.. _fft:
**fft.h**  SchoenhageStrassen FFT
================================================================================
Split/combine FFT coefficients

.. function:: mp_size_t fft_split_limbs(mp_limb_t ** poly, mp_srcptr limbs, mp_size_t total_limbs, mp_size_t coeff_limbs, mp_size_t output_limbs)
Split an integer ``(limbs, total_limbs)`` into coefficients of length
``coeff_limbs`` limbs and store as the coefficients of ``poly``
which are assumed to have space for ``output_limbs + 1`` limbs per
coefficient. The coefficients of the polynomial do not need to be zeroed
before calling this function, however the number of coefficients written
is returned by the function and any coefficients beyond this point are
not touched.
.. function:: mp_size_t fft_split_bits(mp_limb_t ** poly, mp_srcptr limbs, mp_size_t total_limbs, flint_bitcnt_t bits, mp_size_t output_limbs)
Split an integer ``(limbs, total_limbs)`` into coefficients of the
given number of ``bits`` and store as the coefficients of ``poly``
which are assumed to have space for ``output_limbs + 1`` limbs per
coefficient. The coefficients of the polynomial do not need to be zeroed
before calling this function, however the number of coefficients written
is returned by the function and any coefficients beyond this point are
not touched.
.. function:: void fft_combine_limbs(mp_limb_t * res, mp_limb_t ** poly, slong length, mp_size_t coeff_limbs, mp_size_t output_limbs, mp_size_t total_limbs)
Evaluate the polynomial ``poly`` of the given ``length`` at
``B^coeff_limbs``, where ``B = 2^FLINT_BITS``, and add the
result to the integer ``(res, total_limbs)`` throwing away any bits
that exceed the given number of limbs. The polynomial coefficients are
assumed to have at least ``output_limbs`` limbs each, however any
additional limbs are ignored.
If the integer is initially zero the result will just be the evaluation
of the polynomial.
.. function:: void fft_combine_bits(mp_limb_t * res, mp_limb_t ** poly, slong length, flint_bitcnt_t bits, mp_size_t output_limbs, mp_size_t total_limbs)
Evaluate the polynomial ``poly`` of the given ``length`` at
``2^bits`` and add the result to the integer
``(res, total_limbs)`` throwing away any bits that exceed the given
number of limbs. The polynomial coefficients are assumed to have at least
``output_limbs`` limbs each, however any additional limbs are ignored.
If the integer is initially zero the result will just be the evaluation
of the polynomial.
Test helper functions

.. function:: void fermat_to_mpz(mpz_t m, mp_limb_t * i, mp_size_t limbs)
Convert the Fermat number ``(i, limbs)`` modulo ``B^limbs + 1`` to
an ``mpz_t m``. Assumes ``m`` has been initialised. This function
is used only in test code.
Arithmetic modulo a generalised Fermat number

.. function:: void mpn_addmod_2expp1_1(mp_limb_t * r, mp_size_t limbs, mp_limb_signed_t c)
Adds the signed limb ``c`` to the generalised Fermat number ``r``
modulo ``B^limbs + 1``. The compiler should be able to inline
this for the case that there is no overflow from the first limb.
.. function:: void mpn_normmod_2expp1(mp_limb_t * t, mp_size_t limbs)
Given ``t`` a signed integer of ``limbs + 1`` limbs in twos
complement format, reduce ``t`` to the corresponding value modulo the
generalised Fermat number ``B^limbs + 1``, where
``B = 2^FLINT_BITS``.
.. function:: void mpn_mul_2expmod_2expp1(mp_limb_t * t, mp_limb_t * i1, mp_size_t limbs, flint_bitcnt_t d)
Given ``i1`` a signed integer of ``limbs + 1`` limbs in twos
complement format reduced modulo ``B^limbs + 1`` up to some
overflow, compute ``t = i1*2^d`` modulo `p`. The result will not
necessarily be fully reduced. The number of bits ``d`` must be
nonnegative and less than ``FLINT_BITS``. Aliasing is permitted.
.. function:: void mpn_div_2expmod_2expp1(mp_limb_t * t, mp_limb_t * i1, mp_size_t limbs, flint_bitcnt_t d)
Given ``i1`` a signed integer of ``limbs + 1`` limbs in twos
complement format reduced modulo ``B^limbs + 1`` up to some
overflow, compute ``t = i1/2^d`` modulo `p`. The result will not
necessarily be fully reduced. The number of bits ``d`` must be
nonnegative and less than ``FLINT_BITS``. Aliasing is permitted.
Generic butterflies

