# Grape xy.m

Cost function for optimal control using the GRAPE algorithm.

## Syntax

    [fidelity,gradient,hessian]=grape_xy(waveform,spin_system)


## Description

Returns fidelity, gradient and hessian for a given waveform, specified in Cartesian coordinates (x and y channels).

## Arguments

  waveform      - normalised set of control amplitudes.


## Returns

  fidelity      - figure of merit for the overlap of the current state
of the system and the desired state(s). When penalty
methods are specified, fidelity is returned as an ar-
ray separating the penalties from the simulation
fidelity.

sequence. When penalty methods are specified, gradi-
ent is returned as an array separating penalty gra-

hessian       - Hessian of the fidelity with respect to the control
sequence. When penalty methods are specified, gradi-
ent is returned as an array separating penalty Hes-
sians from the fidelity Hessian.


## Examples

A typical call would be from an optimisation function (see examples/optimal_control):

    % Optimisation parameters
optim.method='lbfgs';                   % Optimisation method
optim.extremum='maximum';               % Extremum type

% Run the optimization
fminnewton(@grape_xy,guess,optim,spin_system);


## Notes

The fidelity, the gradient and the Hessian may be supplied to any optimisation routine, including those in the Optimisation Toolbox of Matlab.