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The Regula Falsi Method

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The Regula Falsi Method

Example.

Find all the real solutions to the cubic equation[Graphics:Images/RegulaFalsiMod_gr_10.gif].

Solution.

[Graphics:../Images/RegulaFalsiMod_gr_11.gif] [Graphics:../Images/RegulaFalsiMod_gr_12.gif]

Plot the function.

[Graphics:../Images/RegulaFalsiMod_gr_13.gif]

[Graphics:../Images/RegulaFalsiMod_gr_14.gif]

[Graphics:../Images/RegulaFalsiMod_gr_15.gif]

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

[Graphics:../Images/RegulaFalsiMod_gr_16.gif]

[Graphics:../Images/RegulaFalsiMod_gr_17.gif]

[Graphics:../Images/RegulaFalsiMod_gr_18.gif]

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

[Graphics:../Images/RegulaFalsiMod_gr_19.gif]
k [Graphics:../Images/RegulaFalsiMod_gr_20.gif] [Graphics:../Images/RegulaFalsiMod_gr_21.gif] [Graphics:../Images/RegulaFalsiMod_gr_22.gif] [Graphics:../Images/RegulaFalsiMod_gr_23.gif]
[Graphics:../Images/RegulaFalsiMod_gr_24.gif] [Graphics:../Images/RegulaFalsiMod_gr_25.gif] [Graphics:../Images/RegulaFalsiMod_gr_26.gif] [Graphics:../Images/RegulaFalsiMod_gr_27.gif] [Graphics:../Images/RegulaFalsiMod_gr_28.gif]
[Graphics:../Images/RegulaFalsiMod_gr_29.gif] [Graphics:../Images/RegulaFalsiMod_gr_30.gif] [Graphics:../Images/RegulaFalsiMod_gr_31.gif] [Graphics:../Images/RegulaFalsiMod_gr_32.gif] [Graphics:../Images/RegulaFalsiMod_gr_33.gif]
[Graphics:../Images/RegulaFalsiMod_gr_34.gif] [Graphics:../Images/RegulaFalsiMod_gr_35.gif] [Graphics:../Images/RegulaFalsiMod_gr_36.gif] [Graphics:../Images/RegulaFalsiMod_gr_37.gif] [Graphics:../Images/RegulaFalsiMod_gr_38.gif]
[Graphics:../Images/RegulaFalsiMod_gr_39.gif] [Graphics:../Images/RegulaFalsiMod_gr_40.gif] [Graphics:../Images/RegulaFalsiMod_gr_41.gif] [Graphics:../Images/RegulaFalsiMod_gr_42.gif] [Graphics:../Images/RegulaFalsiMod_gr_43.gif]
[Graphics:../Images/RegulaFalsiMod_gr_44.gif] [Graphics:../Images/RegulaFalsiMod_gr_45.gif] [Graphics:../Images/RegulaFalsiMod_gr_46.gif] [Graphics:../Images/RegulaFalsiMod_gr_47.gif] [Graphics:../Images/RegulaFalsiMod_gr_48.gif]
[Graphics:../Images/RegulaFalsiMod_gr_49.gif] [Graphics:../Images/RegulaFalsiMod_gr_50.gif] [Graphics:../Images/RegulaFalsiMod_gr_51.gif] [Graphics:../Images/RegulaFalsiMod_gr_52.gif] [Graphics:../Images/RegulaFalsiMod_gr_53.gif]
[Graphics:../Images/RegulaFalsiMod_gr_54.gif] [Graphics:../Images/RegulaFalsiMod_gr_55.gif] [Graphics:../Images/RegulaFalsiMod_gr_56.gif] [Graphics:../Images/RegulaFalsiMod_gr_57.gif] [Graphics:../Images/RegulaFalsiMod_gr_58.gif]
[Graphics:../Images/RegulaFalsiMod_gr_59.gif] [Graphics:../Images/RegulaFalsiMod_gr_60.gif] [Graphics:../Images/RegulaFalsiMod_gr_61.gif] [Graphics:../Images/RegulaFalsiMod_gr_62.gif] [Graphics:../Images/RegulaFalsiMod_gr_63.gif]
[Graphics:../Images/RegulaFalsiMod_gr_64.gif] [Graphics:../Images/RegulaFalsiMod_gr_65.gif] [Graphics:../Images/RegulaFalsiMod_gr_66.gif] [Graphics:../Images/RegulaFalsiMod_gr_67.gif] [Graphics:../Images/RegulaFalsiMod_gr_68.gif]
[Graphics:../Images/RegulaFalsiMod_gr_69.gif] [Graphics:../Images/RegulaFalsiMod_gr_70.gif] [Graphics:../Images/RegulaFalsiMod_gr_71.gif] [Graphics:../Images/RegulaFalsiMod_gr_72.gif] [Graphics:../Images/RegulaFalsiMod_gr_73.gif]
[Graphics:../Images/RegulaFalsiMod_gr_74.gif] [Graphics:../Images/RegulaFalsiMod_gr_75.gif] [Graphics:../Images/RegulaFalsiMod_gr_76.gif] [Graphics:../Images/RegulaFalsiMod_gr_77.gif] [Graphics:../Images/RegulaFalsiMod_gr_78.gif]