.. function:: void fft_adjust(mp_limb_t * r, mp_limb_t * i1, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w)
Set ``r`` to ``i1`` times `z^i` modulo ``B^limbs + 1`` where
`z` corresponds to multiplication by `2^w`. This can be thought of as part
of a butterfly operation. We require `0 \leq i < n` where `nw =`
``limbs*FLINT_BITS``. Aliasing is not supported.
.. function:: void fft_adjust_sqrt2(mp_limb_t * r, mp_limb_t * i1, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w, mp_limb_t * temp)
Set ``r`` to ``i1`` times `z^i` modulo ``B^limbs + 1`` where
`z` corresponds to multiplication by `\sqrt{2}^w`. This can be thought of
as part of a butterfly operation. We require `0 \leq i < 2*n` and odd
where `nw =` ``limbs*FLINT_BITS``.
.. function:: void butterfly_lshB(mp_limb_t * t, mp_limb_t * u, mp_limb_t * i1, mp_limb_t * i2, mp_size_t limbs, mp_size_t x, mp_size_t y)
We are given two integers ``i1`` and ``i2`` modulo
``B^limbs + 1`` which are not necessarily normalised. We compute
``t = (i1 + i2)*B^x`` and ``u = (i1  i2)*B^y`` modulo `p`. Aliasing
between inputs and outputs is not permitted. We require ``x`` and
``y`` to be less than ``limbs`` and nonnegative.
.. function:: void butterfly_rshB(mp_limb_t * t, mp_limb_t * u, mp_limb_t * i1, mp_limb_t * i2, mp_size_t limbs, mp_size_t x, mp_size_t y)
We are given two integers ``i1`` and ``i2`` modulo
``B^limbs + 1`` which are not necessarily normalised. We compute
``t = (i1 + i2)/B^x`` and ``u = (i1  i2)/B^y`` modulo `p`. Aliasing
between inputs and outputs is not permitted. We require ``x`` and
``y`` to be less than ``limbs`` and nonnegative.
Radix 2 transforms

.. function:: void fft_butterfly(mp_limb_t * s, mp_limb_t * t, mp_limb_t * i1, mp_limb_t * i2, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w)
Set ``s = i1 + i2``, ``t = z1^i*(i1  i2)`` modulo
``B^limbs + 1`` where ``z1 = exp(Pi*I/n)`` corresponds to
multiplication by `2^w`. Requires `0 \leq i < n` where `nw =`
``limbs*FLINT_BITS``.
.. function:: void ifft_butterfly(mp_limb_t * s, mp_limb_t * t, mp_limb_t * i1, mp_limb_t * i2, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w)
Set ``s = i1 + z1^i*i2``, ``t = i1  z1^i*i2`` modulo
``B^limbs + 1`` where\\ ``z1 = exp(Pi*I/n)`` corresponds to
division by `2^w`. Requires `0 \leq i < 2n` where `nw =`
``limbs*FLINT_BITS``.
.. function:: void fft_radix2(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2)
The radix 2 DIF FFT works as follows:
Input: ``[i0, i1, ..., i(m1)]``, for `m = 2n` a power of `2`.
Output: ``[r0, r1, ..., r(m1)]`` `` = FFT[i0, i1, ..., i(m1)]``.
Algorithm:
 `\bullet` Recursively compute ``[r0, r2, r4, ...., r(m2)]``
 ``= FFT[i0+i(m/2), i1+i(m/2+1), ..., i(m/21)+i(m1)]``

 `\bullet` Let ``[t0, t1, ..., t(m/21)]``
 ``= [i0i(m/2), i1i(m/2+1), ..., i(m/21)i(m1)]``

 `\bullet` Let ``[u0, u1, ..., u(m/21)]``
 ``= [z1^0*t0, z1^1*t1, ..., z1^(m/21)*t(m/21)]``
 where ``z1 = exp(2*Pi*I/m)`` corresponds to multiplication by `2^w`.

