o iJ@stdZddlZddZddZddZd d Zdd dZdddZddZdddZ dddZ dddZ dddZ dS) z8 Routines for manipulating partial fraction expansions. NcCs&ddl}t|}t||S)zEnp.roots replacement. XXX: calls into NumPy, then converts back. rN)numpycupyasarraygetroots)arrnpr Z/home/ubuntu/veenaModal/venv/lib/python3.10/site-packages/cupyx/scipy/signal/_polyutils.pyrsrcCs|j}t|dkr|d|dkr|ddkstdddl}|j|}|j}|jd|d}|D]}|j ||j d| fdd }q3t |jj t jrg|j|td}|||||krg|j}t |S) z7np.poly replacement for 2D A. Otherwise, use cupy.poly.rz*input must be a non-empty square 2d array.N)r dtypefull)mode)shapelen ValueErrorrlinalgeigvalsrronesconvolver_ issubclasstypercomplexfloatingrcomplexallsort conjugaterealcopy)Ashr seq_of_zerosdtazerorr r r polys(  r(cCs"tt|}t||d|fS)z#Sort roots based on magnitude. r)rargsortabstake)pindxr r r _cmplx_sort)sr.c Cst|d}t|d}|d|d}t|d}t|d}d|d}tt||ddf|j}||j}td||dD]}|||} | ||<||||d| |8<qFtj|ddddr|j ddkr|dd}tj|ddddr|j ddkst||fS)Ngrr g?g+=)rtol) r atleast_1drzerosmaxrastyperangeallcloser) uvwmnscaleqrkdr r r _polydiv1s      "" "rAMbP?minc CsF|dvrtj}n|dvrtj}n |dvrtj}ntdt|jddf}t||dddf<t||dddf<tj j |dddddf|dddddfd d }g}g}tj |jdt d }t |D].\} } || rsqj| |k|@} | | } | jdkr|||| || jdd || <qjt|t|fS) aDetermine unique roots and their multiplicities from a list of roots. Parameters ---------- p : array_like The list of roots. tol : float, optional The tolerance for two roots to be considered equal in terms of the distance between them. Default is 1e-3. Refer to Notes about the details on roots grouping. rtype : {'max', 'maximum', 'min', 'minimum', 'avg', 'mean'}, optional How to determine the returned root if multiple roots are within `tol` of each other. - 'max', 'maximum': pick the maximum of those roots - 'min', 'minimum': pick the minimum of those roots - 'avg', 'mean': take the average of those roots When finding minimum or maximum among complex roots they are compared first by the real part and then by the imaginary part. Returns ------- unique : ndarray The list of unique roots. multiplicity : ndarray The multiplicity of each root. See Also -------- scipy.signal.unique_roots Notes ----- If we have 3 roots ``a``, ``b`` and ``c``, such that ``a`` is close to ``b`` and ``b`` is close to ``c`` (distance is less than `tol`), then it doesn't necessarily mean that ``a`` is close to ``c``. It means that roots grouping is not unique. In this function we use "greedy" grouping going through the roots in the order they are given in the input `p`. This utility function is not specific to roots but can be used for any sequence of values for which uniqueness and multiplicity has to be determined. For a more general routine, see `numpy.unique`. )r3maximum)rCminimum)avgmeanzJ`rtype` must be one of {'max', 'maximum', 'min', 'minimum', 'avg', 'mean'}rr Nr r0)axisr T)rr3rCrGremptyrr imagrnormr2bool enumeratesizeappendr) r,tolrtypereducepointsdistp_uniquep_multiplicityusedidsmaskgroupr r r unique_rootsDs0.8  r\Fc Cstdg}|g}t|ddd|dddD] \}}tjd| f}tt|D]}t||}q*||q|ddd}g} tdg}t|||D]4\}}} tjd| f}g} tt|D]} | dksk|rt| t|| t||}qc| t | qN| |fS)z>Compute the total polynomial divided by factors for each root.r r0rN) rarrayziprr5intpolymulrOextendreversed) r multiplicityinclude_powerscurrentsuffixespolemultmonomial_factorssuffixblockrXr r r _compute_factorss& &   rncCst||\}}||j}g}t|||D]Z\}}}|dkr.|t||t||q|} tjd| f} t || \}} g} t t |D]}t | | \} } | d| d}t | ||} | |qI| t| qt|S)Nr r)rnr4rr^rOrpolyvalr!rrAr5r_polysubrarbr)polesrc numeratordenominator_factorsrjresiduesrgrhfactornumerrir@rmr;r>r r r _compute_residuess*    rwrFc Cst|}t|}tt|d}t|||\}}t||dd\}}t|dkr-d} nt||} t||D] \} } t| | | } q8| |fS)aCompute b(s) and a(s) from partial fraction expansion. If `M` is the degree of numerator `b` and `N` the degree of denominator `a`:: b(s) b[0] s**(M) + b[1] s**(M-1) + ... + b[M] H(s) = ------ = ------------------------------------------ a(s) a[0] s**(N) + a[1] s**(N-1) + ... + a[N] then the partial-fraction expansion H(s) is defined as:: r[0] r[1] r[-1] = -------- + -------- + ... + --------- + k(s) (s-p[0]) (s-p[1]) (s-p[-1]) If