4.4.1 Advanced Complex DFTs

fftw_plan fftw_plan_many_dft(int rank, const int *n, int howmany,
                             fftw_complex *in, const int *inembed,
                             int istride, int idist,
                             fftw_complex *out, const int *onembed,
                             int ostride, int odist,
                             int sign, unsigned flags);

This routine plans multiple multidimensional complex DFTs, and it extends the fftw_plan_dft routine (see Complex DFTs) to compute howmany transforms, each having rank rank and size n. In addition, the transform data need not be contiguous, but it may be laid out in memory with an arbitrary stride. To account for these possibilities, fftw_plan_many_dft adds the new parameters howmany, {i,o}nembed, {i,o}stride, and {i,o}dist. The FFTW basic interface (see Complex DFTs) provides routines specialized for ranks 1, 2, and 3, but the advanced interface handles only the general-rank case.

howmany is the number of transforms to compute. The resulting plan computes howmany transforms, where the input of the k-th transform is at location in+k*idist (in C pointer arithmetic), and its output is at location out+k*odist. Plans obtained in this way can often be faster than calling FFTW multiple times for the individual transforms. The basic fftw_plan_dft interface corresponds to howmany=1 (in which case the dist parameters are ignored).

Each of the howmany transforms has rank rank and size n, as in the basic interface. In addition, the advanced interface allows the input and output arrays of each transform to be row-major subarrays of larger rank-rank arrays, described by inembed and onembed parameters, respectively. {i,o}nembed must be arrays of length rank, and n should be elementwise less than or equal to {i,o}nembed. Passing NULL for an nembed parameter is equivalent to passing n (i.e. same physical and logical dimensions, as in the basic interface.)

The stride parameters indicate that the j-th element of the input or output arrays is located at j*istride or j*ostride, respectively. (For a multi-dimensional array, j is the ordinary row-major index.) When combined with the k-th transform in a howmany loop, from above, this means that the (j,k)-th element is at j*stride+k*dist. (The basic fftw_plan_dft interface corresponds to a stride of 1.)

For in-place transforms, the input and output stride and dist parameters should be the same; otherwise, the planner may return NULL.

Arrays n, inembed, and onembed are not used after this function returns. You can safely free or reuse them.

Examples: One transform of one 5 by 6 array contiguous in memory:

   int rank = 2;
   int n[] = {5, 6};
   int howmany = 1;
   int idist = odist = 0; /* unused because howmany = 1 */
   int istride = ostride = 1; /* array is contiguous in memory */
   int *inembed = n, *onembed = n;

Transform of three 5 by 6 arrays, each contiguous in memory, stored in memory one after another:

   int rank = 2;
   int n[] = {5, 6};
   int howmany = 3;
   int idist = odist = n[0]*n[1]; /* = 30, the distance in memory
                                     between the first element
                                     of the first array and the
                                     first element of the second array */
   int istride = ostride = 1; /* array is contiguous in memory */
   int *inembed = n, *onembed = n;

Transform each column of a 2d array with 10 rows and 3 columns:

   int rank = 1; /* not 2: we are computing 1d transforms */
   int n[] = {10}; /* 1d transforms of length 10 */
   int howmany = 3;
   int idist = odist = 1;
   int istride = ostride = 3; /* distance between two elements in 
                                 the same column */
   int *inembed = n, *onembed = n;