 `\bullet` Recursively compute ``[r1, r3, ..., r(m1)]``
 ``= FFT[u0, u1, ..., u(m/21)]``
The parameters are as follows:
`\bullet` ``2*n`` is the length of the input and output arrays
`\bullet` `w` is such that `2^w` is an `2n`th root of unity in the ring `\mathbf{Z}/p\mathbf{Z}` that we are working in, i.e. `p = 2^{wn} + 1` (here `n` is divisible by
``GMP_LIMB_BITS``)
`\bullet` ``ii`` is the array of inputs (each input is an
array of limbs of length ``wn/GMP_LIMB_BITS + 1`` (the
extra limbs being a "carry limb"). Outputs are written
inplace.
We require `nw` to be at least 64 and the two temporary space pointers to
point to blocks of size ``n*w + FLINT_BITS`` bits.
.. function:: void fft_truncate(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_size_t trunc)
As for ``fft_radix2`` except that only the first ``trunc``
coefficients of the output are computed and the input is regarded as
having (implied) zero coefficients from coefficient ``trunc`` onwards.
The coefficients must exist as the algorithm needs to use this extra
space, but their value is irrelevant. The value of ``trunc`` must be
divisible by 2.
.. function:: void fft_truncate1(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_size_t trunc)
As for ``fft_radix2`` except that only the first ``trunc``
coefficients of the output are computed. The transform still needs all
`2n` input coefficients to be specified.
.. function:: void ifft_radix2(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2)
The radix 2 DIF IFFT works as follows:
Input: ``[i0, i1, ..., i(m1)]``, for `m = 2n` a power of `2`.
Output: ``[r0, r1, ..., r(m1)]``\\
`` = IFFT[i0, i1, ..., i(m1)]``.
Algorithm:
`\bullet` Recursively compute ``[s0, s1, ...., s(m/21)]``
``= IFFT[i0, i2, ..., i(m2)]``
`\bullet` Recursively compute ``[t(m/2), t(m/2+1), ..., t(m1)]``
``= IFFT[i1, i3, ..., i(m1)]``
`\bullet` Let ``[r0, r1, ..., r(m/21)]``
``= [s0+z1^0*t0, s1+z1^1*t1, ..., s(m/21)+z1^(m/21)*t(m/21)]`` where ``z1 = exp(2*Pi*I/m)`` corresponds to division by `2^w`.
`\bullet` Let ``[r(m/2), r(m/2+1), ..., r(m1)]``
``= [s0z1^0*t0, s1z1^1*t1, ..., s(m/21)z1^(m/21)*t(m/21)]``
The parameters are as follows:
`\bullet` ``2*n`` is the length of the input and output
arrays
`\bullet` `w` is such that `2^w` is an `2n`th root of unity in the ring `\mathbf{Z}/p\mathbf{Z}` that we are working in, i.e. `p = 2^{wn} + 1` (here `n` is divisible by
``GMP_LIMB_BITS``)
`\bullet` ``ii`` is the array of inputs (each input is an array of limbs of length ``wn/GMP_LIMB_BITS + 1`` (the extra limbs being a "carry limb"). Outputs are written inplace.
We require `nw` to be at least 64 and the two temporary space pointers
to point to blocks of size ``n*w + FLINT_BITS`` bits.
.. function:: void ifft_truncate(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_size_t trunc)
As for ``ifft_radix2`` except that the output is assumed to have
zeros from coefficient trunc onwards and only the first trunc
coefficients of the input are specified. The remaining coefficients need
to exist as the extra space is needed, but their value is irrelevant.
The value of ``trunc`` must be divisible by 2.
Although the implementation does not require it, we assume for simplicity
that ``trunc`` is greater than `n`. The algorithm begins by computing
the inverse transform of the first `n` coefficients of the input array.
The unspecified coefficients of the second half of the array are then
written: coefficient ``trunc + i`` is computed as a twist of
coefficient ``i`` by a root of unity. The values of these coefficients
are then equal to what they would have been if the inverse transform of
the right hand side of the input array had been computed with full data
from the start. The function ``ifft_truncate1`` is then called on the
entire right half of the input array with this auxiliary data filled in.
Finally a single layer of the IFFT is completed on all the coefficients
up to ``trunc`` being careful to note that this involves doubling the
coefficients from ``trunc  n`` up to ``n``.
.. function:: void ifft_truncate1(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_size_t trunc)
Computes the first ``trunc`` coefficients of the radix 2 inverse
transform assuming the first ``trunc`` coefficients are given and that
the remaining coefficients have been set to the value they would have if
an inverse transform had already been applied with full data.
The algorithm is the same as for ``ifft_truncate`` except that the
coefficients from ``trunc`` onwards after the inverse transform are
not inferred to be zero but the supplied values.
.. function:: void fft_butterfly_sqrt2(mp_limb_t * s, mp_limb_t * t, mp_limb_t * i1, mp_limb_t * i2, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w, mp_limb_t * temp)
Let `w = 2k + 1`, `i = 2j + 1`. Set ``s = i1 + i2``,
``t = z1^i*(i1  i2)`` modulo ``B^limbs + 1`` where
``z1^2 = exp(Pi*I/n)`` corresponds to multiplication by `2^w`. Requires
`0 \leq i < 2n` where `nw =` ``limbs*FLINT_BITS``.
Here ``z1`` corresponds to multiplication by `2^k` then multiplication
by\\ ``(2^(3nw/4)  2^(nw/4))``. We see ``z1^i`` corresponds to
multiplication by ``(2^(3nw/4)  2^(nw/4))*2^(j+ik)``.
We first multiply by ``2^(j + ik + wn/4)`` then multiply by an
additional ``2^(nw/2)`` and subtract.
.. function:: void ifft_butterfly_sqrt2(mp_limb_t * s, mp_limb_t * t, mp_limb_t * i1, mp_limb_t * i2, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w, mp_limb_t * temp)
Let `w = 2k + 1`, `i = 2j + 1`. Set ``s = i1 + z1^i*i2``,
``t = i1  z1^i*i2`` modulo ``B^limbs + 1`` where
``z1^2 = exp(Pi*I/n)`` corresponds to division by `2^w`. Requires
`0 \leq i < 2n` where `nw =` ``limbs*FLINT_BITS``.
Here ``z1`` corresponds to division by `2^k` then division by
``(2^(3nw/4)  2^(nw/4))``. We see ``z1^i`` corresponds to division
by ``(2^(3nw/4)  2^(nw/4))*2^(j+ik)`` which is the same as division
by ``2^(j+ik + 1)`` then multiplication by
``(2^(3nw/4)  2^(nw/4))``.
Of course, division by ``2^(j+ik + 1)`` is the same as multiplication
by ``2^(2*wn  j  ik  1)``. The exponent is positive as
`i \leq 2*n`, `j < n`, `k < w/2`.
We first multiply by ``2^(2*wn  j  ik  1 + wn/4)`` then multiply by
an additional ``2^(nw/2)`` and subtract.
.. function:: void fft_truncate_sqrt2(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** temp, mp_size_t trunc)
As per ``fft_truncate`` except that the transform is twice the usual
length, i.e. length `4n` rather than `2n`. This is achieved by making use
of twiddles by powers of a square root of 2, not powers of 2 in the first
layer of the transform.
We require `nw` to be at least 64 and the three temporary space pointers
to point to blocks of size ``n*w + FLINT_BITS`` bits.
.. function:: void ifft_truncate_sqrt2(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** temp, mp_size_t trunc)
As per ``ifft_truncate`` except that the transform is twice the usual
length, i.e. length `4n` instead of `2n`. This is achieved by making use
of twiddles by powers of a square root of 2, not powers of 2 in the final
layer of the transform.
We require `nw` to be at least 64 and the three temporary space pointers
to point to blocks of size ``n*w + FLINT_BITS`` bits.
Matrix Fourier Transforms

