Ecuciones Algebrics lineles
An eqution of the form x+by+c=0 or equivlently x+by=c is clled liner eqution in x nd y vribles. x+by+cz=d is liner eqution in three vribles, x, y, nd z. Thus, liner eqution in n vribles is 1 x 1 + 2 x 2 + + n x n = b A solution of such n eqution consists of rel numbers c 1, c 2, c 3,, c n. If you need to work more thn one liner equtions, system of liner equtions must be solved simultneously.
Mtrices ij = elementos de un mtriz i=número del renglón j=número de l column Vector column Vector renglón
Mtriz cudrd m=n Número de ecuciones Digonl principl Número de incóngnits
Regls de operciones con mtrices
Representción de ecuciones lgebrics lineles en form mtricil Solving for X
Noncomputer Methods for Solving Systems of Equtions For smll number of equtions (n 3) liner equtions cn be solved redily by simple techniques such s method of elimintion. Liner lgebr provides the tools to solve such systems of liner equtions. Nowdys, esy ccess to computers mkes the solution of lrge sets of liner lgebric equtions possible nd prcticl. Prt 3 10
Guss Elimintion Chpter 9 Solving Smll Numbers of Equtions There re mny wys to solve system of liner equtions: Grphicl method Crmer s rule For n 3 Method of elimintion Computer methods Prt 3 11
Grphicl Method For two equtions: + = x + x b 11 1 12 2 1 21 x 1 + 22 x 2 = b Solve both equtions for x 2: x b + 2 11 1 2 = x1 + x2 = (slope)x1 intercept 12 12 x 2 = x + b 21 2 1 22 22 Prt 3 12
Plot x 2 vs. x 1 Fig. 9.1 on rectiliner pper, the intersection of the lines present the solution. Prt 3 13
Grphicl Method Grphicl Method Or equte nd solve for x 1 + = + = 2 21 1 11 b x b x x + = + = 22 1 22 12 1 12 2 b b x x x = + 22 2 12 1 1 12 11 22 21 0 b b x 12 1 22 2 22 2 12 1 b b b b x = = 12 11 22 21 12 11 22 21 1 x Prt 3 14 12 22 12 22
Figure 9.2 No solution Infinite solutions Ill conditioned (Slopes re too close) Prt 3 15
Determinnts nd Crmer s Rule Determinnt cn be illustrted for set of three equtions: Ax = b Where A is the coefficient mtrix: A 11 12 13 = 21 22 23 31 32 33 Prt 3 16
Assuming ll mtrices re squre mtrices, there is number ssocited with ech squre mtrix A clled the determinnt, D, of A. (D=det (A)). If [A] is order 1, then [A] hs one element: A=[ 11 ] D= 11 For squre mtrix of order 2, A= 11 12 the determinnt t is D= 11 22-21 12 21 22 Prt 3 17
For squre mtrix of order 3, the minor of n element ij is the determinnt of the mtrix of order 2 by deleting row i nd column j of A. Prt 3 18
13 12 11 D 33 32 31 23 22 21 D = 23 32 33 22 23 22 11 D = = 23 21 33 32 D = = 23 31 33 21 33 31 12 D = = 22 31 32 21 32 31 22 21 13 D = = Prt 3 19
22 23 21 23 D = 11 12 + 32 33 31 33 13 21 31 22 32 Crmer s rule expresses the solution of systems of liner equtions in terms of rtios of determinnts of the rry of coefficients of the equtions. For exmple, x 1 would be computed s: b b 1 2 12 22 13 23 x 1 b3 32 33 D Prt 3 20
Method of Elimintion The bsic strtegy is to successively solve one of the equtions of the set for one of the unknowns nd to eliminte tht vrible from the remining equtions by substitution. The elimintion of unknowns cn be extended to systems with more thn two or three equtions; however, the method becomes extremely tedious to solve by hnd. Prt 3 23
Relción con Crmer
Nive Guss Elimintion Extension of method of elimintion to lrge sets of equtions by developing systemtic scheme or lgorithm to eliminte unknowns nd to bck substitute. As in the cse of the solution of two equtions, the technique for n equtions sconsists sssof two phses: Forwrd elimintion of unknowns Bck substitution Prt 3 25
Fig. 9.3 Prt 3 26
Generlizndo Elemento pivote Multiplicndo ec 1 32 /22 = nuevo elemento pivote Restndo ec2 de l nuev ec1 Reescribiendo ec nterior
Pitflls of Elimintion Methods Division i i by zero. It is possible tht t during both elimintion nd bck-substitution phses division by zero cn occur. Round-off errors. Ill-conditioned systems. Systems where smll chnges in coefficients result in lrge chnges in the solution. Alterntively, it hppens when two or more equtions re nerly identicl, resulting wide rnges of nswers to pproximtely stisfy the equtions. Since round off errors cn induce smll chnges in the coefficients, these chnges cn led to lrge solution errors. Prt 3 31
Singulr systems. When two equtions re identicl, we would loose one degree of freedom nd be deling with the impossible cse of n-1 equtions for n unknowns. For lrge sets of equtions, it my not be obvious however. The fct tht the determinnt of singulr system is zero cn be used nd tested by computer lgorithm fter the elimintion stge. If zero digonl element is creted, clcultion cu is terminted. ed. Prt 3 32
Techniques for Improving Solutions Use of more significnt ifi figures. Pivoting. If pivot element is zero, normliztion i step leds to division i i by zero. The sme problem my rise, when the pivot element is close to zero. Problem cn be voided: Prtil pivoting. Switching the rows so tht the lrgest element is the pivot element. Complete pivoting. Serching for the lrgest element in ll rows nd columns then switching. Prt 3 33
Crmer o sustituciòn
Determinnt Evlution Using Guss Elimintion
Csi cero!!! Depende del numero de cifrs significtivs
SCALING
Guss-Jordn It is vrition of Guss elimintion. The mjor differences re: When n unknown is eliminted, it is eliminted from ll other equtions rther thn just the subsequent ones. All rows re normlized by dividing them by their pivot elements. Elimintion step results in n identity mtrix. Consequently, it is not necessry to employ bck substitution to obtin solution. Prt 3 44
Descomposición LU e inversión de Mti Mtrices [A]{X}={B} [A]{X}-{B}=0 [U]{X}-{D}=0 Guss Elimintion
De l eliminción hci delnte de Guss tenemos : Finlmente
Encontrndo d plicndo l eliminción hci delnte pero solo sobre el vector B Encontrndo X plicndo l sustitución hci trás
Mtriz Invers
Homework