Numerical methods

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Bisection method, Newton method, Falsi method and Secant method with examples.

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Numerical methods

  1. 1. By Ahmed Haider Ahmed Pre-M.Sc. Physics – ASU
  2. 2. To My father , May Allah enter you his paradise
  3. 3. 1 – Bisection Method  If F(wi+1) is negative we put wi+1 instead of ai or bi 2 1 ii i ba w
  4. 4. 2 – Falsie Method )()( )()( 1 ii iiii i bfaf afbbfa x
  5. 5. 3 – Newton Method )( )( 1 i i ii xf xf xx
  6. 6. 4 – Secant Method )()( ))(( 1 1 1 ii iii ii xfxf xxxf xx
  7. 7. Examples
  8. 8. Find the positive root of x – cos x = 0 using bisection method.  We will take points that give positive value with the negative one i.e 0 and 1 )(1)2( )(00015.0)1( )(1)0( cos)( vef vef vef xxxf
  9. 9. )(249.0)75.0( 75.0 2 15.0 150 )(5.0)5.0( )(5.0 2 10 10 2 1 vef x and.enroot betweHence the vef vex andenroot betweHence the
  10. 10. 062.0)9375.0( 9375.0 2 1875.0 18750 )(12488.0)875.0( 875.0 2 75.01 1750 4 3 f x and.enroot betweHence the vef x and.enroot betweHence the
  11. 11. Solve equation x3 – x – 1 = 0 using falsie method. 5128)2( 1111)1( 1)( 3 f f xxxf 2,1 )()( )()( 00 00 0000 1 ba bfaf afbbfa x
  12. 12. )(279351.0)2547.1( 2547.1 5684.05 5684.02517.1 217.1from )(5684.0)17.1( 17.1 6 25 51 1251 2 2 1 vef x tox vef x
  13. 13. )(0578.0)311.1( 311.1 127.05 127.025294.1 2294.1from )(127.0)294.1( 294.1 279351.05 279351.0252547.1 22547.1from 4 4 3 3 vef x tox vef x tox
  14. 14. 3238.1 0109.05 0109.0253223.1 23223.1from 0109.00)3223.1( 3223.1 024255.05 024255.025319.1 2319.1from )(024255.0)319.1( 319.1 0578.05 0578.025311.1 2311.1from 7 7 6 6 5 5 x tox f x tox vef x tox
  15. 15. Solve equation x3 – x – 1 = 0 using Newton method at x0 = 1 213)( 1111)( 13)( 1)( )( )( 0 0 2 3 1 xf xf xxf xxxf xf xf xx i i ii
  16. 16. 4497.4)3478.1( 1006.0)3478.1( 3478.1 75.5 875.0 5.1 )( )( 75.5)5.1( 875.0)5.1( 5.1 2 1 1 )( )( 1 1 12 0 0 01 f f xf xf xx f f xf xf xx
  17. 17. 000077.0)3247.1( 3247.1 269.4 00206.0 3252.1 )( )( 269.4)3252.1( 00206.0)3252.1( 3252.1 4497.4 1006.0 3478.1 )( )( 3 3 34 2 2 23 f xf xf xx f f xf xf xx
  18. 18. REFERANCES  Lecture notes on numerical methods , Dr. Shemi 2011, Minia university written by Aya Hassan.  Lecture notes on computational physics for Pre-M.Sc. Students , Prof. Dr. S.Hendawi 2013, Ain Shams University.

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