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The document describes recursive sequences defined by initial conditions and a recurrence relation. It provides examples of linear homogeneous recurrence relations with constant coefficients, including their characteristic equations and solutions. It notes that any linear combination of solutions to a linear homogeneous recurrence is also a solution, and discusses finding general solutions when the characteristic equation has repeated roots.
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Daa linear recurrences
1) The document discusses solving linear recurrence relations, which are relations where each term is a linear combination of previous terms.
2) It provides methods for solving both homogeneous relations, where the right side is only previous terms, and non-homogeneous relations, where the right side also includes an additional term.
3) The key steps are finding the characteristic equation, which determines the possible solutions, then using the initial conditions to find the specific solution. Examples are worked through to demonstrate the process.
LINEAR RECURRENCE RELATIONS WITH CONSTANT COEFFICIENTS
This document discusses linear recurrence relations with constant coefficients. It covers homogeneous solutions, particular solutions, and the total solution. It also discusses solving recurrence relations using generating functions. Key points:
- Homogeneous solutions are found by solving the characteristic equation.
- Particular solutions are found for homogeneous and non-homogeneous equations.
- The total solution is the sum of the homogeneous and particular solutions.
- Generating functions can also be used to solve recurrence relations.
Solving linear homogeneous recurrence relations
The document discusses solving recurrence relations through the following steps:
1) Write the recurrence relation and initial conditions as a characteristic equation.
2) Find the solutions to the characteristic equation.
3) Use the solutions to write the general solution as a linear combination with constants.
4) Determine the constants using the initial conditions.
Two examples are provided to demonstrate solving both regular and similar solutions to linear homogeneous recurrence relations.
Modeling with Recurrence Relations
This document discusses using recurrence relations to model problems involving counting techniques. It provides examples of modeling problems related to bacteria population growth, rabbit population growth, the Tower of Hanoi puzzle, and valid codeword enumeration. For each problem, it defines the recurrence relation and initial conditions, derives a closed-form solution, and proves its correctness using mathematical induction. Recurrence relations provide a way to define sequences and solve problems recursively by relating terms to previous terms in the sequence.
Solving recurrences
This document discusses solving linear homogeneous recurrence relations with constant coefficients. It begins by defining such a recurrence relation as one where the terms are expressed as a linear combination of previous terms. It then explains that these types of relations can be solved by finding the characteristic roots of the characteristic equation. The document provides an example of solving a degree two recurrence relation and outlines the basic approach of finding a solution of the form an = rn. It also discusses solving coupled recurrence relations by eliminating variables to obtain a single recurrence relation that can be solved. Finally, it revisits the Martian DNA problem and shows its solution is a Fibonacci number.
recurrence relations
The document discusses recurrence relations and their applications. It begins by defining a recurrence relation as an equation that expresses the terms of a sequence in terms of previous terms. It provides examples of recurrence relations and their solutions. It then discusses solving linear homogeneous recurrence relations with constant coefficients by finding the characteristic roots and obtaining an explicit formula. Applications discussed include financial recurrence relations, the partition function, binary search, and the Fibonacci numbers. It concludes by discussing the case when the characteristic equation has a single root.
recurence solutions
The document discusses recurrence relations and methods for solving them. It defines a recurrence relation as an equation that expresses the terms of a sequence in terms of previous terms. It provides examples of homogeneous and non-homogeneous recurrence relations. For homogeneous relations, it describes guessing a solution of the form T(n)=x^n and finding the characteristic equation. For non-homogeneous relations, it explains adding the non-recursive term to the characteristic equation. It then works through examples of solving both homogeneous and non-homogeneous recurrence relations.
Recurrence relations
This document discusses recurrence relations and their use in defining sequences. It introduces key concepts like recurrence relations, initial conditions, explicit formulas, and solving recurrence relations using techniques like backtracking or finding the characteristic equation. As examples, it examines the Fibonacci sequence and linear homogeneous recurrence relations of varying degrees.
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Daa linear recurrences
1) The document discusses solving linear recurrence relations, which are relations where each term is a linear combination of previous terms.
2) It provides methods for solving both homogeneous relations, where the right side is only previous terms, and non-homogeneous relations, where the right side also includes an additional term.