.. function:: void fft_butterfly_twiddle(mp_limb_t * u, mp_limb_t * v, mp_limb_t * s, mp_limb_t * t, mp_size_t limbs, flint_bitcnt_t b1, flint_bitcnt_t b2)
Set ``u = 2^b1*(s + t)``, ``v = 2^b2*(s  t)`` modulo
``B^limbs + 1``. This is used to compute
``u = 2^(ws*tw1)*(s + t)``,\\ ``v = 2^(w+ws*tw2)*(s  t)`` in the
matrix fourier algorithm, i.e. effectively computing an ordinary butterfly
with additional twiddles by ``z1^rc`` for row `r` and column `c` of the
matrix of coefficients. Aliasing is not allowed.
.. function:: void ifft_butterfly_twiddle(mp_limb_t * u, mp_limb_t * v, mp_limb_t * s, mp_limb_t * t, mp_size_t limbs, flint_bitcnt_t b1, flint_bitcnt_t b2)
Set ``u = s/2^b1 + t/2^b1)``, ``v = s/2^b1  t/2^b1`` modulo
``B^limbs + 1``. This is used to compute
``u = 2^(ws*tw1)*s + 2^(ws*tw2)*t)``,\\
``v = 2^(ws*tw1)*s + 2^(ws*tw2)*t)`` in the matrix fourier algorithm,
i.e. effectively computing an ordinary butterfly with additional twiddles
by ``z1^(rc)`` for row `r` and column `c` of the matrix of
coefficients. Aliasing is not allowed.
.. function:: void fft_radix2_twiddle(mp_limb_t ** ii, mp_size_t is, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_size_t ws, mp_size_t r, mp_size_t c, mp_size_t rs)
As for ``fft_radix2`` except that the coefficients are spaced by
``is`` in the array ``ii`` and an additional twist by ``z^c*i``
is applied to each coefficient where `i` starts at `r` and increases by
``rs`` as one moves from one coefficient to the next. Here ``z``
corresponds to multiplication by ``2^ws``.
.. function:: void ifft_radix2_twiddle(mp_limb_t ** ii, mp_size_t is, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_size_t ws, mp_size_t r, mp_size_t c, mp_size_t rs)
As for ``ifft_radix2`` except that the coefficients are spaced by
``is`` in the array ``ii`` and an additional twist by
``z^(c*i)`` is applied to each coefficient where `i` starts at `r`
and increases by ``rs`` as one moves from one coefficient to the next.
Here ``z`` corresponds to multiplication by ``2^ws``.
.. function:: void fft_truncate1_twiddle(mp_limb_t ** ii, mp_size_t is, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_size_t ws, mp_size_t r, mp_size_t c, mp_size_t rs, mp_size_t trunc)
As per ``fft_radix2_twiddle`` except that the transform is truncated
as per\\ ``fft_truncate1``.
.. function:: void ifft_truncate1_twiddle(mp_limb_t ** ii, mp_size_t is, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_size_t ws, mp_size_t r, mp_size_t c, mp_size_t rs, mp_size_t trunc)
As per ``ifft_radix2_twiddle`` except that the transform is truncated
as per\\ ``ifft_truncate1``.
.. function:: void fft_mfa_truncate_sqrt2(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** temp, mp_size_t n1, mp_size_t trunc)
This is as per the ``fft_truncate_sqrt2`` function except that the
matrix fourier algorithm is used for the left and right FFTs. The total
transform length is `4n` where ``n = 2^depth`` so that the left and
right transforms are both length `2n`. We require ``trunc > 2*n`` and
that ``trunc`` is divisible by ``2*n1`` (explained below).
The matrix fourier algorithm, which is applied to each transform of length
`2n`, works as follows. We set ``n1`` to a power of 2 about the square
root of `n`. The data is then thought of as a set of ``n2`` rows each
with ``n1`` columns (so that ``n1*n2 = 2n``).
The length `2n` transform is then computed using a whole pile of short
transforms. These comprise ``n1`` column transforms of length ``n2``
followed by some twiddles by roots of unity (namely ``z^rc`` where `r`
is the row and `c` the column within the data) followed by ``n2``
row transforms of length ``n1``. Along the way the data needs to be
rearranged due to the fact that the short transforms output the data in
binary reversed order compared with what is needed.
The matrix fourier algorithm provides better cache locality by decomposing
the long length `2n` transforms into many transforms of about the square
root of the original length.
For better cache locality the sqrt2 layer of the full length `4n`
transform is folded in with the column FFTs performed as part of the first
matrix fourier algorithm on the left half of the data.
The second half of the data requires a truncated version of the matrix
fourier algorithm. This is achieved by truncating to an exact multiple of
the row length so that the row transforms are full length. Moreover, the
column transforms will then be truncated transforms and their truncated
length needs to be a multiple of 2. This explains the condition on
``trunc`` given above.
To improve performance, the extra twiddles by roots of unity are combined
with the butterflies performed at the last layer of the column transforms.
We require `nw` to be at least 64 and the three temporary space pointers
to point to blocks of size ``n*w + FLINT_BITS`` bits.
.. function:: void ifft_mfa_truncate_sqrt2(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** temp, mp_size_t n1, mp_size_t trunc)
This is as per the ``ifft_truncate_sqrt2`` function except that the
matrix fourier algorithm is used for the left and right IFFTs. The total
transform length is `4n` where ``n = 2^depth`` so that the left and
right transforms are both length `2n`. We require ``trunc > 2*n`` and
that ``trunc`` is divisible by ``2*n1``.
We set ``n1`` to a power of 2 about the square root of `n`.
As per the matrix fourier FFT the sqrt2 layer is folded into the the
final column IFFTs for better cache locality and the extra twiddles that
occur in the matrix fourier algorithm are combined with the butterflied
performed at the first layer of the final column transforms.
We require `nw` to be at least 64 and the three temporary space pointers
to point to blocks of size ``n*w + FLINT_BITS`` bits.
.. function:: void fft_mfa_truncate_sqrt2_outer(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** temp, mp_size_t n1, mp_size_t trunc)
Just the outer layers of ``fft_mfa_truncate_sqrt2``.
.. function:: void fft_mfa_truncate_sqrt2_inner(mp_limb_t ** ii, mp_limb_t ** jj, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** temp, mp_size_t n1, mp_size_t trunc, mp_limb_t * tt)
The inner layers of ``fft_mfa_truncate_sqrt2`` and
``ifft_mfa_truncate_sqrt2`` combined with pointwise mults.
.. function:: void ifft_mfa_truncate_sqrt2_outer(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** temp, mp_size_t n1, mp_size_t trunc)
The outer layers of ``ifft_mfa_truncate_sqrt2`` combined with
normalisation.
Negacyclic multiplication

