 0.8000 2.0000 -4.0000 -3.2000 -3.2806 0.6655 -0.0806 3.0000 -3.2000 -3.2806 -3.3282 -0.0659 -0.0476 4.0000 -3.2806 -3.3282 -3.3239 0.0020 0.0043 secantmethod has converged ans = -3.3239 p.2.20 (a) [-3 -4] secant(inline('x^3-7*x^2+14*x-7'),-3,-4,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -3.0000 -4.0000 -1.6100 -51.8580 2.3900 2.0000 -4.0000 -1.6100 -0.9477 -27.4065 0.6623 3.0000 -1.6100 -0.9477 -0.2054 -10.1796 0.7423 4.0000 -0.9477 -0.2054 0.2332 -4.1027 0.4386 5.0000 -0.2054 0.2332 0.5294 -1.4020 0.2961 6.0000 0.2332 0.5294 0.6831 -0.3841 0.1537 7.0000 0.5294 0.6831 0.7411 -0.0620 0.0580 ans = 0.7411 (b) [3 4] secant(inline('x^3-7*x^2+14*x-7'),3,4,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 3.0000 4.0000 3.5000 -0.8750 -0.5000 2.0000 4.0000 3.5000 3.7333 -0.2634 0.2333 3.0000 3.5000 3.7333 3.8338 0.1364 0.1005 4.0000 3.7333 3.8338 3.7995 -0.0099 -0.0343 5.0000 3.8338 3.7995 3.8019 -0.0003 0.0023 secantmethod has converged ans = 3.8019 (c) [0 1]
>> secant(inline('x^3-7*x^2+14*x-7'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.8750 0.5605 -0.1250 2.0000 1.0000 0.8750 0.7156 -0.2000 -0.1594 3.0000 0.8750 0.7156 0.7575 0.0229 0.0419 4.0000 0.7156 0.7575 0.7532 0.0008 -0.0043 secantmethod has converged ans = 0.7532 REGULA FALSI METHOD Falsi method Q1 [s,y]=falsi(inline('x^2-2'),1,2,0.005,5) step a b s y 1.0000 1.0000 2.0000 1.3333 -0.2222 2.0000 1.3333 2.0000 1.4000 -0.0400 3.0000 1.4000 2.0000 1.4118 -0.0069 4.0000 1.4118 2.0000 1.4138 -0.0012 regulafalsi methodhasconverged s = 1.4138 y = -0.0012 Q2. [s,y]=falsi(inline('x^2-5'),2,3,0.005,5) step a b s y 1.0000 2.0000 3.0000 2.2000 -0.1600 2.0000 2.2000 3.0000 2.2308 -0.0237 3.0000 2.2308 3.0000 2.2353 -0.0035 regulafalsi methodhasconverged s = 2.2353 y = -0.0035 Q3
>> [s,y]=falsi(inline('x^2-7'),2,3,0.005,5) step a b s y 1.0000 2.0000 3.0000 2.6000 -0.2400 2.0000 2.6000 3.0000 2.6429 -0.0153 3.0000 2.6429 3.0000 2.6456 -0.0010 regulafalsi methodhasconverged s = 2.6456 y = -9.6138e-004 Q4 >> [s,y]=falsi(inline('x^3-3'),1,2,0.005,5) step a b s y 1.0000 1.0000 2.0000 1.2857 -0.8746 2.0000 1.2857 2.0000 1.3921 -0.3024 3.0000 1.3921 2.0000 1.4267 -0.0958 4.0000 1.4267 2.0000 1.4375 -0.0295 5.0000 1.4375 2.0000 1.4408 -0.0090 ZERO NOT FOUND TO DESIRED TOLERANCE s = 1.4408 y = -0.0090 Q5 >> [s,y]=falsi(inline('x^3-4'),1,2,0.005,5) step a b s y 1.0000 1.0000 2.0000 1.4286 -1.0845 2.0000 1.4286 2.0000 1.5505 -0.2728 3.0000 1.5505 2.0000 1.5792 -0.0620 4.0000 1.5792 2.0000 1.5856 -0.0137 5.0000 1.5856 2.0000 1.5870 -0.0030 regulafalsi methodhasconverged s = 1.5870 y = -0.0030 Q6 >> [s,y]=falsi(inline('x^3-6'),1,2,0.005,5) step a b s y 1.0000 1.0000 2.0000 1.7143 -0.9621 2.0000 1.7143 2.0000 1.8071 -0.0988
3.0000 1.8071 2.0000 1.8162 -0.0094 4.0000 1.8162 2.0000 1.8170 -0.0009 regulafalsi methodhasconverged s = 1.8170 y = -8.8557e-004 Q7 >> [s,y]=falsi(inline('x^4-0.45'),0,1,0.005,5) step a b s y 1.0000 0 1.0000 0.4500 -0.4090 2.0000 0.4500 1.0000 0.6846 -0.2304 3.0000 0.6846 1.0000 0.7777 -0.0842 4.0000 0.7777 1.0000 0.8072 -0.0254 5.0000 0.8072 1.0000 0.8157 -0.0072 ZERO NOT FOUND TO DESIRED TOLERANCE s = 0.8157 y = -0.0072 Q8 >> [s,y]=falsi(inline('x^4-0.65'),0,1,0.005,5) step a b s y 1.0000 0 1.0000 0.6500 -0.4715 2.0000 0.6500 1.0000 0.8509 -0.1258 3.0000 0.8509 1.0000 0.8903 -0.0217 4.0000 0.8903 1.0000 0.8967 -0.0034 regulafalsi methodhasconverged s = 0.8967 y = -0.0034 Q9 >> [s,y]=falsi(inline('x^4-0.06'),0,1,0.005,5) step a b s y 1.0000 0 1.0000 0.0600 -0.0600 2.0000 0.0600 1.0000 0.1164 -0.0598 3.0000 0.1164 1.0000 0.1693 -0.0592 4.0000 0.1693 1.0000 0.2185 -0.0577 5.0000 0.2185 1.0000 0.2637 -0.0552 ZERO NOT FOUND TO DESIRED TOLERANCE s =
0.2637 y = -0.0552 Q10 >> [s,y]=falsi(inline('x^4-0.25'),0,1,0.005,5) step a b s y 1.0000 0 1.0000 0.2500 -0.2461 2.0000 0.2500 1.0000 0.4353 -0.2141 3.0000 0.4353 1.0000 0.5607 -0.1512 4.0000 0.5607 1.0000 0.6344 -0.0880 5.0000 0.6344 1.0000 0.6728 -0.0451 ZERO NOT FOUND TO DESIRED TOLERANCE s = 0.6728 y = -0.0451 Q11 [s,y]=falsi(inline('x^3-9*x+2'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.2500 -0.2344 2.0000 0 0.2500 0.2238 -0.0028 regulafalsi methodhasconverged s = 0.2238 y = -0.0028 >> [s,y]=falsi(inline('x^3-9*x+2'),-3,-4,0.005,8) step a b s y 1.0000 -3.0000 -4.0000 -3.0714 0.6680 2.0000 -3.0714 -4.0000 -3.0947 0.2141 3.0000 -3.0947 -4.0000 -3.1021 0.0677 4.0000 -3.1021 -4.0000 -3.1044 0.0213 5.0000 -3.1044 -4.0000 -3.1051 0.0067 6.0000 -3.1051 -4.0000 -3.1054 0.0021 regulafalsi methodhasconverged s = -3.1054 y = 0.0021 >> [s,y]=falsi(inline('x^3-9*x+2'),2,3,0.005,8) step a b s y
1.0000 2.0000 3.0000 2.8000 -1.2480 2.0000 2.8000 3.0000 2.8768 -0.0821 3.0000 2.8768 3.0000 2.8817 -0.0050 4.0000 2.8817 3.0000 2.8820 -0.0003 regulafalsi methodhasconverged s = 2.8820 y = -3.0703e-004 Q12 >> [s,y]=falsi(inline('x^3-2*x^2-5'),2,3,0.005,8) step a b s y 1.0000 2.0000 3.0000 2.5556 -1.3717 2.0000 2.5556 3.0000 2.6691 -0.2338 3.0000 2.6691 3.0000 2.6873 -0.0363 4.0000 2.6873 3.0000 2.6901 -0.0056 5.0000 2.6901 3.0000 2.6906 -0.0008 regulafalsi methodhasconverged s = 2.6906 y = -8.4925e-004 >> [s,y]=falsi(inline('x^3-3*x^2-1'),0,1,0.005,8) ??? Error using==> falsi at 5 functionhassame signat endpoints >> [s,y]=falsi(inline('x^3+3*x^2-1'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.2500 -0.7969 2.0000 0.2500 1.0000 0.4074 -0.4344 3.0000 0.4074 1.0000 0.4824 -0.1897 4.0000 0.4824 1.0000 0.5132 -0.0749 5.0000 0.5132 1.0000 0.5250 -0.0284 6.0000 0.5250 1.0000 0.5295 -0.0106 7.0000 0.5295 1.0000 0.5311 -0.0039 regulafalsi methodhasconverged s = 0.5311 y = -0.0039 >> [s,y]=falsi(inline('x^3+3*x^2-1'),-1,0,0.005,8) step a b s y
1.0000 -1.0000 0 -0.5000 -0.3750 2.0000 -1.0000 -0.5000 -0.6364 -0.0428 3.0000 -1.0000 -0.6364 -0.6513 -0.0037 regulafalsi methodhasconverged s = -0.6513 y = -0.0037 >> [s,y]=falsi(inline('x^3+3*x^2-1'),-2,-3,0.005,8) step a b s y 1.0000 -2.0000 -3.0000 -2.7500 0.8906 2.0000 -2.7500 -3.0000 -2.8678 0.0875 3.0000 -2.8678 -3.0000 -2.8784 0.0074 4.0000 -2.8784 -3.0000 -2.8793 0.0006 regulafalsi methodhasconverged s = -2.8793 y = 6.2341e-004 >> [s,y]=falsi(inline('x^3-4*x+1'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.3333 -0.2963 2.0000 0 0.3333 0.2571 -0.0116 3.0000 0 0.2571 0.2542 -0.0004 regulafalsi methodhasconverged s = 0.2542 y = -3.8225e-004 >> [s,y]=falsi(inline('x^3-4*x+1'),1,2,0.005,8) step a b s y 1.0000 1.0000 2.0000 1.6667 -1.0370 2.0000 1.6667 2.0000 1.8364 -0.1528 3.0000 1.8364 2.0000 1.8581 -0.0175 4.0000 1.8581 2.0000 1.8605 -0.0020 regulafalsi methodhas converged s = 1.8605 y = -0.0020 >> [s,y]=falsi(inline('x^3-4*x+1'),-2,-3,0.005,8) step a b s y
1.0000 -2.0000 -3.0000 -2.0667 0.4397 2.0000 -2.0667 -3.0000 -2.0951 0.1842 3.0000 -2.0951 -3.0000 -2.1068 0.0756 4.0000 -2.1068 -3.0000 -2.1116 0.0308 5.0000 -2.1116 -3.0000 -2.1136 0.0125 6.0000 -2.1136 -3.0000 -2.1144 0.0051 7.0000 -2.1144 -3.0000 -2.1147 0.0020 regulafalsi methodhasconverged s = -2.1147 y = 0.0020 >> [s,y]=falsi(inline('x^3-x^2-4*x-3'),2,3,0.005,8) step a b s y 1.0000 2.0000 3.0000 2.7000 -1.4070 2.0000 2.7000 3.0000 2.7958 -0.1466 3.0000 2.7958 3.0000 2.8053 -0.0141 4.0000 2.8053 3.0000 2.8062 -0.0013 regulafalsi methodhasconverged s = 2.8062 y = -0.0013 >> [s,y]=falsi(inline('x^3-6*x^2+11*x-5'),2,3,0.005,8) ??? Error using==> falsi at 5 functionhassame signat endpoints >> [s,y]=falsi(inline('x^3-6*x^2+11*x-5'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.8333 0.5787 2.0000 0 0.8333 0.7469 0.2854 3.0000 0 0.7469 0.7066 0.1295 4.0000 0 0.7066 0.6887 0.0566 5.0000 0 0.6887 0.6810 0.0243 6.0000 0 0.6810 0.6777 0.0104 7.0000 0 0.6777 0.6763 0.0044 regulafalsi methodhasconverged s = 0.6763 y = 0.0044 >> [s,y]=falsi(inline('6*x^3-23*x^2+20*x'),1,2,0.005,8)
step a b s y 1.0000 1.0000 2.0000 1.4286 -0.8746 2.0000 1.0000 1.4286 1.3318 0.0140 3.0000 1.3318 1.4286 1.3334 -0.0002 regulafalsi methodhasconverged s = 1.3334 y = -2.2752e-004 >> [s,y]=falsi(inline('6*x^3-23*x^2+20*x'),2,3,0.005,8) step a b s y 1.0000 2.0000 3.0000 2.2105 -3.3678 2.0000 2.2105 3.0000 2.3553 -2.0900 3.0000 2.3553 3.0000 2.4341 -1.0590 4.0000 2.4341 3.0000 2.4714 -0.4819 5.0000 2.4714 3.0000 2.4879 -0.2086 6.0000 2.4879 3.0000 2.4949 -0.0883 7.0000 2.4949 3.0000 2.4979 -0.0370 8.0000 2.4979 3.0000 2.4991 -0.0155 ZERO NOT FOUND TO DESIRED TOLERANCE s = 2.4991 y = -0.0155 >> [s,y]=falsi(inline('3*x^3-x^2+18*x+6'),0,1,0.005,8) ??? Error using==> falsi at 5 functionhassame signat endpoints >> [s,y]=falsi(inline('3*x^3-x^2-18*x+6'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.3750 -0.7324 2.0000 0 0.3750 0.3342 -0.0154 3.0000 0 0.3342 0.3333 -0.0003 regulafalsi methodhasconverged s = 0.3333 y = -2.8549e-004 >> [s,y]=falsi(inline('3*x^3-x^2-18*x+6'),2,3,0.005,8) step a b s y 1.0000 2.0000 3.0000 2.2941 -4.3354 2.0000 2.2941 3.0000 2.4021 -1.4263
3.0000 2.4021 3.0000 2.4357 -0.4261 4.0000 2.4357 3.0000 2.4455 -0.1236 5.0000 2.4455 3.0000 2.4483 -0.0356 6.0000 2.4483 3.0000 2.4492 -0.0102 7.0000 2.4492 3.0000 2.4494 -0.0029 regulafalsi methodhasconverged s = 2.4494 y = -0.0029 >> [s,y]=falsi(inline('3*x^3-x^2-18*x+6'),-2,-3,0.005,8) step a b s y 1.0000 -2.0000 -3.0000 -2.3182 4.9798 2.0000 -2.3182 -3.0000 -2.4152 1.3736 3.0000 -2.4152 -3.0000 -2.4408 0.3517 4.0000 -2.4408 -3.0000 -2.4473 0.0883 5.0000 -2.4473 -3.0000 -2.4489 0.0221 6.0000 -2.4489 -3.0000 -2.4494 0.0055 7.0000 -2.4494 -3.0000 -2.4495 0.0014 regulafalsi methodhasconverged s = -2.4495 y = 0.0014 >> [s,y]=falsi(inline('x^3-x^2-24*x-32'),-1,-2,0.005,8) step a b s y 1.0000 -1.0000 -2.0000 -1.7143 1.1662 2.0000 -1.0000 -1.7143 -1.6397 0.2555 3.0000 -1.0000 -1.6397 -1.6238 0.0523 4.0000 -1.0000 -1.6238 -1.6205 0.0106 5.0000 -1.0000 -1.6205 -1.6198 0.0021 regulafalsi methodhasconverged s = -1.6198 y = 0.0021 >> [s,y]=falsi(inline('x^3-x^2-24*x-32'),5,6,0.005,8) step a b s y 1.0000 5.0000 6.0000 5.9286 -1.0565 2.0000 5.9286 6.0000 5.9435 -0.0142 3.0000 5.9435 6.0000 5.9437 -0.0002 regulafalsi methodhasconverged
