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Thread: Newton's Method for matrix-valued functions; finding square root of given matrix

  1. #1

    Newton's Method for matrix-valued functions; finding square root of given matrix

    Please let me know how to solve the question below:



    1. Newton's method can be extended to matrix-functions as well. For example, given a square matrix A and a real number t, the matrix-exponential etA is defined via the Taylor series for the exponential function:

    . . . . .[tex]e^{tA}\, =\, 1\, +\, tA\, +\, \dfrac{(tA)^2}{2!}\, +\, \dfrac{(tA)^3}{3!}\, +\, \dfrac{(tA)^4}{4!}\, +\, ...[/tex]

    Obviously, the matrix A must be square.

    (a) Derive Newton's method for finding the root of an arbitrary matrix-valued function [tex]f =f(X)[/tex], where by "root" we mean that X is a root of f if [tex]f(X)= \mathbf{0}[/tex], where 0 is the matrix of all zeroes. Assume that the matrix arguments of f are square and invertible.

    (b) The square root of a matrix A is a matrix X such that [tex]X^T\, X= A.[/tex] For a symmetric, positive-definite matrix A, derive the Newton iteration for finding [tex]X= \sqrt{\strut A\,}.[/tex]

    (c) Write a program using the Newton iteration you derived above to find the square root of the matrix below:

    . . . . .[tex]A\, =\, \left(\begin{array}{cccc}8&4&2&1\\4&8&4&2\\2&4&8&4 \\1&2&4&8\end{array}\right)[/tex]

    The stopping criterion for your Newton iteration should be when the absolute difference between elements of successive iterations is at most [tex]10^{-10}.[/tex]
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    Last edited by stapel; 02-16-2018 at 04:36 PM. Reason: Typing out the text in the graphic; creating useful subject line.

  2. #2
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    Computer program for Newton's method

    Quote Originally Posted by snehahl View Post
    Please let me know how to solve the question below:



    1. Newton's method can be extended to matrix-functions as well. For example, given a square matrix A and a real number t, the matrix-exponential etA is defined via the Taylor series for the exponential function:

    . . . . .[tex]e^{tA}\, =\, 1\, +\, tA\, +\, \dfrac{(tA)^2}{2!}\, +\, \dfrac{(tA)^3}{3!}\, +\, \dfrac{(tA)^4}{4!}\, +\, ...[/tex]

    Obviously, the matrix A must be square.

    (a) Derive Newton's method for finding the root of an arbitrary matrix-valued function [tex]f =f(X)[/tex], where by "root" we mean that X is a root of f if [tex]f(X)= \mathbf{0}[/tex], where 0 is the matrix of all zeroes. Assume that the matrix arguments of f are square and invertible.

    (b) The square root of a matrix A is a matrix X such that [tex]X^T\, X= A.[/tex] For a symmetric, positive-definite matrix A, derive the Newton iteration for finding [tex]X= \sqrt{\strut A\,}.[/tex]

    (c) Write a program using the Newton iteration you derived above to find the square root of the matrix below:

    . . . . .[tex]A\, =\, \left(\begin{array}{cccc}8&4&2&1\\4&8&4&2\\2&4&8&4 \\1&2&4&8\end{array}\right)[/tex]

    The stopping criterion for your Newton iteration should be when the absolute difference between elements of successive iterations is at most [tex]10^{-10}.[/tex]
    What are your thoughts?

    Please share your work with us ...even if you know it is wrong.

    If you are stuck at the beginning tell us and we'll start with the definitions.

    You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

    http://www.freemathhelp.com/forum/announcement.php?f=33
    Last edited by stapel; 02-16-2018 at 04:36 PM. Reason: Copying typed-out graphical content into reply.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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