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The Bisection Method

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** Gate 2013 Question Papers.. ** CEED 2013 Results.. ** Gate 2013 Question Papers With Solutions.. ** GATE 2013 CUT-OFFs.. ** GATE 2013 Results.. **

The Bisection Method

Example.

Find all the real solutions to the cubic equation[Graphics:Images/BisectionMod_gr_12.gif].

Solution.

[Graphics:../Images/BisectionMod_gr_13.gif]

[Graphics:../Images/BisectionMod_gr_14.gif]

Plot the function.

[Graphics:../Images/BisectionMod_gr_15.gif]

[Graphics:../Images/BisectionMod_gr_16.gif]

[Graphics:../Images/BisectionMod_gr_17.gif]

There appears to be only one real root which lies in the interval [1,2].

[Graphics:../Images/BisectionMod_gr_18.gif]

[Graphics:../Images/BisectionMod_gr_19.gif]

[Graphics:../Images/BisectionMod_gr_20.gif]

Call the Bisection subroutine on the interval [1,2] using 10 iterations

[Graphics:../Images/BisectionMod_gr_21.gif]

k [Graphics:../Images/BisectionMod_gr_22.gif] [Graphics:../Images/BisectionMod_gr_23.gif] [Graphics:../Images/BisectionMod_gr_24.gif] [Graphics:../Images/BisectionMod_gr_25.gif]
[Graphics:../Images/BisectionMod_gr_26.gif] [Graphics:../Images/BisectionMod_gr_27.gif] [Graphics:../Images/BisectionMod_gr_28.gif] [Graphics:../Images/BisectionMod_gr_29.gif] [Graphics:../Images/BisectionMod_gr_30.gif]
[Graphics:../Images/BisectionMod_gr_31.gif] [Graphics:../Images/BisectionMod_gr_32.gif] [Graphics:../Images/BisectionMod_gr_33.gif] [Graphics:../Images/BisectionMod_gr_34.gif] [Graphics:../Images/BisectionMod_gr_35.gif]
[Graphics:../Images/BisectionMod_gr_36.gif] [Graphics:../Images/BisectionMod_gr_37.gif] [Graphics:../Images/BisectionMod_gr_38.gif] [Graphics:../Images/BisectionMod_gr_39.gif] [Graphics:../Images/BisectionMod_gr_40.gif]
[Graphics:../Images/BisectionMod_gr_41.gif] [Graphics:../Images/BisectionMod_gr_42.gif] [Graphics:../Images/BisectionMod_gr_43.gif] [Graphics:../Images/BisectionMod_gr_44.gif] [Graphics:../Images/BisectionMod_gr_45.gif]
[Graphics:../Images/BisectionMod_gr_46.gif] [Graphics:../Images/BisectionMod_gr_47.gif] [Graphics:../Images/BisectionMod_gr_48.gif] [Graphics:../Images/BisectionMod_gr_49.gif] [Graphics:../Images/BisectionMod_gr_50.gif]
[Graphics:../Images/BisectionMod_gr_51.gif] [Graphics:../Images/BisectionMod_gr_52.gif] [Graphics:../Images/BisectionMod_gr_53.gif] [Graphics:../Images/BisectionMod_gr_54.gif] [Graphics:../Images/BisectionMod_gr_55.gif]
[Graphics:../Images/BisectionMod_gr_56.gif] [Graphics:../Images/BisectionMod_gr_57.gif] [Graphics:../Images/BisectionMod_gr_58.gif] [Graphics:../Images/BisectionMod_gr_59.gif] [Graphics:../Images/BisectionMod_gr_60.gif]
[Graphics:../Images/BisectionMod_gr_61.gif] [Graphics:../Images/BisectionMod_gr_62.gif] [Graphics:../Images/BisectionMod_gr_63.gif] [Graphics:../Images/BisectionMod_gr_64.gif] [Graphics:../Images/BisectionMod_gr_65.gif]
[Graphics:../Images/BisectionMod_gr_66.gif] [Graphics:../Images/BisectionMod_gr_67.gif] [Graphics:../Images/BisectionMod_gr_68.gif] [Graphics:../Images/BisectionMod_gr_69.gif] [Graphics:../Images/BisectionMod_gr_70.gif]
[Graphics:../Images/BisectionMod_gr_71.gif] [Graphics:../Images/BisectionMod_gr_72.gif] [Graphics:../Images/BisectionMod_gr_73.gif] [Graphics:../Images/BisectionMod_gr_74.gif] [Graphics:../Images/BisectionMod_gr_75.gif]
[Graphics:../Images/BisectionMod_gr_76.gif] [Graphics:../Images/BisectionMod_gr_77.gif] [Graphics:../Images/BisectionMod_gr_78.gif] [Graphics:../Images/BisectionMod_gr_79.gif] [Graphics:../Images/BisectionMod_gr_80.gif] 

