Analysis Tools: More on Recursive Relations

1
Analysis Tools More on Recursive Relations

• Lecture 4 Analysis Tools Recursive Relations
• Difference Equations
• Existence and Uniqueness of Solutions

• Lecture 5 Analysis Tools
• More on Recursive Relations
• Recap
• Linear Difference Equations
• 4 theorems about the nature of solutions to
  linear problems
• Linear Difference Equations with Constant
  Coefficients
• Characteristic Equation

Reference Introduction to Dynamic Systems,
D.G. Luenberger
2
Recap from Last Time

• Definitions
• Difference equation
• Linear function
• Linear difference equation
• Order
• Coefficients
• Time-invariant equations
• Forcing term
• Solution

• Theorems
• Structure of a linear difference equation
• Existence and uniqueness of solutions

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Linear Difference Equations

• A linear difference equation, given by
• is said to be homogeneous if g(k)0 for all k in
  the set over which the equation is defined.

Note that 1) The difference of two solutions to
the nonhomogeneous equation must satisfy
the associated homogeneous equation, 2)
The sum of a solution of the nonhomogeneous
equation and a solution of the associated
homogeneous equation is a solution of the
nonhomogeneous equation.
4
Linear Difference Equations

• Theorem 1 (A solution is composed of two parts)
  Let y(k) be a given solution to the linear
  difference equation
• Then the collection of all solutions to this
  equation is the collection of all functions of
  the form y(k)y(k)z(k), where z(k) is a
  solution of the corresponding homogeneous
  equation.

5
Linear Difference Equations

• Theorem 2 (Linear combinations of homogeneous
  solutions are solutions) If z1(k), z2(k), ,
  zm(k) are solutions to the homogeneous equation
• then any linear combination of these m solutions
• where c1, c2, , cm are arbitrary constants, is
  also a solution of the homogeneous equation.

This theorem says linear combinations of
solutions are solutions, but it does not say all
solutions are found this way.
Proof Do it!
6
Linear Difference Equations

• Define a special set of solutions to the
  homogeneous equation corresponding to the n
  fundamental initial conditions
• Call this set of solutions z1,,zn the
  fundamental set of solutions.

...
7
Linear Difference Equations

• Theorem 3 (Every homogeneous solution is a
  linear combination of the n fundamental
  solutions) If z(k) is any solution to the
  homogeneous equation,
• then z(k) can be expressed in terms of the n
  fundamental solutions, z1(k),,zn(k), of the form
• for some constants c1, c2, , cn.

This theorem says all solutions of the
homogeneous equation can be computed from the n
solutions generated by the fundamental initial
conditions.
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Linear Difference Equations

• Theorem 3 (Every homogeneous solution is a
  linear combination of the n fundamental
  solutions) If z(k) is any solution to the
  homogeneous equation,
• then z(k) can be expressed in terms of the n
  fundamental solutions, z1(k),,zn(k), of the form
• for some constants c1, c2, , cn.

9
Linear Difference Equations
Definition Given a finite set of functions
z1(k),,zm(k) defined for a set of integers,
say k0, 1, 2, , N, we say that these functions
are linearly independent if it is impossible to
find a relation of the form valid for all k0,
1, 2, , N, except by setting c1c2cn0.

• Theorem 4 (any n linearly independent solutions
  will do) Suppose z1(k),,zn(k) is a linearly
  independent set of solutions to the homogeneous
  equation
• Then any solution z(k) can be expressed as a
  linear combination
• for some constants c1, c2, , cn.

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Linear Difference Equations

• Put it all together for a general method to
  solve a nonhomogeneous equation of the form
• which satisfies a given set of initial
  conditions.

• Find a set of n linearly independent solutions
  to
• the corresponding homogeneous equation,
• 2) Find a particular solution to the
  nonhomogeneous
• equation that does not necessarily satisfy the
  given
• initial conditions, and
• Modify the particular solution by adding a
  linear
• combination of the homogeneous solutions such
  that
• the given initial conditions are satisfied.

11
Linear Equations with Constant Coefficients

• For this special case, we can find all solutions
  to the homogeneous equation
• The key result is that there exists a geometric
  sequence that satisfies the homogeneous equation

Characteristic Polynomial
12
Linear Equations with Constant Coefficients

• Fundamental Theorem of Algebra says any
  polynomial of degree n has exactly n roots.
  Three cases for l
• P(l) has distinct roots. General 2nd order
  solution given by
• P(l) has repeated roots. General 2nd order
  solution given by
• P(l) has complex roots. Will always appear in
  complex conjugate pairs since the coefficients in
  the polynomial are real-valued. General 2nd
  order solution given by

13
Linear Equations with Constant Coefficients

• Example

14
Linear Equations with Constant Coefficients

• Example (Fibonacci Sequence)