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, 4.2Cayley-Hamilton TheoremA 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 MethodZIEBUR'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): 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.SlidesCayley-Hamilton topics, Section 6.3.Cayley-Hamilton TheoremA 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]: 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)Slides