power

.^ 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.

• 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.