[Graphics:../Images/RegulaFalsiMod_gr_79.gif] [Graphics:../Images/RegulaFalsiMod_gr_80.gif]

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

[Graphics:../Images/RegulaFalsiMod_gr_81.gif] [Graphics:../Images/RegulaFalsiMod_gr_82.gif] [Graphics:../Images/RegulaFalsiMod_gr_83.gif]

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

[Graphics:../Images/RegulaFalsiMod_gr_84.gif] [Graphics:../Images/RegulaFalsiMod_gr_85.gif]

The reported root is alleged to be

[Graphics:../Images/RegulaFalsiMod_gr_86.gif] [Graphics:../Images/RegulaFalsiMod_gr_87.gif]

The estimate of "how things are going" is the distance betweencand the nearest endpoint to the interval.

[Graphics:../Images/RegulaFalsiMod_gr_88.gif] [Graphics:../Images/RegulaFalsiMod_gr_89.gif]

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

[Graphics:../Images/RegulaFalsiMod_gr_90.gif]
k [Graphics:../Images/RegulaFalsiMod_gr_91.gif] [Graphics:../Images/RegulaFalsiMod_gr_92.gif] [Graphics:../Images/RegulaFalsiMod_gr_93.gif] [Graphics:../Images/RegulaFalsiMod_gr_94.gif]
[Graphics:../Images/RegulaFalsiMod_gr_95.gif] [Graphics:../Images/RegulaFalsiMod_gr_96.gif] [Graphics:../Images/RegulaFalsiMod_gr_97.gif] [Graphics:../Images/RegulaFalsiMod_gr_98.gif] [Graphics:../Images/RegulaFalsiMod_gr_99.gif]
[Graphics:../Images/RegulaFalsiMod_gr_100.gif] [Graphics:../Images/RegulaFalsiMod_gr_101.gif] [Graphics:../Images/RegulaFalsiMod_gr_102.gif] [Graphics:../Images/RegulaFalsiMod_gr_103.gif] [Graphics:../Images/RegulaFalsiMod_gr_104.gif]
[Graphics:../Images/RegulaFalsiMod_gr_105.gif] [Graphics:../Images/RegulaFalsiMod_gr_106.gif] [Graphics:../Images/RegulaFalsiMod_gr_107.gif] [Graphics:../Images/RegulaFalsiMod_gr_108.gif] [Graphics:../Images/RegulaFalsiMod_gr_109.gif]
[Graphics:../Images/RegulaFalsiMod_gr_110.gif] [Graphics:../Images/RegulaFalsiMod_gr_111.gif] [Graphics:../Images/RegulaFalsiMod_gr_112.gif] [Graphics:../Images/RegulaFalsiMod_gr_113.gif] [Graphics:../Images/RegulaFalsiMod_gr_114.gif]
[Graphics:../Images/RegulaFalsiMod_gr_115.gif] [Graphics:../Images/RegulaFalsiMod_gr_116.gif] [Graphics:../Images/RegulaFalsiMod_gr_117.gif] [Graphics:../Images/RegulaFalsiMod_gr_118.gif] [Graphics:../Images/RegulaFalsiMod_gr_119.gif]
[Graphics:../Images/RegulaFalsiMod_gr_120.gif] [Graphics:../Images/RegulaFalsiMod_gr_121.gif] [Graphics:../Images/RegulaFalsiMod_gr_122.gif] [Graphics:../Images/RegulaFalsiMod_gr_123.gif] [Graphics:../Images/RegulaFalsiMod_gr_124.gif]
[Graphics:../Images/RegulaFalsiMod_gr_125.gif] [Graphics:../Images/RegulaFalsiMod_gr_126.gif] [Graphics:../Images/RegulaFalsiMod_gr_127.gif] [Graphics:../Images/RegulaFalsiMod_gr_128.gif] [Graphics:../Images/RegulaFalsiMod_gr_129.gif]
[Graphics:../Images/RegulaFalsiMod_gr_130.gif] [Graphics:../Images/RegulaFalsiMod_gr_131.gif] [Graphics:../Images/RegulaFalsiMod_gr_132.gif] [Graphics:../Images/RegulaFalsiMod_gr_133.gif] [Graphics:../Images/RegulaFalsiMod_gr_134.gif]
