Please send proof of theorem.... Should short and Authentic....
0, for each i= 1,2, ...,n, then one and only one of the following statements holds: (i) Os P(T,) < p(t)="">< i:="" (i)="" p(t)="p(T,)" =="" 0;="" (ii)="">< p(t)="">< p(t):="" (iv)="" p(t)="p(T,)" =="" 1.="" for="" the="" special="" case="" described="" in="" theorem="" 7.22,="" we="" see="" from="" part="" (i)="" that="" when="" one="" method="" gives="" convergence,="" then="" both="" give="" convergence,="" and="" the="" gauss-seidel="" method="" con-="" verges="" faster="" than="" the="" jacobi="" method.="" part="" (ii)="" indicates="" that="" when="" one="" method="" diverges="" then="" both="" diverge,="" and="" the="" divergence="" is="" more="" pronounced="" for="" the="" gauss-seidel="" method.="" ercise="" set="" 73="" 3="" 1.="" find="" the="" first="" two="" iterations="" of="" the="" jacobi="" method="" for="" the="" following="" linear="" systems,="" using="" x"="0" a.="" 3="" -="" +="" -1,="" 3="" +="" 6r="" +="" 2ky="0." 3="" +="" 3x="" +7x)="4." 10="" -="" -a+="" 10r="" -="" 2,7="" -="" 2="" +="" 10r,="6." =="" 9,="" .="" i0,="" +="" s="" sx,="" +="" 10="" -="" 4xy="" -6,="" d.="" 4="" +="" +="" +="" x="" 6.="" -="" 25.="" -="" -="" 3k="" +="" +="" 4="" -6,="" 4x="" +="" 8="" -="" xa="-11." 2="" +="" +="" sa="" -="" 4-="" as="6," -a-="" ay-="" +="" 4="" -="" +="" s="" -1.="" -6,="" 2="" -="" +="" a+="" 4xs="6." find="" the="" first="" two="" iterations="" of="" the="" jacobi="" method="" for="" the="" following="" linear="" systems,="" using="" x="" e="" 2.="" h="" -2+="" +="" a-4,="" x-2="" -="" a="-4," a.="" 4x="" +="" a="" -="5," -="" +="" 3="" +="-4," 2="" +="" 2x="" +="" sx="1." n+="" 2-0.="" c.="" xi="" +="" 4="" -="" -="-I," 4x="" +="" -="" +="" -2.="" d.="" -4="" -0,="" -n="" +4x="" -="" -="" 5.="" -="" -="" +="" s+="" 0,="" -="" a="" +="" 4="" -="" 0,="" +="" 4-="" -="" 4+="" 4r,="" -="" -2,="" -="" a+="" 4-="" 6.="" x-+="" +="" 3="1." +="" l="" -6,="" -="" "/="">
Extracted text: (1) x-x| S ITI|x - x|l: di) |k-x®I< –="" we="" have="" seen="" that="" the="" jacobi="" and="" gauss-seidel="" iterative="" techniques="" can="" be="" written="" x="Tx-+e" and="" x="T,x"-+e." using="" the="" matrices="" t,="D(L+" u)="" and="" t,-="" (d-="" l)-u.="" if="" p(t,)="" or="" p(t,)="" is="" less="" than="" 1,="" then="" the="" corresponding="" sequence="" (x="" will="" converge="" to="" the="" solution="" x="" of="" ax="b." for="" example,="" the="" jacobi="" scheme="" has="" `="D'(L" +="" u)xª-d="" +="" d-'b,="" and,="" if="" (x1="" converges="" to="" x,="" then="" x="DL+U)x" +db.="" this="" implies="" that="" dx="(L+" u)x="" +b="" and="" (d="" -="" l="" -="" u)x="b." .="" since="" d-l-u="A," the="" solution="" x="" satisfies="" ax="b." we="" can="" now="" give="" easily="" verified="" sufficiency="" conditions="" for="" convergence="" of="" the="" jacobi="" and="" gauss-seidel="" methods.="" (to="" prove="" convergence="" for="" the="" jacobi="" scheme="" see="" exercise="" 14,="" and="" for="" the="" gauss-seidel="" scheme="" see="" (or2),="" p.="" 120.)