there are any repeated roots (closer together than `tol`), then H(s) has terms like:: r[i] r[i+1] r[i+n-1] -------- + ----------- + ... + ----------- (s-p[i]) (s-p[i])**2 (s-p[i])**n This function is used for polynomials in positive powers of s or z, such as analog filters or digital filters in controls engineering. For negative powers of z (typical for digital filters in DSP), use `invresz`. Parameters ---------- r : array_like Residues corresponding to the poles. For repeated poles, the residues must be ordered to correspond to ascending by power fractions. p : array_like Poles. Equal poles must be adjacent. k : array_like Coefficients of the direct polynomial term. tol : float, optional The tolerance for two roots to be considered equal in terms of the distance between them. Default is 1e-3. See `unique_roots` for further details. rtype : {'avg', 'min', 'max'}, optional Method for computing a root to represent a group of identical roots. Default is 'avg'. See `unique_roots` for further details. Returns ------- b : ndarray Numerator polynomial coefficients. a : ndarray Denominator polynomial coefficients. See Also -------- scipy.signal.invres residue, invresz, unique_roots fTrdr rr1 trim_zerosr\rnrr`r^polyadd r>r,r?rPrQ unique_polesrcrk denominatorrrresiduerur r r invress 9   rc Cst|}t|}tt|d}t|||\}}t||dd\}}t|dkr-d} nt|ddd|ddd} t||D]\} } t| | | ddd} qB| ddd|fS)aCompute b(z) and a(z) from partial fraction expansion. If `M` is the degree of numerator `b` and `N` the degree of denominator `a`:: b(z) b[0] + b[1] z**(-1) + ... + b[M] z**(-M) H(z) = ------ = ------------------------------------------ a(z) a[0] + a[1] z**(-1) + ... + a[N] z**(-N) then the partial-fraction expansion H(z) is defined as:: r[0] r[-1] = --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ... (1-p[0]z**(-1)) (1-p[-1]z**(-1)) If there are any repeated roots (closer than `tol`), then the partial fraction expansion has terms like:: r[i] r[i+1] r[i+n-1] -------------- + ------------------ + ... + ------------------ (1-p[i]z**(-1)) (1-p[i]z**(-1))**2 (1-p[i]z**(-1))**n This function is used for polynomials in negative powers of z, such as digital filters in DSP. For positive powers, use `invres`. Parameters ---------- r : array_like Residues corresponding to the poles. For repeated poles, the residues must be ordered to correspond to ascending by power fractions. p : array_like Poles. Equal poles must be adjacent. k : array_like Coefficients of the direct polynomial term. tol : float, optional The tolerance for two roots to be considered equal in terms of the distance between them. Default is 1e-3. See `unique_roots` for further details. rtype : {'avg', 'min', 'max'}, optional Method for computing a root to represent a group of identical roots. Default is 'avg'. See `unique_roots` for further details. Returns ------- b : ndarray Numerator polynomial coefficients. a : ndarray Denominator polynomial coefficients. See Also -------- scipy.signal.invresz residuez, unique_roots, invres bTryrNr0rzr}r r r invreszs 8   rc CsPt|jtjst|jtjr|t}|t}n |t}|t}tt|d}tt|d}|j dkr@t dt |}|j dkrYt |j t|dtgfSt|t|krgtd}nt||\}}t|||d\}}t|\}}||}t|||} d} t||D]\} } | || | | <| | 7} q| |d||fS)a Compute partial-fraction expansion of b(s) / a(s). If `M` is the degree of numerator `b` and `N` the degree of denominator `a`:: b(s) b[0] s**(M) + b[1] s**(M-1) + ... + b[M] H(s) = ------ = ------------------------------------------ a(s) a[0] s**(N) + a[1] s**(N-1) + ... + a[N] then the partial-fraction expansion H(s) is defined as:: r[0] r[1] r[-1] = -------- + -------- + ... + --------- + k(s) (s-p[0]) (s-p[1]) (s-p[-1]) If there are any repeated roots (closer together than `tol`), then H(s) has terms like:: r[i] r[i+1] r[i+n-1] -------- + ----------- + ... + ----------- (s-p[i]) (s-p[i])**2 (s-p[i])**n This function is used for polynomials in positive powers of s or z, such as analog filters or digital filters in controls engineering. For negative powers of z (typical for digital filters in DSP), use `residuez`. See Notes