.. function:: void fft_negacyclic(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** temp)
As per ``fft_radix2`` except that it performs a sqrt2 negacyclic
transform of length `2n`. This is the same as the radix 2 transform
except that the `i`th coefficient of the input is first multiplied by
`\sqrt{2}^{iw}`.
We require `nw` to be at least 64 and the two temporary space pointers to
point to blocks of size ``n*w + FLINT_BITS`` bits.
.. function:: void ifft_negacyclic(mp_limb_t ** ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** temp)
As per ``ifft_radix2`` except that it performs a sqrt2 negacyclic
inverse transform of length `2n`. This is the same as the radix 2 inverse
transform except that the `i`th coefficient of the output is finally
divided by `\sqrt{2}^{iw}`.
We require `nw` to be at least 64 and the two temporary space pointers to
point to blocks of size ``n*w + FLINT_BITS`` bits.
.. function:: void fft_naive_convolution_1(mp_limb_t * r, mp_limb_t * ii, mp_limb_t * jj, mp_size_t m)
Performs a naive negacyclic convolution of ``ii`` with ``jj``,
both of length `m` and sets `r` to the result. This is essentially
multiplication of polynomials modulo `x^m + 1`.
.. function:: void _fft_mulmod_2expp1(mp_limb_t * r1, mp_limb_t * i1, mp_limb_t * i2, mp_size_t r_limbs, flint_bitcnt_t depth, flint_bitcnt_t w)
Multiply ``i1`` by ``i2`` modulo ``B^r_limbs + 1`` where
``r_limbs = nw/FLINT_BITS`` with ``n = 2^depth``. Uses the
negacyclic FFT convolution CRT'd with a 1 limb naive convolution. We
require that ``depth`` and ``w`` have been selected as per the
wrapper ``fft_mulmod_2expp1`` below.
.. function:: slong fft_adjust_limbs(mp_size_t limbs)
Given a number of limbs, returns a new number of limbs (no more than
the next power of 2) which will work with the Nussbaumer code. It is only
necessary to make this adjustment if
``limbs > FFT_MULMOD_2EXPP1_CUTOFF``.
.. function:: void fft_mulmod_2expp1(mp_limb_t * r, mp_limb_t * i1, mp_limb_t * i2, mp_size_t n, mp_size_t w, mp_limb_t * tt)
As per ``_fft_mulmod_2expp1`` but with a tuned cutoff below which more
classical methods are used for the convolution. The temporary space is
required to fit ``n*w + FLINT_BITS`` bits. There are no restrictions
on `n`, but if ``limbs = n*w/FLINT_BITS`` then if ``limbs`` exceeds
``FFT_MULMOD_2EXPP1_CUTOFF`` the function ``fft_adjust_limbs`` must
be called to increase the number of limbs to an appropriate value.
Integer multiplication

