This is a heavily modified implementation of FFTPack [1,2], with the following advantages:

• Strictly C++11 compliant
• More accurate twiddle factor computation
• Worst case complexity for transform sizes with large prime factors is N*log(N), because Bluestein's algorithm [3] is used for these cases.
• Supports multidimensional arrays and selection of the axes to be transformed.
• Supports float, double, and long double types.
• Supports fully complex and half-complex (i.e. complex-to-real and real-to-complex) FFTs. For half-complex transforms, several conventions for representing the complex-valued side are supported (reduced-size complex array, FFTPACK-style half-complex format and Hartley transform).
• Supports discrete cosine and sine transforms (Types I-IV)
• Makes use of CPU vector instructions when performing 2D and higher-dimensional transforms, if they are available.
• Has a small internal cache for transform plans, which speeds up repeated transforms of the same length (most significant for 1D transforms).
• Has optional multi-threading support for multidimensional transforms

3-clause BSD (see LICENSE.md)

Twiddle factor computation:

• making use of symmetries to reduce number of sin/cos evaluations
• all angles are reduced to the range [0; pi/4] for higher accuracy
• if n sin/cos pairs are required, the trogonometric functions are only called 2*sqrt(n) times; the remaining values are obtained by evaluating the angle addition theorems in a numerically accurate way.

Efficient codelets are available for the factors:

• 2, 3, 4, 5, 7, 8, 11 for complex-valued FFTs
• 2, 3, 4, 5 for real-valued FFTs

Larger prime factors are handled by somewhat less efficient, generic routines.

For lengths with very large prime factors, Bluestein's algorithm is used, and instead of an FFT of length n, a convolution of length n2 >= 2*n-1 is performed, where n2 is chosen to be highly composite.

[1] Swarztrauber, P. 1982, Vectorizing the Fast Fourier Transforms (New York: Academic Press), 51

[2] https://www.netlib.org/fftpack/

[3] https://en.wikipedia.org/wiki/Chirp_Z-transform

Since this is a header-only library, it can only be configured via preprocessor macros.

POCKETFFT_CACHE_SIZE:
if 0, disable all caching of FFT plans, else use an LRU cache with the requested size. If undefined, assume a cache size of 0.
NOTE: caching is disabled by default because its benefits are only really noticeable for short 1D transforms. When using caching with transforms that have very large axis lengths, it may use up a lot of memory, so only switch this on if you know you really need it! Default: undefined

POCKETFFT_NO_VECTORS:
if defined, disable all support for CPU vector instructions.
Default: undefined

POCKETFFT_NO_MULTITHREADING:
if defined, multi-threading will be disabled.
Default: undefined

All symbols are encapsulated in the namespace pocketfft.

• shape[_*] contains the number of array entries along each axis. For c2c and r2r transforms, shape is identical for input and output arrays. For r2c transforms the shape of the input array must be specified, while for c2r transforms the shape of the output array must be given.

• stride_* describes array strides, i.e. the memory distance (in bytes) between two neighboring array entries along an axis.

• axes is a vector of nonnegative integers, describing the axes along which a transform is to be carried out. The order of axes usually does not matter, except for r2c and c2r transforms, where the last entry of axes is treated specially.

• forward describes the direction of a transform. Generally a forward transform has a minus sign in the complex exponent, while the backward transform has a positive one. Instead if true/false, the symbolic constants FORWARD/BACKWARD can be used. NOTE: Unlike many other libraries, pocketfft also allows a forward argument in r2c and c2r transforms, instead of having hard-wired forward r2c and backward c2r transforms. Calling r2c with forward=false, for example, performs a transform from purely real data in the frequency domain to Hermitian data in the position domain. If you want the "traditional" behavior, call r2c with forward=true and c2r with forward=false.

• fct is a floating-point value which is used to scale the result of a transform. pocketfft's transforms are not normalized, so if normalization is required, an appropriate scaling factor has to be specified.

• data_in and data_out are pointers to the first element of the input and output data arrays.

• nthreads is a nonnegative integer specifying the number of threads to use for the operation. A value of 0 means that the number of logical CPU cores will be used. This value is only a recommendation. If pocketfft is compiled without multi-threading support, it will be silently ignored. For one-dimensional transforms, multi-threading is disabled as well.

