o jg-@sdZddlmZmZmZddlmZddlmZm Z ddl m Z m Z ddl mZmZddlmZmZddlmZd$d d Zd%d dZd%ddZeje_d$ddZddZddZeje_d$ddZddZddZeje_ddZd&d d!Zd&d"d#Z eje _d S)'zd Discrete Fourier Transform, Number Theoretic Transform, Walsh Hadamard Transform, Mobius Transform )SSymbolsympify) expand_mul)piI)sincos)isprimeprimitive_root)ibiniterableas_intFc st|stddd|D}tdd|Drtdt|dkr&|Sd}d@r:|d7}d||tjgt|7}tdD]"}t t ||d d d d d d}||krm||||||<||<qK|rvd t ndt d ur dfddtdD}d}|kr|d|} } td|D]3}t| D],}|||t |||| || |} } | | | | |||<|||| <qq|d9}|ks|rd urfdd|Dnfdd|D}|S)z3Utility function for the Discrete Fourier TransformzAExpected a sequence of numeric coefficients for Fourier TransformcSg|]}t|qSr.0argrrQ/var/www/html/zoom/venv/lib/python3.10/site-packages/sympy/discrete/transforms.py z&_fourier_transform..css|]}|tVqdSN)hasrrxrrr sz%_fourier_transform..z"Expected non-symbolic coefficientsTstrNcs(g|]}t|tt|qSr)r rr)ri)angrrr4s(rcsg|] }|qSr)evalfr)dpsnrrr@scg|]}|qSrrrr(rrrAr)r TypeErrorany ValueErrorlen bit_lengthrZerorangeintr rr&r) seqr'inverseabr$jwhhfutuvr)r%r'r(r_fourier_transformsJ   .(r>NcC t||dS)am Performs the Discrete Fourier Transform (**DFT**) in the complex domain. The sequence is automatically padded to the right with zeros, as the *radix-2 FFT* requires the number of sample points to be a power of 2. This method should be used with default arguments only for short sequences as the complexity of expressions increases with the size of the sequence. Parameters ========== seq : iterable The sequence on which **DFT** is to be applied. dps : Integer Specifies the number of decimal digits for precision. Examples ======== >>> from sympy import fft, ifft >>> fft([1, 2, 3, 4]) [10, -2 - 2*I, -2, -2 + 2*I] >>> ifft(_) [1, 2, 3, 4] >>> ifft([1, 2, 3, 4]) [5/2, -1/2 + I/2, -1/2, -1/2 - I/2] >>> fft(_) [1, 2, 3, 4] >>> ifft([1, 7, 3, 4], dps=15) [3.75, -0.5 - 0.75*I, -1.75, -0.5 + 0.75*I] >>> fft(_) [1.0, 7.0, 3.0, 4.0] References ========== .. [1] https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm .. [2] https://mathworld.wolfram.com/FastFourierTransform.html )r'r>r3r'rrrfftFs .rBcCt||ddS)NT)r'r4r@rArrrifftwrDcs0t|stdt|tstdfdd|D}t|}|dkr'|S|d}||d@r;|d7}d|}d|rEtd|dg|t|7}td|D]"}tt ||d d d d d d}||krw||||||<||<qUt }t |d|} |rt | d} dg|d} td|dD]}| |d| | |<qd} | |kr| d|| } } td|| D]5}t| D].}||||||| | | |}}|||||||<|||| <qq| d9} | |ks|rt |dfd d|D}|S)z3Utility function for the Number Theoretic TransformzJExpected a sequence of integer coefficients for Number Theoretic Transformz5Expected prime modulus for Number Theoretic Transformcsg|]}t|qSrrr)prrrz/_number_theoretic_transform..rrz/Expected prime modulus of the form (m*2**k + 1)rTr Nr"csg|]}|qSrrrrFrvrrrrG) r r+rr r-r.r/r1r2r r pow)r3primer4r5r(r6r$r7prrtr8r9r:r;r<r=rrHr_number_theoretic_transformsT     *0rNcCr?)aR Performs the Number Theoretic Transform (**NTT**), which specializes the Discrete Fourier Transform (**DFT**) over quotient ring `Z/pZ` for prime `p` instead of complex numbers `C`. The sequence is automatically padded to the right with zeros, as the *radix-2 NTT* requires the number of sample points to be a power of 2. Parameters ========== seq : iterable The sequence on which **DFT** is to be applied. prime : Integer Prime modulus of the form `(m 2^k + 1)` to be used for performing **NTT** on the sequence. Examples ======== >>> from sympy import ntt, intt >>> ntt([1, 2, 3, 4], prime=3*2**8 + 1) [10, 643, 767, 122] >>> intt(_, 3*2**8 + 1) [1, 2, 3, 4] >>> intt([1, 2, 3, 