Titus Beu University Babes-Bolyai Department of Theoretical and Computational Physics Cluj-Napoca, Romania

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1 8. Systems of Lier Algeric Equtios Titus Beu Uiversity Bes-Bolyi Deprtmet of Theoreticl d Computtiol Physics Cluj-Npoc, Romi Biliogrphy Itroductio Gussi elimitio method Guss-Jord elimitio method Systems of lier equtios with tridigol mtri

2 Gh. Dodescu, Metode umerice î lgeră, Editur tehică, Bucureşti, 979). B.P. Demidovich şi I.A. Mro, Computtiol Mthemtics(MIR Pulishers, Moskow, 98). R.L. Burde şij.d. Fires, Numericl Alysis, Third Editio(Pridle, Weer & Schmidt, Bosto, 985). W.H. Press, S.A. Teukolsky, W.T. VetterligşiB.P. Flery, Numericl Recipes i C: The Art of Scietific Computig, Secod Editio(Cmridge Uiversity Press, Cmridge, 992). Beu, T. A., Numericl Clculus i C, Third Editio (MicroIformtic Pulishig House, Cluj-Npoc, 2004). Solvig systems of lier equtios plys cetrl role i umericl lysis. Direct methods fiite lgorithms, roud-off errors Guss, Crout, Choleski Itertive methods ifiite processes, tructio errors Jcoi, Guss-Seidel. 2

3 Bsic ide equivlet systemwith upper trigulr mtriy elemetry row trsformtios trsformtio of the system Bckwrd sustitutio solvig i reversed order the equivlet system Emple: 3 lier equtios with 3 ukows A = [ ij ] 33, = [ i ] 3, = [ i ] 3 = = = or =

4 . Elimite from ll equtios ecept the first oe Divide st (pivot) equtio to pivot elemet, Sutrct st equtio multiplied y first coefficiet from ll other equtios: () () () () () () = () () () () () () or = 2 = 2 () () () () () () = with () j = j/, j=,2,3 () = / () () ij = ij ij, j=,2,3, i= 2,3 () () i = i i 2. Elimite 2 from the lst equtio Divide 2 d (pivot) equtio to pivot elemet, 22 () Sutrct 2d equtio multiplied y 32 () from 3 rd equtio: () () () 2 3 (2) (2) = 2 (2) (2) with (2) () () 2j = 2j / 22, j= 2,3 (2) () () 2 = 2 / 22 (2) () () (2) ij = ij i22j, j= 2, 3, i= 3 (2) () () (2) i = i i2 2 4

5 3. Divide 3 d (pivot) equtio to pivot elemet, 33 (2) : with () () () 2 3 (2) (2) = (3) 3 3 (3) (2) (2) 3 = 3 / 33 Bckwrd sustitutio Epress solutio compoets recurretly i reversed order: Determit (3) 3 = 3 (2) (2) 2 = () () () = ( 22 33) Due to successive divisios to the pivot elemets: A A (3) det det = = () (2) () (2) deta= 5

6 Geerliztio: lier equtios with ukows A = [ ij ], = [ i ], = [ i ] Before step k (k =,2,..., ): () () () () 2 k k () (2) (2) (2) 0 2 k 2 k 2 (2) 2 2 ( k ) ( k ) ( k ) 0 0 kk kk ( k ) k k = k ( k ) ( k ) ( k ) 0 0 k k k k ( ) k k k k ( k ) ( k ) ( k ) 0 0 k k ( k ) Step k elimite k from ll equtios elow equtio k: () () () () 2 k k () (2) (2) (2) 0 2 k 2 k 2 () kk k k = k k k k k k k New elemets of pivot lie k: kk = ( k ) ( k ) kj = kj / kk, j= k,, ( k ) ( k ) k = k / kk 6

7 Step k elimite k from ll equtios elow equtio k: () () () () 2 k k () (2) (2) (2) 0 2 k 2 k 2 () kk k k = k k k k k k k New elemets of o-pivot lies elow pivot lie k: ik = 0 ( k ) ( k ) ij = ij ik k j, j= k,,, i= k,, ( k ) ( k ) i = i ik k Step divide lst lie to lst pivot ( ) : or () () () () () 2 k k (2) (2) (2) (2) 0 2 k 2 k ( 0 0 kk k k = k) k ( k ) ( k ) k k k ( ) ( ) ( ) A = 7