.. function:: void mul_truncate_sqrt2(mp_ptr r1, mp_srcptr i1, mp_size_t n1, mp_srcptr i2, mp_size_t n2, flint_bitcnt_t depth, flint_bitcnt_t w)
Integer multiplication using the radix 2 truncated sqrt2 transforms.
Set ``(r1, n1 + n2)`` to the product of ``(i1, n1)`` by
``(i2, n2)``. This is achieved through an FFT convolution of length at
most ``2^(depth + 2)`` with coefficients of size `nw` bits where
``n = 2^depth``. We require ``depth >= 6``. The input data is
broken into chunks of data not exceeding ``(nw  (depth + 1))/2``
bits. If breaking the first integer into chunks of this size results in
``j1`` coefficients and breaking the second integer results in
``j2`` chunks then ``j1 + j2  1 <= 2^(depth + 2)``.
If ``n = 2^depth`` then we require `nw` to be at least 64.
.. function:: void mul_mfa_truncate_sqrt2(mp_ptr r1, mp_srcptr i1, mp_size_t n1, mp_srcptr i2, mp_size_t n2, flint_bitcnt_t depth, flint_bitcnt_t w)
As for ``mul_truncate_sqrt2`` except that the cache friendly matrix
fourier algorithm is used.
If ``n = 2^depth`` then we require `nw` to be at least 64. Here we
also require `w` to be `2^i` for some `i \geq 0`.
.. function:: void flint_mpn_mul_fft_main(mp_ptr r1, mp_srcptr i1, mp_size_t n1, mp_srcptr i2, mp_size_t n2)
The main integer multiplication routine. Sets ``(r1, n1 + n2)`` to
``(i1, n1)`` times ``(i2, n2)``. We require ``n1 >= n2 > 0``.
Convolution

