\h-`SrSSKJrJrJr SSKJr SSKJrJ r SSK J r J r SSK JrJr SSKJrJr SSKJr SS jrSS jrSS jr\R\lSS jrSrSr\R\lSSjrSrSr\R\lSrSSjrSSjr \R\ lg )zd Discrete Fourier Transform, Number Theoretic Transform, Walsh Hadamard Transform, Mobius Transform )SSymbolsympify) expand_mul)piI)sincos)isprimeprimitive_root)ibiniterable)as_intc ~[U5(d [S5eUVs/sHn[U5PM nn[SU55(a [ S5e[ U5nUS:aU$UR 5S- nXUS- -(a US- nSU-nU[R/U[ U5- -- n[SU5H1n[[XvSS9SSS 2S5nXx:dM$XHXGsXG'XH'M3 U(a S [-U- O S[-U- n UbU RUS-5n [US-5Vs/sH%n[X-5[[!X-5--PM' n nSn X::apU S-X[-p[S X[5HLn[U 5H:nXGU-[#XGU-U -XU--5pX-X- sXGU-'XGU-U -'M< MN U S-n X::aMpU(a>Ub%UVs/sHnUU- RU5PM snOUVs/sHnUU- PM snnU$s snfs snfs snfs snf) z3Utility function for the Discrete Fourier TransformzAExpected a sequence of numeric coefficients for Fourier Transformc3J# UHoR[5v M g7fN)hasr).0xs Q/var/www/auris/envauris/lib/python3.13/site-packages/sympy/discrete/transforms.py %_fourier_transform..s $!Q55==!s!#z"Expected non-symbolic coefficientsTstrNr)r TypeErrorrany ValueErrorlen bit_lengthrZerorangeintr revalfr rr r)seqdpsinversearganbijangwhhfutuvrs r_fourier_transformr8sE C==01 1"%%#A% $! $$$=>> AA1u Aa%y Q qD!&&1s1v: A 1a[ Qt$TrT*A . 5qtJAD!$ "R%'!B$q&C iia ,1!q&M:MqSUaCE l "MA: A &aBq!A2YQxA!ebjM!F),C!D1*+%'a%!EBJ-  Q &-0_q )q!ac[[ q )/0!1q!!A#q!1  HO &0 ; *!1sH+?,H02H5H:Nc[XS9$)a Performs the Discrete Fourier Transform (**DFT**) in the complex domain. The sequence is automatically padded to the right with zeros, as the *radix-2 FFT* requires the number of sample points to be a power of 2. This method should be used with default arguments only for short sequences as the complexity of expressions increases with the size of the sequence. Parameters ========== seq : iterable The sequence on which **DFT** is to be applied. dps : Integer Specifies the number of decimal digits for precision. Examples ======== >>> from sympy import fft, ifft >>> fft([1, 2, 3, 4]) [10, -2 - 2*I, -2, -2 + 2*I] >>> ifft(_) [1, 2, 3, 4] >>> ifft([1, 2, 3, 4]) [5/2, -1/2 + I/2, -1/2, -1/2 - I/2] >>> fft(_) [1, 2, 3, 4] >>> ifft([1, 7, 3, 4], dps=15) [3.75, -0.5 - 0.75*I, -1.75, -0.5 + 0.75*I] >>> fft(_) [1.0, 7.0, 3.0, 4.0] References ========== .. [1] https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm .. [2] https://mathworld.wolfram.com/FastFourierTransform.html )r)r8r(r)s rfftr<Fs\ c ++c[XSS9$)NT)r)r*r:r;s rifftr?ws cD 99r=c 6[U5(d [S5e[U5n[U5(d [ S5eUVs/sHn[U5U-PM nn[ U5nUS:aU$UR 5S- nXfS- -(a US- nSU-nUS- U-(a [ S5eUS/U[ U5- -- n[SU5H1n[[XSS9S S S 2S5n X:dM$XYXXsXX'XY'M3 [U5n [XS- U-U5n U(a[XS- U5n S/US--n [SUS-5HnXS- U -U-X'M Sn X::apU S-Xm-p[SXm5HLn[U5H:n XXU -XXU -U-XU --nnUU-U-UU- U-sXXU -'XXU -U-'M< MN U S-n X::aMpU(a&[XcS- U5nUVs/sH oDU-U-PM nnU$s snfs snf) z3Utility function for the Number Theoretic TransformzJExpected a sequence of integer coefficients for Number Theoretic Transformz5Expected prime modulus for Number Theoretic Transformrrz/Expected prime modulus of the form (m*2**k + 1)rTrNr) rrrr r!r"r#r%r&r r pow)r(primer*prr,r-r.r/r0prrtr2r3r4r5r6r7rvs r_number_theoretic_transformrGs\ C==9: : u A 1::56 6!$$1QA$ AA1u Aa%y Q qD A{JKK!a#a&j A 1a[ Qt$TrT*A . 5qtJAD!