점화식

수학노트
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개요

  • 점화식 : 수열의 여러항들이 만족시키는 관계
  • 점화식이 만족하는 수열의 일반항을 알아 내는 문제 등
  • 보통의 경우 초기항이 주어져야 완전한 답을 낼 수 있다.



기본적인 점화식

  • \(a_{n+1} - a_n = c\) : 등차수열
  • \(a_{n+1} / a_n = c\) : 등비수열
  • \(a_{n+1} - a_n = b_n\) : 계차수열 참고.
  • \(a_{n+1} / a_n = b_n\) : 계차수열을 통한 풀이에서, <모든 항을 더하>지 않고 <모든 항을 곱하>면 됨.



기본 점화식의 응용

  • \(a_{n+1} = ka_n + c\)
    • 양변에 적당한 상수를 더하면 \((a_{n+1} + p)= k(a_n + p)\) 꼴로 만들 수 있다.
    • 일반항이 \((a_n + p)\) 인 수열은 공비 \(k\) 인 등비수열, \(a_n + p =(a_1 + p) k^{n-1}\)
    • 적당한 상수 \(p\) 는 어떻게 찾냐고? 생각해 볼 것.
    • ex) \(a_{n+1} = 2a_{n} + 3\), 초기항 1 양변에 3을 더하면 \((a_{n+1} +3 )= 2( a_{n} + 3)\), 적당한 상수 \(k\) 에 대하여 \(a_{n} + 3 = k 2^n\) 초항을 만족시키는 \(k\) 값은 2이므로, \(a_n = 2^{n+1} - 3\)


  • \(a_{n+1} = ra_n + b_n\) 꼴의 점화식
    • 양변을 \(r^{n+1}\) 로 나눈 후, \(a_n / r^n\) 에 대한 점화식을 푸는 것이 한 방법. \(b_n\) 이 등비수열인 경우 효과적이다.
    • 양변에 적당히 \(n\) 에 대한 식을 더해서 공비 \(r\) 에 대한 등비수열 꼴로 만들 수 있는 경우가 많다.
      • \(b_n\) 이 다항식인 경우, 양변에 \(b_n\) 과 같은 차수의 다항식을 (계수를 문자로 두고) 더해서, 등비수열 꼴로 만든 후에 계수 비교를 통해 문자를 찾는다.
      • ex ) \(a_{n+1} = 2a_n + 3n + 5\) 인 경우, 양변에 \(pn + q\) 를 더하면\[a_{n+1} + pn + q = 2a_n + (3+p)n + (5+q)\] 우변이 \(2(a_n+ pn + q)\) 인 경우에 등비수열이 되니까, \(3+p = 2p, \quad 5+q=2q\) 이므로 \(p=3, \quad q=5\). 그러므로\[a_n + 3n + 5 = k2^n\]. 초기항이 주어진 경우 k 를 찾을 수 있다.


  • 점화식에 덧셈 기호가 없을 때
    • 로그를 취하면 도움이 됨. 로그의 밑은 계산이 간단하도록 적절히 선택하기.
    • ex) \(a_{n+1} = 4a_n^2\) : 밑 2 인 로그를 취하면 \(\log a_{n+1} = 2\log a_n + 2\) 이제 \(\log a_n = b_n\)에 대한 점화식을 풀면 됨. (양변에 2를 더해서 …)
  • 점화식이 분수꼴일때
    • 역수를 취하면 도움이 될 때가 많음. (만능은 아님)
    • ex) \({a_{n+1}} = \frac{2a_n}{3a_n+1}\) : 역수를 취하면 \(\frac{1}{a_{n+1}} = \frac{1}{2} \frac{1}{a_n} + \frac{3}{2}\). 이제 \(b_n = \frac{1}{a_n}\) 에 대한 점화식으로 보고 풀면 됨.
  • 점화식에 \(a_n\) 과 \(S_n\) 이 함께 나올 때
    • \(S_n - S_{n-1}=a_n\)\((n \ge 2)\) , \(S_1 = a_1 \) 의 관계를 사용하면 \(a_n\) 만의 점화식으로 만들 수 있다.
    • ex) \( a_n = 2 S_n + 3^n\) 일 때, \( a_1 = 2 a_1 + 3\) 이므로 \(a_1 = -3\)\[a_{n+1} = 2 S_{n+1} + 3^{n+1}\] 식과 \( a_n = 2 S_n + 3^n\) 식을 빼 주면 \(a_{n+1} - a_n = 2 a_{n+1} + 2\cdot 3^{n}\) 이제부터는 알아서 할 것. 부분합과 일반항이 함께 등장하는 점화식에서는 초기 조건에서 실수를 할 가능성이 높으므로, \(a_1, a_2, a_3\) 정도는 점화식으로부터 직접 구해 보는 것이 좋을 것이다. 그리고 구한 일반항 \(a_n\) 은 \((n \ge 2)\) 에서만 성립하는 식일 가능성도 있다.



