(1) Determine if it's practical to use Algorithm 8.3.12 on the following recursions, and explain why or why not. NOTE: YOU DO NOT NEED TO ACTUALLY FIND THE CLOSED FORM. (a) S(k) = S(k) + S(k − 1) + S(k − 2) (b) S(k) = S(k-1) +5 (c) S(k)= 9S(K-2) Algorithm 8.3.12 Algorithm for Solving Homogeneous Finite Order Linear Relations. (a) Write out the characteristic equation of the relation S(k) + C₂S(k-1) + ... + C₂S(k − n) = 0, which is a" +₁1++C-1ª+Cn=0. (b) Find all roots of the characteristic equation, the characteristic roots. (c) If there are n distinct characteristic roots, a, a,...an, then the general solution of the recurrence relation is S(k)=b₁a1* + b₂ª2* ++ b₂ªn*. If there are fewer than n characteristic roots, then at least one root is a multiple root. If a, is a double root, then the bja, term is replaced with (bjo+bjak) a. In general, if a, is a root of multiplicity p, then the b,a, term is replaced with (bjo+bak + + bjp-1)-¹) a. (d) If n initial conditions are given, we get n linear equations in n unknowns (the by's from Step 3) by substitution. If possible, solve these equations to determine a final form for S(k). Although this algorithm is valid for all values of n, there are limits to the size of n for which the algorithm is feasible. Using just a pencil and paper, we can always solve second-order equations. The quadratic formula for the roots of az²+bx+c=0 is -b± √b²-4ac 2a z = The solutions of a² + C₁a + C₂ = 0 are then (-C₁+√C₁²-40₂) and (-C₁ - √C₁² - 40₂) Although cubic and quartie formulas exist, they are too lengthy to introduce here. For this reason, the only higher-order relations (n ≥ 3) that you could be expected to solve by hand are ones for which there is an easy factorization of the characteristic polynomial.
(1) Determine if it's practical to use Algorithm 8.3.12 on the following recursions, and explain why or why not. NOTE: YOU DO NOT NEED TO ACTUALLY FIND THE CLOSED FORM. (a) S(k) = S(k) + S(k − 1) + S(k − 2) (b) S(k) = S(k-1) +5 (c) S(k)= 9S(K-2) Algorithm 8.3.12 Algorithm for Solving Homogeneous Finite Order Linear Relations. (a) Write out the characteristic equation of the relation S(k) + C₂S(k-1) + ... + C₂S(k − n) = 0, which is a" +₁1++C-1ª+Cn=0. (b) Find all roots of the characteristic equation, the characteristic roots. (c) If there are n distinct characteristic roots, a, a,...an, then the general solution of the recurrence relation is S(k)=b₁a1* + b₂ª2* ++ b₂ªn*. If there are fewer than n characteristic roots, then at least one root is a multiple root. If a, is a double root, then the bja, term is replaced with (bjo+bjak) a. In general, if a, is a root of multiplicity p, then the b,a, term is replaced with (bjo+bak + + bjp-1)-¹) a. (d) If n initial conditions are given, we get n linear equations in n unknowns (the by's from Step 3) by substitution. If possible, solve these equations to determine a final form for S(k). Although this algorithm is valid for all values of n, there are limits to the size of n for which the algorithm is feasible. Using just a pencil and paper, we can always solve second-order equations. The quadratic formula for the roots of az²+bx+c=0 is -b± √b²-4ac 2a z = The solutions of a² + C₁a + C₂ = 0 are then (-C₁+√C₁²-40₂) and (-C₁ - √C₁² - 40₂) Although cubic and quartie formulas exist, they are too lengthy to introduce here. For this reason, the only higher-order relations (n ≥ 3) that you could be expected to solve by hand are ones for which there is an easy factorization of the characteristic polynomial.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please help solve and explain. Thank you.
Transcribed Image Text:(1) Determine if it's practical to use Algorithm 8.3.12 on the following recursions, and explain why or why
not. NOTE: YOU DO NOT NEED TO ACTUALLY FIND THE CLOSED FORM.
(a) S(k) = S(k) + S(k − 1) + S(k − 2)
(b) S(k) = S(k-1) + 5
(c) S(k)= 95(k - 2)
U
Algorithm 8.3.12 Algorithm for Solving Homogeneous Finite Order
Linear Relations.
(a) Write out the characteristic equation of the relation S(k) + C₁S(k − 1) +
... + C₂S(kn) = 0, which is a" +C₁a-¹++Cn-1a+Cn=0.
(b) Find all roots of the characteristic equation, the characteristic roots.
(c) If there are n distinct characteristic roots, a1,02,...an, then the general
solution of the recurrence relation is S(k)= bịaik + b₂aşk +...+ b₂ªn*.
If there are fewer than n characteristic roots, then at least one root is a
multiple root. If a; is a double root, then the bjajk term is replaced with
(bj.o + bj.1k) a. In general, if a, is a root of multiplicity p, then the b,a,k
term is replaced with (bo+b₁.₁k++ bj.(p-1) k²−¹) aħj.
(d) If n initial conditions are given, we get n linear equations in n unknowns
(the b's from Step 3) by substitution. If possible, solve these equations
to determine a final form for S(k).
Although this algorithm is valid for all values of n, there are limits to the
size of n for which the algorithm is feasible. Using just a pencil and paper, we
can always solve second-order equations. The quadratic formula for the roots
of az²+bx+c=0 is
-b± √b²-4ac
2a
The solutions of a² + Cia + C₂ = 0 are then
(-C₁+√C₁²-40₂) and 1/(-0₁-√C₁²-40₂)
Although cubic and quartic formulas exist, they are too lengthy to introduce
here. For this reason, the only higher-order relations (n ≥ 3) that you could
be expected to solve by hand are ones for which there is an easy factorization
of the characteristic polynomial.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 5 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
