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 matrixAand a real numbert, the matrix-exponentialeis defined via the Taylor series for the exponential function:^{tA}

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

Obviously, the matrixAmust 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 thatXis a root offif [tex]f(X)= \mathbf{0}[/tex], where0is the matrix of all zeroes. Assume that the matrix arguments offare square and invertible.

(b) The square root of a matrixAis a matrixXsuch that [tex]X^T\, X= A.[/tex] For a symmetric, positive-definite matrixA, 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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