s = 5.9437 y = -1.9042e-004 >> [s,y]=falsi(inline('x^3-x^2-24*x-32'),-3,-4,0.005,8) step a b s y 1.0000 -3.0000 -4.0000 -3.2000 1.7920 2.0000 -3.2000 -4.0000 -3.2806 0.6655 3.0000 -3.2806 -4.0000 -3.3093 0.2300 4.0000 -3.3093 -4.0000 -3.3191 0.0775 5.0000 -3.3191 -4.0000 -3.3224 0.0259 6.0000 -3.3224 -4.0000 -3.3235 0.0086 7.0000 -3.3235 -4.0000 -3.3238 0.0029 regulafalsi methodhasconverged s = -3.3238 y = 0.0029 >> [s,y]=falsi(inline('x^3-7*x^2+14*x-7'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.8750 0.5605 2.0000 0 0.8750 0.8101 0.2793 3.0000 0 0.8101 0.7790 0.1310 4.0000 0 0.7790 0.7647 0.0597 5.0000 0 0.7647 0.7583 0.0269 6.0000 0 0.7583 0.7554 0.0120 7.0000 0 0.7554 0.7541 0.0054 8.0000 0 0.7541 0.7535 0.0024 regulafalsi methodhasconverged s = 0.7535 y = 0.0024 >> [s,y]=falsi(inline('x^3-7*x^2+14*x-7'),3,4,0.005,8) step a b s y 1.0000 3.0000 4.0000 3.5000 -0.8750 2.0000 3.5000 4.0000 3.7333 -0.2634 3.0000 3.7333 4.0000 3.7889 -0.0531 4.0000 3.7889 4.0000 3.7996 -0.0098 5.0000 3.7996 4.0000 3.8015 -0.0018 regulafalsi methodhasconverged s =
3.8015 y = -0.0018 THOMAS METHOD a=[2,4,3,0] a = 2 4 3 0 >> b=[0,2,1,2] b = 0 2 1 2 >> d=[2,4,3,5] d = 2 4 3 5 >> r=[4,6,7,10] r = 4 6 7 10 >> x=thomas(a,d,b,r) x = 1 1 0 2 >> x = 10.0000 -5.8000 2.2000
Question1 >> a=[2,3,0] a = 2 3 0 >> b=[0,1,3] b = 0 1 3 >> d=[1,3,10] d = 1 3 10 >> x=thomas(a,d,b,r) x = 2 4 1 Question3.22 d=[-2,-2,-2,-2] d = -2 -2 -2 -2 >> b=[0,1,1,1] b = 0 1 1 1 >> a=[1,1,1,0]
a = 1 1 1 0 >> r=[-1,0,0,0] r = -1 0 0 0 >> x=thomas(a,d,b,r) x = 0.8000 0.6000 0.4000 0.2000 >> r=[33,26,30,15] question3.23 r = 33 26 30 15 >> d=[5,5,5,5] d = 5 5 5 5 >> a=[1,1,1,0] a = 1 1 1 0 >> b=[0,1,1,1] b = 0 1 1 1
>> x=thomas(a,d,b,r) x = 6.0000 3.0000 5.0000 2.0000 >> Question3.24 r=[14;-36;-6;14;-9;6] r = 14 -36 -6 14 -9 6 >> d=[-3,4,-1,4,1,2] d = -3 4 -1 4 1 2 >> a=[-4,5,-3,-5,-5,0] a = -4 5 -3 -5 -5 0 >> b=[0,-3,1,0,3,-1] b = 0 -3 1 0 3 -1
>> x=thomas(a,d,b,r) x = 2.0000 -5.0000 -2.0000 1.0000 -2.0000 2.0000 >> Question3.25 b=[0,5,5,2,5,1,-2] b = 0 5 5 2 5 1 -2 >> a=[3,-1,-1,1,-1,0,0] a = 3 -1 -1 1 -1 0 0 >> d=[1,-4,-2,3,-3,-1,4] d = 1 -4 -2 3 -3 -1 4 >> r=[19;1;28;0;-25;0;2] r = 19 1 28 0 -25 0 2 >> x=thomas(a,d,b,r) x = 4.0000 5.0000 -1.0000 -1.0000 5.0000 5.0000 3.0000
C3.1 d=[-1,4,1,-1,-2,-2,4,2] d = -1 4 1 -1 -2 -2 4 2 b=[0,-1,4,0,-2,-4,2,0] b = 0 -1 4 0 -2 -4 2 0 >> a=[1,1,3,-2,-2,-2,0,0] a = 1 1 3 -2 -2 -2 0 0 >> r=[7,13,-3,-2,-4,-28,26,10] r = 7 13 -3 -2 -4 -28 26 10 >> x=thomas(a,d,b,r) x = -4.0000 3.0000 -3.0000 -4.0000 3.0000 3.0000 5.0000 5.0000 >>
questionc3.2 r=[-1;19;20;-1;-19;14;0;-4;-2] r = -1 19 20 -1 -19 14 0 -4 -2 >> a=[1,1,-1,4,5,0,-4,-4,0] a = 1 1 -1 4 5 0 -4 -4 0 >> b=[0,2,3,-4,3,-1,-5,-2,-4] b = 0 2 3 -4 3 -1 -5 -2 -4 >> d=[-1,3,-3,3,3,-5,1,2,2] d = -1 3 -3 3 3 -5 1 2 2 >> x=thomas(a,d,b,r) x = 5.0000 4.0000 -3.0000 1.0000 -4.0000 -2.0000 -2.0000 2.0000 3.0000 Questionc3.3 d=[3,3,1,-4,0,-3,0,0,0,1]
d = 3 3 1 -4 0 -3 0 0 0 1 >> a=[-4,5,2,5,-2,-2,-5,-1,1,0] a = -4 5 2 5 -2 -2 -5 -1 1 0 >> b=[0,3,-1,-2,1,5,1,-3,-3,-4] b = 0 3 -1 -2 1 5 1 -3 -3 -4 >> r=[-13,-11,-6,25,6,29,1,0,3,-12] r = -13 -11 -6 25 6 29 1 0 3 -12 >> x=thomas(a,d,b,r) x = Columns1 through9 -3.0000 1.0000 -1.0000 -2.0000 3.0000 -4.0000 -1.0000 -1.0000 3.0000 Column10 -0.0000 >>questiona3.7 a=[1,1,1,1,1,1,0] a = 1 1 1 1 1 1 0
>> b=[0,1,1,1,1,1,1] b = 0 1 1 1 1 1 1 >> d=[4,4,4,4,4,4,4] d = 4 4 4 4 4 4 4 >> r=[7.2,11.82,12,0,-12,-11.82,-7.2] r = 7.2000 11.8200 12.0000 0 -12.0000 -11.8200 -7.2000 >> x=thomas(a,d,b,r) x = 1.2986 2.0057 2.4986 0.0000 -2.4986 -2.0057 -1.2986 >> Questiona3.8 d=[-1.99,-1.99,-1.99,-1.99,-1.99,-1.99,-1.99,-1.99,-1.99] d = -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 >> a=[1,1,1,1,1,1,1,1,0] a = 1 1 1 1 1 1 1 1 0 >> b=[0,1,1,1,1,1,1,1,1] b =
0 1 1 1 1 1 1 1 1 >> r=[-0.99;0.002;0.0031;0.0042;0.0055;0.0068;0.0084;0.0103;-0.6874] r = -0.9900 0.0020 0.0031 0.0042 0.0055 0.0068 0.0084 0.0103 -0.6874 >> x=thomas(a,d,b,r) x = 0.9846 0.9694 0.9465 0.9172 0.8830 0.8454 0.8061 0.7672 0.7310 GAUSS SEIDEL METHOD Q P6.1 a=[10 -2 1; -2 10 -2; -2 -5 10] a = 10 -2 1 -2 10 -2 -2 -5 10 >> b=[9;12;18] b =
9 12 18 >> x0=[0;0;0] x0 = 0 0 0 >> tol=0.0001 tol = 1.0000e-004 >> max_it=7 max_it= 7 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 0.9000 1.3800 2.6700 2.0000 0.9090 1.9158 2.9397 3.0000 0.9892 1.9858 2.9907 4.0000 0.9981 1.9978 2.9985 5.0000 0.9997 1.9996 2.9998 6.0000 1.0000 1.9999 3.0000 gaussseidel methodconverged x =
1.0000 2.0000 3.0000 >> P6.2 max_it=7 max_it= 7 >> tol=0.0001 tol = 1.0000e-004 >> x0=[0;0;0] x0 = 0 0 0 >> b=[8;4;12] b = 8 4 12 >> a=[8 1 -1;-1 7 -2;2 1 9] a = 8 1 -1 -1 7 -2 2 1 9
>> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 1.0000 0.7143 1.0317 2.0000 1.0397 1.0147 0.9895 3.0000 0.9969 0.9966 1.0011 4.0000 1.0006 1.0004 0.9998 5.0000 0.9999 0.9999 1.0000 6.0000 1.0000 1.0000 1.0000 gaussseidel methodconverged x = 1.0000 1.0000 1.0000 >> P6.3 a=[5 -1 0;-1 5 -1;0 -1 5] a = 5 -1 0 -1 5 -1 0 -1 5 >> b=[9;4;-6] b = 9 4 -6 >> x0=[0;0;0]
x0 = 0 0 0 >> tol=0.0001 tol = 1.0000e-004 >> max_it=7 max_it= 7 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 1.8000 1.1600 -0.9680 2.0000 2.0320 1.0128 -0.9974 3.0000 2.0026 1.0010 -0.9998 4.0000 2.0002 1.0001 -1.0000 5.0000 2.0000 1.0000 -1.0000 gaussseidel methodconverged x = 2.0000 1.0000 -1.0000 >> P6.4 max_it=7
max_it= 7 >> tol=0.0001 tol = 1.0000e-004 >> x0=[0;0;0] x0 = 0 0 0 >> b=[3;-4;5] b = 3 -4 5 >> a=[4 1 0;1 3 -1;1 0 2] a = 4 1 0 1 3 -1 1 0 2 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 0.7500 -1.5833 2.1250 2.0000 1.1458 -1.0069 1.9271 3.0000 1.0017 -1.0249 1.9991
4.0000 1.0062 -1.0024 1.9969 5.0000 1.0006 -1.0012 1.9997 6.0000 1.0003 -1.0002 1.9998 7.0000 1.0001 -1.0001 2.0000 gaussseidel methoddidnotconverged x = 1.0001 -1.0001 2.0000 >> P6.5 a=[4 1 0;1 3 -1;0 -1 4] a = 4 1 0 1 3 -1 0 -1 4 >> b=[3;4;5] b = 3 4 5 >> x0=[0;0;0] x0 = 0 0 0
>> tol=0.0001 tol = 1.0000e-004 >> max_it=7 max_it= 7 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 0.7500 1.0833 1.5208 2.0000 0.4792 1.6806 1.6701 3.0000 0.3299 1.7801 1.6950 4.0000 0.3050 1.7967 1.6992 5.0000 0.3008 1.7994 1.6999 6.0000 0.3001 1.7999 1.7000 7.0000 0.3000 1.8000 1.7000 gaussseidel methoddidnotconverged x = 0.3000 1.8000 1.7000 >> P6.6 x0=[0;0;0;0] x0 =
0 0 0 0 max_it=7 max_it= 7 a=[-2 1 0 0 ; 1 -2 1 0; 0 1 -2 1; 0 0 1 -2] a = -2 1 0 0 1 -2 1 0 0 1 -2 1 0 0 1 -2 b=[-1;0;0;0] b = -1 0 0 0 x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 0.5000 0.2500 0.1250 0.0625 2.0000 0.6250 0.3750 0.2188 0.1094 3.0000 0.6875 0.4531 0.2813 0.1406 4.0000 0.7266 0.5039 0.3223 0.1611 5.0000 0.7520 0.5371 0.3491 0.1746 6.0000 0.7686 0.5588 0.3667 0.1833
7.0000 0.7794 0.5731 0.3782 0.1891 gaussseidel methoddidnotconverged x = 0.7794 0.5731 0.3782 0.1891 >> P6.7 max_it=7 max_it= 7 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> b=[33;26;30;15] b = 33 26 30 15 a=[5 1 0 0;1 5 1 0;0 1 5 1;0 0 1 5] a = 5 1 0 0 1 5 1 0
0 1 5 1 0 0 1 5 >> tol=0.0001 tol = 1.0000e-004 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 6.6000 3.8800 5.2240 1.9552 2.0000 5.8240 2.9904 5.0109 1.9978 3.0000 6.0019 2.9974 5.0009 1.9998 4.0000 6.0005 2.9997 5.0001 2.0000 5.0000 6.0001 3.0000 5.0000 2.0000 gaussseidel methodconverged x = 6.0000 3.0000 5.0000 2.0000 >> P6.8 a=[1 2 0 0;2 6 8 0;0 8 35 18;0 0 18 112] a = 1 2 0 0 2 6 8 0 0 8 35 18 0 0 18 112 >> b=[2 ;6;-10;-112]
b = 2 6 -10 -112 tol=0.0001 tol = 1.0000e-004 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 2.0000 0.3333 -0.3619 -0.9418 2.0000 1.3333 1.0381 -0.0386 -0.9938 3.0000 -0.0762 1.0769 -0.0208 -0.9967 4.0000 -0.1538 1.0789 -0.0198 -0.9968 5.0000 -0.1579 1.0790 -0.0197 -0.9968 6.0000 -0.1580 1.0789 -0.0197 -0.9968 7.0000 -0.1578 1.0788 -0.0196 -0.9968 gaussseidel methoddidnotconverged x = -0.1578 1.0788 -0.0196 -0.9968 >> P6.9 b=[-3;5;2;3.5]
b = -3.0000 5.0000 2.0000 3.5000 >> a=[1 -2 0 0;-2 5 -1 0;0 -1 2 -0.5;0 0 -0.5 1.25] a = 1.0000 -2.0000 0 0 -2.0000 5.0000 -1.0000 0 0 -1.0000 2.0000 -0.5000 0 0 -0.5000 1.2500 >> x=seidel(a,b,x0,tol,100) i x1 x2 x3 1.0000 -3.0000 -0.2000 0.9000 3.1600 2.0000 -3.4000 -0.1800 1.7000 3.4800 3.0000 -3.3600 -0.0040 1.8680 3.5472 4.0000 -3.0080 0.1704 1.9720 3.5888 5.0000 -2.6592 0.3307 2.0626 3.6250 6.0000 -2.3386 0.4771 2.1448 3.6579 gaussseidel methodconverged at95th iteration x = 0.9991 1.9996 2.9998 3.9999 p.6.10a=[4 -8 0 0;-8 18 -2 0;0 -2 5 -1.5;0 0 -1.5 1.75] a =
4.0000 -8.0000 0 0 -8.0000 18.0000 -2.0000 0 0 -2.0000 5.0000 -1.5000 0 0 -1.5000 1.7500 >> b=[-12;22;5;2] b = -12 22 5 2 >> max_it=7 max_it= 7 >> tol=0.0001 tol = 1.0000e-004 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 -3.0000 -0.1111 0.9556 1.9619 2.0000 -3.2222 -0.1037 1.5471 2.4689
3.0000 -3.2074 -0.0314 1.7281 2.6241 4.0000 -3.0628 0.0530 1.8084 2.6929 5.0000 -2.8940 0.1369 1.8627 2.7394 6.0000 -2.7261 0.2176 1.9089 2.7790 7.0000 -2.5649 0.2944 1.9515 2.8155 gaussseidel methoddidnotconverged x = -2.5649 0.2944 1.9515 2.8155 p.6.11 >> a=[4 8 0 0;8 18 2 0;0 2 5 1.5;0 0 1.5 1.75] a = 4.0000 8.0000 0 0 8.0000 18.0000 2.0000 0 0 2.0000 5.0000 1.5000 0 0 1.5000 1.7500 >> b=[8;18;0.5;-1.75] b = 8.0000 18.0000 0.5000 -1.7500 >> tol=0.0001 tol =
1.0000e-004 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> max_it=7 max_it= 7 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 2.0000 0.1111 0.0556 -1.0476 2.0000 1.7778 0.2037 0.3328 -1.2853 3.0000 1.5926 0.2552 0.3835 -1.3287 4.0000 1.4896 0.2953 0.3805 -1.3261 5.0000 1.4093 0.3314 0.3653 -1.3131 6.0000 1.3373 0.3651 0.3479 -1.2982 7.0000 1.2699 0.3970 0.3307 -1.2834 gaussseidel methoddid notconverged x = 1.2699 0.3970 0.3307 -1.2834
p.6.12 a=[1 -2 0 0 0;-2 5 1 0 0;0 1 2 -2 0;0 0 -2 5 1;0 0 0 1 2] a = 1 -2 0 0 0 -2 5 1 0 0 0 1 2 -2 0 0 0 -2 5 1 0 0 0 1 2 >> b=[5;-9;0;3;0] b = 5 -9 0 3 0 >> x0=[0;0;0;0;0] x0 = 0 0 0 0 0 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 5.0000 0.2000 -0.1000 0.5600 -0.2800 2.0000 5.4000 0.3800 0.3700 0.8040 -0.4020 3.0000 5.7600 0.4300 0.5890 0.9160 -0.4580 4.0000 5.8600 0.4262 0.7029 0.9728 -0.4864 5.0000 5.8524 0.4004 0.7726 1.0063 -0.5032
6.0000 5.8008 0.3658 0.8234 1.0300 -0.5150 7.0000 5.7316 0.3280 0.8660 1.0494 -0.5247 gaussseidel methoddidnotconverged x = 5.7316 0.3280 0.8660 1.0494 -0.5247 >>p.6.13>> a=[1 -2 0 0 0;-2 6 4 0 0;0 4 9 -0.5 0;0 0 -0.5 1.25 0.5;0 0 0 0.5 3.25] a = 1.0000 -2.0000 0 0 0 -2.0000 6.0000 4.0000 0 0 0 4.0000 9.0000 -0.5000 0 0 0 -0.5000 1.2500 0.5000 0 0 0 0.5000 3.2500 >> b=[5;-2;18;0.5;-2.25] b = 5.0000 -2.0000 18.0000 0.5000 -2.2500 >> max_it=260 max_it= 260 >> x0=[0;0;0;0;0]
