Ed-,dZddlmZmZmZddlmZddlmZm Z ddl m Z m Z ddl mZmZddlmZmZddlmZdd Zdd Zdd Zeje_ddZdZdZeje_ddZdZdZeje_dZddZddZ eje _d S)zd Discrete Fourier Transform, Number Theoretic Transform, Walsh Hadamard Transform, Mobius Transform )SSymbolsympify) expand_mul)piI)sincos)isprimeprimitive_root)ibiniterableas_intFc 0 t|stdd|D}td|Drtdt |dkr|Sdz }dz zr |dz }d|z|t jgt |z zz }tdD]H}tt||dd d d d}||kr||||c||<||<I|r d tzz n dtzz  dz fd tdzD}d}|kr|dz|z} } td |D]`}t| D]N}|||zt|||z| z|| |zz} } | | z| | z c|||z<|||z| z<Oa|dz}|k|rfd|Dn fd|D}|S)z3Utility function for the Discrete Fourier TransformzAExpected a sequence of numeric coefficients for Fourier Transformc,g|]}t|Sr.0args 9lib/python3.11/site-packages/sympy/discrete/transforms.py z&_fourier_transform..%%%#%%%c3JK|]}|tVdSN)hasr)rxs r z%_fourier_transform..s, $ $Q155== $ $ $ $ $ $rz"Expected non-symbolic coefficientsTstrNcjg|]/}t|ztt|zzz0Sr)r rr )riangs rrz&_fourier_transform..4s6:::qSUaCE l ":::rrc@g|]}|z Sr)evalf)rrdpsns rrz&_fourier_transform..@s) ) ) )!ac[[   ) ) )rcg|]}|z Srrrrr-s rrz&_fourier_transform..As!1!1!1!!A#!1!1!1r)r TypeErrorany ValueErrorlen bit_lengthrZerorangeintr rr+r)seqr,inverseabr(jwhhfutuvr)r-s ` @@r_fourier_transformrCs C==1011 1 &%%%%A $ $! $ $ $$$?=>>> AA1u A!a%y Q qD!&1s1vv: A 1a[[$$ Qt$$$TTrT*A . . q5 $1qtJAaD!A$ ("R%''!B$q&C !iia  ::::E!q&MM:::A A q&aaBq!Q 7 7A2YY 7 7QxAa!ebjM!BF),C!D!D1*+a%Q'!a%!AEBJ-- 7 Q q&2-0 2 ) ) ) ) )q ) ) ) )!1!1!1!1q!1!1!1  HrNc$t||S)al Performs the Discrete Fourier Transform (**DFT**) in the complex domain. The sequence is automatically padded to the right with zeros, as the *radix-2 FFT* requires the number of sample points to be a power of 2. This method should be used with default arguments only for short sequences as the complexity of expressions increases with the size of the sequence. Parameters ========== seq : iterable The sequence on which **DFT** is to be applied. dps : Integer Specifies the number of decimal digits for precision. Examples ======== >>> from sympy import fft, ifft >>> fft([1, 2, 3, 4]) [10, -2 - 2*I, -2, -2 + 2*I] >>> ifft(_) [1, 2, 3, 4] >>> ifft([1, 2, 3, 4]) [5/2, -1/2 + I/2, -1/2, -1/2 - I/2] >>> fft(_) [1, 2, 3, 4] >>> ifft([1, 7, 3, 4], dps=15) [3.75, -0.5 - 0.75*I, -1.75, -0.5 + 0.75*I] >>> fft(_) [1.0, 7.0, 3.0, 4.0] References ========== .. [1] https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm .. [2] http://mathworld.wolfram.com/FastFourierTransform.html )r,rCr8r,s rfftrGFs\ cs + + ++rc&t||dS)NT)r,r9rErFs rifftrIws csD 9 9 99rct|stdt|tst dfd|D}t |}|dkr|S|dz }||dz zr |dz }d|z}dz |zrt d|dg|t |z zz }td|D]H}tt||d d d d d}||kr||||c||<||<It}t|dz |z} |rt| dz } dg|dzz} td|dzD]}| |dz | zz| |<d} | |kr| dz|| z} } td|| D]Y}t| D]G}|||z|||z| z| | |zz}}||zz||z zc|||z<|||z| z<HZ| dz} | |k|r#t|dz fd |D}|S) z3Utility function for the Number Theoretic TransformzJExpected a sequence of integer coefficients for Number Theoretic Transformz5Expected prime modulus for Number Theoretic Transformc4g|]}t|zSrr)rrps rrz/_number_theoretic_transform..s#$$$1Q$$$rr"r!z/Expected prime modulus of the form (m*2**k + 1)rTr#Nr%c g|] }|zz Srr)rrrLrvs rrz/_number_theoretic_transform..s! ! ! !!QrTAX ! ! !r) rr0rr r2r3r4r6r7r r pow)r8primer9r:r-r;r(r<prrtr=r>r?r@rArBrLrNs @@r_number_theoretic_transformrSs C==:9:: : u A 1::6566 6 %$$$$$$A AA1u A!a%y Q qD A{LJKKK!a#a&&j A 1a[[$$ Qt$$$TTrT*A . . q5 $1qtJAaD!A$   B R!a%Aq ! !B QUA   Q!V A 1a1f  Qx{Q! A q&aaBq!Q C CA2YY C CQx1q52:qay!81+,q5A+A{'!a%!AEBJ-- C Q q&" AE1   ! ! ! ! !q ! ! ! Hrc$t||S)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 )rPrSr8rPs rnttrWsP 's% 8 8 88rc&t||dS)NT)rPr9rUrVs rinttrYs &s% F F FFrc t|stdd|D}t| dkr|S dz zrd z |tjg t|z zz }d}| kri|dz}t d |D]G}t |D]5}|||z|||z|z}}||z||z c|||z<|||z|z<6H|dz}| ki|r fd|D}|S)z1Utility function for the Walsh Hadamard Transformz@Expected a sequence of coefficients for Walsh Hadamard Transformc,g|]}t|Srrrs rrz-_walsh_hadamard_transform..rrr!r"rcg|]}|z Srrr/s rrz-_walsh_hadamard_transform..s   QQqS   rrr0r3r4rr5r6) r8r9r:r>r?r(r<rArBr-s @r_walsh_hadamard_transformr^so C==8788 8 &%%%%A AA1u!a%y q||~~ !&1s1vv: A A q& !Vq!Q 7 7A2YY 7 7Qx1q52:1*+a%Q'!a%!AEBJ-- 7 Q q&    !    Hrc t|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^r8s rfwhtrbsH %S ) ))rc$t|dS)NT)r9r`ras rifwhtrd:s $S$ 7 7 77rc,t|stdd|D}t|}|dkr|S||dz zrd|z}|tjg|t|z zz }|rGd}||kr>t |D]#}||zr||xx||||z zz cc<$|dz}||k>nGd}||kr?t |D]$}||zr||xx||||z zz cc<%|dz}||k?|S)z\Utility function for performing Mobius Transform using Yate's Dynamic Programming methodz#Expected a sequence of coefficientsc,g|]}t|Srrrs rrz%_mobius_transform..Mrrr!r"r])r8sgnsubsetr:r-r(r<s r_mobius_transformriFs} C==?=>>>%%%%%A AA1u!a%y q||~~ !&1s1vv: A  !e 1XX ) )q5)aDDDC!a%L(DDD FA !e  !e 1XX % %q5!Aa!eH $ FA !e  HrTc&t|d|S)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 r"rgrhrir8rhs rmobius_transformrnlsp Sb 8 8 88rc&t|d|S)Nr%rkrlrms rinverse_mobius_transformrps Sb 8 8 88r)Fr)T)!__doc__ sympy.corerrrsympy.core.functionrsympy.core.numbersrr(sympy.functions.elementary.trigonometricr r sympy.ntheoryr r sympy.utilities.iterablesr rsympy.utilities.miscrrCrGrIrSrWrYr^rbrdrirnrprrrrys *)))))))))******$$$$$$$$========1111111144444444''''''. . . . b.,.,.,.,b::::{ 7 7 7 7 t(9(9(9VGGG{     >$*$*$*N888  # # # L89898989t9999$4#; r