power (^,.^) power operation

## Syntax

                t = A ^ b
                t = A ** b
                t = A .^ b

## Arguments

A, t
  a scalar, vector, or matrix of encoded integers, decimal or complex numbers, polynomials, or rationals.

b
  a scalar, vector, or matrix of encoded integers, decimal or complex numbers.

If an operand are encoded integers, the other one can be only encoded integers or real numbers.

If `A` are polynomials or rationals, `b` can only be a single decimal (positive or negative) integer.

## Description

### .^ by-element power

If `A` or `b` is scalar, it is first replicated to the size of the other, with A*ones(b) or b*ones(A). Otherwise, `A` and `b` must have the same size.

Then, for each element of index i, `t(i) = A(i)^b(i)` is computed.

### ^ matricial power

The ^ operator is equivalent to the .^ by-element power in the following cases:

- `A` is scalar and `b` is a vector.
- `A` is a vector and `b` is scalar.

Otherwise, `A` or `b` must be a scalar, and the other one must be a square matrix:

- If `A` is scalar and `b` is a square matrix, then `A^b` is the matrix `expm(log(A) * b)`

- If `A` is a square matrix and `b` is scalar, then `A^b` is the matrix `A` to the power `b`.

### Remarks

1. For square matrices `A^p` is computed through successive matrices multiplications if `p` is a positive integer, and by diagonalization if not (see "note 2 and 3" below for details).

2. If `A` is a square and Hermitian matrix and `p` is a non-integer scalar, `A^p` is computed as:

   `A^p = u*diag(diag(s).^p)*u'` (For real matrix `A`, only the real part of the answer is taken into account).

   `u` and `s` are determined by `[u,s] = schur(A)` .

3. If `A` is not a Hermitian matrix and `p` is a non-integer scalar, `A^p` is computed as:

   `A^p = v*diag(diag(d).^p)*inv(v)` (For real matrix `A`, only the real part of the answer is taken into account).

   `d` and `v` are determined by `[d,v] = bdiag(A+0*%i)`.

4. If `A` and `p` are real or complex numbers, `A^p` is the principal value determined by

   `A^p = exp(p*log(A))`

   (or `A^p = exp(p*(log(abs(A))+ %i*atan(imag(A)/real(A))))` ).

5. If `A` is a square matrix and `p` is a real or complex number, `A.^p` is the principal value computed as:

   `A.^p = exp(p*log(A))` (same as case 4 above).

6. `**` and `^` operators are synonyms.

Exponentiation is right-associative in Scilab, contrarily to Matlab® and Octave. For example 2^3^4 is equal to 2^(3^4) in Scilab, but to (2^3)^4 in Matlab® and Octave.

## Examples

    A = [1 2 ; 3 4];
    A ^ 2.5,
    A .^ 2.5
    (1:10) ^ 2
    (1:10) .^ 2

    A ^ %i
    A .^ %i
    exp(%i*log(A))

    s = poly(0,'s')
    s ^ (1:10)

## See also

- exp
- expm
- hat
- inv

## History

6.0.0 With decimal or complex numbers, `scalar ^ squareMat` now yields `expm(log(scalar)*squareMat)` instead of `scalar .^ squareMat`