x0 = 0 0 0 0 0 >> >> max_it=1000 max_it= 1000 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 5.0000 1.3333 1.4074 0.9630 -0.8405 2.0000 7.6667 1.2840 1.4829 1.3293 -0.8968 3.0000 7.5679 1.2007 1.5402 1.3748 -0.9038 4.0000 7.4015 1.1070 1.5844 1.3953 -0.9070 5.0000 7.2141 1.0151 1.6264 1.4133 -0.9097 6.0000 7.0302 0.9258 1.6670 1.4307 -0.9124 gaussseidel methodconverged at260th iteration x = 1.0028 -1.9986 2.9994 1.9997 -1.0000 p.6.14 > a=[1 -2 0 0 0 0;-2 6 4 0 0 0;0 4 9 -0.5 0 0;0 0 -0.5 3.25 1.5 0;0 0 0 1.5 1.75 -3;0 0 0 0 -3 13]
a = 1.0000 -2.0000 0 0 0 0 -2.0000 6.0000 4.0000 0 0 0 0 4.0000 9.0000 -0.5000 0 0 0 0 -0.5000 3.2500 1.5000 0 0 0 0 1.5000 1.7500 -3.0000 0 0 0 0 -3.0000 13.0000 >> b=[-3;22;35.5;-7.75;4;-33] b = -3.0000 22.0000 35.5000 -7.7500 4.0000 -33.0000 >> x0=[0;0;0;0;0;0] x0 = 0 0 0 0 0 0 >> x=seidel(a,b,x0,tol,300) i x1 x2 x3 1.0000 -3.0000 2.6667 2.7593 -1.9601 3.9658 -1.6233 2.0000 2.3333 2.6049 2.6778 -3.8030 2.7627 -1.9009 3.0000 2.2099 2.6181 2.5696 -3.2644 1.8250 -2.1173 4.0000 2.2362 2.6990 2.5635 -2.8326 1.0840 -2.2883
5.0000 2.3980 2.7570 2.5618 -2.4908 0.4979 -2.4236 6.0000 2.5140 2.7968 2.5630 -2.2201 0.0339 -2.5306 gaussseidel methodconvergedat234th iteration x = 1.0029 2.0014 2.9993 -1.0003 -1.9995 -2.9999 Jacobi method P.6.1 a=[10 -2 1;-2 10 -2;-2 -5 10] a = 10 -2 1 -2 10 -2 -2 -5 10 >> b=[9;12;18] b = 9 12 18 >> xo=[0;0;0] xo= 0 0 0 >> x=jacobi(a,b,xo,tol,20)
i x1 x2 x3 1.0000 0.9000 1.2000 1.8000 2.0000 0.9600 1.7400 2.5800 3.0000 0.9900 1.9080 2.8620 4.0000 0.9954 1.9704 2.9520 5.0000 0.9989 1.9895 2.9843 6.0000 0.9995 1.9966 2.9945 7.0000 0.9999 1.9988 2.9982 8.0000 0.9999 1.9996 2.9994 9.0000 1.0000 1.9999 2.9998 10.0000 1.0000 2.0000 2.9999 jacobi methodhasconverged x = 1.0000 2.0000 3.0000 P.6.2 > a=[8 1 -1;-1 7 -2;2 1 9] a = 8 1 -1 -1 7 -2 2 1 9 >> b=[8;4;12] b =
8 4 12 >> x=jacobi(a,b,xo,tol,20) i x1 x2 x3 1.0000 1.0000 0.5714 1.3333 2.0000 1.0952 1.0952 1.0476 3.0000 0.9940 1.0272 0.9683 4.0000 0.9926 0.9901 0.9983 5.0000 1.0010 0.9985 1.0027 6.0000 1.0005 1.0009 0.9999 7.0000 0.9999 1.0001 0.9998 8.0000 1.0000 0.9999 1.0000 jacobi methodhasconverged x = 1.0000 1.0000 1.0000 >>p6.3 b=[9;4;-6] b = 9 4 -6 >> a=[5 -1 0;-1 5 -1;0 -1 5]
a = 5 -1 0 -1 5 -1 0 -1 5 >> x=jacobi(a,b,xo,tol,20) i x1 x2 x3 1.0000 1.8000 0.8000 -1.2000 2.0000 1.9600 0.9200 -1.0400 3.0000 1.9840 0.9840 -1.0160 4.0000 1.9968 0.9936 -1.0032 5.0000 1.9987 0.9987 -1.0013 6.0000 1.9997 0.9995 -1.0003 7.0000 1.9999 0.9999 -1.0001 8.0000 2.0000 1.0000 -1.0000 jacobi methodhasconverged x = 2.0000 1.0000 -1.0000 >> >>p6.4 a=[4 1 0;1 3 -1;1 0 2] a = 4 1 0 1 3 -1 1 0 2
>> b=[3;-4;5] b = 3 -4 5 >> x=jacobi(a,b,xo,tol,20) i x1 x2 x3 1.0000 0.7500 -1.3333 2.5000 2.0000 1.0833 -0.7500 2.1250 3.0000 0.9375 -0.9861 1.9583 4.0000 0.9965 -0.9931 2.0313 5.0000 0.9983 -0.9884 2.0017 6.0000 0.9971 -0.9988 2.0009 7.0000 0.9997 -0.9987 2.0014 8.0000 0.9997 -0.9994 2.0001 9.0000 0.9999 -0.9998 2.0002 10.0000 1.0000 -0.9999 2.0001 jacobi methodhasconverged x = 1.0000 -1.0000 2.0000 >> P6.5 a=[4 1 0;1 3 -1;0 -1 4]
a = 4 1 0 1 3 -1 0 -1 4 >> b=[3;4;5] b = 3 4 5 >> x=jacobi(a,b,xo,tol,20) i x1 x2 x3 1.0000 0.7500 1.3333 1.2500 2.0000 0.4167 1.5000 1.5833 3.0000 0.3750 1.7222 1.6250 4.0000 0.3194 1.7500 1.6806 5.0000 0.3125 1.7870 1.6875 6.0000 0.3032 1.7917 1.6968 7.0000 0.3021 1.7978 1.6979 8.0000 0.3005 1.7986 1.6995 9.0000 0.3003 1.7996 1.6997 10.0000 0.3001 1.7998 1.6999 11.0000 0.3001 1.7999 1.6999 jacobi methodhasconverged x =
0.3000 1.8000 1.7000 >> P6.6 b=[-1;0;0;0] b = -1 0 0 0 >> a=[-2 1 0 0;1 -2 1 0;0 1 -2 1;0 0 1 -2] a = -2 1 0 0 1 -2 1 0 0 1 -2 1 0 0 1 -2 >> x0=[0;0;0;0] x0 = 0 0 0 0 x=jacobi(a,b,x0,tol,max_it) i x1 x2 x3 x4 1.0000 0.5000 0 0 0 2.0000 0.5000 0.2500 0 0 3.0000 0.6250 0.2500 0.1250 0 4.0000 0.6250 0.3750 0.1250 0.0625
5.0000 0.6875 0.3750 0.2188 0.0625 6.0000 0.6875 0.4531 0.2188 0.1094 7.0000 0.7266 0.4531 0.2813 0.1094 jacobi methoddidnotconverged resultaftermax no. of iterations x = 0.7266 0.4531 0.2813 0.1094 >> P6.7 =[5 1 0 0;1 5 1 0;0 1 5 1;0 0 1 5] a = 5 1 0 0 1 5 1 0 0 1 5 1 0 0 1 5 >> b=[33;26;30;15] b = 33 26 30 15 >> x0=[0;0;0;0] x0 = 0
0 0 0 x=jacobi(a,b,x0,tol,20) i x1 x2 x3 x4 1.0000 6.6000 5.2000 6.0000 3.0000 2.0000 5.5600 2.6800 4.3600 1.8000 3.0000 6.0640 3.2160 5.1040 2.1280 4.0000 5.9568 2.9664 4.9312 1.9792 5.0000 6.0067 3.0224 5.0109 2.0138 6.0000 5.9955 2.9965 4.9928 1.9978 7.0000 6.0007 3.0023 5.0011 2.0014 8.0000 5.9995 2.9996 4.9992 1.9998 9.0000 6.0001 3.0002 5.0001 2.0002 jacobi methodhasconverged x = 6.0000 3.0000 4.9999 2.0000 >>p6.8 a=[1 2 0 0;2 6 8 0;0 8 35 18] a = 1 2 0 0 2 6 8 0 0 8 35 18 >> b=[2;6;-10;-112]
b = 2 6 -10 -112 >> x=jacobi(a,b,x0,tol,10) i x1 x2 x3 x4 1.0000 2.0000 1.0000 -0.2857 -1.0000 2.0000 0 0.7143 0 -0.9541 3.0000 0.5714 1.0000 0.0417 -1.0000 4.0000 0 0.7539 0 -1.0067 5.0000 0.4921 1.0000 0.0597 -1.0000 6.0000 0 0.7564 0 -1.0096 7.0000 0.4873 1.0000 0.0606 -1.0000 8.0000 0 0.7568 0 -1.0097 9.0000 0.4865 1.0000 0.0606 -1.0000 10.0000 0 0.7570 0 -1.0097 jacobi methoddidnotconverged resultaftermax no. of iterations x = 0 0.7570 0 -1.0097
P6.9 a=[1 -2 0 0;-2 5 -1 0;0 -1 2 -0.5;0 0 -0.5 1.25] a = 1.0000 -2.0000 0 0 -2.0000 5.0000 -1.0000 0 0 -1.0000 2.0000 -0.5000 0 0 -0.5000 1.2500 >> b=[-3;5;2;3.5] b = -3.0000 5.0000 2.0000 3.5000 >> tol=0.001 tol = 1.0000e-003 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> x=jacobi(a,b,x0,tol,200) i x1 x2 x3 x4 1.0000 -3.0000 1.0000 1.0000 2.8000 2.0000 -1.0000 -0.0000 2.2000 3.2000 3.0000 -3.0000 1.0400 1.8000 3.6800
4.0000 -0.9200 0.1600 2.4400 3.5200 5.0000 -2.6800 1.1200 1.9600 3.7760 6.0000 -0.7600 0.3200 2.5040 3.5840 jacobi methodhasconverged at172nd iteration x = 0.9983 1.9996 2.9995 3.9999 >>p6.10 a=[4 -8 0 0;-8 18 -2 0;0 -2 5 -1.5;0 0 -1.5 1.75] a = 4.0000 -8.0000 0 0 -8.0000 18.0000 -2.0000 0 0 -2.0000 5.0000 -1.5000 0 0 -1.5000 1.7500 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> tol=0.001 tol = 1.0000e-003 >> b=[-12;22;5;2]
b = -12 22 5 2 x=jacobi(a,b,x0,tol,500) i x1 x2 x3 x4 1.0000 -3.0000 1.2222 1.0000 1.1429 2.0000 -0.5556 0.0000 1.8317 2.0000 3.0000 -3.0000 1.1788 1.6000 2.7129 4.0000 -0.6423 0.0667 2.2854 2.5143 5.0000 -2.8667 1.1907 1.7810 3.1018 6.0000 -0.6186 0.1460 2.4068 2.6694 jacobi methodhasconverged at310th iteration x = 0.4987 1.7498 2.7496 3.4999 >> P6.11 b=[8;18;0.5;-1.75] b = 8.0000 18.0000 0.5000 -1.7500 >> tol=0.001
tol = 1.0000e-003 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> a=[4 8 0 0;8 18 2 0;0 2 5 1.5;0 0 1.5 1.75] a = 4.0000 8.0000 0 0 8.0000 18.0000 2.0000 0 0 2.0000 5.0000 1.5000 0 0 1.5000 1.7500 >> x=jacobi(a,b,x0,tol,500) i x1 x2 x3 x4 1.0000 2.0000 1.0000 0.1000 -1.0000 2.0000 0 0.1000 -0.0000 -1.0857 3.0000 1.8000 1.0000 0.3857 -1.0000 4.0000 0 0.1571 -0.0000 -1.3306 5.0000 1.6857 1.0000 0.4363 -1.0000 6.0000 0 0.2023 -0.0000 -1.3740 jacobi methodhasconverge at 300th iteration x =
0.0008 1.0000 0.0002 -1.0000 >>p6.12 a=[1 -2 0 0 0;-2 5 1 0 0;0 1 2 -2 0;0 0 -2 5 1;0 0 0 1 2] a = 1 -2 0 0 0 -2 5 1 0 0 0 1 2 -2 0 0 0 -2 5 1 0 0 0 1 2 >> b=[5;-9;0;3;0] b = 5 -9 0 3 0 x=jacobi(a,b,x0,tol,1000) i x1 x2 x3 x4 1.0000 5.0000 -1.8000 0 0.6000 0 2.0000 1.4000 0.2000 1.5000 0.6000 -0.3000 3.0000 5.4000 -1.5400 0.5000 1.2600 -0.3000 4.0000 1.9200 0.2600 2.0300 0.8600 -0.6300 5.0000 5.5200 -1.4380 0.7300 1.5380 -0.4300 6.0000 2.1240 0.2620 2.2570 0.9780 -0.7690 jacobi methodhasconverged at968th iteration
x = 1.0011 -1.9998 2.9995 1.9999 -0.9999 p.6.13 >> a=[1 -2 0 0 0;-2 6 4 0 0;0 4 9 -0.5 0;0 0 -0.5 1.25 0.5;0 0 0 0.5 3.25] a = 1.0000 -2.0000 0 0 0 -2.0000 6.0000 4.0000 0 0 0 4.0000 9.0000 -0.5000 0 0 0 -0.5000 1.2500 0.5000 0 0 0 0.5000 3.2500 >> b=[5;-2;18;0.5;-2.25] b = 5.0000 -2.0000 18.0000 0.5000 -2.2500. x=jacobi(a,b,x0,tol,500) i x1 x2 x3 x4 1.0000 5.0000 -0.3333 2.0000 0.4000 -0.6923 2.0000 4.3333 -0.0000 2.1704 1.4769 -0.7538 3.0000 5.0000 -0.3358 2.0821 1.5697 -0.9195 4.0000 4.3284 -0.0547 2.2365 1.6006 -0.9338 5.0000 4.8906 -0.3815 2.1132 1.6681 -0.9386 6.0000 4.2370 -0.1120 2.2622 1.6207 -0.9489
jacobi methodhasconverged at436th iteration x = 1.0059 -1.9975 2.9987 1.9995 -0.9999 >>p6.14 a=[1 -2 0 0 0 0;-2 6 4 0 0 0;0 4 9 -0.5 0 0;0 0 -0.5 3.25 1.5 0;0 0 0 1.5 1.75 -3;0 0 0 0 -3 13] a = 1.0000 -2.0000 0 0 0 0 -2.0000 6.0000 4.0000 0 0 0 0 4.0000 9.0000 -0.5000 0 0 0 0 -0.5000 3.2500 1.5000 0 0 0 0 1.5000 1.7500 -3.0000 0 0 0 0 -3.0000 13.0000 >> b=[-3;22;35.5;-7.75;4;-33] b = -3.0000 22.0000 35.5000 -7.7500 4.0000 -33.0000 >> x0=[0;0;0;0;0;0] x0 = 0 0 0 0
0 0 >> x=jacobi(a,b,x0,tol,500) i x1 x2 x3 x4 1.0000 -3.0000 3.6667 3.9444 -2.3846 2.2857 -2.5385 2.0000 4.3333 0.0370 2.1823 -2.8327 -0.0220 -2.0110 3.0000 -2.9259 3.6562 3.7706 -2.0387 1.2664 -2.5435 4.0000 4.3124 0.1776 2.2062 -2.3890 -0.3271 -2.2462 5.0000 -2.6448 3.6334 3.7328 -1.8942 0.4827 -2.6140 6.0000 4.2667 0.2966 2.2244 -2.0331 -0.5717 -2.4271 jacobi methoddidnotconverged resultaftermax no. of iterations x = 1.0028 1.9989 2.9994 -0.9997 -1.9995 -3.0001 jacobi methodhasconverged at622nd iteration x = 0.9996 2.0002 3.0001 -1.0001 -2.0001 -3.0000 p.6.18
>> a=[10 0 1 0 0 0 0 0;0 10 0 0 0 0 -1 0;0 0 10 0 0 -2 0 0;2 0 0 10 0 0 0 0;0 0 1 0 10 0 0 0;0 0 0 -3 0 10 0 0;0 3 0 0 0 0 10 0;0 0 0 0 1 0 0 10] a = 10 0 1 0 0 0 0 0 0 10 0 0 0 0 -1 0 0 0 10 0 0 -2 0 0 2 0 0 10 0 0 0 0 0 0 1 0 10 0 0 0 0 0 0 -3 0 10 0 0 0 3 0 0 0 0 10 0 0 0 0 0 1 0 0 10 > b=[13;13;18;42;53;48;76;85] b = 13 13 18 42 53 48 76 85 >> x0=[0;0;0;0;0;0;0;0] x0 = 0 0 0 0 0 0 0 0 >> x=jacobi(a,b,x0,tol,1000) i x1 x2 x3 x4 1.0000 1.3000 1.3000 1.8000 4.2000 5.3000 4.8000 7.6000 8.5000
2.0000 1.1200 2.0600 2.7600 3.9400 5.1200 6.0600 7.2100 7.9700 3.0000 1.0240 2.0210 3.0120 3.9760 5.0240 5.9820 6.9820 7.9880 4.0000 0.9988 1.9982 2.9964 3.9952 4.9988 5.9928 6.9937 7.9976 5.0000 1.0004 1.9994 2.9986 4.0002 5.0004 5.9986 7.0005 8.0001 6.0000 1.0001 2.0001 2.9997 3.9999 5.0001 6.0001 7.0002 8.0000 jacobi methodhasconverged x = 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000

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numerical method solutions

  • 1.
    Laurence V. Fausett,Applied Numerical Analysis, Using MATLAB, Pearson, BISECTION METHOD p.2.1 [1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 0.2500 0.5000 2.0000 1.0000 1.5000 1.2500 -0.4375 0.2500 3.0000 1.2500 1.5000 1.3750 -0.1094 0.1250 4.0000 1.3750 1.5000 1.4375 0.0664 0.0625 5.0000 1.3750 1.4375 1.4063 -0.0225 0.0313 6.0000 1.4063 1.4375 1.4219 0.0217 0.0156 7.0000 1.4063 1.4219 1.4141 -0.0004 0.0078 zero not foundto desired tolerance p.2.2 [ 2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 1.2500 0.5000 2.0000 2.0000 2.5000 2.2500 0.0625 0.2500 3.0000 2.0000 2.2500 2.1250 -0.4844 0.1250 4.0000 2.1250 2.2500 2.1875 -0.2148 0.0625 5.0000 2.1875 2.2500 2.2188 -0.0771 0.0313 6.0000 2.2188 2.2500 2.2344 -0.0076 0.0156 7.0000 2.2344 2.2500 2.2422 0.0274 0.0078 8.0000 2.2344 2.2422 2.2383 0.0099 0.0039 zero not foundto desiredtolerence p.2.3 [ 2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 -0.7500 0.5000 2.0000 2.5000 3.0000 2.7500 0.5625 0.2500 3.0000 2.5000 2.7500 2.6250 -0.1094 0.1250 4.0000 2.6250 2.7500 2.6875 0.2227 0.0625 5.0000 2.6250 2.6875 2.6563 0.0557 0.0313 6.0000 2.6250 2.6563 2.6406 -0.0271 0.0156 zero not foundto desired tolerance
  • 2.
    p.2.4 [1 2] step ab m ym bound 1.0000 1.0000 2.0000 1.5000 0.3750 0.5000 2.0000 1.0000 1.5000 1.2500 -1.0469 0.2500 3.0000 1.2500 1.5000 1.3750 -0.4004 0.1250 4.0000 1.3750 1.5000 1.4375 -0.0295 0.0625 5.0000 1.4375 1.5000 1.4688 0.1684 0.0313 6.0000 1.4375 1.4688 1.4531 0.0684 0.0156 7.0000 1.4375 1.4531 1.4453 0.0192 0.0078 8.0000 1.4375 1.4453 1.4414 -0.0053 0.0039 9.0000 1.4414 1.4453 1.4434 0.0069 0.0020 10.0000 1.4414 1.4434 1.4424 0.0008 0.0010 zero not foundto desired tolerance p.2.5 [1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 -0.6250 0.5000 2.0000 1.5000 2.0000 1.7500 1.3594 0.2500 3.0000 1.5000 1.7500 1.6250 0.2910 0.1250 4.0000 1.5000 1.6250 1.5625 -0.1853 0.0625 5.0000 1.5625 1.6250 1.5938 0.0482 0.0313 6.0000 1.5625 1.5938 1.5781 -0.0697 0.0156 7.0000 1.5781 1.5938 1.5859 -0.0111 0.0078 zero not foundto desiredtolerance p.2.6 [1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 -2.6250 0.5000 2.0000 1.5000 2.0000 1.7500 -0.6406 0.2500 3.0000 1.7500 2.0000 1.8750 0.5918 0.1250 4.0000 1.7500 1.8750 1.8125 -0.0457 0.0625 5.0000 1.8125 1.8750 1.8438 0.2677 0.0313 6.0000 1.8125 1.8438 1.8281 0.1097 0.0156 7.0000 1.8125 1.8281 1.8203 0.0317 0.0078 zero not foundto desiredtolerance
  • 3.
    p.2.7 [ 0 1] stepa b m ym bound 1.0000 0 1.0000 0.5000 -0.3875 0.5000 2.0000 0.5000 1.0000 0.7500 -0.1336 0.2500 3.0000 0.7500 1.0000 0.8750 0.1362 0.1250 4.0000 0.7500 0.8750 0.8125 -0.0142 0.0625 5.0000 0.8125 0.8750 0.8438 0.0568 0.0313 6.0000 0.8125 0.8438 0.8281 0.0203 0.0156 7.0000 0.8125 0.8281 0.8203 0.0028 0.0078 zero not foundto desiredtolerance p.2.8 [ 0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.5875 0.5000 2.0000 0.5000 1.0000 0.7500 -0.3336 0.2500 3.0000 0.7500 1.0000 0.8750 -0.0638 0.1250 4.0000 0.8750 1.0000 0.9375 0.1225 0.0625 5.0000 0.8750 0.9375 0.9063 0.0245 0.0313 6.0000 0.8750 0.9063 0.8906 -0.0208 0.0156 7.0000 0.8906 0.9063 0.8984 0.0016 0.0078 zero not foundto desired tolerance p.2.9 [ 0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 0.0025 0.5000 2.0000 0 0.5000 0.2500 -0.0561 0.2500 3.0000 0.2500 0.5000 0.3750 -0.0402 0.1250 4.0000 0.3750 0.5000 0.4375 -0.0234 0.0625 5.0000 0.4375 0.5000 0.4688 -0.0117 0.0313 6.0000 0.4688 0.5000 0.4844 -0.0050 0.0156 7.0000 0.4844 0.5000 0.4922 -0.0013 0.0078 zero not foundto desiredtolerence p.2.10
  • 4.
    [ 0 1] stepa b m ym bound 1.0000 0 1.0000 0.5000 -0.1875 0.5000 2.0000 0.5000 1.0000 0.7500 0.0664 0.2500 3.0000 0.5000 0.7500 0.6250 -0.0974 0.1250 4.0000 0.6250 0.7500 0.6875 -0.0266 0.0625 5.0000 0.6875 0.7500 0.7188 0.0169 0.0313 6.0000 0.6875 0.7188 0.7031 -0.0056 0.0156 7.0000 0.7031 0.7188 0.7109 0.0055 0.0078 zero not foundto desired tolerance p.2.11 (a) [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -2.3750 0.5000 2.0000 0 0.5000 0.2500 -0.2344 0.2500 3.0000 0 0.2500 0.1250 0.8770 0.1250 4.0000 0.1250 0.2500 0.1875 0.3191 0.0625 5.0000 0.1875 0.2500 0.2188 0.0417 0.0313 6.0000 0.2188 0.2500 0.2344 -0.0965 0.0156 7.0000 0.2188 0.2344 0.2266 -0.0274 0.0078 8.0000 0.2188 0.2266 0.2227 0.0071 0.0039 9.0000 0.2227 0.2266 0.2246 -0.0102 0.0020 10.0000 0.2227 0.2246 0.2236 -0.0015 0.0010 zero not foundto desired tolerance (b) [2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 -4.8750 0.5000 2.0000 2.5000 3.0000 2.7500 -1.9531 0.2500 3.0000 2.7500 3.0000 2.8750 -0.1113 0.1250 4.0000 2.8750 3.0000 2.9375 0.9099 0.0625 5.0000 2.8750 2.9375 2.9063 0.3908 0.0313 6.0000 2.8750 2.9063 2.8906 0.1376 0.0156 7.0000 2.8750 2.8906 2.8828 0.0126 0.0078 8.0000 2.8750 2.8828 2.8789 -0.0495 0.0039 9.0000 2.8789 2.8828 2.8809 -0.0185 0.0020 10.0000 2.8809 2.8828 2.8818 -0.0029 0.0010 zero not foundto desired tolerance (c) [-3 -4] step a b m ym bound 1.0000 -3.0000 -4.0000 -3.5000 -9.3750 -0.5000 2.0000 -3.0000 -3.5000 -3.2500 -3.0781 -0.2500 3.0000 -3.0000 -3.2500 -3.1250 -0.3926 -0.1250
  • 5.
    4.0000 -3.0000 -3.1250-3.0625 0.8396 -0.0625 5.0000 -3.0625 -3.1250 -3.0938 0.2326 -0.0313 6.0000 -3.0938 -3.1250 -3.1094 -0.0777 -0.0156 7.0000 -3.0938 -3.1094 -3.1016 0.0780 -0.0078 8.0000 -3.1016 -3.1094 -3.1055 0.0003 -0.0039 zero not foundto desired tolerance p.2.12 one real and 2 imaginary roots step a b m ym bound 1.0000 2.0000 3.0000 2.5000 -1.8750 0.5000 2.0000 2.5000 3.0000 2.7500 0.6719 0.2500 3.0000 2.5000 2.7500 2.6250 -0.6934 0.1250 4.0000 2.6250 2.7500 2.6875 -0.0344 0.0625 5.0000 2.6875 2.7500 2.7188 0.3127 0.0313 6.0000 2.6875 2.7188 2.7031 0.1377 0.0156 7.0000 2.6875 2.7031 2.6953 0.0512 0.0078 8.0000 2.6875 2.6953 2.6914 0.0083 0.0039 9.0000 2.6875 2.6914 2.6895 -0.0131 0.0020 10.0000 2.6895 2.6914 2.6904 -0.0024 0.0010 zero not foundto desiredtolerence p.2.13 (a) [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.1250 0.5000 2.0000 0.5000 1.0000 0.7500 1.1094 0.2500 3.0000 0.5000 0.7500 0.6250 0.4160 0.1250 4.0000 0.5000 0.6250 0.5625 0.1272 0.0625 5.0000 0.5000 0.5625 0.5313 -0.0034 0.0313 6.0000 0.5313 0.5625 0.5469 0.0608 0.0156 7.0000 0.5313 0.5469 0.5391 0.0284 0.0078 8.0000 0.5313 0.5391 0.5352 0.0124 0.0039 9.0000 0.5313 0.5352 0.5332 0.0045 0.0020 10.0000 0.5313 0.5332 0.5322 0.0006 0.0010 zero not foundto desiredtolerence (b) [0 -1] step a b m ym bound 1.0000 0 -1.0000 -0.5000 -0.3750 -0.5000 2.0000 -0.5000 -1.0000 -0.7500 0.2656 -0.2500 3.0000 -0.5000 -0.7500 -0.6250 -0.0723 -0.1250 4.0000 -0.6250 -0.7500 -0.6875 0.0930 -0.0625
  • 6.
    5.0000 -0.6250 -0.6875-0.6563 0.0094 -0.0313 6.0000 -0.6250 -0.6563 -0.6406 -0.0317 -0.0156 7.0000 -0.6406 -0.6563 -0.6484 -0.0112 -0.0078 8.0000 -0.6484 -0.6563 -0.6523 -0.0009 -0.0039 9.0000 -0.6523 -0.6563 -0.6543 0.0042 -0.0020 10.0000 -0.6523 -0.6543 -0.6533 0.0016 -0.0010 (c) [-2 -3] step a b m ym bound 1.0000 -2.0000 -3.0000 -2.5000 2.1250 -0.5000 2.0000 -2.5000 -3.0000 -2.7500 0.8906 -0.2500 3.0000 -2.7500 -3.0000 -2.8750 0.0332 -0.1250 4.0000 -2.8750 -3.0000 -2.9375 -0.4607 -0.0625 5.0000 -2.8750 -2.9375 -2.9063 -0.2082 -0.0313 6.0000 -2.8750 -2.9063 -2.8906 -0.0861 -0.0156 7.0000 -2.8750 -2.8906 -2.8828 -0.0261 -0.0078 8.0000 -2.8750 -2.8828 -2.8789 0.0036 -0.0039 9.0000 -2.8789 -2.8828 -2.8809 -0.0112 -0.0020 10.0000 -2.8789 -2.8809 -2.8799 -0.0038 -0.0010 zero not foundto desired tolerance p.2.14 (a) [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.8750 0.5000 2.0000 0 0.5000 0.2500 0.0156 0.2500 3.0000 0.2500 0.5000 0.3750 -0.4473 0.1250 4.0000 0.2500 0.3750 0.3125 -0.2195 0.0625 5.0000 0.2500 0.3125 0.2813 -0.1028 0.0313 6.0000 0.2500 0.2813 0.2656 -0.0438 0.0156 7.0000 0.2500 0.2656 0.2578 -0.0141 0.0078 8.0000 0.2500 0.2578 0.2539 0.0007 0.0039 9.0000 0.2539 0.2578 0.2559 -0.0067 0.0020 10.0000 0.2539 0.2559 0.2549 -0.0030 0.0010 zero not foundto desired tolerance (b) [1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 -1.6250 0.5000 2.0000 1.5000 2.0000 1.7500 -0.6406 0.2500 3.0000 1.7500 2.0000 1.8750 0.0918 0.1250 4.0000 1.7500 1.8750 1.8125 -0.2957 0.0625 5.0000 1.8125 1.8750 1.8438 -0.1073 0.0313 6.0000 1.8438 1.8750 1.8594 -0.0091 0.0156
  • 7.
    7.0000 1.8594 1.87501.8672 0.0410 0.0078 8.0000 1.8594 1.8672 1.8633 0.0158 0.0039 9.0000 1.8594 1.8633 1.8613 0.0033 0.0020 10.0000 1.8594 1.8613 1.8604 -0.0029 0.0010 zero not foundto desired tolerance (c) [-2 -3] step a b m ym bound 1.0000 -2.0000 -3.0000 -2.5000 -4.6250 -0.5000 2.0000 -2.0000 -2.5000 -2.2500 -1.3906 -0.2500 3.0000 -2.0000 -2.2500 -2.1250 -0.0957 -0.1250 4.0000 -2.0000 -2.1250 -2.0625 0.4763 -0.0625 5.0000 -2.0625 -2.1250 -2.0938 0.1964 -0.0313 6.0000 -2.0938 -2.1250 -2.1094 0.0519 -0.0156 7.0000 -2.1094 -2.1250 -2.1172 -0.0215 -0.0078 8.0000 -2.1094 -2.1172 -2.1133 0.0153 -0.0039 9.0000 -2.1133 -2.1172 -2.1152 -0.0031 -0.0020 10.0000 -2.1133 -2.1152 -2.1143 0.0061 -0.0010 zero not foundto desiredtolerence >> p.2.15 [2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 -3.6250 0.5000 2.0000 2.5000 3.0000 2.7500 -0.7656 0.2500 3.0000 2.7500 3.0000 2.8750 0.9980 0.1250 4.0000 2.7500 2.8750 2.8125 0.0872 0.0625 5.0000 2.7500 2.8125 2.7813 -0.3464 0.0313 6.0000 2.7813 2.8125 2.7969 -0.1314 0.0156 7.0000 2.7969 2.8125 2.8047 -0.0226 0.0078 8.0000 2.8047 2.8125 2.8086 0.0322 0.0039 9.0000 2.8047 2.8086 2.8066 0.0048 0.0020 10.0000 2.8047 2.8066 2.8057 -0.0089 0.0010 zero not foundto desired tolerance p.2.16 [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -0.8750 0.5000 2.0000 0.5000 1.0000 0.7500 0.2969 0.2500 3.0000 0.5000 0.7500 0.6250 -0.2246 0.1250 4.0000 0.6250 0.7500 0.6875 0.0515 0.0625
  • 8.
    5.0000 0.6250 0.68750.6563 -0.0826 0.0313 6.0000 0.6563 0.6875 0.6719 -0.0146 0.0156 7.0000 0.6719 0.6875 0.6797 0.0187 0.0078 8.0000 0.6719 0.6797 0.6758 0.0021 0.0039 9.0000 0.6719 0.6758 0.6738 -0.0062 0.0020 10.0000 0.6738 0.6758 0.6748 -0.0020 0.0010 zero not foundto desiredtolerence p.2.17 (a) [ 1 2] step a b m ym bound 1.0000 1.0000 2.0000 1.5000 -1.5000 0.5000 2.0000 1.0000 1.5000 1.2500 0.7813 0.2500 3.0000 1.2500 1.5000 1.3750 -0.3867 0.1250 4.0000 1.2500 1.3750 1.3125 0.1948 0.0625 5.0000 1.3125 1.3750 1.3438 -0.0971 0.0313 6.0000 1.3125 1.3438 1.3281 0.0486 0.0156 7.0000 1.3281 1.3438 1.3359 -0.0243 0.0078 8.0000 1.3281 1.3359 1.3320 0.0122 0.0039 9.0000 1.3320 1.3359 1.3340 -0.0061 0.0020 10.0000 1.3320 1.3340 1.3330 0.0030 0.0010 zero not foundto desiredtolerence (b) [2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 0 0.5000 bisectionhas converged (c) 0 p.2.18 (a) [0 1] step a b m ym bound 1.0000 0 1.0000 0.5000 -2.8750 0.5000 2.0000 0 0.5000 0.2500 1.4844 0.2500 3.0000 0.2500 0.5000 0.3750 -0.7324 0.1250 4.0000 0.2500 0.3750 0.3125 0.3689 0.0625 5.0000 0.3125 0.3750 0.3438 -0.1838 0.0313 6.0000 0.3125 0.3438 0.3281 0.0921 0.0156 7.0000 0.3281 0.3438 0.3359 -0.0460 0.0078 8.0000 0.3281 0.3359 0.3320 0.0230 0.0039 9.0000 0.3320 0.3359 0.3340 -0.0115 0.0020
  • 9.
    10.0000 0.3320 0.33400.3330 0.0058 0.0010 zero not foundto desiredtolerence (b) [-2 -3] step a b m ym bound 1.0000 -2.0000 -3.0000 -2.5000 -2.1250 -0.5000 2.0000 -2.0000 -2.5000 -2.2500 7.2656 -0.2500 3.0000 -2.2500 -2.5000 -2.3750 2.9199 -0.1250 4.0000 -2.3750 -2.5000 -2.4375 0.4871 -0.0625 5.0000 -2.4375 -2.5000 -2.4688 -0.7963 -0.0313 6.0000 -2.4375 -2.4688 -2.4531 -0.1490 -0.0156 7.0000 -2.4375 -2.4531 -2.4453 0.1704 -0.0078 8.0000 -2.4453 -2.4531 -2.4492 0.0111 -0.0039 9.0000 -2.4492 -2.4531 -2.4512 -0.0689 -0.0020 10.0000 -2.4492 -2.4512 -2.4502 -0.0289 -0.0010 zero not foundto desired tolerance (c) [2 3] step a b m ym bound 1.0000 2.0000 3.0000 2.5000 1.6250 0.5000 2.0000 2.0000 2.5000 2.2500 -5.3906 0.2500 3.0000 2.2500 2.5000 2.3750 -2.2012 0.1250 4.0000 2.3750 2.5000 2.4375 -0.3699 0.0625 5.0000 2.4375 2.5000 2.4688 0.6068 0.0313 6.0000 2.4375 2.4688 2.4531 0.1133 0.0156 7.0000 2.4375 2.4531 2.4453 -0.1295 0.0078 8.0000 2.4453 2.4531 2.4492 -0.0084 0.0039 9.0000 2.4492 2.4531 2.4512 0.0524 0.0020 10.0000 2.4492 2.4512 2.4502 0.0220 0.0010 zero not foundto desired tolerance p.2.19 (a) [-1 -2] step a b m ym bound 1.0000 -1.0000 -2.0000 -1.5000 -1.6250 -0.5000 2.0000 -1.5000 -2.0000 -1.7500 1.5781 -0.2500 3.0000 -1.5000 -1.7500 -1.6250 0.0684 -0.1250 4.0000 -1.5000 -1.6250 -1.5625 -0.7561 -0.0625 5.0000 -1.5625 -1.6250 -1.5938 -0.3382 -0.0313 6.0000 -1.5938 -1.6250 -1.6094 -0.1335 -0.0156 7.0000 -1.6094 -1.6250 -1.6172 -0.0322 -0.0078 8.0000 -1.6172 -1.6250 -1.6211 0.0182 -0.0039 9.0000 -1.6172 -1.6211 -1.6191 -0.0070 -0.0020 10.0000 -1.6191 -1.6211 -1.6201 0.0056 -0.0010 zero not foundto desiredtolerence
  • 10.
