6.11 Global functions

These functions are not methods of mglData class. However it provide additional functionality to handle data. So I put it in this chapter.

MGL command: transform DAT 'type' real imag

Global function: mglData mglTransform (const mglDataA &real, const mglDataA &imag, const char *type)

C function: HMDT mgl_transform (HCDT real, HCDT imag, const char *type)

Does integral transformation of complex data real, imag on specified direction. The order of transformations is specified in string type: first character for x-dimension, second one for y-dimension, third one for z-dimension. The possible character are: ‘f’ is forward Fourier transformation, ‘i’ is inverse Fourier transformation, ‘s’ is Sine transform, ‘c’ is Cosine transform, ‘h’ is Hankel transform, ‘n’ or ‘ ’ is no transformation.

MGL command: transforma DAT 'type' ampl phase

Global function: mglData mglTransformA const mglDataA &ampl, const mglDataA &phase, const char *type)

C function: HMDT mgl_transform_a HCDT ampl, HCDT phase, const char *type)

The same as previous but with specified amplitude ampl and phase phase of complex numbers.

MGL command: fourier reDat imDat 'dir'

Global function: void mglFourier const mglDataA &re, const mglDataA &im, const char *dir)

C function: void mgl_data_fourier HCDT re, HCDT im, const char *dir)

Does Fourier transform of complex data re+i*im in directions dir. Result is placed back into re and im data arrays.

MGL command: stfad RES real imag dn ['dir'='x']

Global function: mglData mglSTFA (const mglDataA &real, const mglDataA &imag, int dn, char dir='x')

C function: HMDT mgl_data_stfa (HCDT real, HCDT imag, int dn,char dir)

Short time Fourier transformation for real and imaginary parts. Output is amplitude of partial Fourier of length dn. For example if dir=‘x’, result will have size {int(nx/dn), dn, ny} and it will contain res[i,j,k]=|\sum_d^dn exp(I*j*d)*(real[i*dn+d,k]+I*imag[i*dn+d,k])|/dn.

MGL command: pde RES 'ham' ini_re ini_im [dz=0.1 k0=100]

Global function: mglData mglPDE (HMGL gr, const char *ham, const mglDataA &ini_re, const mglDataA &ini_im, mreal dz=0.1, mreal k0=100, const char *opt="")

C function: HMDT mgl_pde_solve (HMGL gr, const char *ham, HCDT ini_re, HCDT ini_im, mreal dz, mreal k0, const char *opt)

Solves equation du/dz = i*k0*ham(p,q,x,y,z,|u|)[u], where p=-i/k0*d/dx, q=-i/k0*d/dy are pseudo-differential operators. Parameters ini_re, ini_im specify real and imaginary part of initial field distribution. Parameters Min, Max set the bounding box for the solution. Note, that really this ranges are increased by factor 3/2 for purpose of reducing reflection from boundaries. Parameter dz set the step along evolutionary coordinate z. At this moment, simplified form of function ham is supported – all “mixed” terms (like ‘x*p’->x*d/dx) are excluded. For example, in 2D case this function is effectively ham = f(p,z) + g(x,z,u). However commutable combinations (like ‘x*q’->x*d/dy) are allowed. Here variable ‘u’ is used for field amplitude |u|. This allow one solve nonlinear problems – for example, for nonlinear Shrodinger equation you may set ham="p^2 + q^2 - u^2". You may specify imaginary part for wave absorption, like ham = "p^2 + i*x*(x>0)", but only if dependence on variable ‘i’ is linear (i.e. ham = hre+i*him). See section PDE solving hints, for sample code and picture.

MGL command: ray RES 'ham' x0 y0 z0 p0 q0 v0 [dt=0.1 tmax=10]

Global function: mglData mglRay (const char *ham, mglPoint r0, mglPoint p0, mreal dt=0.1, mreal tmax=10)

C function: HMDT mgl_ray_trace (const char *ham, mreal x0, mreal y0, mreal z0, mreal px, mreal py, mreal pz, mreal dt, mreal tmax)

Solves GO ray equation like dr/dt = d ham/dp, dp/dt = -d ham/dr. This is Hamiltonian equations for particle trajectory in 3D case. Here ham is Hamiltonian which may depend on coordinates ‘x’, ‘y’, ‘z’, momentums ‘p’=px, ‘q’=py, ‘v’=pz and time ‘t’: ham = H(x,y,z,p,q,v,t). The starting point (at t=0) is defined by variables r0, p0. Parameters dt and tmax specify the integration step and maximal time for ray tracing. Result is array of {x,y,z,p,q,v,t} with dimensions {7 * int(tmax/dt+1) }.

MGL command: qo2d RES 'ham' ini_re ini_im ray [r=1 k0=100 xx yy]

Global function: mglData mglQO2d (const char *ham, const mglDataA &ini_re, const mglDataA &ini_im, const mglDataA &ray, mreal r=1, mreal k0=100, mglData *xx=0, mglData *yy=0)

Global function: mglData mglQO2d (const char *ham, const mglDataA &ini_re, const mglDataA &ini_im, const mglDataA &ray, mglData &xx, mglData &yy, mreal r=1, mreal k0=100)

C function: HMDT mgl_qo2d_solve (const char *ham, HCDT ini_re, HCDT ini_im, HCDT ray, mreal r, mreal k0, HMDT xx, HMDT yy)

Solves equation du/dt = i*k0*ham(p,q,x,y,|u|)[u], where p=-i/k0*d/dx, q=-i/k0*d/dy are pseudo-differential operators (see mglPDE() for details). Parameters ini_re, ini_im specify real and imaginary part of initial field distribution. Parameters ray set the reference ray, i.e. the ray around which the accompanied coordinate system will be maked. You may use, for example, the array created by mglRay() function. Note, that the reference ray must be smooth enough to make accompanied coodrinates unambiguity. Otherwise errors in the solution may appear. If xx and yy are non-zero then Cartesian coordinates for each point will be written into them. See also mglPDE(). See section PDE solving hints, for sample code and picture.

MGL command: jacobian RES xdat ydat [zdat]

Global function: mglData mglJacobian (const mglDataA &x, const mglDataA &y)

Global function: mglData mglJacobian (const mglDataA &x, const mglDataA &y, const mglDataA &z)

C function: HMDT mgl_jacobian_2d (HCDT x, HCDT y)

C function: HMDT mgl_jacobian_3d (HCDT x, HCDT y, HCDT z)

Computes the Jacobian for transformation {i,j,k} to {x,y,z} where initial coordinates {i,j,k} are data indexes normalized in range [0,1]. The Jacobian is determined by formula det||dr_\alpha/d\xi_\beta|| where r={x,y,z} and \xi={i,j,k}. All dimensions must be the same for all data arrays. Data must be 3D if all 3 arrays {x,y,z} are specified or 2D if only 2 arrays {x,y} are specified.

MGL command: triangulation RES xdat ydat

Global function: mglData mglTriangulation (const mglDataA &x, const mglDataA &y)

C function: HMDT mgl_triangulation_2d (HCDT x, HCDT y)

Computes triangulation for arbitrary placed points with coordinates {x,y} (i.e. finds triangles which connect points). MathGL use s-hull code for triangulation. The sizes of 1st dimension must be equal for all arrays x.nx=y.nx. Resulting array can be used in triplot or tricont functions for visualization of reconstructed surface. See section Making regular data, for sample code and picture.

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