Root

sqrt(x)

Return $\sqrt{x}$.

Throws DomainError for negative Real arguments. Use complex negative arguments instead. Note that sqrt has a branch cut along the negative real axis.

The prefix operator √ is equivalent to sqrt.

See also: hypot.

Examples

julia> sqrt(big(81))
9.0

julia> sqrt(big(-81))
ERROR: DomainError with -81.0:
NaN result for non-NaN input.
Stacktrace:
 [1] sqrt(::BigFloat) at ./mpfr.jl:501
[...]

julia> sqrt(big(complex(-81)))
0.0 + 9.0im

julia> sqrt(-81 - 0.0im)  # -0.0im is below the branch cut
0.0 - 9.0im

julia> .√(1:4)
4-element Vector{Float64}:
 1.0
 1.4142135623730951
 1.7320508075688772
 2.0

sqrt(A::AbstractMatrix)

If A has no negative real eigenvalues, compute the principal matrix square root of A, that is the unique matrix $X$ with eigenvalues having positive real part such that $X^2 = A$. Otherwise, a nonprincipal square root is returned.

If A is real-symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the square root. For such matrices, eigenvalues λ that appear to be slightly negative due to roundoff errors are treated as if they were zero. More precisely, matrices with all eigenvalues ≥ -rtol*(max |λ|) are treated as semidefinite (yielding a Hermitian square root), with negative eigenvalues taken to be zero. rtol is a keyword argument to sqrt (in the Hermitian/real-symmetric case only) that defaults to machine precision scaled by size(A,1).

Otherwise, the square root is determined by means of the Björck-Hammarling method ^[BH83], which computes the complex Schur form (schur) and then the complex square root of the triangular factor. If a real square root exists, then an extension of this method ^[H87] that computes the real Schur form and then the real square root of the quasi-triangular factor is instead used.

Examples

julia> A = [4 0; 0 4]
2×2 Matrix{Int64}:
 4  0
 0  4

julia> sqrt(A)
2×2 Matrix{Float64}:
 2.0  0.0
 0.0  2.0

isqrt(n::Integer)

Integer square root: the largest integer m such that m*m <= n.

julia> isqrt(5)
2

cbrt(x::Real)

Return the cube root of x, i.e. $x^{1/3}$. Negative values are accepted (returning the negative real root when $x < 0$).

The prefix operator ∛ is equivalent to cbrt.

Examples

julia> cbrt(big(27))
3.0

julia> cbrt(big(-27))
-3.0

cbrt(A::AbstractMatrix{<:Real})

Computes the real-valued cube root of a real-valued matrix A. If T = cbrt(A), then we have T*T*T ≈ A, see example given below.

If A is symmetric, i.e., of type HermOrSym{<:Real}, then (eigen) is used to find the cube root. Otherwise, a specialized version of the p-th root algorithm ^[S03] is utilized, which exploits the real-valued Schur decomposition (schur) to compute the cube root.

Examples

julia> A = [0.927524 -0.15857; -1.3677 -1.01172]
2×2 Matrix{Float64}:
  0.927524  -0.15857
 -1.3677    -1.01172

julia> T = cbrt(A)
2×2 Matrix{Float64}:
  0.910077  -0.151019
 -1.30257   -0.936818

julia> T*T*T ≈ A
true