    (b) [5 6] stepa b m ym bound 1.0000 5.0000 6.0000 5.5000 -27.8750 0.5000 2.0000 5.5000 6.0000 5.7500 -12.9531 0.2500 3.0000 5.7500 6.0000 5.8750 -4.7363 0.1250 4.0000 5.8750 6.0000 5.9375 -0.4338 0.0625 5.0000 5.9375 6.0000 5.9688 1.7666 0.0313 6.0000 5.9375 5.9688 5.9531 0.6623 0.0156 7.0000 5.9375 5.9531 5.9453 0.1132 0.0078 8.0000 5.9375 5.9453 5.9414 -0.1606 0.0039 9.0000 5.9414 5.9453 5.9434 -0.0238 0.0020 10.0000 5.9434 5.9453 5.9443 0.0447 0.0010 zero not foundto desiredtolerence (c) [-3 -4] step a b m ym bound 1.0000 -3.0000 -4.0000 -3.5000 -3.1250 -0.5000 2.0000 -3.0000 -3.5000 -3.2500 1.1094 -0.2500 3.0000 -3.2500 -3.5000 -3.3750 -0.8340 -0.1250 4.0000 -3.2500 -3.3750 -3.3125 0.1804 -0.0625 5.0000 -3.3125 -3.3750 -3.3438 -0.3160 -0.0313 6.0000 -3.3125 -3.3438 -3.3281 -0.0651 -0.0156 7.0000 -3.3125 -3.3281 -3.3203 0.0583 -0.0078 8.0000 -3.3203 -3.3281 -3.3242 -0.0032 -0.0039 9.0000 -3.3203 -3.3242 -3.3223 0.0276 -0.0020 10.0000 -3.3223 -3.3242 -3.3232 0.0122 -0.0010 zero not foundto desiredtolerence p.2.20 (a) [3 4] step a b m ym bound 1.0000 3.0000 4.0000 3.5000 -0.8750 0.5000 2.0000 3.5000 4.0000 3.7500 -0.2031 0.2500 3.0000 3.7500 4.0000 3.8750 0.3262 0.1250 4.0000 3.7500 3.8750 3.8125 0.0442 0.0625 5.0000 3.7500 3.8125 3.7813 -0.0837 0.0313 6.0000 3.7813 3.8125 3.7969 -0.0208 0.0156 7.0000 3.7969 3.8125 3.8047 0.0114 0.0078 8.0000 3.7969 3.8047 3.8008 -0.0048 0.0039 9.0000 3.8008 3.8047 3.8027 0.0033 0.0020 10.0000 3.8008 3.8027 3.8018 -0.0007 0.0010 zero not foundto desiredtolerence (b) [0 1]
  • 11.
    step a bm ym bound 1.0000 0 1.0000 0.5000 -1.6250 0.5000 2.0000 0.5000 1.0000 0.7500 -0.0156 0.2500 3.0000 0.7500 1.0000 0.8750 0.5605 0.1250 4.0000 0.7500 0.8750 0.8125 0.2903 0.0625 5.0000 0.7500 0.8125 0.7813 0.1419 0.0313 6.0000 0.7500 0.7813 0.7656 0.0643 0.0156 7.0000 0.7500 0.7656 0.7578 0.0246 0.0078 8.0000 0.7500 0.7578 0.7539 0.0046 0.0039 9.0000 0.7500 0.7539 0.7520 -0.0055 0.0020 10.0000 0.7520 0.7539 0.7529 -0.0005 0.0010 zero not foundto desiredtolerence >> SECANT METHOD p.2.1 [-1 2] secant(inline('x^2-2'),-1,2,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1 -1 2 0 -2 -2 2 2 0 1 -1 1 3 0 1 2 2 1 4 1.0000 2.0000 1.3333 -0.2222 -0.6667 5 2.0000 1.3333 1.4000 -0.0400 0.0667 6 1.3333 1.4000 1.4146 0.0012 0.0146 secantmethod has converged ans = 1.4146 p.2.2 [ 2 3] secant(inline('x^2-5'),2,3,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1 2.0000 3.0000 2.2000 -0.1600 -0.8000 2 3.0000 2.2000 2.2308 -0.0237 0.0308 3 2.2000 2.2308 2.2361 0.0002 0.0053 secantmethod has converged ans = 2.2361
  • 12.
    p.2.3 [ 2 3] >>secant(inline('x^2-7'),2,3,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1 2.0000 3.0000 2.6000 -0.2400 -0.4000 2 3.0000 2.6000 2.6429 -0.0153 0.0429 3 2.6000 2.6429 2.6458 0.0001 0.0029 secantmethod has converged ans = 2.6458 p.2.4 [1 2] secant(inline('x^3-6'),1,2,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1 1.0000 2.0000 1.7143 -0.9621 -0.2857 2 2.0000 1.7143 1.8071 -0.0988 0.0928 3 1.7143 1.8071 1.8177 0.0059 0.0106 4. 1.8071 1.8177 1.8171 -0.0000 -0.0006 secantmethod has converged ans = 1.8171 p.2.5 [1 2] secant(inline('x^3-4'),1,2,0.005,7) step x(k-1) x (k) x(k+1) y(k+1) dx(k+1) 1 1.0000 2.0000 1.4286 -1.0845 -0.5714 2 2.0000 1.4286 1.5505 -0.2728 0.1219 3 1.4286 1.5505 1.5914 0.0305 0.0410 4 1.5505 1.5914 1.5873 -0.0007 -0.0041 secantmethod has converged ans = 1.5873 p.2.6
  • 13.
    [1 2] secant(inline('x^3-6'),1,2,0.005,7) step x(k-1)x(k) x(k+1) y(k+1) dx(k+1) 1.0000 1.0000 2.0000 1.7143 -0.9621 -0.2857 2.0000 2.0000 1.7143 1.8071 -0.0988 0.0928 3.0000 1.7143 1.8071 1.8177 0.0059 0.0106 4.0000 1.8071 1.8177 1.8171 -0.0000 -0.0006 secantmethod has converged ans = 1.8171 p.2.7 [ 0 1] secant(inline('x^4-0.45'),0,1,0.005,7) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.4500 -0.4090 -0.5500 2.0000 1.0000 0.4500 0.6846 -0.2304 0.2346 3.0000 0.4500 0.6846 0.9871 0.4995 0.3026 4.0000 0.6846 0.9871 0.7801 -0.0797 -0.2071 5.0000 0.9871 0.7801 0.8086 -0.0226 0.0285 6.0000 0.7801 0.8086 0.8198 0.0017 0.0113 secantmethod has converged ans = 0.8198 p.2.8 [ 0 1] secant(inline('x^4-0.65'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.6500 -0.4715 -0.3500 2.0000 1.0000 0.6500 0.8509 -0.1258 0.2009 3.0000 0.6500 0.8509 0.9240 0.0789 0.0731 4.0000 0.8509 0.9240 0.8958 -0.0060 -0.0282 5.0000 0.9240 0.8958 0.8978 -0.0003 0.0020 secantmethod has converged ans = 0.8978
  • 14.
    >> p.2.9 sm prob ofe p.2.10 [ 0 1] secant(inline('x^4-0.25'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.2500 -0.2461 -0.7500 2.0000 1.0000 0.2500 0.4353 -0.2141 0.1853 3.0000 0.2500 0.4353 1.6751 7.6241 1.2398 4.0000 0.4353 1.6751 0.4692 -0.2016 -1.2060 5.0000 1.6751 0.4692 0.5002 -0.1874 0.0311 6.0000 0.4692 0.5002 0.9112 0.4395 0.4110 7.0000 0.5002 0.9112 0.6231 -0.0993 -0.2881 ans = 0.6231 >> p.2.11 (a) [0 1] secant(inline('x^3-9*x+2'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.2500 -0.2344 -0.7500 2.0000 1.0000 0.2500 0.2195 0.0350 -0.0305 3.0000 0.2500 0.2195 0.2235 -0.0001 0.0040 secantmethod has converged ans = 0.2235 (b) [-3 -4] secant(inline('x^3-9*x+2'),-3,-4,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -3.0000 -4.0000 -3.0714 0.6680 0.9286 2.0000 -4.0000 -3.0714 -3.0947 0.2141 -0.0233 3.0000 -3.0714 -3.0947 -3.1057 -0.0035 -0.0110 secantmethod has converged ans =
  • 15.
    -3.1057 (c) [2 3] secant(inline('x^3-9*x+2'),2,3,0.005,8) stepx(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.8000 -1.2480 -0.2000 2.0000 3.0000 2.8000 2.8768 -0.0821 0.0768 3.0000 2.8000 2.8768 2.8823 0.0038 0.0054 secantmethod has converged ans = 2.8823 >> p.2.12 secant(inline('x^3-2*x^2-5'),2,3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.5556 -1.3717 -0.4444 2.0000 3.0000 2.5556 2.6691 -0.2338 0.1135 3.0000 2.5556 2.6691 2.6924 0.0189 0.0233 4.0000 2.6691 2.6924 2.6906 -0.0002 -0.0017 secantmethod has converged ans = 2.6906 p.2.13 (a) [-2 -3] secant(inline('x^3+3*x^2-1'),-2,-3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -2.0000 -3.0000 -2.7500 0.8906 0.2500 2.0000 -3.0000 -2.7500 -2.8678 0.0875 -0.1178 3.0000 -2.7500 -2.8678 -2.8806 -0.0092 -0.0128 4.0000 -2.8678 -2.8806 -2.8794 0.0001 0.0012 secantmethod has converged ans = -2.8794 (b) [0 -1] secant(inline('x^3+3*x^2-1'),0,-1,0.005,8)
  • 16.
    step x(k-1) x(k)x(k+1) y(k+1) dx(k+1) 1.0000 0 -1.0000 -0.5000 -0.3750 0.5000 2.0000 -1.0000 -0.5000 -0.6364 -0.0428 -0.1364 3.0000 -0.5000 -0.6364 -0.6539 0.0033 -0.0176 secantmethod has converged ans = -0.6539 (c) [0 1] secant(inline('x^3+3*x^2-1'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.2500 -0.7969 -0.7500 2.0000 1.0000 0.2500 0.4074 -0.4344 0.1574 3.0000 0.2500 0.4074 0.5961 0.2777 0.1887 4.0000 0.4074 0.5961 0.5225 -0.0383 -0.0736 5.0000 0.5961 0.5225 0.5314 -0.0027 0.0089 secantmethod has converged ans = 0.5314 p.2.14 (a) [0 1] secant(inline('x^3-4*x+1'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.3333 -0.2963 -0.6667 2.0000 1.0000 0.3333 0.2174 0.1407 -0.1159 3.0000 0.3333 0.2174 0.2547 -0.0024 0.0373 secantmethod has converged ans = 0.2547 (b) [1 2] secant(inline('x^3-4*x+1'),1,2,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 1.0000 2.0000 1.6667 -1.0370 -0.3333 2.0000 2.0000 1.6667 1.8364 -0.1528 0.1697 3.0000 1.6667 1.8364 1.8657 0.0313 0.0293 4.0000 1.8364 1.8657 1.8607 -0.0007 -0.0050 secantmethod has converged ans = 1.8607
  • 17.
    (c) [-2 -3] secant(inline('x^3-4*x+1'),-2,-3,0.005,8) stepx(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -2.0000 -3.0000 -2.0667 0.4397 0.9333 2.0000 -3.0000 -2.0667 -2.0951 0.1842 -0.0284 3.0000 -2.0667 -2.0951 -2.1156 -0.0063 -0.0205 4.0000 -2.0951 -2.1156 -2.1149 0.0001 0.0007 secantmethod has converged ans = -2.1149 p.2.15 [2 3] secant(inline('x^3-x^2-4*x-3'),2,3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.7000 -1.4070 -0.3000 2.0000 3.0000 2.7000 2.7958 -0.1466 0.0958 3.0000 2.7000 2.7958 2.8069 0.0087 0.0111 4.0000 2.7958 2.8069 2.8063 -0.0000 -0.0006 secantmethod has converged ans = 2.8063 p.2.16 [0 1] secant(inline('x^3-6*x^2+11*x-5'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.8333 0.5787 -0.1667 2.0000 1.0000 0.8333 0.6044 -0.3226 -0.2289 3.0000 0.8333 0.6044 0.6863 0.0467 0.0819 4.0000 0.6044 0.6863 0.6760 0.0030 -0.0104 secantmethod has converged ans = 0.6760
  • 18.
    p.2.17 (a) [ 12] secant(inline('6*x^3-23*x^2+20*x'),1,2,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 1.0000 2.0000 1.4286 -0.8746 -0.5714 2.0000 2.0000 1.4286 1.2687 0.6062 -0.1599 3.0000 1.4286 1.2687 1.3341 -0.0073 0.0655 4.0000 1.2687 1.3341 1.3333 -0.0000 -0.0008 secantmethod has converged ans = 1.3333 (b) [2 3] Secant(inline('6*x^3-23*x^2+20*x'),2,3,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.2105 -3.3678 -0.7895 2.0000 3.0000 2.2105 2.3553 -2.0900 0.1448 3.0000 2.2105 2.3553 2.5920 1.8018 0.2368 4.0000 2.3553 2.5920 2.4824 -0.3007 -0.1096 5.0000 2.5920 2.4824 2.4981 -0.0330 0.0157 6.0000 2.4824 2.4981 2.5000 0.0007 0.0019 secantmethod has converged ans = 2.5000 p.2.18 (a) [2 3] secant(inline('3*x^3-x^2-18*x+6'),2,3,0.005,8) step x(k-1) x(k) x (k+1) y(k+1) dx(k+1) 1.0000 2.0000 3.0000 2.2941 -4.3354 -0.7059 2.0000 3.0000 2.2941 2.4021 -1.4263 0.1080 3.0000 2.2941 2.4021 2.4551 0.1743 0.0530 4.0000 2.4021 2.4551 2.4493 -0.0057 -0.0058 5.0000 2.4551 2.4493 2.4495 -0.0000 0.0002 secantmethod has converged ans = 2.4495
  • 19.
    (b) [-2 -3] secant(inline('3*x^3-x^2-18*x+6'),-2,-3,0.005,8) stepx(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -2.0000 -3.0000 -2.3182 4.9798 0.6818 2.0000 -3.0000 -2.3182 -2.4152 1.3736 -0.0971 3.0000 -2.3182 -2.4152 -2.4522 -0.1118 -0.0370 4.0000 -2.4152 -2.4522 -2.4494 0.0022 0.0028 secantmethod has converged ans = -2.4494 (c) [0 1] secant(inline('3*x^3-x^2-18*x+6'),0,1,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.3750 -0.7324 -0.6250 2.0000 1.0000 0.3750 0.3256 0.1366 -0.0494 3.0000 0.3750 0.3256 0.3334 -0.0007 0.0078 secantmethod has converged ans = 0.3334 p.2.19 (a) [-1 -2] secant(inline('x^3-x^2-24*x-32'),-1,-2,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -1.0000 -2.0000 -1.7143 1.1662 0.2857 2.0000 -2.0000 -1.7143 -1.5967 -0.2993 0.1176 3.0000 -1.7143 -1.5967 -1.6207 0.0133 -0.0240 4.0000 -1.5967 -1.6207 -1.6197 0.0001 0.0010 secantmethod has converged ans = -1.6197 (b) [5 6] secant(inline('x^3-x^2-24*x-32'),5,6,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 5.0000 6.0000 5.9286 -1.0565 -0.0714 2.0000 6.0000 5.9286 5.9435 -0.0142 0.0149 3.0000 5.9286 5.9435 5.9437 0.0001 0.0002 secantmethod has converged ans =
  • 20.