3) The key steps are finding the characteristic equation, which determines the possible solutions, then using the initial conditions to find the specific solution. Examples are worked through to demonstrate the process.
LINEAR RECURRENCE RELATIONS WITH CONSTANT COEFFICIENTS
This document discusses linear recurrence relations with constant coefficients. It covers homogeneous solutions, particular solutions, and the total solution. It also discusses solving recurrence relations using generating functions. Key points:
- Homogeneous solutions are found by solving the characteristic equation.
- Particular solutions are found for homogeneous and non-homogeneous equations.
- The total solution is the sum of the homogeneous and particular solutions.
- Generating functions can also be used to solve recurrence relations.
Solving linear homogeneous recurrence relations
The document discusses solving recurrence relations through the following steps:
1) Write the recurrence relation and initial conditions as a characteristic equation.
2) Find the solutions to the characteristic equation.
3) Use the solutions to write the general solution as a linear combination with constants.
4) Determine the constants using the initial conditions.
Two examples are provided to demonstrate solving both regular and similar solutions to linear homogeneous recurrence relations.
Modeling with Recurrence Relations
This document discusses using recurrence relations to model problems involving counting techniques. It provides examples of modeling problems related to bacteria population growth, rabbit population growth, the Tower of Hanoi puzzle, and valid codeword enumeration. For each problem, it defines the recurrence relation and initial conditions, derives a closed-form solution, and proves its correctness using mathematical induction. Recurrence relations provide a way to define sequences and solve problems recursively by relating terms to previous terms in the sequence.
Solving recurrences
This document discusses solving linear homogeneous recurrence relations with constant coefficients. It begins by defining such a recurrence relation as one where the terms are expressed as a linear combination of previous terms. It then explains that these types of relations can be solved by finding the characteristic roots of the characteristic equation. The document provides an example of solving a degree two recurrence relation and outlines the basic approach of finding a solution of the form an = rn. It also discusses solving coupled recurrence relations by eliminating variables to obtain a single recurrence relation that can be solved. Finally, it revisits the Martian DNA problem and shows its solution is a Fibonacci number.
recurrence relations
The document discusses recurrence relations and their applications. It begins by defining a recurrence relation as an equation that expresses the terms of a sequence in terms of previous terms. It provides examples of recurrence relations and their solutions. It then discusses solving linear homogeneous recurrence relations with constant coefficients by finding the characteristic roots and obtaining an explicit formula. Applications discussed include financial recurrence relations, the partition function, binary search, and the Fibonacci numbers. It concludes by discussing the case when the characteristic equation has a single root.
recurence solutions
The document discusses recurrence relations and methods for solving them. It defines a recurrence relation as an equation that expresses the terms of a sequence in terms of previous terms. It provides examples of homogeneous and non-homogeneous recurrence relations. For homogeneous relations, it describes guessing a solution of the form T(n)=x^n and finding the characteristic equation. For non-homogeneous relations, it explains adding the non-recursive term to the characteristic equation. It then works through examples of solving both homogeneous and non-homogeneous recurrence relations.
Recurrence relations
This document discusses recurrence relations and their use in defining sequences. It introduces key concepts like recurrence relations, initial conditions, explicit formulas, and solving recurrence relations using techniques like backtracking or finding the characteristic equation. As examples, it examines the Fibonacci sequence and linear homogeneous recurrence relations of varying degrees.
Recurrence Relation
The document discusses recurrence relations and their applications. It can be summarized as follows:
1. A recurrence relation expresses the terms of a sequence in terms of previous terms, providing a rule to generate all terms. It does not specify initial values.
2. Recurrence relations are useful for modeling many real-world phenomena like financial growth, partitions of sets, and algorithm analysis.
3. Linear homogeneous recurrence relations of constant coefficients can be solved by finding the characteristic roots from the characteristic equation. The general solution is a linear combination of terms with the roots.
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1) The document discusses modeling and applications of second order differential equations. It provides examples of second order differential equations that model vibrating springs and electric current circuits.
2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
3) An example solves a second order differential equation modeling a vibrating spring to find the position of a mass attached to the spring at any time.
