2, 4, 6, 8, 10, 12, Two ways: Formula for the generic term s n involving index n only: closed form

Transcription

1 CSC Discrete Structures Chapter 8: Recurreces Recurreces Part II CSC Discrete Structures Villaova Uiversity Defiig a Numerical Sequece,, 6, 8,,, Two ways: Formula for the geeric term s ivolvig idex oly: s = for EquaNo relang the geeric term s to oe or more precedig terms of the sequece s = s - + for >, s = Villaova CSC - Dr Papalaskari Solvig Recurrece RelaNos Method of forward subsntunos To solve a rela,o subject to a ii,al codi,o meas to fid a formula expressig its geeric term as a fucno of, the idex of the sequece (referred to as the ). Example: s = s - + & s = relano with iinal codino Usig the, geerate a few terms: x x x x x Try to discer a pa[er Prove the formula s validity (iducno) Example: x = x - + for >, x = s = Villaova CSC - Dr Papalaskari Villaova CSC - Dr Papalaskari Dr Papalaskari

2 CSC Discrete Structures Chapter 8: Recurreces Method of backward subsntunos Usig the, subsntute for previous terms (i.e., terms precedig x ) i a hope to see a pa[er. Cosider what would happe if you keep subsntung unl the iinal codino is reached. SubsNtute iinal codino to obtai the closed form. Prove the formula s validity (iducno) Example: x = x - + for >, x = Method of fiite differeces This is useful if you have a of the form x = x - + p() for x = c Compute differeces of subsequet terms. if differece is costat, you are doe otherwise, compute d differeces, etc. This method is described i some detail i SecNo 8.6 Example: x = x - + for >, x = Villaova CSC - Dr Papalaskari 5 Villaova CSC - Dr Papalaskari 6 F = F - + F Fiboacci sequece what if we had differet iinal codinos? F = F - + F a = ba - a = b c c b c = c cb b c = bc cb b c cb b c cb b c Geometric Sequeces Villaova CSC - Dr Papalaskari 8 Dr Papalaskari

3 CSC Discrete Structures Chapter 8: Recurreces Ca the sequece a = () be a soluno for the a = a - +a -? Ca the sequece a = () be a soluno for the a = a - +a -? a = () a = a - +a - a = () a = a - +a - iinal codinos iinal codinos Villaova CSC - Dr Papalaskari 9 Villaova CSC - Dr Papalaskari Ca the sequece a = (-) be a soluno for the a = a - +a -? Ca the sequece a = () be a soluno for the a = a - +a -? a = (-) a = a - +a - a = () a = a - +a - iinal codinos iinal codinos Villaova CSC - Dr Papalaskari Villaova CSC - Dr Papalaskari Dr Papalaskari

4 CSC Discrete Structures Chapter 8: Recurreces Ca the sequece a = 5 () be a soluno for the a = a - +a -? Aother soluno for the a = a - +a -? a = 5 () a = a - +a - a = a - +a - Villaova CSC - Dr Papalaskari Villaova CSC - Dr Papalaskari Ca the sequece a = 5 (-) be a soluno for the a = a - +a -? More solunos for a = a - +a -? a = 5 () a = a - +a - a = a - +a - Villaova CSC - Dr Papalaskari 5 Villaova CSC - Dr Papalaskari 6 Dr Papalaskari

5 CSC Discrete Structures Chapter 8: Recurreces More solunos for a = a - +a -? a = a - +a - d order liear homogeeous with costat coefficiets A that ca be wri[e i the form: ax + bx - + cx - = where a, b, c are real umbers (called the coefficiets), a. Examples: Villaova CSC - Dr Papalaskari 7 Villaova CSC - Dr Papalaskari 8 d order liear homogeeous with costat coefficiets Uless b = c =, the ax + bx - + cx - = has ifiitely may solunos (sequeces), called the geeral solu,o to the. All of them ca be obtaied by a sigle formula; the type of this formula depeds o the roots of the quadranc equano called the characteris,c equa,o for the above : ar + br + c = d order liear homogeeous with costat coefficiets Theorem If the characterisnc equano has two disnct real roots r, r the the solunos will be of the form: x = q r + q r If the characterisnc equano has two equal real roots r = r = r, the the solunos will be of the form: x = q r + q r (q ad q are ay two real umbers) Villaova CSC - Dr Papalaskari 9 Villaova CSC - Dr Papalaskari Dr Papalaskari 5

6 CSC Discrete Structures Chapter 8: Recurreces Example Try a few terms to verify the soluno Recurrece: a = a - +a - CharacterisNc equano: a = a - +a - 7 SoluNos of the form: SoluNo for iinal codinos: a = 7, a = Villaova CSC - Dr Papalaskari Villaova CSC - Dr Papalaskari Example (agai) Try a few terms to verify the soluno Recurrece: a = a - +a - CharacterisNc equano: a = a - +a - SoluNos of the form: 5 SoluNo for iinal codinos: a =, a = 5 Villaova CSC - Dr Papalaskari Villaova CSC - Dr Papalaskari Dr Papalaskari 6

7 CSC Discrete Structures Chapter 8: Recurreces Example Try a few terms to verify the soluno Recurrece: x = 5 x x - x = 5 x x - CharacterisNc equano: SoluNos of the form: SoluNo for iinal codinos: x =, x = Villaova CSC - Dr Papalaskari 5 Villaova CSC - Dr Papalaskari 6 Example Try a few terms to verify the soluno Recurrece: x = x - - x - x = x - - x - CharacterisNc equano: SoluNos of the form: SoluNo for iinal codinos: x =, x = Villaova CSC - Dr Papalaskari 7 Villaova CSC - Dr Papalaskari 8 Dr Papalaskari 7