    5.9437 (c) [-3 -4] secant(inline('x^3-x^2-24*x-32'),-3,-4,0.005,8) stepx(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -3.0000 -4.0000 -3.2000 1.7920 0.8000 2.0000 -4.0000 -3.2000 -3.2806 0.6655 -0.0806 3.0000 -3.2000 -3.2806 -3.3282 -0.0659 -0.0476 4.0000 -3.2806 -3.3282 -3.3239 0.0020 0.0043 secantmethod has converged ans = -3.3239 p.2.20 (a) [-3 -4] secant(inline('x^3-7*x^2+14*x-7'),-3,-4,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 -3.0000 -4.0000 -1.6100 -51.8580 2.3900 2.0000 -4.0000 -1.6100 -0.9477 -27.4065 0.6623 3.0000 -1.6100 -0.9477 -0.2054 -10.1796 0.7423 4.0000 -0.9477 -0.2054 0.2332 -4.1027 0.4386 5.0000 -0.2054 0.2332 0.5294 -1.4020 0.2961 6.0000 0.2332 0.5294 0.6831 -0.3841 0.1537 7.0000 0.5294 0.6831 0.7411 -0.0620 0.0580 ans = 0.7411 (b) [3 4] secant(inline('x^3-7*x^2+14*x-7'),3,4,0.005,8) step x(k-1) x(k) x(k+1) y(k+1) dx(k+1) 1.0000 3.0000 4.0000 3.5000 -0.8750 -0.5000 2.0000 4.0000 3.5000 3.7333 -0.2634 0.2333 3.0000 3.5000 3.7333 3.8338 0.1364 0.1005 4.0000 3.7333 3.8338 3.7995 -0.0099 -0.0343 5.0000 3.8338 3.7995 3.8019 -0.0003 0.0023 secantmethod has converged ans = 3.8019 (c) [0 1]
  • 21.
    >> secant(inline('x^3-7*x^2+14*x-7'),0,1,0.005,8) step x(k-1)x(k) x(k+1) y(k+1) dx(k+1) 1.0000 0 1.0000 0.8750 0.5605 -0.1250 2.0000 1.0000 0.8750 0.7156 -0.2000 -0.1594 3.0000 0.8750 0.7156 0.7575 0.0229 0.0419 4.0000 0.7156 0.7575 0.7532 0.0008 -0.0043 secantmethod has converged ans = 0.7532 REGULA FALSI METHOD Falsi method Q1 [s,y]=falsi(inline('x^2-2'),1,2,0.005,5) step a b s y 1.0000 1.0000 2.0000 1.3333 -0.2222 2.0000 1.3333 2.0000 1.4000 -0.0400 3.0000 1.4000 2.0000 1.4118 -0.0069 4.0000 1.4118 2.0000 1.4138 -0.0012 regulafalsi methodhasconverged s = 1.4138 y = -0.0012 Q2. [s,y]=falsi(inline('x^2-5'),2,3,0.005,5) step a b s y 1.0000 2.0000 3.0000 2.2000 -0.1600 2.0000 2.2000 3.0000 2.2308 -0.0237 3.0000 2.2308 3.0000 2.2353 -0.0035 regulafalsi methodhasconverged s = 2.2353 y = -0.0035 Q3
  • 22.
    >> [s,y]=falsi(inline('x^2-7'),2,3,0.005,5) step ab s y 1.0000 2.0000 3.0000 2.6000 -0.2400 2.0000 2.6000 3.0000 2.6429 -0.0153 3.0000 2.6429 3.0000 2.6456 -0.0010 regulafalsi methodhasconverged s = 2.6456 y = -9.6138e-004 Q4 >> [s,y]=falsi(inline('x^3-3'),1,2,0.005,5) step a b s y 1.0000 1.0000 2.0000 1.2857 -0.8746 2.0000 1.2857 2.0000 1.3921 -0.3024 3.0000 1.3921 2.0000 1.4267 -0.0958 4.0000 1.4267 2.0000 1.4375 -0.0295 5.0000 1.4375 2.0000 1.4408 -0.0090 ZERO NOT FOUND TO DESIRED TOLERANCE s = 1.4408 y = -0.0090 Q5 >> [s,y]=falsi(inline('x^3-4'),1,2,0.005,5) step a b s y 1.0000 1.0000 2.0000 1.4286 -1.0845 2.0000 1.4286 2.0000 1.5505 -0.2728 3.0000 1.5505 2.0000 1.5792 -0.0620 4.0000 1.5792 2.0000 1.5856 -0.0137 5.0000 1.5856 2.0000 1.5870 -0.0030 regulafalsi methodhasconverged s = 1.5870 y = -0.0030 Q6 >> [s,y]=falsi(inline('x^3-6'),1,2,0.005,5) step a b s y 1.0000 1.0000 2.0000 1.7143 -0.9621 2.0000 1.7143 2.0000 1.8071 -0.0988
  • 23.
    3.0000 1.8071 2.00001.8162 -0.0094 4.0000 1.8162 2.0000 1.8170 -0.0009 regulafalsi methodhasconverged s = 1.8170 y = -8.8557e-004 Q7 >> [s,y]=falsi(inline('x^4-0.45'),0,1,0.005,5) step a b s y 1.0000 0 1.0000 0.4500 -0.4090 2.0000 0.4500 1.0000 0.6846 -0.2304 3.0000 0.6846 1.0000 0.7777 -0.0842 4.0000 0.7777 1.0000 0.8072 -0.0254 5.0000 0.8072 1.0000 0.8157 -0.0072 ZERO NOT FOUND TO DESIRED TOLERANCE s = 0.8157 y = -0.0072 Q8 >> [s,y]=falsi(inline('x^4-0.65'),0,1,0.005,5) step a b s y 1.0000 0 1.0000 0.6500 -0.4715 2.0000 0.6500 1.0000 0.8509 -0.1258 3.0000 0.8509 1.0000 0.8903 -0.0217 4.0000 0.8903 1.0000 0.8967 -0.0034 regulafalsi methodhasconverged s = 0.8967 y = -0.0034 Q9 >> [s,y]=falsi(inline('x^4-0.06'),0,1,0.005,5) step a b s y 1.0000 0 1.0000 0.0600 -0.0600 2.0000 0.0600 1.0000 0.1164 -0.0598 3.0000 0.1164 1.0000 0.1693 -0.0592 4.0000 0.1693 1.0000 0.2185 -0.0577 5.0000 0.2185 1.0000 0.2637 -0.0552 ZERO NOT FOUND TO DESIRED TOLERANCE s =
  • 24.
    0.2637 y = -0.0552 Q10 >> [s,y]=falsi(inline('x^4-0.25'),0,1,0.005,5) stepa b s y 1.0000 0 1.0000 0.2500 -0.2461 2.0000 0.2500 1.0000 0.4353 -0.2141 3.0000 0.4353 1.0000 0.5607 -0.1512 4.0000 0.5607 1.0000 0.6344 -0.0880 5.0000 0.6344 1.0000 0.6728 -0.0451 ZERO NOT FOUND TO DESIRED TOLERANCE s = 0.6728 y = -0.0451 Q11 [s,y]=falsi(inline('x^3-9*x+2'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.2500 -0.2344 2.0000 0 0.2500 0.2238 -0.0028 regulafalsi methodhasconverged s = 0.2238 y = -0.0028 >> [s,y]=falsi(inline('x^3-9*x+2'),-3,-4,0.005,8) step a b s y 1.0000 -3.0000 -4.0000 -3.0714 0.6680 2.0000 -3.0714 -4.0000 -3.0947 0.2141 3.0000 -3.0947 -4.0000 -3.1021 0.0677 4.0000 -3.1021 -4.0000 -3.1044 0.0213 5.0000 -3.1044 -4.0000 -3.1051 0.0067 6.0000 -3.1051 -4.0000 -3.1054 0.0021 regulafalsi methodhasconverged s = -3.1054 y = 0.0021 >> [s,y]=falsi(inline('x^3-9*x+2'),2,3,0.005,8) step a b s y
  • 25.
    1.0000 2.0000 3.00002.8000 -1.2480 2.0000 2.8000 3.0000 2.8768 -0.0821 3.0000 2.8768 3.0000 2.8817 -0.0050 4.0000 2.8817 3.0000 2.8820 -0.0003 regulafalsi methodhasconverged s = 2.8820 y = -3.0703e-004 Q12 >> [s,y]=falsi(inline('x^3-2*x^2-5'),2,3,0.005,8) step a b s y 1.0000 2.0000 3.0000 2.5556 -1.3717 2.0000 2.5556 3.0000 2.6691 -0.2338 3.0000 2.6691 3.0000 2.6873 -0.0363 4.0000 2.6873 3.0000 2.6901 -0.0056 5.0000 2.6901 3.0000 2.6906 -0.0008 regulafalsi methodhasconverged s = 2.6906 y = -8.4925e-004 >> [s,y]=falsi(inline('x^3-3*x^2-1'),0,1,0.005,8) ??? Error using==> falsi at 5 functionhassame signat endpoints >> [s,y]=falsi(inline('x^3+3*x^2-1'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.2500 -0.7969 2.0000 0.2500 1.0000 0.4074 -0.4344 3.0000 0.4074 1.0000 0.4824 -0.1897 4.0000 0.4824 1.0000 0.5132 -0.0749 5.0000 0.5132 1.0000 0.5250 -0.0284 6.0000 0.5250 1.0000 0.5295 -0.0106 7.0000 0.5295 1.0000 0.5311 -0.0039 regulafalsi methodhasconverged s = 0.5311 y = -0.0039 >> [s,y]=falsi(inline('x^3+3*x^2-1'),-1,0,0.005,8) step a b s y
  • 26.
    1.0000 -1.0000 0-0.5000 -0.3750 2.0000 -1.0000 -0.5000 -0.6364 -0.0428 3.0000 -1.0000 -0.6364 -0.6513 -0.0037 regulafalsi methodhasconverged s = -0.6513 y = -0.0037 >> [s,y]=falsi(inline('x^3+3*x^2-1'),-2,-3,0.005,8) step a b s y 1.0000 -2.0000 -3.0000 -2.7500 0.8906 2.0000 -2.7500 -3.0000 -2.8678 0.0875 3.0000 -2.8678 -3.0000 -2.8784 0.0074 4.0000 -2.8784 -3.0000 -2.8793 0.0006 regulafalsi methodhasconverged s = -2.8793 y = 6.2341e-004 >> [s,y]=falsi(inline('x^3-4*x+1'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.3333 -0.2963 2.0000 0 0.3333 0.2571 -0.0116 3.0000 0 0.2571 0.2542 -0.0004 regulafalsi methodhasconverged s = 0.2542 y = -3.8225e-004 >> [s,y]=falsi(inline('x^3-4*x+1'),1,2,0.005,8) step a b s y 1.0000 1.0000 2.0000 1.6667 -1.0370 2.0000 1.6667 2.0000 1.8364 -0.1528 3.0000 1.8364 2.0000 1.8581 -0.0175 4.0000 1.8581 2.0000 1.8605 -0.0020 regulafalsi methodhas converged s = 1.8605 y = -0.0020 >> [s,y]=falsi(inline('x^3-4*x+1'),-2,-3,0.005,8) step a b s y
  • 27.
    1.0000 -2.0000 -3.0000-2.0667 0.4397 2.0000 -2.0667 -3.0000 -2.0951 0.1842 3.0000 -2.0951 -3.0000 -2.1068 0.0756 4.0000 -2.1068 -3.0000 -2.1116 0.0308 5.0000 -2.1116 -3.0000 -2.1136 0.0125 6.0000 -2.1136 -3.0000 -2.1144 0.0051 7.0000 -2.1144 -3.0000 -2.1147 0.0020 regulafalsi methodhasconverged s = -2.1147 y = 0.0020 >> [s,y]=falsi(inline('x^3-x^2-4*x-3'),2,3,0.005,8) step a b s y 1.0000 2.0000 3.0000 2.7000 -1.4070 2.0000 2.7000 3.0000 2.7958 -0.1466 3.0000 2.7958 3.0000 2.8053 -0.0141 4.0000 2.8053 3.0000 2.8062 -0.0013 regulafalsi methodhasconverged s = 2.8062 y = -0.0013 >> [s,y]=falsi(inline('x^3-6*x^2+11*x-5'),2,3,0.005,8) ??? Error using==> falsi at 5 functionhassame signat endpoints >> [s,y]=falsi(inline('x^3-6*x^2+11*x-5'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.8333 0.5787 2.0000 0 0.8333 0.7469 0.2854 3.0000 0 0.7469 0.7066 0.1295 4.0000 0 0.7066 0.6887 0.0566 5.0000 0 0.6887 0.6810 0.0243 6.0000 0 0.6810 0.6777 0.0104 7.0000 0 0.6777 0.6763 0.0044 regulafalsi methodhasconverged s = 0.6763 y = 0.0044 >> [s,y]=falsi(inline('6*x^3-23*x^2+20*x'),1,2,0.005,8)
  • 28.
    step a bs y 1.0000 1.0000 2.0000 1.4286 -0.8746 2.0000 1.0000 1.4286 1.3318 0.0140 3.0000 1.3318 1.4286 1.3334 -0.0002 regulafalsi methodhasconverged s = 1.3334 y = -2.2752e-004 >> [s,y]=falsi(inline('6*x^3-23*x^2+20*x'),2,3,0.005,8) step a b s y 1.0000 2.0000 3.0000 2.2105 -3.3678 2.0000 2.2105 3.0000 2.3553 -2.0900 3.0000 2.3553 3.0000 2.4341 -1.0590 4.0000 2.4341 3.0000 2.4714 -0.4819 5.0000 2.4714 3.0000 2.4879 -0.2086 6.0000 2.4879 3.0000 2.4949 -0.0883 7.0000 2.4949 3.0000 2.4979 -0.0370 8.0000 2.4979 3.0000 2.4991 -0.0155 ZERO NOT FOUND TO DESIRED TOLERANCE s = 2.4991 y = -0.0155 >> [s,y]=falsi(inline('3*x^3-x^2+18*x+6'),0,1,0.005,8) ??? Error using==> falsi at 5 functionhassame signat endpoints >> [s,y]=falsi(inline('3*x^3-x^2-18*x+6'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.3750 -0.7324 2.0000 0 0.3750 0.3342 -0.0154 3.0000 0 0.3342 0.3333 -0.0003 regulafalsi methodhasconverged s = 0.3333 y = -2.8549e-004 >> [s,y]=falsi(inline('3*x^3-x^2-18*x+6'),2,3,0.005,8) step a b s y 1.0000 2.0000 3.0000 2.2941 -4.3354 2.0000 2.2941 3.0000 2.4021 -1.4263
  • 29.
    3.0000 2.4021 3.00002.4357 -0.4261 4.0000 2.4357 3.0000 2.4455 -0.1236 5.0000 2.4455 3.0000 2.4483 -0.0356 6.0000 2.4483 3.0000 2.4492 -0.0102 7.0000 2.4492 3.0000 2.4494 -0.0029 regulafalsi methodhasconverged s = 2.4494 y = -0.0029 >> [s,y]=falsi(inline('3*x^3-x^2-18*x+6'),-2,-3,0.005,8) step a b s y 1.0000 -2.0000 -3.0000 -2.3182 4.9798 2.0000 -2.3182 -3.0000 -2.4152 1.3736 3.0000 -2.4152 -3.0000 -2.4408 0.3517 4.0000 -2.4408 -3.0000 -2.4473 0.0883 5.0000 -2.4473 -3.0000 -2.4489 0.0221 6.0000 -2.4489 -3.0000 -2.4494 0.0055 7.0000 -2.4494 -3.0000 -2.4495 0.0014 regulafalsi methodhasconverged s = -2.4495 y = 0.0014 >> [s,y]=falsi(inline('x^3-x^2-24*x-32'),-1,-2,0.005,8) step a b s y 1.0000 -1.0000 -2.0000 -1.7143 1.1662 2.0000 -1.0000 -1.7143 -1.6397 0.2555 3.0000 -1.0000 -1.6397 -1.6238 0.0523 4.0000 -1.0000 -1.6238 -1.6205 0.0106 5.0000 -1.0000 -1.6205 -1.6198 0.0021 regulafalsi methodhasconverged s = -1.6198 y = 0.0021 >> [s,y]=falsi(inline('x^3-x^2-24*x-32'),5,6,0.005,8) step a b s y 1.0000 5.0000 6.0000 5.9286 -1.0565 2.0000 5.9286 6.0000 5.9435 -0.0142 3.0000 5.9435 6.0000 5.9437 -0.0002 regulafalsi methodhasconverged
  • 30.
    s = 5.9437 y = -1.9042e-004 >>[s,y]=falsi(inline('x^3-x^2-24*x-32'),-3,-4,0.005,8) step a b s y 1.0000 -3.0000 -4.0000 -3.2000 1.7920 2.0000 -3.2000 -4.0000 -3.2806 0.6655 3.0000 -3.2806 -4.0000 -3.3093 0.2300 4.0000 -3.3093 -4.0000 -3.3191 0.0775 5.0000 -3.3191 -4.0000 -3.3224 0.0259 6.0000 -3.3224 -4.0000 -3.3235 0.0086 7.0000 -3.3235 -4.0000 -3.3238 0.0029 regulafalsi methodhasconverged s = -3.3238 y = 0.0029 >> [s,y]=falsi(inline('x^3-7*x^2+14*x-7'),0,1,0.005,8) step a b s y 1.0000 0 1.0000 0.8750 0.5605 2.0000 0 0.8750 0.8101 0.2793 3.0000 0 0.8101 0.7790 0.1310 4.0000 0 0.7790 0.7647 0.0597 5.0000 0 0.7647 0.7583 0.0269 6.0000 0 0.7583 0.7554 0.0120 7.0000 0 0.7554 0.7541 0.0054 8.0000 0 0.7541 0.7535 0.0024 regulafalsi methodhasconverged s = 0.7535 y = 0.0024 >> [s,y]=falsi(inline('x^3-7*x^2+14*x-7'),3,4,0.005,8) step a b s y 1.0000 3.0000 4.0000 3.5000 -0.8750 2.0000 3.5000 4.0000 3.7333 -0.2634 3.0000 3.7333 4.0000 3.7889 -0.0531 4.0000 3.7889 4.0000 3.7996 -0.0098 5.0000 3.7996 4.0000 3.8015 -0.0018 regulafalsi methodhasconverged s =
  • 31.
