Naïve Gauss Elimination

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1 Nïve Guss Elmnton Ch.9 Nïve Guss Elmnton Lner Alger Revew Elementry Mtr Opertons Needed for Elmnton Methods: Multply n equton n the system y nonzero rel numer. Interchnge the postons of two equton n the system. Replce n equton y the sum of tself nd multple of nother equton of the system. Nïve Guss Elmnton Smlr to Elmnton of Unknowns. Forwrd Elmnton. Bckwrd Susttuton Nïve ecuse we don t consder dvson y zero to e posslty

2 Nïve Guss Elmnton Smlr to Elmnton of Unknowns. Forwrd Elmnton of Unknowns. Reduce the coefcent mtr [A] to n upper trngulr system. Elmnte from the nd to nth Eqns.. Elmnte from the rd to nth Eqns. 4. Contnue process untl the nth equton hs only Non-Zero coeffcent Nïve Guss Elmnton. Forwrd Elmnton + + () + + () + + () Elmnte from equton (). Multply () y /, then sutrct the result from () () Nïve Guss Elmnton. Forwrd Elmnton + + () Pvot Equton + + () Elmnton Row + + () Elmnte from equton (). Multply () y /, then sutrct the result from () + + Pvot Element ()

3 Nïve Guss Elmnton. Forwrd Elmnton + + () + + () + + () Elmnton Row Elmnte from (). Multply () y /, then sutrct the result from () Nïve Guss Elmnton. Forwrd Elmnton + + () + + () Pvot Equton + + () Elmnton Row Elmnte from ( ). Multply ( ) y /, then sutrct the result from ( ) Nïve Guss Elmnton ). Forwrd Elmnton + + () + () + ( Solve for

4 Nïve Guss Elmnton. Bckwrds susttuton: + + From ( ) + + () () () From () Nïve Guss Elmnton In Generl, the lst equton should reduce to: ( n) n n ( n ) nn n ( ) ( ) j j j + ( ) Generl form s how we wll numerclly mplement. Note: Snce we normlze wth the pvot element, f t s zero, we hve prolem Nïve method Emple: Nïve Guss Elmnton () () () 4

5 5 Nïve Guss Elmnton Numerclly Implementng Mn Loops: Forwrd Elmnton. Pvot Row from st row to the n- row, move down, we wll cll the pvot row, row k.. Elmnton Row Rows elow Pvot row, where elmntons tke plce (top down), cll ths the th row.. Element trnsform Loop columns, jth column. Move left to rght. Nïve Guss Elmnton Numerclly Implementng Wht we do: Forwrd Elmnton A. Normlzton Step: multply the kth row elements kj y (- k / kk ) B. Add the result of step A, to j C. Clculte the new s or rght hnd sde terms kj kk k j j Insted of svng j we sve s j kk k k Nïve Guss Elmnton Numerclly Implementng Wht we do: Bck Susttuton For the nth Row: Now, work ckwrds row y row, rght to left (n- row nd n column) nn n n n j j j +

6 Nïve Guss Elmnton Pseudocode %Forwrd Elmnton to uld n upper trngulr mtr for k:n- for k+:n fctor (,k)/(k,k); %normlzng fctor (Step A) for jk+:n %move ccross the columns loop (,j) (,j) - fctor*(k,j); %(Step B) ()()-fctor*(k); %(Step C) %Bckwrd Susttuton (n)(n)/(n,n); %solve for the lst vlue for n-:-: sum ; for j+:n sum sum + (,j)*(j); ()(()-sum)/(,); Prolems wth Nïve Elmnton Methods. Dvson y zero f, then the st elmnton step yelds dvson y zero. pvotng technque wll e used to vod ths prolem. R.O. Error Every result s depnt on prevous results RO error cn propgte Rule of Thum f n > Doule precson wll help. Ill Condtoned Systems (D~) Smll chnges n the coeffcent ( j ) mtr result n lrge chnges n the soluton Or, lterntvely wde rnge of nswers ( s) stsfy the equtons RO error cn produce smll chnges n coeffcents tht cn led to lrge errors, (Check y slghtly chngng the coeffcents nd seeng the effect on the results) Prolems wth Nïve Elmnton Methods 4. Sngulr Systems: (D) One or more equtons re dentcl We hve (n-) equtons nd n unknowns QUICK wy to check D After the forwrd elmnton evlute the determnnt of the modfed coeffcent mtr D 6

7 Prolems wth Nïve Elmnton Methods When checkng D, how smll s too smll? Soluton: Stndrdze the determnnt. Scle Equtons such tht the mmum coeffcent for ny equton s. + + D Dvde () y nd () y D. Prolems wth Nïve Elmnton Methods Now for mtr D tkes on Vlues: + + D D Methods for Improvng Solutons. Use More Sgnfcnt dgts. Prtl Pvotng Avod dvson y zero or vry smll numers ) Before normlzng n Guss elmnton, fnd the lrgest element (solute vlue)n the frst column ) Reorder the equtons so tht the lrgest element s the pvot element c) Repet for ech elmnton step I.e., nd pplcton would fnd the lrgest element n the nd column (elow the st Equton) nd seek the lrgest pvot element. 7

8 Methods for Improvng Solutons Prtl Pvotng code pk; gs((k,k)) %ssume row wth lrgest coeffcent %ssume the dgnol term s lrgest for k+:n dummys((,k)); f dummy > g gdummy; p; %move down the rows to check elements %f the elent s gger swp t out %renme the lrgest row % for loop %f p s not equl to k, we need to swp row k wth row p %f p s equl to k, then we dont do nythng f p~k for jjk:n %move cross columns to swp coeffcent vlues dummy(p,jj); %temporrly store the element (p,jj)(k,jj); (k,jj)dummy; dummy (p); %now swy rght hnd sde vlues (p) (k); (k) dummy; Show Mtl emple Methods for Improvng Solutons. Sclng helps mke pvotng decsons + 7, +. scle 4, Wht prolem could Arse here Pvot Methods for Improvng Solutons. Sclng We hve seen tht ddng nd sutrctng of numers wth very dfferent mgntudes cn result n RO error Scle ll rows so tht the mmum coeffcent vlue n ny row s one. NOTE: sclng y very lrge numers cn potentlly ntroduce RO error Suggeston: Employ sclng only to mke decson regrdng pvotng Comprson & row swtchng re not suject to RO error Complete soluton usng orgnl coeffcents 8

9 Guss-Jordn Elmnton Vrton of Guss Elmnton When n unknown s elmnted t s elmnted from ll equtons, not just susequent ones (Dgonl Mtr Results) All rows re normlzed y ther pvot element Identty Mtr results [ A] []{} I {} Is ugmented Almost dentcl to Guss Elmnton ut, more opertons re requred No ck susttuton step Methods for Improvng Solutons Prtl Pvotng Overhed Guss Jordn emple 9

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