[Graphics:../Images/RegulaFalsiMod_gr_135.gif] [Graphics:../Images/RegulaFalsiMod_gr_136.gif] [Graphics:../Images/RegulaFalsiMod_gr_137.gif] [Graphics:../Images/RegulaFalsiMod_gr_138.gif] [Graphics:../Images/RegulaFalsiMod_gr_139.gif]
[Graphics:../Images/RegulaFalsiMod_gr_140.gif] [Graphics:../Images/RegulaFalsiMod_gr_141.gif] [Graphics:../Images/RegulaFalsiMod_gr_142.gif] [Graphics:../Images/RegulaFalsiMod_gr_143.gif] [Graphics:../Images/RegulaFalsiMod_gr_144.gif]
[Graphics:../Images/RegulaFalsiMod_gr_145.gif] [Graphics:../Images/RegulaFalsiMod_gr_146.gif] [Graphics:../Images/RegulaFalsiMod_gr_147.gif] [Graphics:../Images/RegulaFalsiMod_gr_148.gif] [Graphics:../Images/RegulaFalsiMod_gr_149.gif]
[Graphics:../Images/RegulaFalsiMod_gr_150.gif] [Graphics:../Images/RegulaFalsiMod_gr_151.gif] [Graphics:../Images/RegulaFalsiMod_gr_152.gif] [Graphics:../Images/RegulaFalsiMod_gr_153.gif] [Graphics:../Images/RegulaFalsiMod_gr_154.gif]
[Graphics:../Images/RegulaFalsiMod_gr_155.gif] [Graphics:../Images/RegulaFalsiMod_gr_156.gif] [Graphics:../Images/RegulaFalsiMod_gr_157.gif] [Graphics:../Images/RegulaFalsiMod_gr_158.gif] [Graphics:../Images/RegulaFalsiMod_gr_159.gif]
[Graphics:../Images/RegulaFalsiMod_gr_160.gif] [Graphics:../Images/RegulaFalsiMod_gr_161.gif] [Graphics:../Images/RegulaFalsiMod_gr_162.gif] [Graphics:../Images/RegulaFalsiMod_gr_163.gif] [Graphics:../Images/RegulaFalsiMod_gr_164.gif]
[Graphics:../Images/RegulaFalsiMod_gr_165.gif] [Graphics:../Images/RegulaFalsiMod_gr_166.gif] [Graphics:../Images/RegulaFalsiMod_gr_167.gif] [Graphics:../Images/RegulaFalsiMod_gr_168.gif] [Graphics:../Images/RegulaFalsiMod_gr_169.gif]
[Graphics:../Images/RegulaFalsiMod_gr_170.gif] [Graphics:../Images/RegulaFalsiMod_gr_171.gif] [Graphics:../Images/RegulaFalsiMod_gr_172.gif] [Graphics:../Images/RegulaFalsiMod_gr_173.gif] [Graphics:../Images/RegulaFalsiMod_gr_174.gif]
[Graphics:../Images/RegulaFalsiMod_gr_175.gif] [Graphics:../Images/RegulaFalsiMod_gr_176.gif] [Graphics:../Images/RegulaFalsiMod_gr_177.gif] [Graphics:../Images/RegulaFalsiMod_gr_178.gif] [Graphics:../Images/RegulaFalsiMod_gr_179.gif]
[Graphics:../Images/RegulaFalsiMod_gr_180.gif] [Graphics:../Images/RegulaFalsiMod_gr_181.gif] [Graphics:../Images/RegulaFalsiMod_gr_182.gif] [Graphics:../Images/RegulaFalsiMod_gr_183.gif] [Graphics:../Images/RegulaFalsiMod_gr_184.gif]
[Graphics:../Images/RegulaFalsiMod_gr_185.gif] [Graphics:../Images/RegulaFalsiMod_gr_186.gif] [Graphics:../Images/RegulaFalsiMod_gr_187.gif] [Graphics:../Images/RegulaFalsiMod_gr_188.gif] [Graphics:../Images/RegulaFalsiMod_gr_189.gif]
[Graphics:../Images/RegulaFalsiMod_gr_190.gif] [Graphics:../Images/RegulaFalsiMod_gr_191.gif] [Graphics:../Images/RegulaFalsiMod_gr_192.gif] [Graphics:../Images/RegulaFalsiMod_gr_193.gif] [Graphics:../Images/RegulaFalsiMod_gr_194.gif]