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This document discusses solutions to higher order differential equations with constant coefficients. It begins by introducing homogeneous linear equations of order two or higher of the form y'' + ay' + by + c = 0. It then presents the method of solving such equations by finding the roots of the characteristic or auxiliary equation. Depending on whether the roots are real/distinct, real/equal, or complex conjugates, the general solution will take different forms involving exponential or trigonometric functions. Several examples are worked through. The document also discusses solving higher order differential equations and presents solutions to sample third and fourth order equations.
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1. Generating functions can be used to represent sequences as power series and solve recurrence relations.
2. Common examples of generating functions are presented for various sequences like the constant sequence {1,1,...}, the sequence of powers of 2, and binomial coefficients.
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2) Equations with complex roots have solutions involving sine and cosine of the complex terms.
3) Equations with repeated roots have solutions involving higher powers of the variable.
4) Examples are provided to illustrate the different cases.
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2) First order differential equations can sometimes be solved by separation of variables, or by finding an integrating factor. Homogeneous equations can be transformed by substitution.
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On the Fixed Point Extension Results in the Differential Systems of Ordinary ...
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1) The document presents information on ordinary differential equations including definitions, types, order, degree, and solution methods.
2) Differential equations can be written in derivative, differential, and differential operator forms. Common solution methods covered are variable separable, homogeneous, linear, and exact differential equations.
3) Applications of differential equations include physics, astronomy, meteorology, chemistry, biology, ecology, and economics for modeling various real-world systems.
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This document provides an overview of methods for solving higher order ordinary differential equations (ODEs), including variation of parameters and Laplace transforms. It presents the general steps for each method and works through examples of applying each method to solve higher order nonhomogeneous ODEs. The document also discusses applications of higher order ODEs to model physical systems like a bungee jumper attached to a cord. It works through several problems modeling the bungee jumper's motion to determine properties like the maximum jump height.
First order linear differential equation
1. A differential equation relates an unknown function and its derivatives, and can be ordinary (involving one variable) or partial (involving partial derivatives).
2. Linear differential equations have dependent variables and derivatives that are of degree one, and coefficients that do not depend on the dependent variable.
3. Common methods for solving first-order linear differential equations include separation of variables, homogeneous equations, and exact equations.
formulation of first order linear and nonlinear 2nd order differential equation
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
Differential equations
1. Differential equations are equations involving derivatives of an unknown function and can be of different orders. Separable differential equations can be expressed as the product of a function of x and a function of y.
2. The general solution or family of solutions to a differential equation represents all possible solutions as determined by initial or boundary conditions. Initial value problems find a particular solution satisfying given initial conditions.
3. Models of natural growth and decay can be represented by differential equations where the rate of change is proportional to the amount present, with solutions in the form of exponential functions. The logistic growth model accounts for limiting factors with a carrying capacity.
Ordinary differential equation
1. The document defines ordinary and partial differential equations and discusses the order and degree of differential equations.
2. Examples of common second order linear differential equations with constant coefficients are given, including equations for free fall, spring displacement, and RLC circuits.
3. The document also discusses homogeneous linear equations and Newton's law of cooling as examples of differential equations.
METHOD OF JACOBI
This document discusses iterative methods for solving systems of equations. It introduces the Jacobi iteration method and the Successive Over-Relaxation (SOR) method. SOR can accelerate the convergence compared to Jacobi by introducing an optimal relaxation parameter. Pseudocode is provided to implement SOR to iteratively solve a system of equations until the solution converges within a specified tolerance.
Solving recurrences
This is a Research and introductory document related to recurrence relations. Beneficial for Computer Science Students and Professionals.
Impact of Linear Homogeneous Recurrent Relation Analysis
A linear homogeneous recurrence relation of degree k with constant coefficients is a recurrence. A recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given each further term of the sequence or array is defined as a function of the preceding terms. Thidar Hlaing "Impact of Linear Homogeneous Recurrent Relation Analysis" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26662.pdfPaper URL: https://www.ijtsrd.com/computer-science/other/26662/impact-of-linear-homogeneous-recurrent-relation-analysis/thidar-hlaing
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The document discusses recurrence relations and their applications. It can be summarized as follows:
1. A recurrence relation expresses the terms of a sequence in terms of previous terms, providing a rule to generate all terms. It does not specify initial values.
2. Recurrence relations are useful for modeling many real-world phenomena like financial growth, partitions of sets, and algorithm analysis.
