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

[snehahl]

Please let me know how to solve the question below:

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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 e^tA is defined via the Taylor series for the exponential function:

. . . . .$\displaystyle e^{tA}\, =\, 1\, +\, tA\, +\, \dfrac{(tA)^2}{2!}\, +\, \dfrac{(tA)^3}{3!}\, +\, \dfrac{(tA)^4}{4!}\, +\, ...$

Obviously, the matrix A must be square.

(a) Derive Newton's method for finding the root of an arbitrary matrix-valued function $\displaystyle f =f(X)$, where by "root" we mean that X is a root of f if $\displaystyle f(X)= \mathbf{0}$, 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 $\displaystyle X^T\, X= A.$ For a symmetric, positive-definite matrix A, derive the Newton iteration for finding \(\displaystyle X= \sqrt{\strut A\,}.\)

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

. . . . .$\displaystyle A\, =\, \left(\begin{array}{cccc}8&4&2&1\\4&8&4&2\\2&4&8&4\\1&2&4&8\end{array}\right)$

The stopping criterion for your Newton iteration should be when the absolute difference between elements of successive iterations is at most $\displaystyle 10^{-10}.$

 

[Deleted member 4993]

Computer program for Newton's method

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 e^tA is defined via the Taylor series for the exponential function:

. . . . .$\displaystyle e^{tA}\, =\, 1\, +\, tA\, +\, \dfrac{(tA)^2}{2!}\, +\, \dfrac{(tA)^3}{3!}\, +\, \dfrac{(tA)^4}{4!}\, +\, ...$

Obviously, the matrix A must be square.

(a) Derive Newton's method for finding the root of an arbitrary matrix-valued function $\displaystyle f =f(X)$, where by "root" we mean that X is a root of f if $\displaystyle f(X)= \mathbf{0}$, 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 $\displaystyle X^T\, X= A.$ For a symmetric, positive-definite matrix A, derive the Newton iteration for finding \(\displaystyle X= \sqrt{\strut A\,}.\)

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

. . . . .$\displaystyle A\, =\, \left(\begin{array}{cccc}8&4&2&1\\4&8&4&2\\2&4&8&4\\1&2&4&8\end{array}\right)$

The stopping criterion for your Newton iteration should be when the absolute difference between elements of successive iterations is at most $\displaystyle 10^{-10}.$

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