[Graphics:../Images/RegulaFalsiMod_gr_195.gif] [Graphics:../Images/RegulaFalsiMod_gr_196.gif] [Graphics:../Images/RegulaFalsiMod_gr_197.gif] [Graphics:../Images/RegulaFalsiMod_gr_198.gif] [Graphics:../Images/RegulaFalsiMod_gr_199.gif]
[Graphics:../Images/RegulaFalsiMod_gr_200.gif] [Graphics:../Images/RegulaFalsiMod_gr_201.gif] [Graphics:../Images/RegulaFalsiMod_gr_202.gif] [Graphics:../Images/RegulaFalsiMod_gr_203.gif] [Graphics:../Images/RegulaFalsiMod_gr_204.gif]
[Graphics:../Images/RegulaFalsiMod_gr_205.gif] [Graphics:../Images/RegulaFalsiMod_gr_206.gif] [Graphics:../Images/RegulaFalsiMod_gr_207.gif] [Graphics:../Images/RegulaFalsiMod_gr_208.gif] [Graphics:../Images/RegulaFalsiMod_gr_209.gif]
[Graphics:../Images/RegulaFalsiMod_gr_210.gif] [Graphics:../Images/RegulaFalsiMod_gr_211.gif] [Graphics:../Images/RegulaFalsiMod_gr_212.gif] [Graphics:../Images/RegulaFalsiMod_gr_213.gif] [Graphics:../Images/RegulaFalsiMod_gr_214.gif]
[Graphics:../Images/RegulaFalsiMod_gr_215.gif] [Graphics:../Images/RegulaFalsiMod_gr_216.gif] [Graphics:../Images/RegulaFalsiMod_gr_217.gif] [Graphics:../Images/RegulaFalsiMod_gr_218.gif] [Graphics:../Images/RegulaFalsiMod_gr_219.gif]
[Graphics:../Images/RegulaFalsiMod_gr_220.gif] [Graphics:../Images/RegulaFalsiMod_gr_221.gif] [Graphics:../Images/RegulaFalsiMod_gr_222.gif] [Graphics:../Images/RegulaFalsiMod_gr_223.gif] [Graphics:../Images/RegulaFalsiMod_gr_224.gif]
[Graphics:../Images/RegulaFalsiMod_gr_225.gif] [Graphics:../Images/RegulaFalsiMod_gr_226.gif] [Graphics:../Images/RegulaFalsiMod_gr_227.gif] [Graphics:../Images/RegulaFalsiMod_gr_228.gif] [Graphics:../Images/RegulaFalsiMod_gr_229.gif]
[Graphics:../Images/RegulaFalsiMod_gr_230.gif] [Graphics:../Images/RegulaFalsiMod_gr_231.gif] [Graphics:../Images/RegulaFalsiMod_gr_232.gif] [Graphics:../Images/RegulaFalsiMod_gr_233.gif] [Graphics:../Images/RegulaFalsiMod_gr_234.gif]
[Graphics:../Images/RegulaFalsiMod_gr_235.gif] [Graphics:../Images/RegulaFalsiMod_gr_236.gif] [Graphics:../Images/RegulaFalsiMod_gr_237.gif] [Graphics:../Images/RegulaFalsiMod_gr_238.gif] [Graphics:../Images/RegulaFalsiMod_gr_239.gif]
[Graphics:../Images/RegulaFalsiMod_gr_240.gif] [Graphics:../Images/RegulaFalsiMod_gr_241.gif] [Graphics:../Images/RegulaFalsiMod_gr_242.gif] [Graphics:../Images/RegulaFalsiMod_gr_243.gif] [Graphics:../Images/RegulaFalsiMod_gr_244.gif]
[Graphics:../Images/RegulaFalsiMod_gr_245.gif] [Graphics:../Images/RegulaFalsiMod_gr_246.gif] [Graphics:../Images/RegulaFalsiMod_gr_247.gif] [Graphics:../Images/RegulaFalsiMod_gr_248.gif] [Graphics:../Images/RegulaFalsiMod_gr_249.gif]

[Graphics:../Images/RegulaFalsiMod_gr_250.gif] [Graphics:../Images/RegulaFalsiMod_gr_251.gif]

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

[Graphics:../Images/RegulaFalsiMod_gr_252.gif]
[Graphics:../Images/RegulaFalsiMod_gr_253.gif]
[Graphics:../Images/RegulaFalsiMod_gr_254.gif]
[Graphics:../Images/RegulaFalsiMod_gr_255.gif]

Back.



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