$  B Ra%Aq !B UA  Q!V A 1a1f Qx{Q A &aBq!A2YQxq52:qay!81+,q5A+A{'a%!EBJ-  Q & E1  !q!rTAXq ! HW %R "s H<Hc[XS9$)a Performs the Number Theoretic Transform (**NTT**), which specializes the Discrete Fourier Transform (**DFT**) over quotient ring `Z/pZ` for prime `p` instead of complex numbers `C`. The sequence is automatically padded to the right with zeros, as the *radix-2 NTT* requires the number of sample points to be a power of 2. Parameters ========== seq : iterable The sequence on which **DFT** is to be applied. prime : Integer Prime modulus of the form `(m 2^k + 1)` to be used for performing **NTT** on the sequence. Examples ======== >>> from sympy import ntt, intt >>> ntt([1, 2, 3, 4], prime=3*2**8 + 1) [10, 643, 767, 122] >>> intt(_, 3*2**8 + 1) [1, 2, 3, 4] >>> intt([1, 2, 3, 4], prime=3*2**8 + 1) [387, 415, 384, 353] >>> ntt(_, prime=3*2**8 + 1) [1, 2, 3, 4] References ========== .. [1] http://www.apfloat.org/ntt.html .. [2] https://mathworld.wolfram.com/NumberTheoreticTransform.html .. [3] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29 )rBrGr(rBs rnttrKsP 's 88r=c[XSS9$)NT)rBr*rIrJs rinttrMs &s FFr=c,[U5(d [S5eUVs/sHn[U5PM nn[U5nUS:aU$XDS- -(aSUR 5-nU[ R /U[U5- -- nSnXT::a\US-n[SXE5H;n[U5H)nX7U-X7U-U-pX-X- sX7U-'X7U-U-'M+ M= US-nXT::aM\U(aUV s/sHoU- PM nn U$s snfs sn f)z1Utility function for the Walsh Hadamard Transformz@Expected a sequence of coefficients for Walsh Hadamard Transformrrrrrrr"r#rr$r%) r(r*r+r,r-r3r4r/r0r6r7rs r_walsh_hadamard_transformrPs( C==78 8"%%#A% AA1ua%y q||~ !&&1s1v: A A & !Vq!A2YQxq52:1*+%'a%!EBJ-  Q & !QqS!  H+ && s D :Dc[U5$)a 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 rPr(s rfwhtrTsH %S ))r=c[USS9$)NT)r*rRrSs rifwhtrV:s $S$ 77r=cD[U5(d [S5eUVs/sHn[U5PM nn[U5nUS:aU$XUS- -(aSUR 5-nU[ R /U[U5- -- nU(aESnXe:a<[U5H!nXv-(dMXG==XXv- -- ss'M# US-nXe:aM<U$SnXe:a<[U5H!nXv-(aMXG==XXv- -- ss'M# US-nXe:aM<U$s snf)zXUtility function for performing Mobius Transform using Yate's Dynamic Programming methodz#Expected a sequence of coefficientsrrrO)r(sgnsubsetr+r,r-r/r0s r_mobius_transformrZFs C===>>!$%#A% AA1ua%y q||~ !&&1s1v: A e1X55DC!%L(D FA e H e1X5aeH $ FA e H9 &sDc[USUS9$)af Performs the Mobius Transform for subset lattice with indices of sequence as bitmasks. The indices of each argument, considered as bit strings, correspond to subsets of a finite set. The sequence is automatically padded to the right with zeros, as the definition of subset/superset based on bitmasks (indices) requires the size of sequence to be a power of 2. Parameters ========== seq : iterable The sequence on which Mobius Transform is to be applied. subset : bool Specifies if Mobius Transform is applied by enumerating subsets or supersets of the given set. Examples ======== >>> from sympy import symbols >>> from sympy import mobius_transform, inverse_mobius_transform >>> x, y, z = symbols('x y z') >>> mobius_transform([x, y, z]) [x, x + y, x + z, x + y + z] >>> inverse_mobius_transform(_) [x, y, z, 0] >>> mobius_transform([x, y, z], subset=False) [x + y + z, y, z, 0] >>> inverse_mobius_transform(_, subset=False) [x, y, z, 0] >>> mobius_transform([1, 2, 3, 4]) [1, 3, 4, 10] >>> inverse_mobius_transform(_) [1, 2, 3, 4] >>> mobius_transform([1, 2, 3, 4], subset=False) [10, 6, 7, 4] >>> inverse_mobius_transform(_, subset=False) [1, 2, 3, 4] References ========== .. [1] https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula .. [2] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf .. [3] https://arxiv.org/pdf/1211.0189.pdf rrXrYrZr(rYs rmobius_transformr_lsp Sb 88r=c[USUS9$)Nrr\r]r^s rinverse_mobius_transformras Sb 88r=)Fr)T)!__doc__ sympy.corerrrsympy.core.functionrsympy.core.numbersrr(sympy.functions.elementary.trigonometricr r sympy.ntheoryr r sympy.utilities.iterablesr rsympy.utilities.miscrr8r<r?rGrKrMrPrTrVrZr_rar=rrks *)*$=14'. b.,b:{{ 7 t(9VG{{  >$*N8  # L89t9$4#;#; r=