8 Bckwrd sustitutio Epress solutio compoets recurretly i reversed order: Determit ( ) = k = k kii, k=,, i= k Due to successive divisios to the pivot elemets: A A ( ) det det = = () ( ) 22 () ( ) 22 deta= Geerliztio: Mtri equtio A= [ ij ], B = [ ij ] m, = [ ij ] m ( k ) ( k ) kj = kj / kk, j= k,, ( k ) ( k ) kj = kj / kk, j=,, m ( k ) ( k ) ij = ij ik kj, j= k,,, i= k,, ( k ) ( k ) ij = ij ik kj, j=,, m Bckwrd sustitutio ( ) j = j, j=,, m kj = kj kiij, j=,, m, k=,, i= k Mtri iversio B= E [ δ ] X= A ij 8

9 Pivotig Rerrgemet of lies d/or columsto hve mimum pivot elemett ech elimitio step Merits Avoids divisios y 0 Miimizes roud-off errors Vrits Prtil pivotig (o colums) t step koe seeks the mimum elemet lk (k-) o colum kd lies l kd iterchges lies kd l. Totl pivotig (method of pricipl elemet) oe seeks mimum pivot o ll lies d colums of A (k ) o which o pivotig ws doe yet //=========================================================================== it Guss(flot **, flot **, it, it m, flot &det) // Solves mtri equtio = y Gussi elimitio, replcig o // output y. Uses prtil pivotig o colums. // - mtri of system ( ) // - mtri of r.h.s. terms ( m); solutio o output // det determit of the system mtri (output) // Error flg: 0 orml eecutio // sigulr mtri { #defie Swp(,) { t = ; = ; = t; flot m, sum, t; it i, im, j, k; det =.0; for (k=; k<=; k) { // ELIMINATION m = 0.0; // determies pivot lie hvig for (i=k; i<=; i) // mimum elemet o colum k if (m < fs([i][k])) {m = fs([i][k]); im = i; if (m == 0.0) {pritf("guss: sigulr mtri!\"); retur ; // iterchges lies im d k if (im!= k) { // to put pivot o digol det = -det; for (j=k; j<=; j) Swp([im][j],[k][j]) for (j=; j<=m; j) Swp([im][j],[k][j]) 9

10 det *= [k][k]; // multiplies determit with pivot t =.0/[k][k]; for (j=k; j<=; j) [k][j] *= t; for (j= ; j<=m; j) [k][j] *= t; // divides pivot lie y pivot for (i=k; i<=; i) { // reduces o-pivot lies t = [i][k]; for (j=k; j<=; j) [i][j] -= [k][j]*t; for (j= ; j<=m; j) [i][j] -= [k][j]*t; for (k=-; k>=; k--) for (j=; j<=m; j) { sum = [k][j]; for (i=k; i<=; i) sum -= [k][i]*[i][j]; [k][j] = sum; retur 0; // BACK SUBSTITUTION Bsic ide equivlet systemwith digol mtriy elemetry row trsformtios trsformtio of the system more elorte Bckwrd sustitutio ieistet Vrit system mtri is trsformed ito the uit mtri Eles simulteous clcultio d storge of iverse 0

11 Before step k (k =,2,..., ): ( k ) ( k ) ( k ) k k ( k ) ( k ) ( k ) 2k 2k 2 ( k ) ( k ) ( k ) kk kk k ( k ) ( k ) ( k ) k k k k k ( k ) ( k ) ( k ) k k ( k ) ( k ) 2 2 ( k ) k = k ( k ) k k ( k ) Relevt iformtio of mtri A (k) eclusively i colums j = k,..., Colums j =,...,kcoti trivil dt c e used to produce the iverse A Step k elimite k from ll equtios: 0 0 k ( k 2 k) kk k k = k k k k k k k New elemets of pivot lie k: kk = ( k ) ( k ) kj = kj / kk, j= k,, ( k ) ( k ) k = k / kk

12 Step k elimite k from ll equtios: 0 0 k ( k 2 k) kk k k = k k k k k k k New elemets o-pivot lies: ik = 0 ( k ) ( k ) ij = ij ik kj, j= k,,, i=,,, i k ( k ) ( k ) i = i ik k Step divide lst lie to lst pivot ( ) : Solutio ( ) ( ) ( ) k = k ( ) k k ( ) Determit = ( ) () ( ) 22 deta= 2