[Graphics:../Images/BisectionMod_gr_81.gif]
[Graphics:../Images/BisectionMod_gr_82.gif]
[Graphics:../Images/BisectionMod_gr_83.gif]

After 10 iterations, the interval has been reduced to [a,b] where

[Graphics:../Images/BisectionMod_gr_84.gif]

[Graphics:../Images/BisectionMod_gr_85.gif]
[Graphics:../Images/BisectionMod_gr_86.gif]
[Graphics:../Images/BisectionMod_gr_87.gif]
[Graphics:../Images/BisectionMod_gr_88.gif]

The root lies somewhere in the interval [a,b] the width of which is

[Graphics:../Images/BisectionMod_gr_89.gif]

[Graphics:../Images/BisectionMod_gr_90.gif]

The reported root is alleged to be

[Graphics:../Images/BisectionMod_gr_91.gif]

[Graphics:../Images/BisectionMod_gr_92.gif]

The accuracy we can guarantee is one half of the interval width.

[Graphics:../Images/BisectionMod_gr_93.gif]

[Graphics:../Images/BisectionMod_gr_94.gif]

Is this the desired accuracy you want ?If not, more iterations are required.

[Graphics:../Images/BisectionMod_gr_95.gif]

k [Graphics:../Images/BisectionMod_gr_96.gif] [Graphics:../Images/BisectionMod_gr_97.gif] [Graphics:../Images/BisectionMod_gr_98.gif] [Graphics:../Images/BisectionMod_gr_99.gif]
[Graphics:../Images/BisectionMod_gr_100.gif] [Graphics:../Images/BisectionMod_gr_101.gif] [Graphics:../Images/BisectionMod_gr_102.gif] [Graphics:../Images/BisectionMod_gr_103.gif] [Graphics:../Images/BisectionMod_gr_104.gif]
[Graphics:../Images/BisectionMod_gr_105.gif] [Graphics:../Images/BisectionMod_gr_106.gif] [Graphics:../Images/BisectionMod_gr_107.gif] [Graphics:../Images/BisectionMod_gr_108.gif] [Graphics:../Images/BisectionMod_gr_109.gif]
[Graphics:../Images/BisectionMod_gr_110.gif] [Graphics:../Images/BisectionMod_gr_111.gif] [Graphics:../Images/BisectionMod_gr_112.gif] [Graphics:../Images/BisectionMod_gr_113.gif] [Graphics:../Images/BisectionMod_gr_114.gif]
[Graphics:../Images/BisectionMod_gr_115.gif] [Graphics:../Images/BisectionMod_gr_116.gif] [Graphics:../Images/BisectionMod_gr_117.gif] [Graphics:../Images/BisectionMod_gr_118.gif] [Graphics:../Images/BisectionMod_gr_119.gif]
[Graphics:../Images/BisectionMod_gr_120.gif] [Graphics:../Images/BisectionMod_gr_121.gif] [Graphics:../Images/BisectionMod_gr_122.gif] [Graphics:../Images/BisectionMod_gr_123.gif] [Graphics:../Images/BisectionMod_gr_124.gif]
[Graphics:../Images/BisectionMod_gr_125.gif] [Graphics:../Images/BisectionMod_gr_126.gif] [Graphics:../Images/BisectionMod_gr_127.gif] [Graphics:../Images/BisectionMod_gr_128.gif] [Graphics:../Images/BisectionMod_gr_129.gif]
[Graphics:../Images/BisectionMod_gr_130.gif] [Graphics:../Images/BisectionMod_gr_131.gif] [Graphics:../Images/BisectionMod_gr_132.gif] [Graphics:../Images/BisectionMod_gr_133.gif] [Graphics:../Images/BisectionMod_gr_134.gif]
[Graphics:../Images/BisectionMod_gr_135.gif] [Graphics:../Images/BisectionMod_gr_136.gif] [Graphics:../Images/BisectionMod_gr_137.gif] [Graphics:../Images/BisectionMod_gr_138.gif] [Graphics:../Images/BisectionMod_gr_139.gif]
