+ i-bRt^RIHtHtHt^RIHt^RIHtH t ^RI H t H t ^RI HtHt^RIHtHt^RIHtRRltRR ltRR lt]P]nRR ltR tRt]P]nRRltRtRt]P]nRtRRltRRlt ]P] nR #)zd Discrete Fourier Transform, Number Theoretic Transform, Walsh Hadamard Transform, Mobius Transform )SSymbolsympify) expand_mul)piI)sincos)isprimeprimitive_root)ibiniterable)as_intc \V4'g \R4hVUu.uFp\V4NK pp\;QJdRV4F 'gK RM RM !RV44'd \ R4h\ V4pV^8dV#VP 4^, pWU^, ,'dV^, p^V,pV\P.V\ V4, ,, p\^V4FAp\\WvRR7RRR1,^4pWx8gK,WH,WG,uWG&WH&KC V'dR \,V, M^\,V, p VeV PV^,4p \V^,4Uu.uF5p\W,4\\!W,4,,NK7 p p^p W8:dV ^,W[,r\^W[4Fp\V 4FnpWGV,,\#WGV,V ,,WV,,,4rW,W, uWGV,&WGV,V ,&Kp K V ^,p KV'dFVe)VUu.uFpVV, PV4NK upMVUu.uF pVV, NK uppV#uupiuupiuupiuupi) z3Utility function for the Discrete Fourier TransformzAExpected a sequence of numeric coefficients for Fourier Transformc3J"TFqP\4xK R#5iN)hasr).0xs& y/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/discrete/transforms.py %_fourier_transform..s $!Q55==!s!#TFz"Expected non-symbolic coefficientsstrN)r TypeErrorrany ValueErrorlen bit_lengthrZerorangeintr revalfr rrr)seqdpsinversearganbijangwhhfutuvrs&&& r_fourier_transformr5sS C==01 1"%%#A% s $! $sss $! $$$=>> AA1u Aa%yy Q qD!&&1s1v: A 1a[ Qt$TrT*A . 5qtJAD!$ "R%'!B$q&C iia ,1!q&M:MqSUaCE l " "MA: A &aBq!A2YQxA!ebjM!F),C!D1*+%'a%!EBJ-  Q-0_q )q!ac[[ q )/0!1q!!A##q!1  HO &0 ; *!1sK-.;K2.!K7K<Nc\WR7#)a Performs the Discrete Fourier Transform (**DFT**) in the complex domain. The sequence is automatically padded to the right with zeros, as the *radix-2 FFT* requires the number of sample points to be a power of 2. This method should be used with default arguments only for short sequences as the complexity of expressions increases with the size of the sequence. Parameters ========== seq : iterable The sequence on which **DFT** is to be applied. dps : Integer Specifies the number of decimal digits for precision. Examples ======== >>> from sympy import fft, ifft >>> fft([1, 2, 3, 4]) [10, -2 - 2*I, -2, -2 + 2*I] >>> ifft(_) [1, 2, 3, 4] >>> ifft([1, 2, 3, 4]) [5/2, -1/2 + I/2, -1/2, -1/2 - I/2] >>> fft(_) [1, 2, 3, 4] >>> ifft([1, 7, 3, 4], dps=15) [3.75, -0.5 - 0.75*I, -1.75, -0.5 + 0.75*I] >>> fft(_) [1.0, 7.0, 3.0, 4.0] References ========== .. [1] https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm .. [2] https://mathworld.wolfram.com/FastFourierTransform.html )r&r5r%r&s&&rfftr9Fs\ c ++c\WRR7#)T)r&r'r7r8s&&rifftr<ws cD 99r:c \V4'g \R4h\V4p\V4'g \ R4hVUu.uFp\V4V,NK pp\ V4pV^8dV#VP 4^, pWf^, ,'dV^, p^V,pV^, V,'d \ R4hV^.V\ V4, ,, p\^V4FAp\\WRR7RRR1,^4p W8gK,WY,WX,uWX&WY&KC \V4p \W^, V,V4p V'd\W^, V4p ^.V^,,p \^V^,4F"pW^, ,V ,V,W&K$ ^p W8:dV ^,Wm,r\^Wm4Fp\V4Fvp WXV ,,WXV ,V,,WV ,,,ppVV,V,VV, V,uWXV ,&WXV ,V,&Kx K V ^,p KV'd2\Wc^, V4pVUu.uFqDV,V,NK ppV#uupiuupi)z3Utility function for the Number Theoretic TransformzJExpected a sequence of integer coefficients for Number Theoretic Transformz5Expected prime modulus for Number Theoretic Transformz/Expected prime modulus of the form (m*2**k + 1)TrNr) r rrr rrr r"r#r r pow)r%primer'prr)r*r+r,r-prrtr/r0r1r2r3r4rvs&&& r_number_theoretic_transformrDs[ C==9: : u A 1::56 6!$$1QA$ AA1u Aa%yy 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!rTAXXq ! HW %R "s K .Kc\WR7#)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 )r?rDr%r?s&&rnttrHsP 's 88r:c\WRR7#)T)r?r'rFrGs&&rinttrJs &s FFr:c\V4'g \R4hVUu.uFp\V4NK pp\V4pV^8dV#WD^, ,'d^VP 4,pV\ P .V\V4, ,, p^pWT8:dV^,p\^WE4Fcp\V4FQpW7V,,W7V,V,,rW,W, uW7V,&W7V,V,&KS Ke V^,pKV'dVU u.uF qV, NK pp V#uupiuup i)z1Utility function for the Walsh Hadamard Transformz@Expected a sequence of coefficients for Walsh Hadamard Transformr rrrr rr!r") r%r'r(r)r*r0r1r,r-r3r4rs && r_walsh_hadamard_transformrMs# C==78 8"%%#A% AA1ua%yy q||~ !&&1s1v: A A & !Vq!A2YQxq52:1*+%'a%!EBJ-  Q !QqSS!  H+ && s EEc\V4#)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 rMr%s&rfwhtrQsH %S ))r:c\VRR7#)T)r'rOrPs&rifwhtrS:s $S$ 77r:c\V4'g \R4hVUu.uFp\V4NK pp\V4pV^8dV#WU^, ,'d^VP 4,pV\ P .V\V4, ,, pV'd`^pWe8dU\V4F:pWv,'gKWG;;,WWv, ,,, uu&K< V^,pKZV#^pWe8dU\V4F:pWv,'dKWG;;,WWv, ,,, uu&K< V^,pKZV#uupi)zXUtility function for performing Mobius Transform using Yate's Dynamic Programming methodz#Expected a sequence of coefficientsrL)r%sgnsubsetr(r)r*r,r-s&&& r_mobius_transformrWFs C===>>!$%#A% AA1ua%yy q||~ !&&1s1v: A e1X55DC!%L(D FA  H e1X55aeH $ FA H9 &sE+c\V^VR7#)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 rUrVrWr%rVs&&rmobius_transformr\lsp Sb 88r:c\VRVR7#)rYrrZr[s&&rinverse_mobius_transformr_s Sb 88r:)Fr)T)!__doc__ sympy.corerrrsympy.core.functionrsympy.core.numbersrr(sympy.functions.elementary.trigonometricrr sympy.ntheoryr r sympy.utilities.iterablesr r sympy.utilities.miscrr5r9r<rDrHrJrMrQrSrWr\r_r:rris *)*$=14'. b.,b:{{ 7 t(9VG{{  >$*N8  # L89t9$4#;#; r: