B [.@sdZddlmZmZmZddlmZmZmZddl m Z m Z m Z ddl mZddlmZmZddlmZmZddlmZmZdd lmZd%d d Zd&ddZd'ddZeje_d(ddZddZddZeje_d)ddZ ddZ!ddZ"e!je"_ddZ#d*d!d"Z$d+d#d$Z%e$je%_d S),zd Discrete Fourier Transform, Number Theoretic Transform, Walsh Hadamard Transform, Mobius Transform )print_functiondivisionunicode_literals)SSymbolsympify)as_intrangeiterable) expand_mul)piI)sincos)isprimeprimitive_root)ibinFc st|stddd|D}tdd|Dr8tdt|dkrL|Sd}d@rt|d7}d||tjgt|7}xRtdD]D}t t ||d d d d d d}||kr||||||<||<qW|rd t n dt d k r dfddtdD}d}x|kr|d|} } x~td|D]n}xft| D]Z}|||t |||| || |} } | | | | |||<|||| <qlWq^W|d9}q4W|rd k rfdd|Dnfdd|D}|S)z3Utility function for the Discrete Fourier TransformzAExpected a sequence of numeric coefficients for Fourier TransformcSsg|] }t|qS)r).0argrr8lib/python3.7/site-packages/sympy/discrete/transforms.py sz&_fourier_transform..css|]}|tVqdS)N)Zhasr)rxrrr sz%_fourier_transform..z"Expected non-symbolic coefficientsT)strNcs(g|] }t|tt|qSr)rr r)ri)angrrr7srcsg|]}|qSr)evalf)rr)dpsnrrrCscsg|] }|qSrr)rr)r#rrrDs)r TypeErrorany ValueErrorlen bit_lengthrZeror intrr r!r ) seqr"inverseabrjwhhfutuvr)r r"r#r_fourier_transformsB    .2r6NcCs t||dS)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")r6)r+r"rrrfftIs.r7cCst||ddS)NT)r"r,)r6)r+r"rrrifftzsr8csHt|stdt|ts(tdfdd|D}t|}|dkrN|S|d}||d@rv|d7}d|}d|rtd|dg|t|7}xRtd|D]D}tt ||d d d d d d}||kr||||||<||<qWt }t |d|} |r&t | d} dg|d} x0td|dD]}| |d| | |<qDWd} x| |kr| d|| } } xtd|| D]r}xjt| D]^}||||||| | | |}}|||||||<|||| <qWqW| d9} qlW|rDt |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|qSr)r)rr)prrrsz/_number_theoretic_transform..rrz/Expected prime modulus of the form (m*2**k + 1)rT)rNrcsg|]}|qSrr)rr)r9rvrrrs) r r$rrr&r'r(r r*rrpow)r+primer,r-r#r.rr/ZprZrtr0r1r2r3r4r5r)r9r:r_number_theoretic_transformsN    *:r=cCs t||dS)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 )r<)r=)r+r<rrrntts(r>cCst||ddS)NT)r<r,)r=)r+r<rrrinttsr?c st|stddd|D}t|dkr2|Sd@rJd|tjgt|7}d}x|kr|d|}}xjtd|D]Z}xTt|D]H}|||||||}} || || |||<||||<qWqW|d9}qhW|rfdd|D}|S)z1Utility function for the Walsh Hadamard Transformz@Expected a sequence of coefficients for Walsh Hadamard TransformcSsg|] }t|qSr)r)rrrrrrsz-_walsh_hadamard_transform..rrrcsg|] }|qSrr)rr)r#rrrs)r r$r'r(rr)r ) r+r,r-r1r2r3rr/r4r5r)r#r_walsh_hadamard_transforms(   . r@cCst|S)aN Performs the Walsh Hadamard Transform (**WHT**), and uses Hadamard ordering for the sequence. The sequence is automatically padded to the right with zeros, as the *radix-2 FWHT* requires the number of sample points to be a power of 2. Parameters ========== seq : iterable The sequence on which WHT is to be applied. Examples ======== >>> from sympy import fwht, ifwht >>> fwht([4, 2, 2, 0, 0, 2, -2, 0]) [8, 0, 8, 0, 8, 8, 0, 0] >>> ifwht(_) [4, 2, 2, 0, 0, 2, -2, 0] >>> ifwht([19, -1, 11, -9, -7, 13, -15, 5]) [2, 0, 4, 0, 3, 10, 0, 0] >>> fwht(_) [19, -1, 11, -9, -7, 13, -15, 5] References ========== .. [1] https://en.wikipedia.org/wiki/Hadamard_transform .. [2] https://en.wikipedia.org/wiki/Fast_Walsh%E2%80%93Hadamard_transform )r@)r+rrrfwhts$rAcCs t|ddS)NT)r,)r@)r+rrrifwht=srBcCst|stddd|D}t|}|dkr2|S||d@rJd|}|tjg|t|7}|rd}x||krx4t|D](}||@r~||||||A7<q~W|d9}qlWnTd}xN||kr x6t|D]*}||@rq||||||A7<qW|d9}qW|S)u]Utility function for performing Möbius Transform using Yate's Dynamic Programming methodz#Expected a sequence of coefficientscSsg|] }t|qSr)r)rrrrrrPsz%_mobius_transform..rr)r r$r'r(rr)r )r+sgnsubsetr-r#rr/rrr_mobius_transformIs0       rETcCst|d|dS)u  Performs the Möbius 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 Möbius Transform is to be applied. subset : bool Specifies if Möbius 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 r)rCrD)rE)r+rDrrrmobius_transformos8rFcCst|d|dS)Nr)rCrD)rE)r+rDrrrinverse_mobius_transformsrG)F)N)N)F)F)T)T)&__doc__Z __future__rrrZ sympy.corerrrZsympy.core.compatibilityrr r Zsympy.core.functionr Zsympy.core.numbersr r Z(sympy.functions.elementary.trigonometricrrZ sympy.ntheoryrrZsympy.utilities.iterablesrr6r7r8r=r>r?r@rArBrErFrGrrrrs0  1 1  :+ ' & :