13 //=========================================================================== it GussJord0(flot **, flot **, it, it m, flot &det) // Solves mtri equtio = y Guss-Jord elimitio, replcig o // output y. Uses prtil pivotig o colums. // - mtri of system ( ) // - mtri of r.h.s. terms ( m); solutio o output // det determit of the system mtri (output) // Error flg: 0 orml eecutio // sigulr mtri { #defie Swp(,) { t = ; = ; = t; flot m, t; it i, im, j, k; det =.0; for (k=; k<=; k) { // ELIMINATION m = 0.0; // determies pivot lie hvig for (i=k; i<=; i) // mimum elemet o colum k if (m < fs([i][k])) {m = fs([i][k]); im = i; if (m == 0.0) {pritf("gussjord0: sigulr mtri!\"); retur ; // iterchges lies im d k if (im!= k) { // to put pivot o digol det = -det; for (j=k; j<=; j) Swp([im][j],[k][j]) for (j=; j<=m; j) Swp([im][j],[k][j]) det *= [k][k]; // multiplies determit with pivot t =.0/[k][k]; for (j=k; j<=; j) [k][j] *= t; for (j= ; j<=m; j) [k][j] *= t; // divides pivot lie y pivot for (i=; i<=; i) // reduces o-pivot lies if (i!= k) { t = [i][k]; for (j=; j<=; j) [i][j] -= [k][j]*t; for (j=; j<=m; j) [i][j] -= [k][j]*t; retur 0; 3

14 //=========================================================================== it GussJord(flot **, flot **, it, it m, flot &det) // Solves mtri equtio = y Guss-Jord elimitio, replcig o // output y ^(-) d y. Uses prtil pivotig o colums. // - mtri of system ( ) // - mtri of r.h.s. terms ( m); solutio o output // det determit of the system mtri (output) // Error flg: 0 orml eecutio // sigulr mtri { #defie Swp(,) { t = ; = ; = t; flot m, t; it i, im, j, k; it *ipivot; ipivot = IVector(,); // stores pivot lie det =.0; for (k=; k<=; k) { // ELIMINATION m = 0.0; // determies pivot lie hvig for (i=k; i<=; i) // mimum elemet o colum k if (m < fs([i][k])) {m = fs([i][k]); im = i; if (m == 0.0) {pritf("gussjord: sigulr mtri!\"); retur ; ipivot[k] = im; // stores pivot lie // iterchges lies im d k if (im!= k) { // to put pivot o digol det = -det; for (j=; j<=; j) Swp([im][j],[k][j]) for (j=; j<=m; j) Swp([im][j],[k][j]) det *= [k][k]; t =.0/[k][k]; [k][k] =.0; for (j=; j<=; j) [k][j] *= t; for (j=; j<=m; j) [k][j] *= t; // multiplies determit with pivot // divides pivot lie y pivot // digol elemet of uit mtri for (i=; i<=; i) // reduces o-pivot lies if (i!= k) { t = [i][k]; [i][k] = 0.0; // o-digol elemet of uit mtri for (j=; j<=; j) [i][j] -= [k][j]*t; for (j=; j<=m; j) [i][j] -= [k][j]*t; for (k=; k>=; k--) // rerrge colums of iverse if (ipivot[k]!= k) for (i=; i<=; i) Swp([i][ipivot[k]],[i][k]) FreeIVector(ipivot,); retur 0; 4

