+ LV-j@>Rt.ROt^RIt^RIHtHtHtHtHtH t H t H t ^RI H t ^RIHt]P !4t^R]3RltR]3RltR]3Rlt]3Rlt]3R ltR]3R ltR]3R ltR]3R ltR]3R ltR]3RltR#)z1 Differential and pseudo-differential operators. N)piasarraysincossinhcoshtanh iscomplexobj)convolve) _datacopiedc\V\P4'd&\VR4'g/VnVPp\ V4pV^8XdV#\ V4'd;\VPWV4R\VPWV4,,#Ve^\,V, pMRp\V4pVPWaV34pVfP\V4^8dV'dVP4KW3Rlp\P!WhV^R7pWsWaV3&\!W@4p \P!WGV^,V R7#)a Return kth derivative (or integral) of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = pow(sqrt(-1)*j*2*pi/period, order) * x_j y_0 = 0 if order is not 0. Parameters ---------- x : array_like Input array. order : int, optional The order of differentiation. Default order is 1. If order is negative, then integration is carried out under the assumption that ``x_0 == 0``. period : float, optional The assumed period of the sequence. Default is ``2*pi``. Notes ----- If ``sum(x, axis=0) = 0`` then ``diff(diff(x, k), -k) == x`` (within numerical accuracy). For odd order and even ``len(x)``, the Nyquist mode is taken zero. diff_cache??c:V'd\W ,V4#^#)pow)kordercs&&&l/Volumes/fast/ai/experiments/ui-tars-smoke/.venv/lib/python3.14/site-packages/scipy/fftpack/_pseudo_diffs.pykerneldiff..kernelIs13u~%d zero_nyquistswap_real_imag overwrite_x) isinstance threadinglocalhasattrr rr diffrealimagrlengetpopitemr init_convolution_kernelr ) xrperiod_cachetmprnomegarr s &&&& rr%r%s/:&)//**v|,, "F "" !*C z CCHHeV4R HHeV9-6-- -  bDK  AA JJ{ #E } v;    00E>?A#{c%K   Seai)4 66rc\V\P4'd&\VR4'g/VnVPp\ V4p\ V4'd;\VPWV4R\VPWV4,,#VeV^,\,V, p\V4pVPWQ34pVfN\V4^8dV'dVP4KV3Rlp\P!WW^R7pWcWQ3&\!W@4p\P!WF^VR7#)a' Return h-Tilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like The input array to transform. h : float Defines the parameter of the Tilbert transform. period : float, optional The assumed period of the sequence. Default period is ``2*pi``. Returns ------- tilbert : ndarray The result of the transform. Notes ----- If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd, then ``tilbert(itilbert(x)) == x``. If ``2 * pi * h / period`` is approximately 10 or larger, then numerically ``tilbert == hilbert`` (theoretically oo-Tilbert == Hilbert). For even ``len(x)``, the Nyquist mode of ``x`` is taken zero. tilbert_cachercFV'dR\W,4, #^#)rrrhs&&rrtilbert..kernels49}$rrr)r!r"r#r$r3rr tilbertr&r'rr(r)r*r r+r r,r7r-r.r/r0r1rr s &&&& rr:r:UsH&)//**v//#%F %% !*CCsxxF3GCHHa889 9 EBJ  AA JJv E } v;    00a@u c%K   SaK PPrc\V\P4'd&\VR4'g/VnVPp\ V4p\ V4'd;\VPWV4R\VPWV4,,#VeV^,\,V, p\V4pVPWQ34pVfN\V4^8dV'dVP4KV3Rlp\P!WW^R7pWcWQ3&\!W@4p\P!WF^VR7#)z Return inverse h-Tilbert transform of a periodic sequence x. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*tanh(j*h*2*pi/period) * x_j y_0 = 0 For more details, see `tilbert`. itilbert_cacherc:V'd\W,4)#^#rr5r6s&&rritilbert..kernelsQS z!rr9r)r!r"r#r$r=rr itilbertr&r'rr(r)r*r r+r r;s &&&& rr@r@s &)//**v/00$&F !