3. Linear homogeneous recurrence relations of constant coefficients can be solved by finding the characteristic roots from the characteristic equation. The general solution is a linear combination of terms with the roots.
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1) The document discusses modeling and applications of second order differential equations. It provides examples of second order differential equations that model vibrating springs and electric current circuits.
2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
3) An example solves a second order differential equation modeling a vibrating spring to find the position of a mass attached to the spring at any time.
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This document discusses higher order differential equations and their applications. It introduces second order homogeneous differential equations and their solutions based on the nature of the roots. Non-homogeneous differential equations are also discussed, along with their general solution being the sum of the solution to the homogeneous equation and a particular solution. Methods for solving non-homogeneous equations are presented, including undetermined coefficients and reduction of order. Applications to problems in various domains like physics, engineering, and circuits are also outlined.
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1. Generating functions can be used to represent sequences as power series and solve recurrence relations.
2. Common examples of generating functions are presented for various sequences like the constant sequence {1,1,...}, the sequence of powers of 2, and binomial coefficients.
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This document discusses methods for solving ordinary differential equations (ODEs) using power series solutions and the Frobenius method. The power series method assumes solutions of the form of a power series centered at an ordinary point. The Frobenius method extends this to regular singular points by assuming solutions of the form of a power series multiplied by (x-x0)^r, where r is determined from the indicial equation. The document outlines the steps for both methods, which involve substituting the assumed series into the ODE and equating coefficients of like powers of (x-x0).
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The document summarizes solutions to homogeneous linear differential equations with constant coefficients. It discusses:
1) Equations with real distinct roots have n distinct exponential solutions.
2) Equations with complex roots have solutions involving sine and cosine of the complex terms.
3) Equations with repeated roots have solutions involving higher powers of the variable.
4) Examples are provided to illustrate the different cases.
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The document discusses series solutions to second order linear differential equations near ordinary points. It provides an example of finding the series solution to the differential equation y'' + y = 0 near x0 = 0. The solution is found to be a cosine series which represents the cosine function, a fundamental solution. A second example finds the series solution to Airy's equation near x0 = 0, obtaining fundamental solutions related to Airy functions.
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This document discusses power series solutions and the Frobenius method for solving ordinary differential equations with variable coefficients. It explains that power series can be used to find solutions around ordinary points, while the Frobenius method extends this approach to regular singular points through generalized power series involving an index term. The Frobenius method involves making an ansatz for the solution as a power series with an unknown index, then determining the index and coefficients by substituting into the differential equation and setting terms of different powers of x equal to zero.
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2) First order differential equations can sometimes be solved by separation of variables, or by finding an integrating factor. Homogeneous equations can be transformed by substitution.
3) Second order linear differential equations can be reduced to a system of two first order equations. The complementary function and particular solutions combine to form the general solution. Unequal or equal roots of the characteristic equation determine the form of the complementary function.
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The document discusses higher order differential equations. It defines nth order differential equations and describes their general forms. For homogeneous equations, the general solution method involves making an operator form, constructing an auxiliary equation, solving for roots, and finding the complementary solution. For non-homogeneous equations, the method of undetermined coefficients is used to find a particular solution and the general solution is the sum of the complementary and particular solutions. Examples are provided to illustrate the solution methods.
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1) The document presents information on ordinary differential equations including definitions, types, order, degree, and solution methods.
2) Differential equations can be written in derivative, differential, and differential operator forms. Common solution methods covered are variable separable, homogeneous, linear, and exact differential equations.
3) Applications of differential equations include physics, astronomy, meteorology, chemistry, biology, ecology, and economics for modeling various real-world systems.
Ordinary Differential Equation
This document provides an overview of methods for solving higher order ordinary differential equations (ODEs), including variation of parameters and Laplace transforms. It presents the general steps for each method and works through examples of applying each method to solve higher order nonhomogeneous ODEs. The document also discusses applications of higher order ODEs to model physical systems like a bungee jumper attached to a cord. It works through several problems modeling the bungee jumper's motion to determine properties like the maximum jump height.
First order linear differential equation
1. A differential equation relates an unknown function and its derivatives, and can be ordinary (involving one variable) or partial (involving partial derivatives).
