powerplus (version 3.1)

Matpow: Matrix Power

Description

Raises a valid Matrix to any power (even complex). Valid matrices are square matrices that are diagonalizable or whose real eigenvalues are positive.

Usage

Matpow(M, numer, denom = 1, expmethod = "Higham08.b",
  logmethod = "Higham08", tol = 1e-12)

Arguments

M

a square matrix

numer

numerator of exponent. Can be a decimal or complex number.

denom

denominator of exponent (1 by default). Can be a decimal or complex number.

expmethod

method chosen to compute the matrix exponential if matrix is known to be non-diagonalizable. The default method is the same as in function expm from package expm.

logmethod

method chosen to compute the matrix logarithm if matrix is known to be non-diagonalizable. The default method is the same as in function logm from package expm.

tol

a tolerance, \(10^{-12}\) by default. Prevents possible numerical problems. Can be set to 0 if desired.

Value

The solution to the exponentiation operation supplied. For diagonalizable matrices, Matpow returns a real-valued root whenever possible (otherwise, the principal complex root).

Details

If the matrix is diagonalizable, the method used is based on spectral decomposition; if the matrix is not diagonalizable, the method used is based on matrix exponentials and logarithms, calling functions matexp and matlog, both from package complexplus. The particular method used to compute the matrix exponential and logarithm may be chosen from the options available in functions expm and logm respectively, both from package expm. Note that Matpow, by extension, allows one to compute roots and the matrix inverse (if invertible).

References

For more on spectral decomposition (also known as eigendecomposition), visit http://mathworld.wolfram.com/EigenDecomposition.html

See Also

matexp matlog expm logm

Examples

A <- matrix(1:4, ncol = 2)

Matpow(A, 3)
Matpow(A, 0.5)
Matpow(A, 0.2)
Matpow(A, 1, 5)
Matpow(A, 2, 4, expmethod = "Pade", logmethod = "Eigen") #inocuous, as A is diagonalizable
Matpow(A, -1)
Matpow(A, 2+5i)
Matpow(A, 3i)
Matpow(A, 1+2i)
Matpow(A, 3i, 2+7i)

B <- matrix(sample(1:100, 81), ncol = 9)
Matpow(B, 2)
Matpow(B, 0.5)
Matpow(B, 7+2i)
Matpow(B, 4i, 1+3i)

C <- matrix(c(1, 0, 1, 1), ncol = 2) # A non-diagonalizable matrix
Matpow(C, 3)
Matpow(C, 0.5)
Matpow(C, 4, 8, expmethod = "Taylor", logmethod = "Eigen")
Matpow(C, 0.5*pi)
Matpow(C, 0.24)
Matpow(C, -2)
Matpow(C, 3+5i)
Matpow(C, 2i, 1+9i)