&& !*CC!V4(388Q778 8  aCF6M AA JJu E } v;   00A>u c%K   SaK PPrcN\V\P4'd&\VR4'g/VnVPp\ V4p\ V4'd9\VPV4R\VPV4,,#\V4pVPV4pVfJ\V4^8dV'dVP4KRp\P!W5^R7pWAV&\W 4p\P!W$^VR7#)a Return Hilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*sign(j) * x_j y_0 = 0 Parameters ---------- x : array_like The input array, should be periodic. _cache : dict, optional Dictionary that contains the kernel used to do a convolution with. Returns ------- y : ndarray The transformed input. See Also -------- scipy.signal.hilbert : Compute the analytic signal, using the Hilbert transform. Notes ----- If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``. For even len(x), the Nyquist mode of x is taken zero. The sign of the returned transform does not have a factor -1 that is more often than not found in the definition of the Hilbert transform. Note also that `scipy.signal.hilbert` does have an extra -1 factor compared to this function. hilbert_cacherc*V^8dR#V^8dR#R#)rrgg)rs&rrhilbert..kernels1uQ rr9r)r!r"r#r$rBrr hilbertr&r'r(r)r*r r+r )r,r.r/r0r1rr s&& rrFrFsN&)//**v//#%F %% !*CCsxx(2&0I+III AA JJqME } v;    00A>q c%K   SaK PPrc\V\P4'd&\VR4'g/VnVPp\ W4)#)z Return inverse Hilbert transform of a periodic sequence x. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*sign(j) * x_j y_0 = 0 ihilbert_cache)r!r"r#r$rHrF)r,r.s&&rihilbertrIsD&)//**v/00$&F !&& A  rc \V\P4'd&\VR4'g/VnVPp\ V4p\ V4'd;\VPWW44R\VPWW44,,#Ve7V^,\,V, pV^,\,V, p\V4pVPWaV34pVfO\V4^8dV'dVP4KW3Rlp\P!Wh^R7pWtWaV3&\!WP4p \P!WW^V R7#)a Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like The array to take the pseudo-derivative from. a, b : float Defines the parameters of the cosh/sinh pseudo-differential operator. period : float, optional The period of the sequence. Default period is ``2*pi``. Returns ------- cs_diff : ndarray Pseudo-derivative of periodic sequence `x`. Notes ----- For even len(`x`), the Nyquist mode of `x` is taken as zero. cs_diff_cachercfV'd)\W,4)\W ,4, #^#r)rrrabs&&&rrcs_diff..kernelHs!QS z$qs)++rr9r)r!r"r#r$rKrr cs_diffr&r'rr(r)r*r r+r r,rNrOr-r.r/r0r1rr s &&&&& rrQrQs<&)//**v//#%F %% !*CCsxxv6'#((A&99: :  aCF6M aCF6M AA JJAw E } v;   00A>Awc%K   SaK PPrc \V\P4'd&\VR4'g/VnVPp\ V4p\ V4'd;\VPWW44R\VPWW44,,#Ve7V^,\,V, pV^,\,V, p\V4pVPWaV34pVfO\V4^8dV'dVP4KW3Rlp\P!Wh^R7pWtWaV3&\!WP4p \P!WW^V R7#)a< Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*sinh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like Input array. a,b : float Defines the parameters of the sinh/cosh pseudo-differential operator. period : float, optional The period of the sequence x. Default is 2*pi. Notes ----- ``sc_diff(cs_diff(x,a,b),b,a) == x`` For even ``len(x)``, the Nyquist mode of x is taken as zero. sc_diff_cachercdV'd(\W,4\W ,4, #^#r)rrrMs&&&rrsc_diff..kernelsACyac**rr9r)r!r"r#r$rTrr sc_diffr&r'rr(r)r*r r+r rRs &&&&& rrWrWRs4&)//**v//#%F %% !