15 //=========================================================================== it GussJord(flot **, flot **, it, it m, flot &det) // Solves mtri equtio = y Guss-Jord elimitio, replcig o // output y ^(-) d y. Uses totl pivotig. // - mtri of system ( ) // - mtri of r.h.s. terms ( m); solutio o output // det determit of the system mtri (output) // Error flg: 0 orml eecutio //,2 sigulr mtri { #defie Swp(,) { t = ; = ; = t; flot m, t; it i, im, j, jm, k; it *ipivot, *jpivot, *pivot; ipivot = IVector(,); jpivot = IVector(,); pivot = IVector(,); // stores pivot lie // stores pivot colum // mrks used pivot colum det =.0; for (i=; i<=; i) pivot[i] = 0; for (k=; k<=; k) { // ELIMINATION m = 0.0; // determies pivot for (i=; i<=; i) { // loop over lies if (pivot[i]!= ) for (j=; j<=; j) { // loop over colums if (pivot[j] == 0) // pivotig ot yet doe? if (m < fs([i][j])) {m = fs([i][j]); im = i; jm = j; else if (pivot[j] > ) {pritf("gussjord: sigulr mtri!\"); retur ; ipivot[k] = im; // stores pivot lie jpivot[k] = jm; // stores pivot colum (pivot[jm]); // mrk used pivot colum if (m == 0.0) {pritf("gussjord: sigulr mtri 2!\"); retur 2; // iterchges lies im d jm if (im!= jm) { // to put pivot o digol det = -det; for (j=; j<=; j) Swp([im][j],[jm][j]) for (j=; j<=m; j) Swp([im][j],[jm][j]) det *= [jm][jm]; // multiplies determit with pivot 5

16 t =.0/[jm][jm]; // divides pivot lie y pivot [jm][jm] =.0; // digol elemet of uit mtri for (j=; j<=; j) [jm][j] *= t; for (j=; j<=m; j) [jm][j] *= t; for (i=; i<=; i) // reduces o-pivot lies if (i!= jm) { t = [i][jm]; [i][jm] = 0.0; // o-digol elemet of uit mtri for (j=; j<=; j) [i][j] -= [jm][j]*t; for (j=; j<=m; j) [i][j] -= [jm][j]*t; for (k=; k>=; k--) // rerrge colums of iverse if (ipivot[k]!= jpivot[k]) for (i=; i<=; i) Swp([i][ipivot[k]],[i][jpivot[k]]) FreeIVector(ipivot,); FreeIVector(jpivot,); FreeIVector(pivot,); retur 0; Tridigol mtrices sprse mtrices most of the o-digol elemets 0. Geerl methods re ot efficiet. Typiclly discretiztio of differetil equtios y fiite differece schemes. Geerl form: A = d c d 2 2 c2 0 2 d 2 i i c i i d i = i i ci i di 0 c d d 6

17 Fctoriztio of mtri A: A= L U β γ α2 β2 0 γ2 0 αi β i γ i A= αi βi γi 0 α β 0 γ α β Elemets of mtrices Ld U y idetifictio(β i 0): β =, γ = c β αi = i, βi = i αγ i i, γi = ci βi, i= 2,3,, α =, β = αγ Iitil system ecomes: L y d L ( U ) d = = U = y First system: β y d 2 β2 0 y 2 d 2 i β i y i d i i β i y = i d i 0 β y d β y d Itermedite solutio y y idetifictio(β i 0): y = d/ β yi = ( di y i i )/ βi, i= 2,3,, 7

18 Iitil system ecomes: L y d L ( U ) d = = U = y Secod system: γ y γ2 0 2 y 2 γ i i y i γ i = i y i 0 γ y y Fil solutio y idetifictio: = y i = yi γ ii, i=,, Algorithm Fctoriztio solutio of system L y = d: β =, γ = c/ β, y = d/ β βi = i iγi, γi = ci/ βi, yi = ( di y i i )/ βi i= 2,3,, Bck sustitutio solutio of system U = y: = ( d y )/( γ ) i = yi γii, i=,, Four rrys (,, c d d) re sufficiet i the implemettio Elemets γ i c e stored i rry c, overwritig elemets c i For d i, y i d i the sme rry d my e used, which filly returs the solutio No eed for pivotig digol domice. 8

19 //=========================================================================== void TriDig(flot [], flot [], flot c[], flot d[], it ) // Solves y LU fctoriztio lier system with tridigol mtri // - lower co-digol (i=2..) // - mi digol // c - upper co-digol (i=..-) // d - r.h.s. terms; solutio o output // - order of system { flot et; it i; // fctoriztio c[] /= []; d[] /= []; for (i=2; i<=(-); i) { et = [i] - [i]*c[i-]; c[i] /= et; d[i] = (d[i] - [i]*d[i-])/et; d[] = (d[] - []*d[-])/([] - []*c[-]); for (i=(-); i>=; i--) d[i] -= c[i]*d[i]; // ck sustitutio 9

Autar Kaw Benjamin Rigsby. Transforming Numerical Methods Education for STEM Undergraduates

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