trapz(x::AbstractMesh, y::AbstractVector{T}) where {T<:N64}

Perform numerical integration by using the composite trapezoidal rule.

Arguments

• x -> Real frequency mesh.
• y -> Function values at real axis.

Returns

• ℐ -> The final value.

See also: simpson.

trapz(
    x::AbstractVector{S},
    y::AbstractVector{T},
    linear::Bool = false
) where {S<:Number, T<:Number}

Perform numerical integration by using the composite trapezoidal rule. Note that it supports arbitrary precision via BigFloat.

Arguments

• x -> Real frequency mesh.
• y -> Function values at real axis.
• linear -> Whether the given mesh is linear?

Returns

• ℐ -> The final value.

See also: simpson.

simpson(
    x::AbstractVector{S},
    y::AbstractVector{T}
) where {S<:Number, T<:Number}

Perform numerical integration by using the simpson rule. Note that the length of x and y must be odd numbers. And x must be a linear and uniform mesh.

Arguments

• x -> Real frequency mesh.
• y -> Function values at real axis.

Returns

• ℐ -> The final value.

See also: trapz.

@einsum(ex)

Perform Einstein summation like operations on Julia Arrays.

Examples

Basic matrix multiplication can be implemented as:

@einsum A[i, j] := B[i, k] * C[k, j]

If the destination array is preallocated, then use =:

A = ones(5, 6, 7) # Preallocated space
X = randn(5, 2)
Y = randn(6, 2)
Z = randn(7, 2)

# Store the result in A, overwriting as necessary:
@einsum A[i, j, k] = X[i, r] * Y[j, r] * Z[k, r]

If destination is not preallocated, then use := to automatically create a new array for the result:

X = randn(5, 2)
Y = randn(6, 2)
Z = randn(7, 2)

# Create new array B with appropriate dimensions:
@einsum B[i, j, k] := X[i, r] * Y[j, r] * Z[k, r]