[Graphics:../Images/BisectionMod_gr_140.gif] [Graphics:../Images/BisectionMod_gr_141.gif] [Graphics:../Images/BisectionMod_gr_142.gif] [Graphics:../Images/BisectionMod_gr_143.gif] [Graphics:../Images/BisectionMod_gr_144.gif]
[Graphics:../Images/BisectionMod_gr_145.gif] [Graphics:../Images/BisectionMod_gr_146.gif] [Graphics:../Images/BisectionMod_gr_147.gif] [Graphics:../Images/BisectionMod_gr_148.gif] [Graphics:../Images/BisectionMod_gr_149.gif]
[Graphics:../Images/BisectionMod_gr_150.gif] [Graphics:../Images/BisectionMod_gr_151.gif] [Graphics:../Images/BisectionMod_gr_152.gif] [Graphics:../Images/BisectionMod_gr_153.gif] [Graphics:../Images/BisectionMod_gr_154.gif]
[Graphics:../Images/BisectionMod_gr_155.gif] [Graphics:../Images/BisectionMod_gr_156.gif] [Graphics:../Images/BisectionMod_gr_157.gif] [Graphics:../Images/BisectionMod_gr_158.gif] [Graphics:../Images/BisectionMod_gr_159.gif]
[Graphics:../Images/BisectionMod_gr_160.gif] [Graphics:../Images/BisectionMod_gr_161.gif] [Graphics:../Images/BisectionMod_gr_162.gif] [Graphics:../Images/BisectionMod_gr_163.gif] [Graphics:../Images/BisectionMod_gr_164.gif]
[Graphics:../Images/BisectionMod_gr_165.gif] [Graphics:../Images/BisectionMod_gr_166.gif] [Graphics:../Images/BisectionMod_gr_167.gif] [Graphics:../Images/BisectionMod_gr_168.gif] [Graphics:../Images/BisectionMod_gr_169.gif]
[Graphics:../Images/BisectionMod_gr_170.gif] [Graphics:../Images/BisectionMod_gr_171.gif] [Graphics:../Images/BisectionMod_gr_172.gif] [Graphics:../Images/BisectionMod_gr_173.gif] [Graphics:../Images/BisectionMod_gr_174.gif]
[Graphics:../Images/BisectionMod_gr_175.gif] [Graphics:../Images/BisectionMod_gr_176.gif] [Graphics:../Images/BisectionMod_gr_177.gif] [Graphics:../Images/BisectionMod_gr_178.gif] [Graphics:../Images/BisectionMod_gr_179.gif]
[Graphics:../Images/BisectionMod_gr_180.gif] [Graphics:../Images/BisectionMod_gr_181.gif] [Graphics:../Images/BisectionMod_gr_182.gif] [Graphics:../Images/BisectionMod_gr_183.gif] [Graphics:../Images/BisectionMod_gr_184.gif]
[Graphics:../Images/BisectionMod_gr_185.gif] [Graphics:../Images/BisectionMod_gr_186.gif] [Graphics:../Images/BisectionMod_gr_187.gif] [Graphics:../Images/BisectionMod_gr_188.gif] [Graphics:../Images/BisectionMod_gr_189.gif]
[Graphics:../Images/BisectionMod_gr_190.gif] [Graphics:../Images/BisectionMod_gr_191.gif] [Graphics:../Images/BisectionMod_gr_192.gif] [Graphics:../Images/BisectionMod_gr_193.gif] [Graphics:../Images/BisectionMod_gr_194.gif]
[Graphics:../Images/BisectionMod_gr_195.gif] [Graphics:../Images/BisectionMod_gr_196.gif] [Graphics:../Images/BisectionMod_gr_197.gif] [Graphics:../Images/BisectionMod_gr_198.gif] [Graphics:../Images/BisectionMod_gr_199.gif]
[Graphics:../Images/BisectionMod_gr_200.gif] [Graphics:../Images/BisectionMod_gr_201.gif] [Graphics:../Images/BisectionMod_gr_202.gif] [Graphics:../Images/BisectionMod_gr_203.gif] [Graphics:../Images/BisectionMod_gr_204.gif]

[Graphics:../Images/BisectionMod_gr_205.gif]

[Graphics:../Images/BisectionMod_gr_206.gif]
[Graphics:../Images/BisectionMod_gr_207.gif]

Compare our result with Mathematica's built in root finder.

