ó ¡¼™\c@ sídZddlmZmZddlmZmZddlmZm Z m Z ddl m Z ddl mZmZmZmZmZmZmZmZdddddd„Zdd „Zd „Zd „Zd „Zd „Zd„ZdS(sl Convolution (using **FFT**, **NTT**, **FWHT**), Subset Convolution, Covering Product, Intersecting Product iÿÿÿÿ(tprint_functiontdivision(tStsympify(trangetas_inttiterable(t expand_mul(tffttiffttntttintttfwhttifwhttmobius_transformtinverse_mobius_transformic C s`t|ƒ}|dkr'tdƒ‚n|r3tnd}|rEtnd}td„||||fDƒƒdkr‚tdƒ‚n|dk rát||d|ƒ}|s­|Sgt|ƒD]#} t|| d|…ƒ|^qºS|rùt||ƒ}n-|rt ||ƒ}nt ||d|ƒ}|s0|Sgt|ƒD]} t|| d|…ƒ^q=S( s Performs convolution by determining the type of desired convolution using hints. Exactly one of ``dps``, ``prime``, ``dyadic``, ``subset`` arguments should be specified explicitly for identifying the type of convolution, and the argument ``cycle`` can be specified optionally. For the default arguments, linear convolution is performed using **FFT**. Parameters ========== a, b : iterables The sequences for which convolution is performed. cycle : Integer Specifies the length for doing cyclic convolution. dps : Integer Specifies the number of decimal digits for precision for performing **FFT** on the sequence. prime : Integer Prime modulus of the form `(m 2^k + 1)` to be used for performing **NTT** on the sequence. dyadic : bool Identifies the convolution type as dyadic (*bitwise-XOR*) convolution, which is performed using **FWHT**. subset : bool Identifies the convolution type as subset convolution. Examples ======== >>> from sympy import convolution, symbols, S, I >>> u, v, w, x, y, z = symbols('u v w x y z') >>> convolution([1 + 2*I, 4 + 3*I], [S(5)/4, 6], dps=3) [1.25 + 2.5*I, 11.0 + 15.8*I, 24.0 + 18.0*I] >>> convolution([1, 2, 3], [4, 5, 6], cycle=3) [31, 31, 28] >>> convolution([111, 777], [888, 444], prime=19*2**10 + 1) [1283, 19351, 14219] >>> convolution([111, 777], [888, 444], prime=19*2**10 + 1, cycle=2) [15502, 19351] >>> convolution([u, v], [x, y, z], dyadic=True) [u*x + v*y, u*y + v*x, u*z, v*z] >>> convolution([u, v], [x, y, z], dyadic=True, cycle=2) [u*x + u*z + v*y, u*y + v*x + v*z] >>> convolution([u, v, w], [x, y, z], subset=True) [u*x, u*y + v*x, u*z + w*x, v*z + w*y] >>> convolution([u, v, w], [x, y, z], subset=True, cycle=3) [u*x + v*z + w*y, u*y + v*x, u*z + w*x] is6The length for cyclic convolution must be non-negativecs s|]}|dk VqdS(N(tNone(t.0tx((s:lib/python2.7/site-packages/sympy/discrete/convolutions.pys Psis0Ambiguity in determining the type of convolutiontprimeNtdps( Rt ValueErrortTrueRtsumt TypeErrortconvolution_nttRtconvolution_fwhttconvolution_subsettconvolution_fft( tatbtcycleRRtdyadictsubsettctlsti((s:lib/python2.7/site-packages/sympy/discrete/convolutions.pyt convolutions :  ( >cC sþ||}}t|ƒt|ƒd}}|dkrZ||d@rZd|jƒ}n|tjg|t|ƒ7}|tjg|t|ƒ7}t||ƒt||ƒ}}gt||ƒD]\}}t||ƒ^qÅ}t||ƒ| }|S(s Performs linear convolution using Fast Fourier Transform. Parameters ========== a, b : iterables The sequences for which convolution is performed. dps : Integer Specifies the number of decimal digits for precision. Examples ======== >>> from