    3.8015 y = -0.0018 THOMAS METHOD a=[2,4,3,0] a= 2 4 3 0 >> b=[0,2,1,2] b = 0 2 1 2 >> d=[2,4,3,5] d = 2 4 3 5 >> r=[4,6,7,10] r = 4 6 7 10 >> x=thomas(a,d,b,r) x = 1 1 0 2 >> x = 10.0000 -5.8000 2.2000
  • 32.
    Question1 >> a=[2,3,0] a = 23 0 >> b=[0,1,3] b = 0 1 3 >> d=[1,3,10] d = 1 3 10 >> x=thomas(a,d,b,r) x = 2 4 1 Question3.22 d=[-2,-2,-2,-2] d = -2 -2 -2 -2 >> b=[0,1,1,1] b = 0 1 1 1 >> a=[1,1,1,0]
  • 33.
    a = 1 11 0 >> r=[-1,0,0,0] r = -1 0 0 0 >> x=thomas(a,d,b,r) x = 0.8000 0.6000 0.4000 0.2000 >> r=[33,26,30,15] question3.23 r = 33 26 30 15 >> d=[5,5,5,5] d = 5 5 5 5 >> a=[1,1,1,0] a = 1 1 1 0 >> b=[0,1,1,1] b = 0 1 1 1
  • 34.
    >> x=thomas(a,d,b,r) x = 6.00003.0000 5.0000 2.0000 >> Question3.24 r=[14;-36;-6;14;-9;6] r = 14 -36 -6 14 -9 6 >> d=[-3,4,-1,4,1,2] d = -3 4 -1 4 1 2 >> a=[-4,5,-3,-5,-5,0] a = -4 5 -3 -5 -5 0 >> b=[0,-3,1,0,3,-1] b = 0 -3 1 0 3 -1
  • 35.
    >> x=thomas(a,d,b,r) x = 2.0000-5.0000 -2.0000 1.0000 -2.0000 2.0000 >> Question3.25 b=[0,5,5,2,5,1,-2] b = 0 5 5 2 5 1 -2 >> a=[3,-1,-1,1,-1,0,0] a = 3 -1 -1 1 -1 0 0 >> d=[1,-4,-2,3,-3,-1,4] d = 1 -4 -2 3 -3 -1 4 >> r=[19;1;28;0;-25;0;2] r = 19 1 28 0 -25 0 2 >> x=thomas(a,d,b,r) x = 4.0000 5.0000 -1.0000 -1.0000 5.0000 5.0000 3.0000
  • 36.
    C3.1 d=[-1,4,1,-1,-2,-2,4,2] d = -1 41 -1 -2 -2 4 2 b=[0,-1,4,0,-2,-4,2,0] b = 0 -1 4 0 -2 -4 2 0 >> a=[1,1,3,-2,-2,-2,0,0] a = 1 1 3 -2 -2 -2 0 0 >> r=[7,13,-3,-2,-4,-28,26,10] r = 7 13 -3 -2 -4 -28 26 10 >> x=thomas(a,d,b,r) x = -4.0000 3.0000 -3.0000 -4.0000 3.0000 3.0000 5.0000 5.0000 >>
  • 37.
    questionc3.2 r=[-1;19;20;-1;-19;14;0;-4;-2] r = -1 19 20 -1 -19 14 0 -4 -2 >> a=[1,1,-1,4,5,0,-4,-4,0] a= 1 1 -1 4 5 0 -4 -4 0 >> b=[0,2,3,-4,3,-1,-5,-2,-4] b = 0 2 3 -4 3 -1 -5 -2 -4 >> d=[-1,3,-3,3,3,-5,1,2,2] d = -1 3 -3 3 3 -5 1 2 2 >> x=thomas(a,d,b,r) x = 5.0000 4.0000 -3.0000 1.0000 -4.0000 -2.0000 -2.0000 2.0000 3.0000 Questionc3.3 d=[3,3,1,-4,0,-3,0,0,0,1]
  • 38.
    d = 3 31 -4 0 -3 0 0 0 1 >> a=[-4,5,2,5,-2,-2,-5,-1,1,0] a = -4 5 2 5 -2 -2 -5 -1 1 0 >> b=[0,3,-1,-2,1,5,1,-3,-3,-4] b = 0 3 -1 -2 1 5 1 -3 -3 -4 >> r=[-13,-11,-6,25,6,29,1,0,3,-12] r = -13 -11 -6 25 6 29 1 0 3 -12 >> x=thomas(a,d,b,r) x = Columns1 through9 -3.0000 1.0000 -1.0000 -2.0000 3.0000 -4.0000 -1.0000 -1.0000 3.0000 Column10 -0.0000 >>questiona3.7 a=[1,1,1,1,1,1,0] a = 1 1 1 1 1 1 0
  • 39.
    >> b=[0,1,1,1,1,1,1] b = 01 1 1 1 1 1 >> d=[4,4,4,4,4,4,4] d = 4 4 4 4 4 4 4 >> r=[7.2,11.82,12,0,-12,-11.82,-7.2] r = 7.2000 11.8200 12.0000 0 -12.0000 -11.8200 -7.2000 >> x=thomas(a,d,b,r) x = 1.2986 2.0057 2.4986 0.0000 -2.4986 -2.0057 -1.2986 >> Questiona3.8 d=[-1.99,-1.99,-1.99,-1.99,-1.99,-1.99,-1.99,-1.99,-1.99] d = -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 -1.9900 >> a=[1,1,1,1,1,1,1,1,0] a = 1 1 1 1 1 1 1 1 0 >> b=[0,1,1,1,1,1,1,1,1] b =
  • 40.
    0 1 11 1 1 1 1 1 >> r=[-0.99;0.002;0.0031;0.0042;0.0055;0.0068;0.0084;0.0103;-0.6874] r = -0.9900 0.0020 0.0031 0.0042 0.0055 0.0068 0.0084 0.0103 -0.6874 >> x=thomas(a,d,b,r) x = 0.9846 0.9694 0.9465 0.9172 0.8830 0.8454 0.8061 0.7672 0.7310 GAUSS SEIDEL METHOD Q P6.1 a=[10 -2 1; -2 10 -2; -2 -5 10] a = 10 -2 1 -2 10 -2 -2 -5 10 >> b=[9;12;18] b =
  • 41.
    9 12 18 >> x0=[0;0;0] x0 = 0 0 0 >>tol=0.0001 tol = 1.0000e-004 >> max_it=7 max_it= 7 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 0.9000 1.3800 2.6700 2.0000 0.9090 1.9158 2.9397 3.0000 0.9892 1.9858 2.9907 4.0000 0.9981 1.9978 2.9985 5.0000 0.9997 1.9996 2.9998 6.0000 1.0000 1.9999 3.0000 gaussseidel methodconverged x =
  • 42.
    1.0000 2.0000 3.0000 >> P6.2 max_it=7 max_it= 7 >> tol=0.0001 tol = 1.0000e-004 >>x0=[0;0;0] x0 = 0 0 0 >> b=[8;4;12] b = 8 4 12 >> a=[8 1 -1;-1 7 -2;2 1 9] a = 8 1 -1 -1 7 -2 2 1 9
  • 43.
    >> x=seidel(a,b,x0,tol,max_it) i x1x2 x3 1.0000 1.0000 0.7143 1.0317 2.0000 1.0397 1.0147 0.9895 3.0000 0.9969 0.9966 1.0011 4.0000 1.0006 1.0004 0.9998 5.0000 0.9999 0.9999 1.0000 6.0000 1.0000 1.0000 1.0000 gaussseidel methodconverged x = 1.0000 1.0000 1.0000 >> P6.3 a=[5 -1 0;-1 5 -1;0 -1 5] a = 5 -1 0 -1 5 -1 0 -1 5 >> b=[9;4;-6] b = 9 4 -6 >> x0=[0;0;0]
  • 44.
    x0 = 0 0 0 >> tol=0.0001 tol= 1.0000e-004 >> max_it=7 max_it= 7 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 1.8000 1.1600 -0.9680 2.0000 2.0320 1.0128 -0.9974 3.0000 2.0026 1.0010 -0.9998 4.0000 2.0002 1.0001 -1.0000 5.0000 2.0000 1.0000 -1.0000 gaussseidel methodconverged x = 2.0000 1.0000 -1.0000 >> P6.4 max_it=7
  • 45.
    max_it= 7 >> tol=0.0001 tol = 1.0000e-004 >>x0=[0;0;0] x0 = 0 0 0 >> b=[3;-4;5] b = 3 -4 5 >> a=[4 1 0;1 3 -1;1 0 2] a = 4 1 0 1 3 -1 1 0 2 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 0.7500 -1.5833 2.1250 2.0000 1.1458 -1.0069 1.9271 3.0000 1.0017 -1.0249 1.9991
  • 46.
    4.0000 1.0062 -1.00241.9969 5.0000 1.0006 -1.0012 1.9997 6.0000 1.0003 -1.0002 1.9998 7.0000 1.0001 -1.0001 2.0000 gaussseidel methoddidnotconverged x = 1.0001 -1.0001 2.0000 >> P6.5 a=[4 1 0;1 3 -1;0 -1 4] a = 4 1 0 1 3 -1 0 -1 4 >> b=[3;4;5] b = 3 4 5 >> x0=[0;0;0] x0 = 0 0 0
  • 47.
    >> tol=0.0001 tol = 1.0000e-004 >>max_it=7 max_it= 7 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 0.7500 1.0833 1.5208 2.0000 0.4792 1.6806 1.6701 3.0000 0.3299 1.7801 1.6950 4.0000 0.3050 1.7967 1.6992 5.0000 0.3008 1.7994 1.6999 6.0000 0.3001 1.7999 1.7000 7.0000 0.3000 1.8000 1.7000 gaussseidel methoddidnotconverged x = 0.3000 1.8000 1.7000 >> P6.6 x0=[0;0;0;0] x0 =
  • 48.
    0 0 0 0 max_it=7 max_it= 7 a=[-2 1 00 ; 1 -2 1 0; 0 1 -2 1; 0 0 1 -2] a = -2 1 0 0 1 -2 1 0 0 1 -2 1 0 0 1 -2 b=[-1;0;0;0] b = -1 0 0 0 x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 0.5000 0.2500 0.1250 0.0625 2.0000 0.6250 0.3750 0.2188 0.1094 3.0000 0.6875 0.4531 0.2813 0.1406 4.0000 0.7266 0.5039 0.3223 0.1611 5.0000 0.7520 0.5371 0.3491 0.1746 6.0000 0.7686 0.5588 0.3667 0.1833
  • 49.
    7.0000 0.7794 0.57310.3782 0.1891 gaussseidel methoddidnotconverged x = 0.7794 0.5731 0.3782 0.1891 >> P6.7 max_it=7 max_it= 7 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> b=[33;26;30;15] b = 33 26 30 15 a=[5 1 0 0;1 5 1 0;0 1 5 1;0 0 1 5] a = 5 1 0 0 1 5 1 0
  • 50.
    0 1 51 0 0 1 5 >> tol=0.0001 tol = 1.0000e-004 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 6.6000 3.8800 5.2240 1.9552 2.0000 5.8240 2.9904 5.0109 1.9978 3.0000 6.0019 2.9974 5.0009 1.9998 4.0000 6.0005 2.9997 5.0001 2.0000 5.0000 6.0001 3.0000 5.0000 2.0000 gaussseidel methodconverged x = 6.0000 3.0000 5.0000 2.0000 >> P6.8 a=[1 2 0 0;2 6 8 0;0 8 35 18;0 0 18 112] a = 1 2 0 0 2 6 8 0 0 8 35 18 0 0 18 112 >> b=[2 ;6;-10;-112]
  • 51.
    b = 2 6 -10 -112 tol=0.0001 tol = 1.0000e-004 >>x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 2.0000 0.3333 -0.3619 -0.9418 2.0000 1.3333 1.0381 -0.0386 -0.9938 3.0000 -0.0762 1.0769 -0.0208 -0.9967 4.0000 -0.1538 1.0789 -0.0198 -0.9968 5.0000 -0.1579 1.0790 -0.0197 -0.9968 6.0000 -0.1580 1.0789 -0.0197 -0.9968 7.0000 -0.1578 1.0788 -0.0196 -0.9968 gaussseidel methoddidnotconverged x = -0.1578 1.0788 -0.0196 -0.9968 >> P6.9 b=[-3;5;2;3.5]
  • 52.
    b = -3.0000 5.0000 2.0000 3.5000 >> a=[1-2 0 0;-2 5 -1 0;0 -1 2 -0.5;0 0 -0.5 1.25] a = 1.0000 -2.0000 0 0 -2.0000 5.0000 -1.0000 0 0 -1.0000 2.0000 -0.5000 0 0 -0.5000 1.2500 >> x=seidel(a,b,x0,tol,100) i x1 x2 x3 1.0000 -3.0000 -0.2000 0.9000 3.1600 2.0000 -3.4000 -0.1800 1.7000 3.4800 3.0000 -3.3600 -0.0040 1.8680 3.5472 4.0000 -3.0080 0.1704 1.9720 3.5888 5.0000 -2.6592 0.3307 2.0626 3.6250 6.0000 -2.3386 0.4771 2.1448 3.6579 gaussseidel methodconverged at95th iteration x = 0.9991 1.9996 2.9998 3.9999 p.6.10a=[4 -8 0 0;-8 18 -2 0;0 -2 5 -1.5;0 0 -1.5 1.75] a =
  • 53.
    4.0000 -8.0000 00 -8.0000 18.0000 -2.0000 0 0 -2.0000 5.0000 -1.5000 0 0 -1.5000 1.7500 >> b=[-12;22;5;2] b = -12 22 5 2 >> max_it=7 max_it= 7 >> tol=0.0001 tol = 1.0000e-004 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 -3.0000 -0.1111 0.9556 1.9619 2.0000 -3.2222 -0.1037 1.5471 2.4689
  • 54.
    3.0000 -3.2074 -0.03141.7281 2.6241 4.0000 -3.0628 0.0530 1.8084 2.6929 5.0000 -2.8940 0.1369 1.8627 2.7394 6.0000 -2.7261 0.2176 1.9089 2.7790 7.0000 -2.5649 0.2944 1.9515 2.8155 gaussseidel methoddidnotconverged x = -2.5649 0.2944 1.9515 2.8155 p.6.11 >> a=[4 8 0 0;8 18 2 0;0 2 5 1.5;0 0 1.5 1.75] a = 4.0000 8.0000 0 0 8.0000 18.0000 2.0000 0 0 2.0000 5.0000 1.5000 0 0 1.5000 1.7500 >> b=[8;18;0.5;-1.75] b = 8.0000 18.0000 0.5000 -1.7500 >> tol=0.0001 tol =
  • 55.
    1.0000e-004 >> x0=[0;0;0;0] x0 = 0 0 0 0 >>max_it=7 max_it= 7 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 2.0000 0.1111 0.0556 -1.0476 2.0000 1.7778 0.2037 0.3328 -1.2853 3.0000 1.5926 0.2552 0.3835 -1.3287 4.0000 1.4896 0.2953 0.3805 -1.3261 5.0000 1.4093 0.3314 0.3653 -1.3131 6.0000 1.3373 0.3651 0.3479 -1.2982 7.0000 1.2699 0.3970 0.3307 -1.2834 gaussseidel methoddid notconverged x = 1.2699 0.3970 0.3307 -1.2834
  • 56.
    p.6.12 a=[1 -2 00 0;-2 5 1 0 0;0 1 2 -2 0;0 0 -2 5 1;0 0 0 1 2] a = 1 -2 0 0 0 -2 5 1 0 0 0 1 2 -2 0 0 0 -2 5 1 0 0 0 1 2 >> b=[5;-9;0;3;0] b = 5 -9 0 3 0 >> x0=[0;0;0;0;0] x0 = 0 0 0 0 0 >> x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 5.0000 0.2000 -0.1000 0.5600 -0.2800 2.0000 5.4000 0.3800 0.3700 0.8040 -0.4020 3.0000 5.7600 0.4300 0.5890 0.9160 -0.4580 4.0000 5.8600 0.4262 0.7029 0.9728 -0.4864 5.0000 5.8524 0.4004 0.7726 1.0063 -0.5032
  • 57.
    6.0000 5.8008 0.36580.8234 1.0300 -0.5150 7.0000 5.7316 0.3280 0.8660 1.0494 -0.5247 gaussseidel methoddidnotconverged x = 5.7316 0.3280 0.8660 1.0494 -0.5247 >>p.6.13>> a=[1 -2 0 0 0;-2 6 4 0 0;0 4 9 -0.5 0;0 0 -0.5 1.25 0.5;0 0 0 0.5 3.25] a = 1.0000 -2.0000 0 0 0 -2.0000 6.0000 4.0000 0 0 0 4.0000 9.0000 -0.5000 0 0 0 -0.5000 1.2500 0.5000 0 0 0 0.5000 3.2500 >> b=[5;-2;18;0.5;-2.25] b = 5.0000 -2.0000 18.0000 0.5000 -2.2500 >> max_it=260 max_it= 260 >> x0=[0;0;0;0;0]
  • 58.
    x0 = 0 0 0 0 0 >> >> max_it=1000 max_it= 1000 >>x=seidel(a,b,x0,tol,max_it) i x1 x2 x3 1.0000 5.0000 1.3333 1.4074 0.9630 -0.8405 2.0000 7.6667 1.2840 1.4829 1.3293 -0.8968 3.0000 7.5679 1.2007 1.5402 1.3748 -0.9038 4.0000 7.4015 1.1070 1.5844 1.3953 -0.9070 5.0000 7.2141 1.0151 1.6264 1.4133 -0.9097 6.0000 7.0302 0.9258 1.6670 1.4307 -0.9124 gaussseidel methodconverged at260th iteration x = 1.0028 -1.9986 2.9994 1.9997 -1.0000 p.6.14 > a=[1 -2 0 0 0 0;-2 6 4 0 0 0;0 4 9 -0.5 0 0;0 0 -0.5 3.25 1.5 0;0 0 0 1.5 1.75 -3;0 0 0 0 -3 13]
  • 59.
    a = 1.0000 -2.00000 0 0 0 -2.0000 6.0000 4.0000 0 0 0 0 4.0000 9.0000 -0.5000 0 0 0 0 -0.5000 3.2500 1.5000 0 0 0 0 1.5000 1.7500 -3.0000 0 0 0 0 -3.0000 13.0000 >> b=[-3;22;35.5;-7.75;4;-33] b = -3.0000 22.0000 35.5000 -7.7500 4.0000 -33.0000 >> x0=[0;0;0;0;0;0] x0 = 0 0 0 0 0 0 >> x=seidel(a,b,x0,tol,300) i x1 x2 x3 1.0000 -3.0000 2.6667 2.7593 -1.9601 3.9658 -1.6233 2.0000 2.3333 2.6049 2.6778 -3.8030 2.7627 -1.9009 3.0000 2.2099 2.6181 2.5696 -3.2644 1.8250 -2.1173 4.0000 2.2362 2.6990 2.5635 -2.8326 1.0840 -2.2883
  • 60.