선형점화식

  • \(pa_{n+2} + qa_{n+1} + ra_n = 0\) 꼴의 점화식
  • \(pa_{n+2} + qa_{n+1} + ra_n = b_n\) 꼴의 점화식
  • 상수계수 선형점화식 항목을 참조


관련된 항목들


사전 형태의 자료

메타데이터

위키데이터

노트

말뭉치

  1. The procedure for finding the terms of a sequence in a recursive manner is called recurrence relation.[1]
  2. We study the theory of linear recurrence relations and their solutions.[1]
  3. Doing so is called solving a recurrence relation .[2]
  4. Recall that the recurrence relation is a recursive definition without the initial conditions.[2]
  5. , 485\ldots\text{.}\) Solution Finding the recurrence relation would be easier if we had some context for the problem (like the Tower of Hanoi, for example).[2]
  6. Remember, the recurrence relation tells you how to get from previous terms to future terms.[2]
  7. Recurrence relations are used to reduce complicated problems to an iterative process based on simpler versions of the problem.[3]
  8. Identifying a candidate problem to solve with a recurrence relation: The problem can be reduced to simpler cases.[3]
  9. The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation.[4]
  10. A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones.[4]
  11. This defines recurrence relation of first order.[4]
  12. The recurrence of order two satisfied by the Fibonacci numbers is the archetype of a homogeneous linear recurrence relation with constant coefficients (see below).[4]
  13. In this article, we will see how we can solve different types of recurrence relations using different approaches.[5]
  14. Therefore, we need to convert the recurrence relation into appropriate form before solving.[5]
  15. There are a variety of methods for solving recurrence relations, with various advantages and disadvantages in particular cases.[6]
  16. One method that works for some recurrence relations involves generating functions.[6]
  17. If we can find an explicit representation for the series for this function, we will have solved the recurrence relation.[6]
  18. We can often solve a recurrence relation in a manner analogous to solving a differential equations by multiplying by an integrating factor and then integrating.[7]
  19. Wolfram|Alpha can solve various kinds of recurrences, find asymptotic bounds and find recurrence relations satisfied by given sequences.[8]
  20. This is called a recurrence relation.[9]
  21. We somehow need to figure out how often the first versus the second branch of this recurrence relation will be taken.[9]
  22. We will first find a recurrence relation for the execution time.[9]
  23. These recurrence relations are shown to be closely related to each other.[10]
  24. Racah has obtained a recurrence relation for the ``inner multiplicity (multiplicity of weights).[10]
  25. Moreover, Kostant's recurrence relation for the partition function as well as Racah's recurrence relation for the inner multiplicity have been generalized.[10]
  26. In this paper, the recurrence relation for negative moments along with negative factorial moments of some discrete distributions can be obtained.[11]
  27. Is `D(4)` easier to calculate using the factorial formula or using the recurrence relation?[12]
  28. Therefore the solution to the recurrence relation will have the form: a n =a2n+b18n.[13]
  29. Therefore the solution to the recurrence relation will have the form: a n =a.6n+b.n.6n.[13]
  30. Explanation: Check for the left side of the equation with all the options into the recurrence relation.[13]
  31. Together with the initial conditions, the recurrence relation provides a recursive definition for the elements of the sequence.[14]
  32. The solution techniques for these two classes of recurrence relation are discussed in detail in the Recurrence page.[15]
  33. We also justified Pascal's Formula as a recurrence relation.[16]
  34. Let's write a recurrence relation for the sum of the elements in column 1 of the triangle.[16]
  35. Write a recurrence relation H(n) to describe the number of moves required when the left tower has n rings on it.[16]
  36. Write a recurrence relation b(n) to describe the balance due on the loan after n payments.[16]
  37. A recurrence relation is an equation which represents a sequence based on some rule.[17]
  38. Once the values of a 0 and a 1 are specified, the whole sequence {a i } i 0 is completely specified by the recurrence relation.[18]
  39. Observe that the difference of the highest index (n+1) and the lowest index (n-2) is exactly the order 6 of the recurrence relation.[18]
  40. Since the highest index in the recurrence relation is n+1 while the lowest index is n-2, their difference (n+1)-(n-2)=3 must be the order m, i.e. m=3.[18]
  41. For each fixed m, these two generating functions share the same denominator, hence the same recurrence relation .[19]
  42. After studying it carefully, it is found that the solutions cannot be given exactly due to the complicated three-term recurrence relation .[19]
  43. Recurrence relations can also be used to calculate some sequences that are usually computed nonrecursively , e.g. via a closed-form formula .[20]
  44. Of course recurrence relations are not limited in application to sequences of integers.[20]

소스

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'recurrence'}, {'LEMMA': 'relation'}]
  • [{'LOWER': 'recurrent'}, {'LEMMA': 'sequence'}]