Scilab Home page | Wiki | Bug tracker | Forge | Mailing list archives | ATOMS | File exchange
Please login or create an account
Change language to: Français - Português - 日本語 - Русский
Scilab Help >> Elementary Functions > Log - exp - power > power

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 — element-wise exponential
  • expm — square matrix exponential
  • hat — (^) exponentiation

History

VersionDescription
6.0.0 With decimal or complex numbers, scalar ^ squareMat now yields expm(log(scalar)*squareMat) instead of scalar .^ squareMat
Scilab Enterprises
Copyright (c) 2011-2017 (Scilab Enterprises)
Copyright (c) 1989-2012 (INRIA)
Copyright (c) 1989-2007 (ENPC)
with contributors
Last updated:
Tue Jul 20 11:10:42 CEST 2021