4], prime=3*2**8 + 1) [387, 415, 384, 353] >>> ntt(_, prime=3*2**8 + 1) [1, 2, 3, 4] References ========== .. [1] http://www.apfloat.org/ntt.html .. [2] https://mathworld.wolfram.com/NumberTheoreticTransform.html .. [3] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29 )rKrNr3rKrrrntts (rQcCrC)NT)rKr4rOrPrrrinttrErRc st|stddd|D}t|dkr|Sd@r%d|tjgt|7}d}|kru|d}td|D]+}t|D]$}|||||||}}|||||||<||||<qGqA|d9}|ks7|rfdd|D}|S)z1Utility function for the Walsh Hadamard Transformz@Expected a sequence of coefficients for Walsh Hadamard TransformcSrrrrrrrrrz-_walsh_hadamard_transform..rrrcr)rrrr*rrrrr r+r.r/rr0r1) r3r4r5r9r:r$r7r<r=rr*r_walsh_hadamard_transforms,   (rTcCst|S)aN Performs the Walsh Hadamard Transform (**WHT**), and uses Hadamard ordering for the sequence. The sequence is automatically padded to the right with zeros, as the *radix-2 FWHT* requires the number of sample points to be a power of 2. Parameters ========== seq : iterable The sequence on which WHT is to be applied. Examples ======== >>> from sympy import fwht, ifwht >>> fwht([4, 2, 2, 0, 0, 2, -2, 0]) [8, 0, 8, 0, 8, 8, 0, 0] >>> ifwht(_) [4, 2, 2, 0, 0, 2, -2, 0] >>> ifwht([19, -1, 11, -9, -7, 13, -15, 5]) [2, 0, 4, 0, 3, 10, 0, 0] >>> fwht(_) [19, -1, 11, -9, -7, 13, -15, 5] References ========== .. [1] https://en.wikipedia.org/wiki/Hadamard_transform .. [2] https://en.wikipedia.org/wiki/Fast_Walsh%E2%80%93Hadamard_transform rTr3rrrfwhts$rWcCs t|ddS)NT)r4rUrVrrrifwht:s rXcCs t|stddd|D}t|}|dkr|S||d@r%d|}|tjg|t|7}|r\d}||krZt|D]}||@rQ||||||A7<q=|d9}||ks9|Sd}||krt|D]}||@rmqf||||||A7<qf|d9}||ksb|S)z\Utility function for performing Mobius Transform using Yate's Dynamic Programming methodz#Expected a sequence of coefficientscSrrrrrrrrMrz%_mobius_transform..rrrS)r3sgnsubsetr5r(r$r7rrr_mobius_transformFs8    r[TcCt|d|dS)a  Performs the Mobius Transform for subset lattice with indices of sequence as bitmasks. The indices of each argument, considered as bit strings, correspond to subsets of a finite set. The sequence is automatically padded to the right with zeros, as the definition of subset/superset based on bitmasks (indices) requires the size of sequence to be a power of 2. Parameters ========== seq : iterable The sequence on which Mobius Transform is to be applied. subset : bool Specifies if Mobius Transform is applied by enumerating subsets or supersets of the given set. Examples ======== >>> from sympy import symbols >>> from sympy import mobius_transform, inverse_mobius_transform >>> x, y, z = symbols('x y z') >>> mobius_transform([x, y, z]) [x, x + y, x + z, x + y + z] >>> inverse_mobius_transform(_) [x, y, z, 0] >>> mobius_transform([x, y, z], subset=False) [x + y + z, y, z, 0] >>> inverse_mobius_transform(_, subset=False) [x, y, z, 0] >>> mobius_transform([1, 2, 3, 4]) [1, 3, 4, 10] >>> inverse_mobius_transform(_) [1, 2, 3, 4] >>> mobius_transform([1, 2, 3, 4], subset=False) [10, 6, 7, 4] >>> inverse_mobius_transform(_, subset=False) [1, 2, 3, 4] References ========== .. [1] https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula .. [2] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf .. [3] https://arxiv.org/pdf/1211.0189.pdf rrYrZr[r3rZrrrmobius_transformls8r`cCr\)Nr"r]r^r_rrrinverse_mobius_transformrEra)Fr)T)!__doc__ sympy.corerrrsympy.core.functionrsympy.core.numbersrr(sympy.functions.elementary.trigonometricrr sympy.ntheoryr r sympy.utilities.iterablesr r sympy.utilities.miscrr>rBrDrNrQrRrTrWrXr[r`rarrrrs0   1 1 :+ ' & :