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The default FFTW interface uses double precision for all
floating-point numbers, and defines a fftw_complex type to hold
complex numbers as:
typedef double fftw_complex[2];
Here, the [0] element holds the real part and the [1]
element holds the imaginary part.
Alternatively, if you have a C compiler (such as gcc) that
supports the C99 revision of the ANSI C standard, you can use C's new
native complex type (which is binary-compatible with the typedef above).
In particular, if you #include <complex.h> before
<fftw3.h>, then fftw_complex is defined to be the native
complex type and you can manipulate it with ordinary arithmetic
(e.g. x = y * (3+4*I), where x and y are
fftw_complex and I is the standard symbol for the
imaginary unit);
C++ has its own complex<T> template class, defined in the
standard <complex> header file. Reportedly, the C++ standards
committee has recently agreed to mandate that the storage format used
for this type be binary-compatible with the C99 type, i.e. an array
T[2] with consecutive real [0] and imaginary [1]
parts. (See report
WG21/N1388.) Although not part of the official standard as of this
writing, the proposal stated that: "This solution has been tested with
all current major implementations of the standard library and shown to
be working." To the extent that this is true, if you have a variable
complex<double> *x, you can pass it directly to FFTW via
reinterpret_cast<fftw_complex*>(x).