Answered: For an n × n matrix A, we can similarly…

For an n × n matrix A, we can similarly define the matrix exponential aseA = I + A + (1/2!)A^2 + (1/3!)A^3 + · · · . a) For the matrix D = 3 0 0 5

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1/2, 1/3, 1/4, 1/5, ... recursively. Include initial condition. аn — аn-1 + n+1' ª1 =1/2 n+1 аn — аn-1 + a1 = 1/2 n n An = -аn-1, aj — 1/2 n+1 n+1 -An-1, a1 =1/2 An
(3) The exponential of a number x can be computed through the Taylor series e=1+2+2/2++"/n!+... e² = 1 + x + x² / 2 + ··· + x^ /n! + ... The exponential of a square matrix A is defined by the Taylor series: A² A³ 2 3! An e₁ = I + A+ + + + +... where I is the identity matrix of the same size as A. n! (a) Find e for 0 2 3 B == 004 000 As you will learn, the solution of the system of first order differential equations d x= Ax dt with initial condition (0) Fois = x(t) = e¹¹ão, where et=I+ At+ (At)² (At)³ (At)" + + + + 2! 3! n! A square matrix A is said to be diagonal, if all of its entries outside of the main diagonal, (a), are equal to zero. If A is a diagonal matrix, then it is very easy to calculate e, For example, let Then eA - (61) + ( 0 -(32) (1+3+ (3)² + .. -(63) 0 A 1-(2) + ( 3 0 02 2! 0 (2)2 + 3! +. 2! 0 3! 0 1+2+(2)²+ -1 (b) What is e, where D = 0 40 ? 0 00
5 (a) Consider the positive integer s = 23n-1 n=2 (i) Express s as a sum without using E-notation. (ii) Give s in base 2.
The Taylor series of the function 1/1+x if x = 0 is given "x" ... = Use this to find the Taylor row if x = 0 for the function: n
6.4 Find the relationship between the A(k), B(k), and X(k), where X(k) are the expo- nential expansion coefficients and A(k) and B(k) are the trigonometric expansion coefficients for the discrete Fourier series of a real periodic sequence.
For each of the time series models below undertake the following steps: • Write each time series in characteristic polynomial form using backshift operator (MA or AR or ARMA polynomials as required) • Factorise the relevant polynomials and find the roots of the polynomials. • You are given the following additional information regarding the relationship between stationarity and the characteristic polynomial roots. An AR(p) process is a stationary process if and only if the modulus of all the roots of the AR characteristic polynomial are greater than one. An MA(q) process is always stationary if the MA coefficients are finite. An ARMA (p,q) model admits a unique stationary solution when the AR characteristic poly- nomial and the MA characteristic polynomial have no common roots and the AR polynomial satisfies that it has no unit roots. Use this information to determine which of the below models is stationary. (Make sure to justify your answer.) (a) VY + ²3 (Yt + 5Y₁-1) = € - €t-1 − −2+ 3…
Use known Taylor series to calculate the In (x³ + 1) dx easily. Write the answer in summation notation.
Q7.4 Find the taylor series for fcx) = centered at a= 2 %3D (A) 1y"-1 (n+ 1) 22 00 (-1y+1 -(x-2)" 2 n=0 (x-2)" (B) n=0 (D) E 2-1 (x- 2) n=0 (-1)*-1 (-1)" (n + 1) (x – 2y" (F) 00 (E) (-1у (x- 2y? 2n-1 n=0 n=0 (G) E (-1)"- (n+ 1) (x - 2y 2-1 n=0 Chose 1 of the given options as your answer!
1. Find the first three nonzero terms for the Taylor series of f(x) = x-² about a = -1. A. 1+2x+3x² OB. 1-2x+3x² OC. 1+ 2(x+1)+3(x + 1)² OD. 1-2(x+1)+3(x + 1)² OE. None of these Entered Answer Preview B B The answer above is NOT correct.