NUMERICAL METHODS & OPTIMISATION

Transcription

1 For updted version, plese click on NUMERICAL METHODS & OPTIMISATION Prt I: s Rihn Edros Fculty of Engineering Technology rzhirh@ump.edu.my

2 Chpter Description Aims Apply numericl methods in solving engineering prolem nd optimistion Expected Outcomes Solve simultneous equtions y using Nïve-Guss nd Guse Jordn methods Apply liner lgeric equtions to solve engineering prolems References Steven C. Chpr nd Rymond P. Cnle (9), Numericl Methods for Engineers, McGrw-Hill, 6 th Edition

3 Overview s This chpter dels with simultneous liner lgeric equtions: x + x +... n xn = x + x +... n xn = x + x +... x = n where s re constnt coefficient nd s re constnts n Guss elimintion involves comining equtions to eliminte unknowns Why Guss elimintion? Algorithm used in populr softwre pckges Most importnt lgorithm Bsis of liner eqution solution 7/8/7 RZE/5/BTP4 3 nn n n

4 Overview s (cont d) Two simultneous equtions cn e solved through: Grphicl method Determinnts & Crmer s Rule Elimintion of unknowns Grphicl method x x x x + + æ = - ç è æ = - ç è x x ö x ø ö x ø = = + + By Picknick 7/8/7 RZE/5/BTP4 4

5 Overview s (cont d) Determinnts & Crmer s Rule Alterntive to grphicl method Used to solve unknowns from smll numer of simultneous equtions Expresses the solution of liner equtions in terms of rtios of D = determinnts [ ]{ X} { B} 3 A = [ A] D = 3 3 é = ë 3 3 D = D = ù û x A - = A - = A - 3 = 3 3 = 3 3 D

6 Overview s (cont d) Elimintion of unknowns Involves two sic steps: ) comining equtions ) Sustitution Used to solve two simultneous equtions Bsis for Nïve-Guss Elimintion method

7 Overview s (cont d) Liner Algeric Eqution Solvers Guss Elimintion LU Decomposition & Mtrix Inversion Guss-Seidel Nive Guss Guss-Jordn

8 Nïve-Guss Elimintion An extension of elimintion of unknowns method to solve higher numer of equtions n sets Solved in two stges: Forwrd elimintion Bck sustitution

9 Nïve-Guss Elimintion: Algorithm Nïve-Guss Elimintion usully suitle to e solved y computers ility to void zero division Mnul clcultions unle to void zero division Nïve Consider the following two equtions: = = = x + x + 3x3 x + x + 3x3 3x + 3x + 33x3 3 Phse : Forwrd elimintion. Multipliction of () y gives: = x + x + 3x3 () () (3) ()

10 Nïve-Guss Elimintion: Algorithm (cont d). Sutrction of () from () gives: = x + x + 3x3 x + 3x3... n xn = x + x... x = n!! æ ö æ ö ç ç n - è ø è ø n3 - x... 3 x3 + + ç - = Which cn e simplified to 3 nn n n x x3 = 3 3. Multipliction of () y nd sutrction of the resulting eqution from (3) yield: () () (3) notifyeditingon=

11 Nïve-Guss Elimintion: Algorithm (cont d) 4. Elimintion of second unknown from (3) y multiplying () to 3 nd sutrction of the resulting eqution from (3) yield: () x + x + 3x3... n xn = x + 3x3... n xn = x... x = n3!! 3 nn n n () (3) 5. The elimintion process cn e repeted until the equtions re trnsformed into tringulr system: = x + x + 3x3 x + 3x3 = '' '' x3 = editingon=

12 Nïve-Guss Elimintion: Algorithm (cont d) Phse : Bck sustitution 6. x 3 cn e solved y using the previous equtions: x = 3 '' 3 '' The process of sustitution is repeted for x nd x. Reference: Exmple 9.

13 Guss-Jordn Elimintion During the forwrd elimintion of Nïve-Guss elimintion, there re instnces where division (.k. pivot coefficient) will get zero Exmple: x + x + 3x3 4x + 6x - 7x3 x + x + 6x 8 = 3 5 = - 3 = Removing x from Eqution & 3 using s pivot coefficient leds to the division of 4 y Guss-Jordn elimintes this possiility y converting the coefficients of equtions into identity mtrix

14 Guss-Jordn Elimintion (cont d) It is vrition of Nïve-Guss elimintion. The mjor differences re:. An unknown is eliminted from ll other equtions.. Elimintion step results in n identity mtrix. = x + x + 3x3 x + x + 3x3 3x + 3x + 33x3 3 = = é ë 3 ² ù ² ² û Identity mtrix

15 Guss-Jordn Elimintion: Algorithm Consider Exmple 9. s reference to explin Guss-Jordn elimintion x -.x -.x =.x + 7x -.3x3 = -.3x -.x + x3 = Express the coefficients nd the equivlences s n ugmented mtrix st row nd row 3 rd row é 3. ë ù û

16 Guss-Jordn Elimintion: Algorithm (cont d). Normlize st row y the pivot coefficient of st row: é. ë ù û 3. Eliminte x from nd row nd 3 rd row. ) Multiply the st row y. nd sutrct it from the nd row ) Multiply the st row y.3 nd sutrct it from the 3 rd row é ë ù û

17 Guss-Jordn Elimintion: Algorithm (cont d) 4. Normlize the nd row y the pivot coefficient of nd row: é ë ù û 5. Eliminte x from st row nd 3 rd row. ) Multiply the nd row y nd sutrct it from the st row ) Multiply the nd row y -.9 nd sutrct it from the 3 rd row é ù ë û 8&notifyeditingon=

18 Guss-Jordn Elimintion: Algorithm (cont d) 6. Normlize 3 rd row y the pivot coefficient of 3 rd row: é ë ù û 7. Eliminte x 3 from st row nd nd row. ) Multiply the 3 rd row y nd sutrct it from the st row ) Multiply the 3 rd row y nd sutrct it from the nd row é ë 3. ù û =68&notifyeditingon=

19 Conclusion The Nïve Guss nd Guss Jordn methods cn e used to solve the simultneous equtions 7/8/7 RZE/5/BTP4 9

20 Min Reference Steven C. Chpr nd Rymond P. Cnle (9), Numericl Methods for Engineers, McGrw-Hill, 6 th Edition Any enquiries kindly contct: Rihn Edros, PhD rzhirh@ump.edu.my