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2250 8:05am Lectures Week 10 S2018

Last Modified: March 26, 2018, 11:35 MDT.    Today: September 22, 2018, 12:38 MDT.
 Numerical Methods for Systems: Section 4.3

 Review of Scalar methods 
   Euler, Heun, RK4

 Change the methods to the vector case, based upon the statement of
 the IVP in Picard's Theorem. It amounts to putting arrows over y and f.

 Vector Methods
   Euler, Heun, RK4
   Manuscript: Vector Methods

 EXAMPLE. Solve the vector problem u'=Au, u(0)=u_0 with A=Matrix([[3,-2],[5,-4]]) and
           u_0 = <3,6> using Euler's method and Heuns's method with step size h=0.1.
           This is the scalar problem x'=3x-2y, y'=5x-4y, x(0)=3, y(0)=6.
Details: PDF: maple Example 1, section 4.3
  Cayley-Hamilton-Ziebur Method: Ch4, sections 4.1, 4.2

 Cayley-Hamilton Theorem
   A matrix satisfies its own characteristic equation.
   ILLUSTRATION: det(A-r I)=0 for the previous example
     is (2-r)(3-r)=0 or r^2 -5r + 6=0. Then C-H says
        (2I-A)(3I-A)=0 or A^2 - 5A + 6I = 0.
Cayley-Hamilton-Ziebur Method
  ZIEBUR'S LEMMA.
        The components of u in u'=Au are linear combinations of
        the atoms created by Euler's theorem applied to the
        roots of the characteristic equation det(A-rI)=0.
  THEOREM. Solve u'=Au without complex numbers or eigenanalysis.
        The solution of u'=Au is a linear combination of atoms
        times certain constant vectors [not arbitrary vectors].
             u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n)
Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (0.0 K, pdf, 31 Dec 1969) PROBLEM: Solve by Cayley-Hamilton-Ziebur the 2x2 dynamical system x' = 2x + y, y' = 3y, x(0)=1, y(0)=2. The characteristic equation is (2-lambda)(3-lambda)=0 with roots lambda = 2,3 Euler's theorem implies the atoms are exp(2t), exp(3t). Ziebur's Theorem says that u(t) = exp(2t) vec(v_1) + exp(3t) vec(v_2) where vectors v_1, uv_2 are to be determined from the matrix A = matrix([[2,1],[0,3]]) and initial conditions x(0)=1, y(0)=2. ZIEBUR ALGORITHM. To solve for v_1, v_2 in the example, differentiate the equation u(t) = exp(2t) v_1 + exp(3t) v_2 and set t=0 in both relations. Then u'=Au implies u_0 = v_1 + v_2, Au_0 = 2 v_1 + 3 v_2. These equations can be solved by elimination. The answer: v_1 = (3 u_0 -Au_0), v_2 = (Au_0 - 2 u_0) = vector([-1,0]) = vector([2,2]) Vectors v_1, v_2 are recognized as eigenvectors of A for lambda=2 and lambda=3, respectively. ZIEBUR SHORTCUT [Edwards-Penney textbook method, Example 5 in 4.1] Start with Ziebur's theorem, which implies that x(t) = k1 exp(2t) + k2 exp(3t). Use the first DE to solve for y(t): y(t) = x'(t) - 2x(t) = 2 k1 exp(2t) + 3 k2 exp(3t) - 2 k1 exp(2t) - 2 k2 exp(3t)) = k2 exp(3t) For example, x(0)=1, y(0)=2 implies k1 and k2 are defined by k1 + k2 = 1, k2 = 2, which implies k1 = -1, k2 = 2, agreeing with a previous solution formula. Cayley-Hamilton topics, Section 6.3. Cayley-Hamilton Theorem A matrix satisfies its own characteristic equation. Proof of the Ziebur Lemma for 2x2 matrices.
Review of Eigenanalysis
  Eigenvalue
  Eigenvector
  Eigenpair
  Fourier's Model
  Diagonalizable matrix

  Main theorem for solving u'=Au by eigenanalysis
  Example 1. Consider the 2x2 system
    x'=x+3y, y'=2y, x(0)=1, y(0)=-1.
    Method 1: Linear integrating factor method for triangular systems
    Method 2: Cayley-Hamilton-Ziebur, textbook shortcuts applied to 2x2
    Method 3: Eigenanalysis
    Method 4: Laplace resolvent
    Method 5: Exponential matrix, maple or Putzer's method [upcoming method]
 
Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (143.5 K, pdf, 16 Mar 2018)
Slides: Laplace resolvent method (0.0 K, pdf, 31 Dec 1969)
Slides: Matrix Exponential, Putzer Formula, Variation Parameters (122.0 K, pdf, 14 Mar 2016)