2. Linear differential equations have dependent variables and derivatives that are of degree one, and coefficients that do not depend on the dependent variable.
3. Common methods for solving first-order linear differential equations include separation of variables, homogeneous equations, and exact equations.
formulation of first order linear and nonlinear 2nd order differential equation
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
Differential equations
1. Differential equations are equations involving derivatives of an unknown function and can be of different orders. Separable differential equations can be expressed as the product of a function of x and a function of y.
2. The general solution or family of solutions to a differential equation represents all possible solutions as determined by initial or boundary conditions. Initial value problems find a particular solution satisfying given initial conditions.
3. Models of natural growth and decay can be represented by differential equations where the rate of change is proportional to the amount present, with solutions in the form of exponential functions. The logistic growth model accounts for limiting factors with a carrying capacity.
Ordinary differential equation
1. The document defines ordinary and partial differential equations and discusses the order and degree of differential equations.
2. Examples of common second order linear differential equations with constant coefficients are given, including equations for free fall, spring displacement, and RLC circuits.
3. The document also discusses homogeneous linear equations and Newton's law of cooling as examples of differential equations.
METHOD OF JACOBI
This document discusses iterative methods for solving systems of equations. It introduces the Jacobi iteration method and the Successive Over-Relaxation (SOR) method. SOR can accelerate the convergence compared to Jacobi by introducing an optimal relaxation parameter. Pseudocode is provided to implement SOR to iteratively solve a system of equations until the solution converges within a specified tolerance.
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Here are the steps to solve this recurrence relation using iteration:
1) Given: T(n) = T(n-1) + 1, T(1) = 1
2) Expand for T(2):
T(2) = T(1) + 1
= 1 + 1 = 2
3) Expand for T(3):
T(3) = T(2) + 1
= 2 + 1 = 3
4) Continue expanding:
T(4) = T(3) + 1 = 3 + 1 = 4
T(5) = T(4) + 1 = 4 + 1 = 5
5) Generalize the pattern:
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This document discusses recurrence relations and their solutions. It begins by defining recurrence relations and providing examples of recurrence relations modeling real-world problems. It then explains that linear homogeneous recurrence relations of degree k can be solved by finding the characteristic roots of the characteristic equation. For relations of degree two, the solution is a linear combination of the characteristic roots raised to successive powers. The document provides examples of solving specific recurrence relations, including deriving an explicit formula for the Fibonacci numbers.
pdf of recurrence relation which is used in statistics
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Maths
- 2. A recursive definition of a sequence specifies ◘ Initial conditions ◘ Recurrence relation a0=0 and a1=3 Initial conditions an = 2an-1 - an-2 Recurrence relation an = 3n Solution
- 3. Each term of a sequence is a linear function of earlier terms in the sequence. a0 = 1 a1 = 6 a2 = 10 an = an-1 + 2an-2 + 3an-3 a3 = a0 + 2a1 + 3a2 = 1 + 2(6) + 3(10) = 43
- 4. 1. Linear homogeneous recurrences 2. Linear non-homogeneous recurrences
- 5. A linear homogenous recurrence relation of degree k with constant coefficients is a recurrence relation of the form an = c1an-1 + c2an-2 + … + ckan-k, where c1, c2, …, ck are real numbers, and ck≠0 an is expressed in terms of the previous k terms of the sequences its degree is k. This recurrence includes k initial conditions. a0 = C0 a1 = C1 … ak = Ck
- 6. Determine if the following recurrence relations are linear homogeneous recurrence relations with constant coefficients. Pn = (1.11)Pn-1 a linear homogeneous recurrence relation of degree one an = an-1 + a2 n-2 not linear fn = fn-1 + fn-2 a linear homogeneous recurrence relation of degree two Hn = 2Hn-1+1 not homogeneous an = an-6 a linear homogeneous recurrence relation of degree six Bn = nBn-1 does not have constant coefficient
- 7. Let an = c1an-1 + c2an-2 + … + ckan-k be a linear homogeneous recurrence. Assume the sequence an satisfies the recurrence. Assume the sequence a’n also satisfies the recurrence. So, bn = an + a’n and dn=an are also sequences that satisfy the recurrence. (α is any constant) bn = an + a’n = (c1an-1 + c2an-2 + … + ckan-k) + (c1a’n-1 + c2a’n-2 + … + cka’n-k) = c1 (an-1+ a’n-1) + c2 (an-2+ a’n-2) + … + ck (an-k + a’n-k) = c1bn-1 + c2bn-2 + … + ckbn-k So, bn is a solution of the recurrence.