[Graphics:../Images/BisectionMod_gr_208.gif]

[Graphics:../Images/BisectionMod_gr_209.gif]
[Graphics:../Images/BisectionMod_gr_210.gif]
[Graphics:../Images/BisectionMod_gr_211.gif]

Question.Why is Mathematica's answer different ?
How many bisections would it take to reduce the interval width to [Graphics:../Images/BisectionMod_gr_212.gif] ?

[Graphics:../Images/BisectionMod_gr_213.gif]

k [Graphics:../Images/BisectionMod_gr_214.gif] [Graphics:../Images/BisectionMod_gr_215.gif] [Graphics:../Images/BisectionMod_gr_216.gif] [Graphics:../Images/BisectionMod_gr_217.gif]
[Graphics:../Images/BisectionMod_gr_218.gif] [Graphics:../Images/BisectionMod_gr_219.gif] [Graphics:../Images/BisectionMod_gr_220.gif] [Graphics:../Images/BisectionMod_gr_221.gif] [Graphics:../Images/BisectionMod_gr_222.gif]
[Graphics:../Images/BisectionMod_gr_223.gif] [Graphics:../Images/BisectionMod_gr_224.gif] [Graphics:../Images/BisectionMod_gr_225.gif] [Graphics:../Images/BisectionMod_gr_226.gif] [Graphics:../Images/BisectionMod_gr_227.gif]
[Graphics:../Images/BisectionMod_gr_228.gif] [Graphics:../Images/BisectionMod_gr_229.gif] [Graphics:../Images/BisectionMod_gr_230.gif] [Graphics:../Images/BisectionMod_gr_231.gif] [Graphics:../Images/BisectionMod_gr_232.gif]
[Graphics:../Images/BisectionMod_gr_233.gif] [Graphics:../Images/BisectionMod_gr_234.gif] [Graphics:../Images/BisectionMod_gr_235.gif] [Graphics:../Images/BisectionMod_gr_236.gif] [Graphics:../Images/BisectionMod_gr_237.gif]
[Graphics:../Images/BisectionMod_gr_238.gif] [Graphics:../Images/BisectionMod_gr_239.gif] [Graphics:../Images/BisectionMod_gr_240.gif] [Graphics:../Images/BisectionMod_gr_241.gif] [Graphics:../Images/BisectionMod_gr_242.gif]
[Graphics:../Images/BisectionMod_gr_243.gif] [Graphics:../Images/BisectionMod_gr_244.gif] [Graphics:../Images/BisectionMod_gr_245.gif] [Graphics:../Images/BisectionMod_gr_246.gif] [Graphics:../Images/BisectionMod_gr_247.gif]
[Graphics:../Images/BisectionMod_gr_248.gif] [Graphics:../Images/BisectionMod_gr_249.gif] [Graphics:../Images/BisectionMod_gr_250.gif] [Graphics:../Images/BisectionMod_gr_251.gif] [Graphics:../Images/BisectionMod_gr_252.gif]
[Graphics:../Images/BisectionMod_gr_253.gif] [Graphics:../Images/BisectionMod_gr_254.gif] [Graphics:../Images/BisectionMod_gr_255.gif] [Graphics:../Images/BisectionMod_gr_256.gif] [Graphics:../Images/BisectionMod_gr_257.gif]
[Graphics:../Images/BisectionMod_gr_258.gif] [Graphics:../Images/BisectionMod_gr_259.gif] [Graphics:../Images/BisectionMod_gr_260.gif] [Graphics:../Images/BisectionMod_gr_261.gif] [Graphics:../Images/BisectionMod_gr_262.gif]
[Graphics:../Images/BisectionMod_gr_263.gif] [Graphics:../Images/BisectionMod_gr_264.gif] [Graphics:../Images/BisectionMod_gr_265.gif] [Graphics:../Images/BisectionMod_gr_266.gif] [Graphics:../Images/BisectionMod_gr_267.gif]
[Graphics:../Images/BisectionMod_gr_268.gif] [Graphics:../Images/BisectionMod_gr_269.gif] [Graphics:../Images/BisectionMod_gr_270.gif] [Graphics:../Images/BisectionMod_gr_271.gif] [Graphics:../Images/BisectionMod_gr_272.gif]
