(UNDER CONSTRUCTION)

This library provides a set of functions for computation derivatives using spectral methods. The lib depends on FFTW library. The library is compatible with W-SLDA Toolkit however, it can be used as standalone lib. It is written in C99 standard.

Name convention

Generic name of a function is wderiv_operation_Nd_t, where:

• wderiv: fixed, function belongs to wderiv lib,
• operation:
  • dfdx : \frac{\partial f}{\partial x}
  • dfdy : \frac{\partial f}{\partial y}
  • dfdz : \frac{\partial f}{\partial z}
  • d2fdx2 : \frac{\partial^2 f}{\partial x^2}
  • d2fdy2 : \frac{\partial^2 f}{\partial y^2}
  • d2fdz2 : \frac{\partial^2 f}{\partial z^2}
  • dnfdxn : \frac{\partial^n f}{\partial x^n}
  • dnfdyn : \frac{\partial^n f}{\partial y^n}
  • dnfdzn : \frac{\partial^n f}{\partial z^n}
  • gradient : (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})
  • gradient2 : |\nabla f|^2, see here for computation details
  • laplace : \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}
  • divergence : \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}
  • curl : (\frac{\partial f_y}{\partial z} - \frac{\partial f_z}{\partial y},\frac{\partial f_z}{\partial x} - \frac{\partial f_x}{\partial z},\frac{\partial f_x}{\partial y} - \frac{\partial f_y}{\partial x})
• N:
  • 1 for 1D problems
  • 2 for 2D problems
  • 3 for 3D problems
• t:
  • r for real problems
  • c for complex problems

Function Documentation

3D derivatives:

Basic 3D derivative function:

int wderiv_derivative_3d_r(int direction, int n, double *f, double *deriv)

Function computes n-th derivative with respect to direction of 3D real function f(x,y,z): \frac{\partial^n f}{\partial t^n}.

• direction - predefined integer corresponding to direction of derivative: WDERIV_DX or WDERIV_DY or WDERIV_DZ
• f - pointer to input function which is array of size [nx*ny*nz]
• deriv - pointer to output derivative function which is array of size [nx*ny*nz]
• return - error code

deriv can be the same pointer as f, then result overwrites input.

First 3D derivative functions:

int wderiv_dfdx_3d_r(double *f, double *dfdx)

Function computes first derivative with respect to x of 3D real function f(x,y,z): \frac{\partial f}{\partial x}.

• f - pointer to input function which is array of size [nx*ny*nz]
• dfdx - pointer to output derivative function which is array of size [nx*ny*nz]
• return - error code

dfdx can be the same pointer as f, then result overwrites input.

int wderiv_dfdy_3d_r(double *f, double *dfdy)

Function computes first derivative with respect to y of 3D real function f(x,y,z): \frac{\partial f}{\partial y}.

• f - pointer to input function which is array of size [nxnynz]
• dfdy - pointer to output derivative function which is array of size [nxnynz]
• return - error code

dfdy can be the same pointer as f, then result overwrites input.

int wderiv_dfdz_3d_r(double *f, double *dfdz)

Function computes first derivative with respect to z of 3D real function f(x,y,z): \frac{\partial f}{\partial x}.

• f - pointer to input function which is array of size [nxnynz]
• dfdz - pointer to output derivative function which is array of size [nxnynz]
• return - error code

dfdz can be the same pointer as f, then result overwrites input.

Second 3D derivative functions:

int wderiv_d2fdx2_3d_r(double *f, double *d2fdx2)

Function computes second derivative with respect to x of 3D real function f(x,y,z): \frac{\partial^2 f}{\partial x^2}.

• f - pointer to input function which is array of size [nx*ny*nz]
• d2fdx2 - pointer to output derivative function which is array of size [nx*ny*nz]
• return - error code

d2fdx2 can be the same pointer as f, then result overwrites input.

int wderiv_d2fdy2_3d_r(double *f, double *d2fdy2)

Function computes second derivative with respect to y of 3D real function f(x,y,z): \frac{\partial^2 f}{\partial y^2}.

• f - pointer to input function which is array of size [nx*ny*nz]
• d2fdy2 - pointer to output derivative function which is array of size [nx*ny*nz]
• return - error code

d2fdy2 can be the same pointer as f, then result overwrites input.

int wderiv_d2fdz2_3d_r(double *f, double *d2fdz2)

Function computes second derivative with respect to z of 3D real function f(x,y,z): (\frac{d^2f}{dz^2}).

• f - pointer to input function which is array of size [nxnynz]
• d2fdz2 - pointer to output derivative function which is array of size [nxnynz]
• return - error code

d2fdz2 can be the same pointer as f, then result overwrites input.

/**

• Function computes gradient
• of 3D real function f(x,y,z)
• @param f pointer to function, array of size [nxnynz] (INPUT)
• @param dfdx pointer to function, array of size [nxnynz] (OUTPUT)
• @param dfdy pointer to function, array of size [nxnynz] (OUTPUT)
• @param dfdz pointer to function, array of size [nxnynz] (OUTPUT)
• @return error code
• */ int wderiv_gradient_3d_r(double *f, double *dfdx, double *dfdy, double *dfdz);

/**

• Function computes second derivative with respect to z (d^2f/dz^2)
• of 3D complex function f(x,y,z)
• @param f pointer to function, array of size [nxnynz] (INPUT)
• @param d2fdz2 pointer to function, array of size [nxnynz] (OUTPUT)

•            NOTE: d2fdz2 can be the same pointer as f, then result overwrites input 

• @return error code
• */ int wderiv_d2fdz2_3d_c(double complex *f, double complex *d2fdz2);

/**

• Function computes n-th derivative with respect to x (d^nf/dx^n)
• of 1D complex function f(x)
• @param n order of derivative
• @param f pointer to function, array of size [nx] (INPUT)
• @param dnfdxn pointer to function, array of size [nx] (OUTPUT)

•            NOTE: dnfdxn can be the same pointer as f, then result overwrites input 

• @return error code
• */ int wderiv_dnfdxn_1d_c(int n, double complex *f, double complex *dnfdxn);