    5.0000 2.3980 2.75702.5618 -2.4908 0.4979 -2.4236 6.0000 2.5140 2.7968 2.5630 -2.2201 0.0339 -2.5306 gaussseidel methodconvergedat234th iteration x = 1.0029 2.0014 2.9993 -1.0003 -1.9995 -2.9999 Jacobi method P.6.1 a=[10 -2 1;-2 10 -2;-2 -5 10] a = 10 -2 1 -2 10 -2 -2 -5 10 >> b=[9;12;18] b = 9 12 18 >> xo=[0;0;0] xo= 0 0 0 >> x=jacobi(a,b,xo,tol,20)
  • 61.
    i x1 x2x3 1.0000 0.9000 1.2000 1.8000 2.0000 0.9600 1.7400 2.5800 3.0000 0.9900 1.9080 2.8620 4.0000 0.9954 1.9704 2.9520 5.0000 0.9989 1.9895 2.9843 6.0000 0.9995 1.9966 2.9945 7.0000 0.9999 1.9988 2.9982 8.0000 0.9999 1.9996 2.9994 9.0000 1.0000 1.9999 2.9998 10.0000 1.0000 2.0000 2.9999 jacobi methodhasconverged x = 1.0000 2.0000 3.0000 P.6.2 > a=[8 1 -1;-1 7 -2;2 1 9] a = 8 1 -1 -1 7 -2 2 1 9 >> b=[8;4;12] b =
  • 62.
    8 4 12 >> x=jacobi(a,b,xo,tol,20) i x1x2 x3 1.0000 1.0000 0.5714 1.3333 2.0000 1.0952 1.0952 1.0476 3.0000 0.9940 1.0272 0.9683 4.0000 0.9926 0.9901 0.9983 5.0000 1.0010 0.9985 1.0027 6.0000 1.0005 1.0009 0.9999 7.0000 0.9999 1.0001 0.9998 8.0000 1.0000 0.9999 1.0000 jacobi methodhasconverged x = 1.0000 1.0000 1.0000 >>p6.3 b=[9;4;-6] b = 9 4 -6 >> a=[5 -1 0;-1 5 -1;0 -1 5]
  • 63.
    a = 5 -10 -1 5 -1 0 -1 5 >> x=jacobi(a,b,xo,tol,20) i x1 x2 x3 1.0000 1.8000 0.8000 -1.2000 2.0000 1.9600 0.9200 -1.0400 3.0000 1.9840 0.9840 -1.0160 4.0000 1.9968 0.9936 -1.0032 5.0000 1.9987 0.9987 -1.0013 6.0000 1.9997 0.9995 -1.0003 7.0000 1.9999 0.9999 -1.0001 8.0000 2.0000 1.0000 -1.0000 jacobi methodhasconverged x = 2.0000 1.0000 -1.0000 >> >>p6.4 a=[4 1 0;1 3 -1;1 0 2] a = 4 1 0 1 3 -1 1 0 2
  • 64.
    >> b=[3;-4;5] b = 3 -4 5 >>x=jacobi(a,b,xo,tol,20) i x1 x2 x3 1.0000 0.7500 -1.3333 2.5000 2.0000 1.0833 -0.7500 2.1250 3.0000 0.9375 -0.9861 1.9583 4.0000 0.9965 -0.9931 2.0313 5.0000 0.9983 -0.9884 2.0017 6.0000 0.9971 -0.9988 2.0009 7.0000 0.9997 -0.9987 2.0014 8.0000 0.9997 -0.9994 2.0001 9.0000 0.9999 -0.9998 2.0002 10.0000 1.0000 -0.9999 2.0001 jacobi methodhasconverged x = 1.0000 -1.0000 2.0000 >> P6.5 a=[4 1 0;1 3 -1;0 -1 4]
  • 65.
    a = 4 10 1 3 -1 0 -1 4 >> b=[3;4;5] b = 3 4 5 >> x=jacobi(a,b,xo,tol,20) i x1 x2 x3 1.0000 0.7500 1.3333 1.2500 2.0000 0.4167 1.5000 1.5833 3.0000 0.3750 1.7222 1.6250 4.0000 0.3194 1.7500 1.6806 5.0000 0.3125 1.7870 1.6875 6.0000 0.3032 1.7917 1.6968 7.0000 0.3021 1.7978 1.6979 8.0000 0.3005 1.7986 1.6995 9.0000 0.3003 1.7996 1.6997 10.0000 0.3001 1.7998 1.6999 11.0000 0.3001 1.7999 1.6999 jacobi methodhasconverged x =
  • 66.
    0.3000 1.8000 1.7000 >> P6.6 b=[-1;0;0;0] b = -1 0 0 0 >> a=[-21 0 0;1 -2 1 0;0 1 -2 1;0 0 1 -2] a = -2 1 0 0 1 -2 1 0 0 1 -2 1 0 0 1 -2 >> x0=[0;0;0;0] x0 = 0 0 0 0 x=jacobi(a,b,x0,tol,max_it) i x1 x2 x3 x4 1.0000 0.5000 0 0 0 2.0000 0.5000 0.2500 0 0 3.0000 0.6250 0.2500 0.1250 0 4.0000 0.6250 0.3750 0.1250 0.0625
  • 67.
    5.0000 0.6875 0.37500.2188 0.0625 6.0000 0.6875 0.4531 0.2188 0.1094 7.0000 0.7266 0.4531 0.2813 0.1094 jacobi methoddidnotconverged resultaftermax no. of iterations x = 0.7266 0.4531 0.2813 0.1094 >> P6.7 =[5 1 0 0;1 5 1 0;0 1 5 1;0 0 1 5] a = 5 1 0 0 1 5 1 0 0 1 5 1 0 0 1 5 >> b=[33;26;30;15] b = 33 26 30 15 >> x0=[0;0;0;0] x0 = 0
  • 68.
    0 0 0 x=jacobi(a,b,x0,tol,20) i x1 x2x3 x4 1.0000 6.6000 5.2000 6.0000 3.0000 2.0000 5.5600 2.6800 4.3600 1.8000 3.0000 6.0640 3.2160 5.1040 2.1280 4.0000 5.9568 2.9664 4.9312 1.9792 5.0000 6.0067 3.0224 5.0109 2.0138 6.0000 5.9955 2.9965 4.9928 1.9978 7.0000 6.0007 3.0023 5.0011 2.0014 8.0000 5.9995 2.9996 4.9992 1.9998 9.0000 6.0001 3.0002 5.0001 2.0002 jacobi methodhasconverged x = 6.0000 3.0000 4.9999 2.0000 >>p6.8 a=[1 2 0 0;2 6 8 0;0 8 35 18] a = 1 2 0 0 2 6 8 0 0 8 35 18 >> b=[2;6;-10;-112]
  • 69.
    b = 2 6 -10 -112 >> x=jacobi(a,b,x0,tol,10) i x1x2 x3 x4 1.0000 2.0000 1.0000 -0.2857 -1.0000 2.0000 0 0.7143 0 -0.9541 3.0000 0.5714 1.0000 0.0417 -1.0000 4.0000 0 0.7539 0 -1.0067 5.0000 0.4921 1.0000 0.0597 -1.0000 6.0000 0 0.7564 0 -1.0096 7.0000 0.4873 1.0000 0.0606 -1.0000 8.0000 0 0.7568 0 -1.0097 9.0000 0.4865 1.0000 0.0606 -1.0000 10.0000 0 0.7570 0 -1.0097 jacobi methoddidnotconverged resultaftermax no. of iterations x = 0 0.7570 0 -1.0097
  • 70.
    P6.9 a=[1 -2 00;-2 5 -1 0;0 -1 2 -0.5;0 0 -0.5 1.25] a = 1.0000 -2.0000 0 0 -2.0000 5.0000 -1.0000 0 0 -1.0000 2.0000 -0.5000 0 0 -0.5000 1.2500 >> b=[-3;5;2;3.5] b = -3.0000 5.0000 2.0000 3.5000 >> tol=0.001 tol = 1.0000e-003 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> x=jacobi(a,b,x0,tol,200) i x1 x2 x3 x4 1.0000 -3.0000 1.0000 1.0000 2.8000 2.0000 -1.0000 -0.0000 2.2000 3.2000 3.0000 -3.0000 1.0400 1.8000 3.6800
  • 71.
    4.0000 -0.9200 0.16002.4400 3.5200 5.0000 -2.6800 1.1200 1.9600 3.7760 6.0000 -0.7600 0.3200 2.5040 3.5840 jacobi methodhasconverged at172nd iteration x = 0.9983 1.9996 2.9995 3.9999 >>p6.10 a=[4 -8 0 0;-8 18 -2 0;0 -2 5 -1.5;0 0 -1.5 1.75] a = 4.0000 -8.0000 0 0 -8.0000 18.0000 -2.0000 0 0 -2.0000 5.0000 -1.5000 0 0 -1.5000 1.7500 >> x0=[0;0;0;0] x0 = 0 0 0 0 >> tol=0.001 tol = 1.0000e-003 >> b=[-12;22;5;2]
  • 72.
    b = -12 22 5 2 x=jacobi(a,b,x0,tol,500) i x1x2 x3 x4 1.0000 -3.0000 1.2222 1.0000 1.1429 2.0000 -0.5556 0.0000 1.8317 2.0000 3.0000 -3.0000 1.1788 1.6000 2.7129 4.0000 -0.6423 0.0667 2.2854 2.5143 5.0000 -2.8667 1.1907 1.7810 3.1018 6.0000 -0.6186 0.1460 2.4068 2.6694 jacobi methodhasconverged at310th iteration x = 0.4987 1.7498 2.7496 3.4999 >> P6.11 b=[8;18;0.5;-1.75] b = 8.0000 18.0000 0.5000 -1.7500 >> tol=0.001
  • 73.
    tol = 1.0000e-003 >> x0=[0;0;0;0] x0= 0 0 0 0 >> a=[4 8 0 0;8 18 2 0;0 2 5 1.5;0 0 1.5 1.75] a = 4.0000 8.0000 0 0 8.0000 18.0000 2.0000 0 0 2.0000 5.0000 1.5000 0 0 1.5000 1.7500 >> x=jacobi(a,b,x0,tol,500) i x1 x2 x3 x4 1.0000 2.0000 1.0000 0.1000 -1.0000 2.0000 0 0.1000 -0.0000 -1.0857 3.0000 1.8000 1.0000 0.3857 -1.0000 4.0000 0 0.1571 -0.0000 -1.3306 5.0000 1.6857 1.0000 0.4363 -1.0000 6.0000 0 0.2023 -0.0000 -1.3740 jacobi methodhasconverge at 300th iteration x =
  • 74.
    0.0008 1.0000 0.0002 -1.0000 >>p6.12 a=[1 -2 00 0;-2 5 1 0 0;0 1 2 -2 0;0 0 -2 5 1;0 0 0 1 2] a = 1 -2 0 0 0 -2 5 1 0 0 0 1 2 -2 0 0 0 -2 5 1 0 0 0 1 2 >> b=[5;-9;0;3;0] b = 5 -9 0 3 0 x=jacobi(a,b,x0,tol,1000) i x1 x2 x3 x4 1.0000 5.0000 -1.8000 0 0.6000 0 2.0000 1.4000 0.2000 1.5000 0.6000 -0.3000 3.0000 5.4000 -1.5400 0.5000 1.2600 -0.3000 4.0000 1.9200 0.2600 2.0300 0.8600 -0.6300 5.0000 5.5200 -1.4380 0.7300 1.5380 -0.4300 6.0000 2.1240 0.2620 2.2570 0.9780 -0.7690 jacobi methodhasconverged at968th iteration
  • 75.
    x = 1.0011 -1.9998 2.9995 1.9999 -0.9999 p.6.13 >> a=[1-2 0 0 0;-2 6 4 0 0;0 4 9 -0.5 0;0 0 -0.5 1.25 0.5;0 0 0 0.5 3.25] a = 1.0000 -2.0000 0 0 0 -2.0000 6.0000 4.0000 0 0 0 4.0000 9.0000 -0.5000 0 0 0 -0.5000 1.2500 0.5000 0 0 0 0.5000 3.2500 >> b=[5;-2;18;0.5;-2.25] b = 5.0000 -2.0000 18.0000 0.5000 -2.2500. x=jacobi(a,b,x0,tol,500) i x1 x2 x3 x4 1.0000 5.0000 -0.3333 2.0000 0.4000 -0.6923 2.0000 4.3333 -0.0000 2.1704 1.4769 -0.7538 3.0000 5.0000 -0.3358 2.0821 1.5697 -0.9195 4.0000 4.3284 -0.0547 2.2365 1.6006 -0.9338 5.0000 4.8906 -0.3815 2.1132 1.6681 -0.9386 6.0000 4.2370 -0.1120 2.2622 1.6207 -0.9489
  • 76.
    jacobi methodhasconverged at436th iteration x= 1.0059 -1.9975 2.9987 1.9995 -0.9999 >>p6.14 a=[1 -2 0 0 0 0;-2 6 4 0 0 0;0 4 9 -0.5 0 0;0 0 -0.5 3.25 1.5 0;0 0 0 1.5 1.75 -3;0 0 0 0 -3 13] a = 1.0000 -2.0000 0 0 0 0 -2.0000 6.0000 4.0000 0 0 0 0 4.0000 9.0000 -0.5000 0 0 0 0 -0.5000 3.2500 1.5000 0 0 0 0 1.5000 1.7500 -3.0000 0 0 0 0 -3.0000 13.0000 >> b=[-3;22;35.5;-7.75;4;-33] b = -3.0000 22.0000 35.5000 -7.7500 4.0000 -33.0000 >> x0=[0;0;0;0;0;0] x0 = 0 0 0 0
  • 77.
    0 0 >> x=jacobi(a,b,x0,tol,500) i x1x2 x3 x4 1.0000 -3.0000 3.6667 3.9444 -2.3846 2.2857 -2.5385 2.0000 4.3333 0.0370 2.1823 -2.8327 -0.0220 -2.0110 3.0000 -2.9259 3.6562 3.7706 -2.0387 1.2664 -2.5435 4.0000 4.3124 0.1776 2.2062 -2.3890 -0.3271 -2.2462 5.0000 -2.6448 3.6334 3.7328 -1.8942 0.4827 -2.6140 6.0000 4.2667 0.2966 2.2244 -2.0331 -0.5717 -2.4271 jacobi methoddidnotconverged resultaftermax no. of iterations x = 1.0028 1.9989 2.9994 -0.9997 -1.9995 -3.0001 jacobi methodhasconverged at622nd iteration x = 0.9996 2.0002 3.0001 -1.0001 -2.0001 -3.0000 p.6.18
  • 78.
    >> a=[10 01 0 0 0 0 0;0 10 0 0 0 0 -1 0;0 0 10 0 0 -2 0 0;2 0 0 10 0 0 0 0;0 0 1 0 10 0 0 0;0 0 0 -3 0 10 0 0;0 3 0 0 0 0 10 0;0 0 0 0 1 0 0 10] a = 10 0 1 0 0 0 0 0 0 10 0 0 0 0 -1 0 0 0 10 0 0 -2 0 0 2 0 0 10 0 0 0 0 0 0 1 0 10 0 0 0 0 0 0 -3 0 10 0 0 0 3 0 0 0 0 10 0 0 0 0 0 1 0 0 10 > b=[13;13;18;42;53;48;76;85] b = 13 13 18 42 53 48 76 85 >> x0=[0;0;0;0;0;0;0;0] x0 = 0 0 0 0 0 0 0 0 >> x=jacobi(a,b,x0,tol,1000) i x1 x2 x3 x4 1.0000 1.3000 1.3000 1.8000 4.2000 5.3000 4.8000 7.6000 8.5000
  • 79.
    2.0000 1.1200 2.06002.7600 3.9400 5.1200 6.0600 7.2100 7.9700 3.0000 1.0240 2.0210 3.0120 3.9760 5.0240 5.9820 6.9820 7.9880 4.0000 0.9988 1.9982 2.9964 3.9952 4.9988 5.9928 6.9937 7.9976 5.0000 1.0004 1.9994 2.9986 4.0002 5.0004 5.9986 7.0005 8.0001 6.0000 1.0001 2.0001 2.9997 3.9999 5.0001 6.0001 7.0002 8.0000 jacobi methodhasconverged x = 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000