="" erle="" arroeet="" mete="" p="" det="" t="" pet="" c="" ep="" d="" iy="" i="" d="" cip="" l="" em="" ia="" d="" ri="" 7.3="" the="" jacobi="" and="" gauss-siedel="" iterative="" techniques="" 459="" theor="" 7.21="" if="" a="" is="" strictly="" diagonally="" dominant,="" then="" for="" any="" choice="" of="" x,="" both="" the="" jacobi="" and="" gauss-scidel="" methods="" give="" sequences="" (x="" ,="" that="" converge="" to="" the="" unique="" solution="" of="" ax-b.="" the="" relations="" ne="" rapidity="" of="" converge="" the="" spectral="" radius="" of="" the="" iteration="" be="" seen="" from="" corollary="" 7.20.="" the="" inequale="" norm,="" so="" it="" follows="" from="" the="" statement="" after="" theorem="" 7.15="" on="" page="" 446="" that="" ix"="" -="" ||="p(Ty'Ix®" –="" x|l.="" (7.12)="" thus="" we="" would="" like="" to="" select="" the="" iterative="" technique="" with="" minimal="" p(t)="">< for="" a="" particular="" system="" ax="b." no="" general="" results="" exist="" to="" tell="" which="" of="" the="" two="" techniques,="" jacobi="" or="" gauss-="" seidel,="" will="" be="" most="" successful="" for="" an="" arbitrary="" linear="" system.="" in="" special="" cases,="" however,="" the="" answer="" is="" known,="" as="" is="" demonstrated="" in="" the="" following="" theorem.="" the="" proof="" of="" this="" result="" can="" be="" found="" in="" [y],="" pp.="" 120-127.="" theorem="" 7.22="" (stein-rosenberg)="" if="" a="" s0,="" for="" each="" i="" +j="" and="" a,=""> 0, for each i= 1,2, ...,n, then one and only one of the following statements holds: (i) Os P(T,) < p(t)="">< i:="" (i)="" p(t)="p(T,)" =="" 0;="" (ii)="">< p(t)="">< p(t):="" (iv)="" p(t)="p(T,)" =="" 1.="" for="" the="" special="" case="" described="" in="" theorem="" 7.22,="" we="" see="" from="" part="" (i)="" that="" when="" one="" method="" gives="" convergence,="" then="" both="" give="" convergence,="" and="" the="" gauss-seidel="" method="" con-="" verges="" faster="" than="" the="" jacobi="" method.="" part="" (ii)="" indicates="" that="" when="" one="" method="" diverges="" then="" both="" diverge,="" and="" the="" divergence="" is="" more="" pronounced="" for="" the="" gauss-seidel="" method.="" ercise="" set="" 73="" 3="" 1.="" find="" the="" first="" two="" iterations="" of="" the="" jacobi="" method="" for="" the="" following="" linear="" systems,="" using="" x"="0" a.="" 3="" -="" +="" -1,="" 3="" +="" 6r="" +="" 2ky="0." 3="" +="" 3x="" +7x)="4." 10="" -="" -a+="" 10r="" -="" 2,7="" -="" 2="" +="" 10r,="6." =="" 9,="" .="" i0,="" +="" s="" sx,="" +="" 10="" -="" 4xy="" -6,="" d.="" 4="" +="" +="" +="" x="" 6.="" -="" 25.="" -="" -="" 3k="" +="" +="" 4="" -6,="" 4x="" +="" 8="" -="" xa="-11." 2="" +="" +="" sa="" -="" 4-="" as="6," -a-="" ay-="" +="" 4="" -="" +="" s="" -1.="" -6,="" 2="" -="" +="" a+="" 4xs="6." find="" the="" first="" two="" iterations="" of="" the="" jacobi="" method="" for="" the="" following="" linear="" systems,="" using="" x="" e="" 2.="" h="" -2+="" +="" a-4,="" x-2="" -="" a="-4," a.="" 4x="" +="" a="" -="5," -="" +="" 3="" +="-4," 2="" +="" 2x="" +="" sx="1." n+="" 2-0.="" c.="" xi="" +="" 4="" -="" -="-I," 4x="" +="" -="" +="" -2.="" d.="" -4="" -0,="" -n="" +4x="" -="" -="" 5.="" -="" -="" +="" s+="" 0,="" -="" a="" +="" 4="" -="" 0,="" +="" 4-="" -="" 4+="" 4r,="" -="" -2,="" -="" a+="" 4-="" 6.="" x-+="" +="" 3="1." +="" l="" -6,="">