[Graphics:../Images/BisectionMod_gr_273.gif] [Graphics:../Images/BisectionMod_gr_274.gif] [Graphics:../Images/BisectionMod_gr_275.gif] [Graphics:../Images/BisectionMod_gr_276.gif] [Graphics:../Images/BisectionMod_gr_277.gif]
[Graphics:../Images/BisectionMod_gr_278.gif] [Graphics:../Images/BisectionMod_gr_279.gif] [Graphics:../Images/BisectionMod_gr_280.gif] [Graphics:../Images/BisectionMod_gr_281.gif] [Graphics:../Images/BisectionMod_gr_282.gif]
[Graphics:../Images/BisectionMod_gr_283.gif] [Graphics:../Images/BisectionMod_gr_284.gif] [Graphics:../Images/BisectionMod_gr_285.gif] [Graphics:../Images/BisectionMod_gr_286.gif] [Graphics:../Images/BisectionMod_gr_287.gif]
[Graphics:../Images/BisectionMod_gr_288.gif] [Graphics:../Images/BisectionMod_gr_289.gif] [Graphics:../Images/BisectionMod_gr_290.gif] [Graphics:../Images/BisectionMod_gr_291.gif] [Graphics:../Images/BisectionMod_gr_292.gif]
[Graphics:../Images/BisectionMod_gr_293.gif] [Graphics:../Images/BisectionMod_gr_294.gif] [Graphics:../Images/BisectionMod_gr_295.gif] [Graphics:../Images/BisectionMod_gr_296.gif] [Graphics:../Images/BisectionMod_gr_297.gif]
[Graphics:../Images/BisectionMod_gr_298.gif] [Graphics:../Images/BisectionMod_gr_299.gif] [Graphics:../Images/BisectionMod_gr_300.gif] [Graphics:../Images/BisectionMod_gr_301.gif] [Graphics:../Images/BisectionMod_gr_302.gif]
[Graphics:../Images/BisectionMod_gr_303.gif] [Graphics:../Images/BisectionMod_gr_304.gif] [Graphics:../Images/BisectionMod_gr_305.gif] [Graphics:../Images/BisectionMod_gr_306.gif] [Graphics:../Images/BisectionMod_gr_307.gif]
[Graphics:../Images/BisectionMod_gr_308.gif] [Graphics:../Images/BisectionMod_gr_309.gif] [Graphics:../Images/BisectionMod_gr_310.gif] [Graphics:../Images/BisectionMod_gr_311.gif] [Graphics:../Images/BisectionMod_gr_312.gif]
[Graphics:../Images/BisectionMod_gr_313.gif] [Graphics:../Images/BisectionMod_gr_314.gif] [Graphics:../Images/BisectionMod_gr_315.gif] [Graphics:../Images/BisectionMod_gr_316.gif] [Graphics:../Images/BisectionMod_gr_317.gif]
[Graphics:../Images/BisectionMod_gr_318.gif] [Graphics:../Images/BisectionMod_gr_319.gif] [Graphics:../Images/BisectionMod_gr_320.gif] [Graphics:../Images/BisectionMod_gr_321.gif] [Graphics:../Images/BisectionMod_gr_322.gif]
[Graphics:../Images/BisectionMod_gr_323.gif] [Graphics:../Images/BisectionMod_gr_324.gif] [Graphics:../Images/BisectionMod_gr_325.gif] [Graphics:../Images/BisectionMod_gr_326.gif] [Graphics:../Images/BisectionMod_gr_327.gif]
[Graphics:../Images/BisectionMod_gr_328.gif] [Graphics:../Images/BisectionMod_gr_329.gif] [Graphics:../Images/BisectionMod_gr_330.gif] [Graphics:../Images/BisectionMod_gr_331.gif] [Graphics:../Images/BisectionMod_gr_332.gif]
[Graphics:../Images/BisectionMod_gr_333.gif] [Graphics:../Images/BisectionMod_gr_334.gif] [Graphics:../Images/BisectionMod_gr_335.gif] [Graphics:../Images/BisectionMod_gr_336.gif] [Graphics:../Images/BisectionMod_gr_337.gif]
[Graphics:../Images/BisectionMod_gr_338.gif] [Graphics:../Images/BisectionMod_gr_339.gif] [Graphics:../Images/BisectionMod_gr_340.gif] [Graphics:../Images/BisectionMod_gr_341.gif] [Graphics:../Images/BisectionMod_gr_342.gif]