*CCsxxv6GCHHaF;;< <  aCF6M aCF6M AA JJAw E } v;   00A>Awc%K   SaK PPrc \V\P4'd&\VR4'g/VnVPp\ V4p\ V4'd;\VPWW44R\VPWW44,,#Ve7V^,\,V, pV^,\,V, p\V4pVPWaV34pVfM\V4^8dV'dVP4KW3Rlp\P!Wh4pWtWaV3&\!WP4p \P!WWV R7#)a Return (a,b)-sinh/sinh pseudo-derivative of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sinh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j y_0 = a/b * x_0 Parameters ---------- x : array_like The array to take the pseudo-derivative from. a,b Defines the parameters of the sinh/sinh pseudo-differential operator. period : float, optional The period of the sequence x. Default is ``2*pi``. Notes ----- ``ss_diff(ss_diff(x,a,b),b,a) == x`` ss_diff_cachercV'd(\W,4\W ,4, #\V4V, #N)rfloatrMs&&&rrss_diff..kernels*ACyac**8A: rr )r!r"r#r$rYrr ss_diffr&r'rr(r)r*r r+r rRs &&&&& rr_r_s2&)//**v//#%F %% !*CCsxxv6'#((A&99: :  aCF6M aCF6M AA JJAw E } v;   00:Awc%K   S; ??rc \V\P4'd&\VR4'g/VnVPp\ V4p\ V4'd;\VPWW44R\VPWW44,,#Ve7V^,\,V, pV^,\,V, p\V4pVPWaV34pVfM\V4^8dV'dVP4KW3Rlp\P!Wh4pWtWaV3&\!WP4p \P!WWV R7#)ab Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = cosh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j Parameters ---------- x : array_like The array to take the pseudo-derivative from. a,b : float Defines the parameters of the sinh/sinh pseudo-differential operator. period : float, optional The period of the sequence x. Default is ``2*pi``. Returns ------- cc_diff : ndarray Pseudo-derivative of periodic sequence `x`. Notes ----- ``cc_diff(cc_diff(x,a,b),b,a) == x`` cc_diff_cachercP\W,4\W ,4, #r[)rrMs&&&rrcc_diff..kernels9T!#Y& &rr^)r!r"r#r$rarr cc_diffr&r'rr(r)r*r r+r rRs &&&&& rrdrds:&)//**v//#%F %% !*CCsxxv6GCHHaF;;< <  aCF6M aCF6M AA JJAw E } v;   '00:Awc%K   S; ??rc\V\P4'd&\VR4'g/VnVPp\ V4p\ V4'd;\VPWV4R\VPWV4,,#VeV^,\,V, p\V4pVPWQ34pVfq\V4^8dV'dVP4KV3RlpV3Rlp\P!WW^^R7p \P!WX^^R7p W3W5V3&MVwr\!W@4p \P"!WIV V R7#)a Shift periodic sequence x by a: y(u) = x(u+a). If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f Parameters ---------- x : array_like The array to take the pseudo-derivative from. a : float Defines the parameters of the sinh/sinh pseudo-differential period : float, optional The period of the sequences x and y. Default period is ``2*pi``. shift_cacherc$\W,4#r[)rrrNs&&r kernel_realshift..kernel_real qs8Orc$\W,4#r[)rrhs&&r kernel_imagshift..kernel_imagrkrrr^)r!r"r#r$rfrr shiftr&r'rr(r)r*r r+r convolve_z) r,rNr-r.r/r0r1rirm omega_real omega_imagr s &&&& rrorosF$&)//**v}--!#F ## !*CCSXXq&1B HHa:)5)) )  aCF6M AA JJu E } v;    55aaCDF 55aaCDF "-!u % c%K   sj+6 88r) r%r:r@rFrIrQrdrWr_ro)__doc____all__r"numpyrrrrrrrr r scipy.fft._pocketfft.helperr r#r.r%r:r@rFrIrQrWr_rdrorDrrrxs   GGG3  $v<6~fBQJV&QR?QD$!8Qv!4Qn!3@l!5@pF28r