for details about the algorithm. Parameters ---------- b : array_like Numerator polynomial coefficients. a : array_like Denominator polynomial coefficients. tol : float, optional The tolerance for two roots to be considered equal in terms of the distance between them. Default is 1e-3. See `unique_roots` for further details. rtype : {'avg', 'min', 'max'}, optional Method for computing a root to represent a group of identical roots. Default is 'avg'. See `unique_roots` for further details. Returns ------- r : ndarray Residues corresponding to the poles. For repeated poles, the residues are ordered to correspond to ascending by power fractions. p : ndarray Poles ordered by magnitude in ascending order. k : ndarray Coefficients of the direct polynomial term. Warning ------- This function may synchronize the device. See Also -------- scipy.signal.residue invres, residuez, numpy.poly, unique_roots Notes ----- The "deflation through subtraction" algorithm is used for computations --- method 6 in [1]_. The form of partial fraction expansion depends on poles multiplicity in the exact mathematical sense. However there is no way to exactly determine multiplicity of roots of a polynomial in numerical computing. Thus you should think of the result of `residue` with given `tol` as partial fraction expansion computed for the denominator composed of the computed poles with empirically determined multiplicity. The choice of `tol` can drastically change the result if there are close poles. References ---------- .. [1] J. F. Mahoney, B. D. Sivazlian, "Partial fractions expansion: a review of computational methodology and efficiency", Journal of Computational and Applied Mathematics, Vol. 9, 1983. rxrDenominator `a` is zero.rPrQ)r issubdtyperrr4rfloatr{r1rNrrr2rr.r]rrIrAr\rwr^) rr&rPrQrqr?r~rcorderrtindexrgrhr r r ras4Q           rcCst|jtjst|jtjr|t}|t}n |t}|t}tt|d}tt|d}|j dkr@t d|ddkrJt dt |}|j dkrct |j t|dtgfS|ddd}|ddd}t|t|krtd}nt||\}}t|||d\}} t|\}} | | } td|| |} d} tjt| td } t|| D]\}}||| | |<dtt|| | | |<| |7} q| | | |d9} | ||dddfS) aSCompute partial-fraction expansion of b(z) / a(z). If `M` is the degree of numerator `b` and `N` the degree of denominator `a`:: b(z) b[0] + b[1] z**(-1) + ... + b[M] z**(-M) H(z) = ------ = ------------------------------------------ a(z) a[0] + a[1] z**(-1) + ... + a[N] z**(-N) then the partial-fraction expansion H(z) is defined as:: r[0] r[-1] = --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ... (1-p[0]z**(-1)) (1-p[-1]z**(-1)) If there are any repeated roots (closer than `tol`), then the partial fraction expansion has terms like:: r[i] r[i+1] r[i+n-1] -------------- + ------------------ + ... + ------------------ (1-p[i]z**(-1)) (1-p[i]z**(-1))**2 (1-p[i]z**(-1))**n This function is used for polynomials in negative powers of z, such as digital filters in DSP. For positive powers, use `residue`. See Notes of `residue` for details about the algorithm. Parameters ---------- b : array_like Numerator polynomial coefficients. a : array_like Denominator polynomial coefficients. tol : float, optional The tolerance for two roots to be considered equal in terms of the distance between them. Default is 1e-3. See `unique_roots` for further details. rtype : {'avg', 'min', 'max'}, optional Method for computing a root to represent a group of identical roots. Default is 'avg'. See `unique_roots` for further details. Returns ------- r : ndarray Residues corresponding to the poles. For repeated poles, the residues are ordered to correspond to ascending by power fractions. p : ndarray Poles ordered by magnitude in ascending order. k : ndarray Coefficients of the direct polynomial term. Warning ------- This function may synchronize the device. See Also -------- scipy.signal.residuez invresz, residue, unique_roots rrrz6First coefficient of determinant `a` must be non-zero.Nr0rr r )rrrrr4rrr{r1rNrrr2rr.r]rrIrAr\rwr_r^arange)rr&rPrQrqb_reva_revk_revr~rcrrtrpowersrgrhr r r residuezsB>           r)rBrC)F)rBrF) __doc__rrr(r.rAr\rnrwrrrrr r r r s   R  L Kv