U Zʹj-@sdZddlmZmZmZddlmZddlmZm Z ddl m Z m Z ddl mZmZddlmZmZddlmZd$d d Zd%d dZd&ddZeje_d'ddZddZddZeje_d(ddZddZddZeje_ddZd)d d!Zd*d"d#Z eje _d S)+zd Discrete Fourier Transform, Number Theoretic Transform, Walsh Hadamard Transform, Mobius Transform )SSymbolsympify) expand_mul)piI)sincos)isprimeprimitive_root)ibiniterableas_intFc st|stddd|D}tdd|Dr8tdt|dkrL|Sd}d@rt|d7}d||tjgt|7}tdD]D}t t ||d d d d d d}||kr||||||<||<q|rd t n dt d k r dfddtdD}d}|kr|d|} } td|D]j}t| D]Z}|||t |||| || |} } | | | | |||<|||| <qbqV|d9}q.|rd k rfdd|Dnfdd|D}|S)z3Utility function for the Discrete Fourier TransformzAExpected a sequence of numeric coefficients for Fourier TransformcSsg|] }t|qSr.0argrrX/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/discrete/transforms.py sz&_fourier_transform..css|]}|tVqdS)N)hasrrxrrr sz%_fourier_transform..z"Expected non-symbolic coefficientsTstrNcs(g|] }t|tt|qSr)r rr)ri)angrrr4srcsg|]}|qSr)evalfr)dpsnrrr@scsg|] }|qSrrrr%rrrAs)r TypeErrorany ValueErrorlen bit_lengthrZerorangeintr rr#r) seqr$inverseabr!jwhhfutuvr)r"r$r%r_fourier_transformsD     .. r:NcCs t||dS)am Performs the Discrete Fourier Transform (**DFT**) in the complex domain. The sequence is automatically padded to the right with zeros, as the *radix-2 FFT* requires the number of sample points to be a power of 2. This method should be used with default arguments only for short sequences as the complexity of expressions increases with the size of the sequence. Parameters ========== seq : iterable The sequence on which **DFT** is to be applied. dps : Integer Specifies the number of decimal digits for precision. Examples ======== >>> from sympy import fft, ifft >>> fft([1, 2, 3, 4]) [10, -2 - 2*I, -2, -2 + 2*I] >>> ifft(_) [1, 2, 3, 4] >>> ifft([1, 2, 3, 4]) [5/2, -1/2 + I/2, -1/2, -1/2 - I/2] >>> fft(_) [1, 2, 3, 4] >>> ifft([1, 7, 3, 4], dps=15) [3.75, -0.5 - 0.75*I, -1.75, -0.5 + 0.75*I] >>> fft(_) [1.0, 7.0, 3.0, 4.0] References ========== .. [1] https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm .. [2] https://mathworld.wolfram.com/FastFourierTransform.html )r$r:r/r$rrrfftFs.r=cCst||ddS)NT)r$r0r;r<rrrifftwsr>cs4t|stdt|ts(tdfdd|D}t|}|dkrN|S|d}||d@rv|d7}d|}d|rtd|dg|t|7}td|D]D}tt ||d d d d d d}||kr||||||<||<qt }t |d|} |r"t | d} dg|d} td|dD]}| |d| | |<q>d} | |kr| d|| } } td|| D]n}t| D]^}||||||| | | |}}|||||||<|||| <qq| d9} qb|r0t |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)prrrsz/_number_theoretic_transform..rrz/Expected prime modulus of the form (m*2**k + 1)rTrNrcsg|]}|qSrrrr?rvrrrs) r r'rr r)r*r+r-r.r r pow)r/primer0r1r%r2r!r3prrtr4r5r6r7r8r9rr@r_number_theoretic_transformsN     *6 rFcCs t||dS)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 )rCrFr/rCrrrntts(rIcCst||ddS)NT)rCr0rGrHrrrinttsrJc st|stddd|D}t|dkr2|Sd@rJd|tjgt|7}d}|kr|d}td|D]V}t|D]H}|||||||}}|||||||<||||<qq|d9}qf|rfdd|D}|S)z1Utility function for the Walsh Hadamard Transformz@Expected a sequence of coefficients for Walsh Hadamard TransformcSsg|] }t|qSrrrrrrrsz-_walsh_hadamard_transform..rrrcsg|] }|qSrrrr&rrrsr r'r*r+rr,r-) r/r0r1r5r6r!r3r8r9rr&r_walsh_hadamard_transforms(   * rLcCst|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 rLr/rrrfwhts$rOcCs t|ddS)NT)r0rMrNrrrifwht:srPcCst|stddd|D}t|}|dkr2|S||d@rJd|}|tjg|t|7}|rd}||krt|D](}||@rz||||||A7<qz|d9}qjnJd}||krt|D]*}||@rq||||||A7<q|d9}q|S)z\Utility function for performing Mobius Transform using Yate's Dynamic Programming methodz#Expected a sequence of coefficientscSsg|] }t|qSrrrrrrrMsz%_mobius_transform..rrrK)r/sgnsubsetr1r%r!r3rrr_mobius_transformFs0      rSTcCst|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 rrQrRrSr/rRrrrmobius_transformls8rWcCst|d|dS)NrrTrUrVrrrinverse_mobius_transformsrX)F)N)N)F)F)T)T)!__doc__Z sympy.corerrrZsympy.core.functionrZsympy.core.numbersrrZ(sympy.functions.elementary.trigonometricrr Z sympy.ntheoryr r Zsympy.utilities.iterablesr r Zsympy.utilities.miscrr:r=r>rFrIrJrLrOrPrSrWrXrrrrs.  1 1  :+ ' & :