[Graphics:../Images/BisectionMod_gr_343.gif] [Graphics:../Images/BisectionMod_gr_344.gif] [Graphics:../Images/BisectionMod_gr_345.gif] [Graphics:../Images/BisectionMod_gr_346.gif] [Graphics:../Images/BisectionMod_gr_347.gif]
[Graphics:../Images/BisectionMod_gr_348.gif] [Graphics:../Images/BisectionMod_gr_349.gif] [Graphics:../Images/BisectionMod_gr_350.gif] [Graphics:../Images/BisectionMod_gr_351.gif] [Graphics:../Images/BisectionMod_gr_352.gif]
[Graphics:../Images/BisectionMod_gr_353.gif] [Graphics:../Images/BisectionMod_gr_354.gif] [Graphics:../Images/BisectionMod_gr_355.gif] [Graphics:../Images/BisectionMod_gr_356.gif] [Graphics:../Images/BisectionMod_gr_357.gif]
[Graphics:../Images/BisectionMod_gr_358.gif] [Graphics:../Images/BisectionMod_gr_359.gif] [Graphics:../Images/BisectionMod_gr_360.gif] [Graphics:../Images/BisectionMod_gr_361.gif] [Graphics:../Images/BisectionMod_gr_362.gif]
[Graphics:../Images/BisectionMod_gr_363.gif] [Graphics:../Images/BisectionMod_gr_364.gif] [Graphics:../Images/BisectionMod_gr_365.gif] [Graphics:../Images/BisectionMod_gr_366.gif] [Graphics:../Images/BisectionMod_gr_367.gif]
[Graphics:../Images/BisectionMod_gr_368.gif] [Graphics:../Images/BisectionMod_gr_369.gif] [Graphics:../Images/BisectionMod_gr_370.gif] [Graphics:../Images/BisectionMod_gr_371.gif] [Graphics:../Images/BisectionMod_gr_372.gif]
[Graphics:../Images/BisectionMod_gr_373.gif] [Graphics:../Images/BisectionMod_gr_374.gif] [Graphics:../Images/BisectionMod_gr_375.gif] [Graphics:../Images/BisectionMod_gr_376.gif] [Graphics:../Images/BisectionMod_gr_377.gif]
[Graphics:../Images/BisectionMod_gr_378.gif] [Graphics:../Images/BisectionMod_gr_379.gif] [Graphics:../Images/BisectionMod_gr_380.gif] [Graphics:../Images/BisectionMod_gr_381.gif] [Graphics:../Images/BisectionMod_gr_382.gif]
[Graphics:../Images/BisectionMod_gr_383.gif] [Graphics:../Images/BisectionMod_gr_384.gif] [Graphics:../Images/BisectionMod_gr_385.gif] [Graphics:../Images/BisectionMod_gr_386.gif] [Graphics:../Images/BisectionMod_gr_387.gif]
[Graphics:../Images/BisectionMod_gr_388.gif] [Graphics:../Images/BisectionMod_gr_389.gif] [Graphics:../Images/BisectionMod_gr_390.gif] [Graphics:../Images/BisectionMod_gr_391.gif] [Graphics:../Images/BisectionMod_gr_392.gif]
[Graphics:../Images/BisectionMod_gr_393.gif] [Graphics:../Images/BisectionMod_gr_394.gif] [Graphics:../Images/BisectionMod_gr_395.gif] [Graphics:../Images/BisectionMod_gr_396.gif] [Graphics:../Images/BisectionMod_gr_397.gif]
[Graphics:../Images/BisectionMod_gr_398.gif] [Graphics:../Images/BisectionMod_gr_399.gif] [Graphics:../Images/BisectionMod_gr_400.gif] [Graphics:../Images/BisectionMod_gr_401.gif] [Graphics:../Images/BisectionMod_gr_402.gif]
[Graphics:../Images/BisectionMod_gr_403.gif] [Graphics:../Images/BisectionMod_gr_404.gif] [Graphics:../Images/BisectionMod_gr_405.gif] [Graphics:../Images/BisectionMod_gr_406.gif] [Graphics:../Images/BisectionMod_gr_407.gif]
[Graphics:../Images/BisectionMod_gr_408.gif] [Graphics:../Images/BisectionMod_gr_409.gif] [Graphics:../Images/BisectionMod_gr_410.gif] [Graphics:../Images/BisectionMod_gr_411.gif] [Graphics:../Images/BisectionMod_gr_412.gif]