.. function:: void fft_convolution(mp_limb_t ** ii, mp_limb_t ** jj, slong depth, slong limbs, slong trunc, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** s1, mp_limb_t * tt)
Perform an FFT convolution of ``ii`` with ``jj``, both of length
``4*n`` where ``n = 2^depth``. Assume that all but the first
``trunc`` coefficients of the output (placed in ``ii``) are zero.
Each coefficient is taken modulo ``B^limbs + 1``. The temporary
spaces ``t1``, ``t2`` and ``s1`` must have ``limbs + 1``
limbs of space and ``tt`` must have ``2*(limbs + 1)`` of free
space.
FFT Precaching

.. function:: void fft_precache(mp_limb_t ** jj, slong depth, slong limbs, slong trunc, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** s1)
Precompute the FFT of ``jj`` for use with precache functions. The
parameters are as for ``fft_convolution``.
.. function:: void fft_convolution_precache(mp_limb_t ** ii, mp_limb_t ** jj, slong depth, slong limbs, slong trunc, mp_limb_t ** t1, mp_limb_t ** t2, mp_limb_t ** s1, mp_limb_t ** tt)
As per ``fft_convolution`` except that it is assumed ``fft_precache`` has
been called on ``jj`` with the same parameters. This will then run faster
than if ``fft_convolution`` had been run with the original ``jj``.