• shape[_*], stride_in and stride_out must have the same size() and must not be empty.
• Entries in shape[_*] must be >=1.
• If data_in==data_out, stride_in and stride_out must have identical content. These in-place transforms are fine for c2c and r2r, but not for r2c/c2r.
• Axes are numbered from 0 to shape.size()-1, inclusively.
• Strides are measured in bytes, to allow maximum flexibility. Negative strides are fine. Strides that lead to multiple accesses of the same memory address are not allowed.
• All memory addresses resulting from a combination of data pointers and strides must have sufficient alignment. On x86 CPUs, badly aligned adresses will only lead to slower execution, but on some other hardware, misaligned memory accesses will cause a crash.
• The same axis must not be specified more than once in an axes argument.
• For r2c and c2r transforms: the length of the complex array along axis (or the last entry in axes) is assumed to be s/2 + 1, where s is the length of the corresponding axis of the real array.

using shape_t = std::vector<std::size_t>;
using stride_t = std::vector<std::ptrdiff_t>;

constexpr bool FORWARD  = true,
               BACKWARD = false;

template<typename T> void c2c(const shape_t &shape, const stride_t &stride_in,
  const stride_t &stride_out, const shape_t &axes, bool forward,
  const complex<T> *data_in, complex<T> *data_out, T fct,
  size_t nthreads=1)

template<typename T> void r2c(const shape_t &shape_in,
  const stride_t &stride_in, const stride_t &stride_out, size_t axis,
  bool forward, const T *data_in, complex<T> *data_out, T fct,
  size_t nthreads=1)

/* This function first carries out an r2c transform along the last axis in axes,
   storing the result in data_out. Then, an in-place c2c transform
   is carried out in data_out along all other axes. */
template<typename T> void r2c(const shape_t &shape_in,
  const stride_t &stride_in, const stride_t &stride_out, const shape_t &axes,
  bool forward, const T *data_in, complex<T> *data_out, T fct,
  size_t nthreads=1)

template<typename T> void c2r(const shape_t &shape_out,
  const stride_t &stride_in, const stride_t &stride_out, size_t axis,
  bool forward, const complex<T> *data_in, T *data_out, T fct,
  size_t nthreads=1)

/* This function first carries out a c2c transform along all axes except the
   last one, storing the result into a temporary array. Then, a c2r transform
   is carried out along the last axis, storing the result in data_out. */
template<typename T> void c2r(const shape_t &shape_out,
  const stride_t &stride_in, const stride_t &stride_out, const shape_t &axes,
  bool forward, const complex<T> *data_in, T *data_out, T fct,
  size_t nthreads=1)

/* This function carries out a FFTPACK-style real-to-halfcomplex or
   halfcomplex-to-real transform (depending on the parameter `real2hermitian`)
   on all specified axes in the given order.
   NOTE: interpreting the result of this function can be complicated when
   transforming more than one axis! */
template<typename T> void r2r_fftpack(const shape_t &shape,
  const stride_t &stride_in, const stride_t &stride_out, const shape_t &axes,
  bool real2hermitian, bool forward, const T *data_in, T *data_out, T fct,
  size_t nthreads=1)

/* For every requested axis, this function carries out a forward Fourier
   transform, and the real and imaginary parts of the result are added before
   the next axis is processed.
   This is analogous to FFTW's implementation of the Hartley transform. */
template<typename T> void r2r_separable_hartley(const shape_t &shape,
  const stride_t &stride_in, const stride_t &stride_out, const shape_t &axes,
  const T *data_in, T *data_out, T fct, size_t nthreads=1);

/* This function carries out a full Fourier transform over the requested axes,
   and the sum of real and imaginary parts of the result is stored in the output
   array. For a single transformed axis, this is identical to
   `r2r_separable_hartley`, but when transforming multiple axes, the results
   are different.

   NOTE: This function allocates temporary working space with a size
   comparable to the input array. */
template<typename T> void r2r_genuine_hartley(const shape_t &shape,
  const stride_t &stride_in, const stride_t &stride_out, const shape_t &axes,
  const T *data_in, T *data_out, T fct, size_t nthreads=1);

/* if ortho==true, the transform is made orthogonal by these additional steps
   in every 1D sub-transform:
   Type 1 : multiply first and last input value by sqrt(2)
            divide first and last output value by sqrt(2)
   Type 2 : divide first output value by sqrt(2)
   Type 3 : multiply first input value by sqrt(2)
   Type 4 : nothing */
template<typename T> void dct(const shape_t &shape,
  const stride_t &stride_in, const stride_t &stride_out, const shape_t &axes,
  int type, const T *data_in, T *data_out, T fct, bool ortho,
  size_t nthreads=1);

/* if ortho==true, the transform is made orthogonal by these additional steps
   in every 1D sub-transform:
   Type 1 : nothing
   Type 2 : divide first output value by sqrt(2)
   Type 3 : multiply first input value by sqrt(2)
   Type 4 : nothing */
template<typename T> void dst(const shape_t &shape,
  const stride_t &stride_in, const stride_t &stride_out, const shape_t &axes,
  int type, const T *data_in, T *data_out, T fct, bool ortho,
  size_t nthreads=1);