[Graphics:../Images/BisectionMod_gr_413.gif] [Graphics:../Images/BisectionMod_gr_414.gif] [Graphics:../Images/BisectionMod_gr_415.gif] [Graphics:../Images/BisectionMod_gr_416.gif] [Graphics:../Images/BisectionMod_gr_417.gif]
[Graphics:../Images/BisectionMod_gr_418.gif] [Graphics:../Images/BisectionMod_gr_419.gif] [Graphics:../Images/BisectionMod_gr_420.gif] [Graphics:../Images/BisectionMod_gr_421.gif] [Graphics:../Images/BisectionMod_gr_422.gif]
[Graphics:../Images/BisectionMod_gr_423.gif] [Graphics:../Images/BisectionMod_gr_424.gif] [Graphics:../Images/BisectionMod_gr_425.gif] [Graphics:../Images/BisectionMod_gr_426.gif] [Graphics:../Images/BisectionMod_gr_427.gif]
[Graphics:../Images/BisectionMod_gr_428.gif] [Graphics:../Images/BisectionMod_gr_429.gif] [Graphics:../Images/BisectionMod_gr_430.gif] [Graphics:../Images/BisectionMod_gr_431.gif] [Graphics:../Images/BisectionMod_gr_432.gif]
[Graphics:../Images/BisectionMod_gr_433.gif] [Graphics:../Images/BisectionMod_gr_434.gif] [Graphics:../Images/BisectionMod_gr_435.gif] [Graphics:../Images/BisectionMod_gr_436.gif] [Graphics:../Images/BisectionMod_gr_437.gif]
[Graphics:../Images/BisectionMod_gr_438.gif] [Graphics:../Images/BisectionMod_gr_439.gif] [Graphics:../Images/BisectionMod_gr_440.gif] [Graphics:../Images/BisectionMod_gr_441.gif] [Graphics:../Images/BisectionMod_gr_442.gif]
[Graphics:../Images/BisectionMod_gr_443.gif] [Graphics:../Images/BisectionMod_gr_444.gif] [Graphics:../Images/BisectionMod_gr_445.gif] [Graphics:../Images/BisectionMod_gr_446.gif] [Graphics:../Images/BisectionMod_gr_447.gif]
[Graphics:../Images/BisectionMod_gr_448.gif] [Graphics:../Images/BisectionMod_gr_449.gif] [Graphics:../Images/BisectionMod_gr_450.gif] [Graphics:../Images/BisectionMod_gr_451.gif] [Graphics:../Images/BisectionMod_gr_452.gif]
[Graphics:../Images/BisectionMod_gr_453.gif] [Graphics:../Images/BisectionMod_gr_454.gif] [Graphics:../Images/BisectionMod_gr_455.gif] [Graphics:../Images/BisectionMod_gr_456.gif] [Graphics:../Images/BisectionMod_gr_457.gif]
[Graphics:../Images/BisectionMod_gr_458.gif] [Graphics:../Images/BisectionMod_gr_459.gif] [Graphics:../Images/BisectionMod_gr_460.gif] [Graphics:../Images/BisectionMod_gr_461.gif] [Graphics:../Images/BisectionMod_gr_462.gif]
[Graphics:../Images/BisectionMod_gr_463.gif] [Graphics:../Images/BisectionMod_gr_464.gif] [Graphics:../Images/BisectionMod_gr_465.gif] [Graphics:../Images/BisectionMod_gr_466.gif] [Graphics:../Images/BisectionMod_gr_467.gif]
[Graphics:../Images/BisectionMod_gr_468.gif] [Graphics:../Images/BisectionMod_gr_469.gif] [Graphics:../Images/BisectionMod_gr_470.gif] [Graphics:../Images/BisectionMod_gr_471.gif] [Graphics:../Images/BisectionMod_gr_472.gif] 

[Graphics:../Images/BisectionMod_gr_473.gif]
[Graphics:../Images/BisectionMod_gr_474.gif]
[Graphics:../Images/BisectionMod_gr_475.gif]

Remember. The bisection method can only be used to find a real root in an interval [a,b] in which f[x] changes sign.


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