ComplexModulus.derivative¶

ComplexModulus.derivative(self, x)[source]¶

Return the derivative operator in the “C = R^2” sense.

The returned operator (self) is the derivative of the operator variant where the complex domain is reinterpreted as a product of two real spaces:

M'(x) = y --> ((Re(x) * Re(y) + Im(x) * Im(y)) /
               sqrt(Re(x)**2 + Re(y) ** 2))

Parameters

xdomain element: Point in which to take the derivative.

Notes

The derivative of the complex modulus

$&M: X(\mathbb{C}) \to X(\mathbb{R}), \\ &M(x) = \sqrt{\Re(x)^2 + \Im(x)^2},$

with $X(\mathbb{F}) = \mathbb{F}^n$ or $L^2(\Omega, \mathbb{F})$ , is given as

$&M'(x): X(\mathbb{C}) \to X(\mathbb{R}), \\ &M'(x)(y) = \frac{\Re(x)\,\Re(y) + \Im(x)\,\Im(y)}{M(x)}.$

It is linear when identifying $\mathbb{C}$ with $\mathbb{R}^2$ , but not complex-linear.

Examples

>>> c2 = odl.cn(2)
>>> op = odl.ComplexModulus(c2)
>>> op([3 + 4j, 2])
rn(2).element([ 5.,  2.])
>>> deriv = op.derivative([3 + 4j, 2])
>>> deriv.domain
cn(2)
>>> deriv.range
rn(2)
>>> deriv([2 + 1j, 4j])  # [(3*2 + 4*1) / 5, (2*0 + 0*4) / 2]
rn(2).element([ 2.,  0.])