a Эмєz&_fourier_transform..css|]}|аtбVqdS)N)Zhasrйr┌xrrr┌ rz%_fourier_transform..z"Expected non-symbolic coefficientsщщTй┌strNщ щ■ cs(g|] }tИ|ГttИ|ГСqSr)r rr)r┌i)┌angrrr4rrcsg|]}|ИаИбСqSr)┌evalfr)┌dps┌nrrr@rcsg|]}|ИСqSrrrйr%rrrAr)r ┌ TypeError┌any┌ ValueError┌len┌ bit_lengthr┌Zero┌range┌intrrr#r) ┌seqr$┌inverse┌a┌br!┌j┌w┌h┌hf┌ut┌u┌vr)r"r$r%r┌_fourier_transformsD .. r:NcCst||dНS)al 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] http://mathworld.wolfram.com/FastFourierTransform.html )r$йr:йr/r$rrr┌fftFs.r=cCst||ddНS)NT)r$r0r;r<rrr┌ifftwsr>cs4t|ГstdГВt|ГЙtИГs(tdГВЗfddД|DГ}t|Г}|dkrN|S|абd}||d@rv|d7}d|}Иd|rКtdГВ|dg|t|Г7}td|ГD]D}tt ||d d НdddЕdГ}||krк||||||<||<qкt ИГ}t|Иd|ИГ} |Рr"t| ИdИГ} dg|d} td|dГD]}| |d| И| |<Рq>d}||kРr|d||}} td||ГD]n}t|ГD]^}|||||||| | |}}||И||И|||<||||<РqЦРqК|d9}Рqb|Рr0t|ИdИГЙЗЗfd dД|DГ}|S)z3Utility function for the Number Theoretic TransformzJExpected a sequence of integer coefficients for Number Theoretic Transformz5Expected prime modulus for Number Theoretic Transformcsg|]}t|ГИСqSrrr)┌prrrПrz/_number_theoretic_transform..rrz/Expected prime modulus of the form (m*2**k + 1)rTrNrcsg|]}|ИИСqSrrrйr?┌rvrrr╕r)r r'rr r)r*r+r-r.rr┌pow)r/┌primer0r1r%r2r!r3┌pr┌rtr4r5r6r7r8r9rr@r┌_number_theoretic_transformГsN *6rFcCst||dНS)aQ 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] http://mathworld.wolfram.com/NumberTheoreticTransform.html .. [3] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29 )rCйrFйr/rCrrr┌ntt╜s(rIcCst||ddНS)NT)rCr0rGrHrrr┌inttшsrJc s■t|ГstdГВddД|DГ}t|ГЙИdkr2|SИИd@rJdИабЙ|tjgИt|Г7}d}|Иkrф|d}tdИ|ГD]V}t|ГD]H}|||||||}}|||||||<||||<qОqВ|d9}qf|r·ЗfddД|DГ}|S)z1Utility function for the Walsh Hadamard Transformz@Expected a sequence of coefficients for Walsh Hadamard TransformcSsg|]}t|ГСqSrrrrrrr√rz-_walsh_hadamard_transform..rrrcsg|]}|ИСqSrrrr&rrrrйr r'r*r+rr,r-) r/r0r1r5r6r!r3r8r9rr&r┌_walsh_hadamard_transformЇs(* rLcCst|ГS)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 йrLйr/rrr┌fwhts$rOcCst|ddНS)NT)r0rMrNrrr┌ifwht:srPcCs■t|ГstdГВddД|DГ}t|Г}|dkr2|S||d@rJd|аб}|tjg|t|Г7}|r░d}||kr·t|ГD](}||@rz||||||A7<qz|d9}qjnJd}||kr·t|ГD]*}||@r╥q─||||||A7<q─|d9}q┤|S)z\Utility function for performing Mobius Transform using Yate's Dynamic Programming methodz#Expected a sequence of coefficientscSsg|]}t|ГСqSrrrrrrrMrz%_mobius_transform..rrrK)r/┌sgn┌subsetr1r%r!r3rrr┌_mobius_transformFs0 rSTcCst|d|dНS)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 rйrQrRйrSйr/rRrrr┌mobius_transformls8rWcCst|d|dНS)NrrTrUrVrrr┌inverse_mobius_transformжsrX)F)N)N)F)F)T)T)!┌__doc__Z sympy.corerrrZsympy.core.functionrZsympy.core.numbersrrZ(sympy.functions.elementary.trigonometricrr Z sympy.ntheoryr rZsympy.utilities.iterablesrr Zsympy.utilities.miscrr:r=r>rFrIrJrLrOrPrSrWrXrrrr┌s. 1 1 :+ ' & :