APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES

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1 Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 3() 0, 5-0 APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES Jolt Borows, Le Łcińs, Jowit Rychlews Istitute of Mthetics, Czestochow Uiversity of Techology Czestochow, Pold jolt.borows@i.pcz.pl, le.lcis@i.pcz.pl, jowit.rychlews@i.pcz.pl Abstrct. I this pper we preset pplictio of secod order hoogeeous lier differece equtios with costt coefficiets to evlute the deterit of tridigol trices. Coprig the obtied results with certi ltertive pproch [] soe forule for the fiite su re derived. Itroductio It is well ow tht the deterits of tridigol trices ply iportt role i y pplictios, for exple, i prllel coputig or i fiite differeces ethod []. Moreover, there re y coectios betwee deterits of tridigol trices d the Fibocci d Lucs ubers [3, 4]. Uder certi coditios the recurrece reltio for coputig the deterit of tridigol trices ws obtied i [5]. I pper [] the uthors cosidered the deterit of tridigol trix of the for A b b b b b 0 0 = 0 0 b b b () It ws show tht the deterit of trix A is give by the forul det A = b + b b + () 3

2 6 J. Borows, L. Łcińs, J. Rychlews Cosidertios of this pper re cocered with the deterit of the for = (3) where,, C 3. It will be show tht the proble of clcultio of the deterit (3) leds to solvig secod order hoogeeous lier differece equtio with costt coefficiets. Moreover, for specil cses, the obtied results will be copred with those of [] i order to derive forule for certi fiite sus.. The i results I this sectio we re goig to give the results cocerig the deterit of tridigol trices of the for (3). Let us observe tht = = (4) Usig the ethod of Lplce expsio with respect to the first colu d subsequetly with respect to the first row we obti =, > (5) The bove expressio c be rewritte i the followig for + =, > 0 (6) Let us observe tht equtio (6) is secod-order hoogeeous lier differece equtio with costt coefficiets together with iitil coditios of for (4). Followig [6] we hve tht the geerl solutio of the equtio (6) is deteried by the roots of the qudrtic equtio which deped o the discriit λ λ 0 + =, λ C (7)

3 Applictio of differece equtios to certi tridigol trices 7 4 = (8) Cse. Let 0. The qudrtic equtio (7) hs two differet coplex roots λ =, λ I this cse the geerl solutio of the equtio (6) hs the for + = (9) = Cλ + Cλ (0) Berig i id the iitil coditios (4) we obti the syste of lier equtios Cλ + Cλ = C C λ + λ = () Fro the bove syste of equtios we hve C = λ d C = λ () Substitutig (9) d () to (0) we fid tht the prticulr solutio of the equtio (6) with the iitil coditios (4) hs the for = (3) Cse. Let = 0. The the qudrtic equtio (7) hs oe double root λ λ = = (4) I cse the geerl solutio of the equtio (6) hs the for λ λ = C + C (5) Fro the iitil coditios (4) we hve the followig syste of equtios Cλ + Cλ = C C λ + λ = (6) The solutio of the bove syste of equtios hs the for

4 8 J. Borows, L. Łcińs, J. Rychlews C= d λ C 3 λ = (7) Tig ito ccout (4) d (7) we obti fro (5) tht the prticulr solutio of the equtio (6) with iitil coditios (4) is give by the forul ( ) + = I the subsequet lysis cosidertios will be restricted to the deterit i which =. (8). Idetities obtied i specil cses The i of this sectio is to derive forule for the fiite su of certi sequeces. To this ed let = 3= b, =, b C, \ { 0}. Tig ito ccout the results obtied i pper [] we c write the deterit of trix () i the for b = b = = 0 = 0 ( ) (9) where the sybol x deotes the lrgest iteger ot greter th x. b Settig c=, c C \ { 0} ito forul (9), we hve = 0 (0) = c O the other hd, fro (8) we hve ( 4 c) It c be esily observed tht if c C \ 0, the 0. I this cse the foru- l (3) yields = = () Coprig (0) d () we hve

5 Applictio of differece equtios to certi tridigol trices c =, = 0 + c C \ 0, () If c= the = 0 d fro (8) we hve 4 + = (3) Fro the copriso of (0) d (3) we derive the followig forul + = (4) = 0 4 Rer If is eve, =, the = = d the forul () is give by c = = 0 + (5) where c C \ 0,. At the se tie the forul (4) tes the for + = (6) = Rer If is odd, = +, the hs the for + = = + = d the forul () c = = 0 +, (7) where c C \ 0,.

6 0 J. Borows, L. Łcińs, J. Rychlews hilst the forul (4) tes the for + + = (8) = Rer 3 Let us observe tht whe is egtive rel or coplex uber the pplictio of the forul () requires usig of the de Moivre forul. For exple if c = the we hve 3 ( + ) π ( ) = ( ) si (9) = Coclusios It ws show tht the secod-order hoogeeous lier differece equtio with costt coefficiets c be used for the clcultio of the deterit of tridigol trix with the se eleets o prticulr digols. It c be observed tht if o prticulr digols we hve differet eleets the the clcultio of deterit leds to secod-order hoogeeous lier differece equtios with vrible coefficiets. The exct solutios of these equtios c be obtied oly i soe specil cses. The lysis of these specil cses will be studied i the forthcoig pper. Refereces [] Biert G., Boryś J., Cłusińs I., Sur I., The three-bd trices, Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 008, (7), 9-. [] Biert G., Siedlec U., Fiite differece ethod i fourier equtio iterl cse - direct foruls, Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 0, (0), -6. [3] Feg J., Fibocci idetities vi the deterit of tridigol trix, Applied Mthetics d Coputtio 0, 7, [4] Nlli A., Civciv H., A geerliztio of tridigol trix deterits, Fibocci d Lus ubers, Chos, Solitos d Frctls 009, 40, [5] El-Miwy M., A ote o three-ter recurrece for tridigol trix, Applied Mthetics d Coputtio 003, 39, [6] Kuchrzewsi M., Piwo J., Rówi różiczowe i różicowe, ydwictwo Politechii Śląsiej, Gliwice 977 (i Polish).

A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD

Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method