Picard Iterations


def picard_iteration(f, a, c, iterations):
    if iterations == 0:
        return c
    if iterations == 1:
        x0 = lambda t: c + integral(f(t=s,x=c), s, a, t)
        return expand(x0(t))
    for i in range(iterations):
        x_old = lambda s: picard_iteration(f, a, c, iterations-1).subs(t=s)
        x0 = lambda t: c + integral(f(t=s,x=x_old(s)), s, a, t)
    return expand(x0(t))

def picarder(n_iterations = slider(0,20,1,default = 2)):
    var('x,t,s')
    f = lambda t,x:x
    html(r"Exact solution to $x' = x$, $x(0) = 1$ is $x = \exp(t)$
")
    pic = picard_iteration(f,0,1,n_iterations)
    html('The Picard iteration approximation after ' + str(n_iterations) + ' iterations is:
')
    html('$'+latex(pic)+'$')
    exact = plot(exp(t),(t,0,2))
    pic_plot = plot(pic,(t,0,2), rgbcolor = (1,0,0))
    show(exact + pic_plot)

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This tutorial corresponds to the matlab “m” files that are posted on the APMA 0340 website. You, as the user, are free to use the m files to your needs for learning how to use the matlab program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately. Any comments and/or contributions for this tutorial are welcome; you can send your remarks to <Vladimir_Dobrushkin@brown.edu>

Two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix S such that
Theorem: If λ is an eigenvalue of a square matrix A, then its algebraic multiplicity is at least as large as its geometric multiplicity.    ▣
Let x1, x2, … , xr be all of the linearly independent eigenvectors associated to λ, so that λ has geometric multiplicity r. Let xr+1, xr+2, … , xn complete this list to a basis for ℜn, and let S be the n×n matrix whose columns are all these vectors xs, s = 1, 2, … , n. As usual, consider the product of two matrices AS. Because the first r columns of S are eigenvectors, we have
\[ {\bf A\,S} = \begin{bmatrix} \vdots & \vdots&& \vdots & \vdots&& \vdots \\ \lambda{\bf x}_1 & \lambda{\bf x}_2 & \cdots & \lambda{\bf x}_r & ?& \cdots & ? \\ \vdots & \vdots&& \vdots & \vdots&& \vdots \end{bmatrix} . \]
Now multiply out S-1AS. Matrix S is invertible because its columns are a basis for ℜn. We get that the first r columns of S-1AS are diagonal with &lambda's on the diagonal, but that the rest of the columns are indeterminable. Now S-1AS has the same characteristic polynomial as A. Indeed,
\[ \det \left( {\bf S}^{-1} {\bf AS} - \lambda\,{\bf I} \right) = \det \left( {\bf S}^{-1} {\bf AS} - {\bf S}^{-1} \lambda\,{\bf I}{\bf S} \right) = \det \left( {\bf S}^{-1} \left( {\bf A} - \lambda\,{\bf I} \right) {\bf S} \right) = \det \left( {\bf S}^{-1} \right) \det \left( {\bf A} - \lambda\,{\bf I} \right) \det \det \left( {\bf S} \right) = \det \left( {\bf A} - \lambda\,{\bf I} \right) \]
because the determinants of S and S-1 cancel. So the characteristic polynomials of A and S-1AS are the same. But since the first few columns of S-1AS has a factor of at least (x - λ)r, so the algebraic multiplicity is at least as large as the geometric.    ◂

 

  2. Keskin, A.U., Ordinary Differential Equations for Engineers: with MATLAB Solutions, Springer; 1st ed. 2019. ISBN-13: 978-3030069995 ISBN-10: 3030069990

Glossary