- 8. Let an = c1an-1 + c2an-2 + … + ckan-k be a linear homogeneous recurrence. Assume the sequence an satisfies the recurrence. Assume the sequence a’n also satisfies the recurrence. So, bn = an + a’n and dn=an are also sequences that satisfy the recurrence. (α is any constant) dn = α an = α (c1an-1 + c2an-2 + … + ckan-k) = c1 (α an-1) + c2 (α an-2) + … + ck (α an-k) = c1dn-1 + c2dn-2 + … + ckdn-k So, dn is a solution of the recurrence.
- 9. It follows from the previous proposition, if we find some solutions to a linear homogeneous recurrence, then any linear combination of them will also be a solution To the linear homogeneous recurrence.
- 10. Geometric sequences come up a lot when solving linear homogeneous recurrences. So, try to find any solution of the form an = rn that satisfies the recurrence relation.
- 11. Recurrence relation: an = c1an-1 + c2an-2 + … + ckan-k Try to find a solution of form rn rn = c1rn-1 + c2rn-2 + … + ckrn-k rn - c1rn-1 - c2rn-2 - … - ckrn-k = 0 rk - c1rk-1 - c2rk-2 - … - ck = 0 (dividing both sides by rn-k) This equation is called the characteristic equation
- 12. What is the solution of the recurrence relation an = an-1 + 2an-2 with a0=2 and a1=7? Since it is linear homogeneous recurrence, first find its characteristic equation r2 - r - 2 = 0 (r+1)(r-2) = 0 r1 = 2 and r2 = -1 So, by theorem an = α 12n + α 2(-1)n is a solution. Now we should find α 1 and α 2 using initial conditions. a0= α 1 + α 2 = 2 a1= α 12 + α 2(-1) = 7 So, α 1= 3 and α 2 = -1. an = 3 . 2n - (-1)n is a solution.
- 13. If the characteristic equation has k distinct solutions r1, r2, …, rk, it can be written as (r - r1)(r - r2)…(r - rk) = 0. If, after factoring, the equation has m+1 factors of (r - r1), for example, r1 is called a solution of the characteristic equation with multiplicity m+1. When this happens, not only r1 n is a solution, but also Nr1n n, n2r1 n, … and nmr1n are solutions of the recurrence.
- 14. When a characteristic equation has fewer than k distinct solutions: We obtain sequences of the form described in Proposition 3. By Proposition 1, we know any combination of these solutions is also a solution to the recurrence. We can find those that satisfies the initial conditions.
- 15. Consider the characteristic equation rk - c1rk-1 - c2rk-2 - … -ck = 0 and the recurrence an = c1an-1 + c2an-2 + … + ckan-k. Assume the characteristic equation has t ≤ k distinct solutions. Let Ɐi (1≤ i≤ t) ri with multiplicity mi be a solution of the equation. Let Ɐ i,j (1 ≤ I ≤ t and 0≤j≤mi-1) ɑij be a constant. So, an = (ɑ 10 + ɑ 11 n+ … + ɑ 1,m1-1 nm1-1 ) r1n + (ɑ 20 + ɑ 21 n+ … + ɑ 2,m2-1 nm2-1 ) r2n + … + (ɑ t0 + ɑ t1 n+ … + ɑ t,mt-1 nmt-1 ) rtn satisfies the recurrence
- 16. What is the solution of the recurrence relation an = 6an-1 - 9an-2 with a0=1 and a1=6? First find its characteristic equation r2 - 6r + 9 = 0 (r - 3)2 = 0 r1 = 3 (Its multiplicity is 2.) So, by theorem an = ( ɑ 10 + ɑ 11n)(3)n is a solution. Now we should find constants using initial conditions. a0= ɑ 10 = 1 a1= 3 ɑ 10 + 3ɑ 11 = 6 So, ɑ 11= 1 and ɑ 10 = 1. an = 3n + n3n is a solution