sympy import S, I >>> from sympy.discrete.convolutions import convolution_fft >>> convolution_fft([2, 3], [4, 5]) [8, 22, 15] >>> convolution_fft([2, 5], [6, 7, 3]) [12, 44, 41, 15] >>> convolution_fft([1 + 2*I, 4 + 3*I], [S(5)/4, 6]) [5/4 + 5*I/2, 11 + 63*I/4, 24 + 18*I] References ========== .. [1] https://en.wikipedia.org/wiki/Convolution_theorem .. [2] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29 iii(tlent bit_lengthRtZeroRtzipRR (RRRtntmRty((s:lib/python2.7/site-packages/sympy/discrete/convolutions.pyRgs!2cC s||t|ƒ}}}t|ƒt|ƒd}}|dkrg||d@rgd|jƒ}n|dg|t|ƒ7}|dg|t|ƒ7}t||ƒt||ƒ}}gt||ƒD]\}}|||^qÌ}t||ƒ| }|S(s= Performs linear convolution using Number Theoretic Transform. Parameters ========== a, b : iterables The sequences for which convolution is performed. prime : Integer Prime modulus of the form `(m 2^k + 1)` to be used for performing **NTT** on the sequence. Examples ======== >>> from sympy.discrete.convolutions import convolution_ntt >>> convolution_ntt([2, 3], [4, 5], prime=19*2**10 + 1) [8, 22, 15] >>> convolution_ntt([2, 5], [6, 7, 3], prime=19*2**10 + 1) [12, 44, 41, 15] >>> convolution_ntt([333, 555], [222, 666], prime=19*2**10 + 1) [15555, 14219, 19404] References ========== .. [1] https://en.wikipedia.org/wiki/Convolution_theorem .. [2] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29 iii(RR&R'R R)R (RRRtpR*R+RR,((s:lib/python2.7/site-packages/sympy/discrete/convolutions.pyRŸs 0cC sô| s| rgS||}}tt|ƒt|ƒƒ}||d@r]d|jƒ}n|tjg|t|ƒ7}|tjg|t|ƒ7}t|ƒt|ƒ}}gt||ƒD]\}}t||ƒ^qÂ}t|ƒ}|S(s Performs dyadic (*bitwise-XOR*) convolution using Fast Walsh Hadamard Transform. The convolution 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 ========== a, b : iterables The sequences for which convolution is performed. Examples ======== >>> from sympy import symbols, S, I >>> from sympy.discrete.convolutions import convolution_fwht >>> u, v, x, y = symbols('u v x y') >>> convolution_fwht([u, v], [x, y]) [u*x + v*y, u*y + v*x] >>> convolution_fwht([2, 3], [4, 5]) [23, 22] >>> convolution_fwht([2, 5 + 4*I, 7], [6*I, 7, 3 + 4*I]) [56 + 68*I, -10 + 30*I, 6 + 50*I, 48 + 32*I] >>> convolution_fwht([S(33)/7, S(55)/6, S(7)/4], [S(2)/3, 5]) [2057/42, 1870/63, 7/6, 35/4] References ========== .. [1] https://www.radioeng.cz/fulltexts/2002/02_03_40_42.pdf .. [2] https://en.wikipedia.org/wiki/Hadamard_transform ii( tmaxR&R'RR(R R)RR (RRR*RR,((s:lib/python2.7/site-packages/sympy/discrete/convolutions.pyRÖs(2 c C s| s| rgSt|ƒ s,t|ƒ r;tdƒ‚ng|D]}t|ƒ^qB}g|D]}t|ƒ^qa}tt|ƒt|ƒƒ}||d@rµd|jƒ}n|tjg|t|ƒ7}|tjg|t|ƒ7}tjg|}x‡t|ƒD]y}|}xD|dkr`||ct |||||Aƒ7<|d|@}qW||ct |||||Aƒ7>> from sympy import symbols, S, I >>> from sympy.discrete.convolutions import convolution_subset >>> u, v, x, y, z = symbols('u v x y z') >>> convolution_subset([u, v], [x, y]) [u*x, u*y + v*x] >>> convolution_subset([u, v, x], [y, z]) [u*y, u*z + v*y, x*y, x*z] >>> convolution_subset([1, S(2)/3], [3, 4]) [3, 6] >>> convolution_subset([1, 3, S(5)/7], [7]) [7, 21, 5, 0] References ========== .. [1] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf s3Expected a sequence of coefficients for convolutioniii( RRRR.R&R'RR(RR(RRtargR*R"tmasktsmask((s:lib/python2.7/site-packages/sympy/discrete/convolutions.pyRs&)&*cC sô| s| rgS||}}tt|ƒt|ƒƒ}||d@r]d|jƒ}n|tjg|t|ƒ7}|tjg|t|ƒ7}t|ƒt|ƒ}}gt||ƒD]\}}t||ƒ^qÂ}t|ƒ}|S(s® Returns the covering product of given sequences. The indices of each argument, considered as bit strings, correspond to subsets of a finite set. The covering product of given sequences is a sequence which contains the sum of products of the elements of the given sequences grouped by the *bitwise-OR* of the corresponding indices. The sequence is automatically padded to the right with zeros, as the definition of subset based on bitmasks (indices) requires the size of sequence to be a power of 2. Parameters ========== a, b : iterables The sequences for which covering product is to be obtained. Examples ======== >>> from sympy import symbols, S, I, covering_product >>> u, v, x, y, z = symbols('u v x y z') >>> covering_product([u, v], [x, y]) [u*x, u*y + v*x + v*y] >>> covering_product([u, v, x], [y, z]) [u*y, u*z + v*y + v*z, x*y, x*z] >>> covering_product([1, S(2)/3], [3, 4 + 5*I]) [3, 26/3 + 25*I/3] >>> covering_product([1, 3, S(5)/7], [7, 8]) [7, 53, 5, 40/7] References ========== .. [1] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf ii( R.R&R'RR(RR)RR(RRR*RR,((s:lib/python2.7/site-packages/sympy/discrete/convolutions.pytcovering_productes,2 cC s| s| rgS||}}tt|ƒt|ƒƒ}||d@r]d|jƒ}n|tjg|t|ƒ7}|tjg|t|ƒ7}t|dtƒt|dtƒ}}gt||ƒD]\}}t||ƒ^qÎ}t |dtƒ}|S(sß Returns the intersecting product of given sequences. The indices of each argument, considered as bit strings, correspond to subsets of a finite set. The intersecting product of given sequences is the sequence which contains the sum of products of the elements of the given sequences grouped by the *bitwise-AND* of the corresponding indices. The sequence is automatically padded to the right with zeros, as the definition of subset based on bitmasks (indices) requires the size of sequence to be a power of 2. Parameters ========== a, b : iterables The sequences for which intersecting product is to be obtained. Examples ======== >>> from sympy import symbols, S, I, intersecting_product >>> u, v, x, y, z = symbols('u v x y z') >>> intersecting_product([u, v], [x, y]) [u*x + u*y + v*x, v*y] >>> intersecting_product([u, v, x], [y, z]) [u*y + u*z + v*y + x*y + x*z, v*z, 0, 0] >>> intersecting_product([1, S(2)/3], [3, 4 + 5*I]) [9 + 5*I, 8/3 + 10*I/3] >>> intersecting_product([1, 3, S(5)/7], [7, 8]) [327/7, 24, 0, 0] References ========== .. [1] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf iiR!( R.R&R'RR(RtFalseR)RR(RRR*RR,((s:lib/python2.7/site-packages/sympy/discrete/convolutions.pytintersecting_product«s,%2N(t__doc__t __future__RRt sympy.coreRRtsympy.core.compatibilityRRRtsympy.core.functionRtsympy.discrete.transformsRR R R R R RRRR%RRRRR2R4(((s:lib/python2.7/site-packages/sympy/discrete/convolutions.pyts:X 8 7 B M F