U Zʹjh @sdZddlZddlmZmZmZmZmZddlm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9ddl:m;Z;mZ>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPmQZQmRZRe efZSeefZTe!efZUe"efZVe#e$fZWe#e$e%fZXddZYd d ZZd d Z[d efddZ\ddZ]efddZ^ddZ_efddZ`efddZadefddZbdefddZcefddZddefd d!Zed"d#Zfefd$d%Zgefd&d'Zhefd(d)Ziefd*d+Zjefd,d-Zkefd.d/Zlefd0d1Zmefd2d3Znefd4d5Zoefd6d7Zpefd8d9Zqefd:d;Zrefdd?Ztefd@dAZudBdCZvefdDdEZwefdFdGZxefdHdIZyefdJdKZzdLdMZ{efdNdOZ|efdPdQZ}efdRdSZ~efdTdUZefdVdWZefdXdYZefdZd[Zefd\d]Zefd^d_Zefd`daZefdbdcZefdddeZefdfdgZefdhdiZefdjdkZedlZedmZdndoZefdpdqZefdrdsZefdtduZefdvdwZefdxdyZefdzd{Zdd}d~ZdddZdddZdddZedkrz(ddlmmmZej~Z~ejzZzWn"eefk redYnXdS)z- Low-level functions for complex arithmetic. N)MPZMPZ_ZEROMPZ_ONEMPZ_TWOBACKEND)1 round_floor round_ceiling round_downround_up round_nearest round_fastbitcountbctable normalize normalize1reciprocal_rndrshiftlshift giant_steps negative_rndto_strto_fixed from_man_exp from_floatto_floatfrom_intto_intfzerofoneftwofhalffinffninffnanfnonempf_absmpf_posmpf_negmpf_addmpf_submpf_mulmpf_div mpf_mul_int mpf_shiftmpf_sqrt mpf_hypot mpf_rdiv_int mpf_floormpf_ceilmpf_nintmpf_fracmpf_signmpf_hash ComplexResult)mpf_pimpf_expmpf_log mpf_cos_sin mpf_cosh_sinhmpf_tan mpf_pow_int mpf_log_hypotmpf_cos_sin_pimpf_phimpf_cosmpf_sin mpf_cos_pi mpf_sin_pimpf_atan mpf_atan2mpf_coshmpf_sinhmpf_tanhmpf_asinmpf_acos mpf_acosh mpf_nthroot mpf_fibonaccicCs$|\}}|tkrdS|tkr dSdS)z2Check if either real or imaginary part is infiniteTF)_infszreimrVR/var/www/html/bwave/python/venv/lib/python3.8/site-packages/mpmath/libmp/libmpc.py mpc_is_inf)s rXcCs$|\}}|tkrdS|tkr dSdS)z9Check if either real or imaginary part is infinite or nanTF) _infs_nanrRrVrVrW mpc_is_infnan0s rZcKsV|\}}t||}|dr8|dtt||f|dS|dt||f|dSdS)Nrz - jz + )rr()rSZdpskwargsrTrUrsrVrVrW mpc_to_str7s  r^FcCs"|\}}tt|||t|||SN)complexr)rSstrictrndrTrUrVrVrWmpc_to_complex?srccCsvtjdkrB|\}}t|tjjt|}|dtjj}t|Sztt|ddWSt k rpt|YSXdS)N)reT)ra) sys version_infor7 hash_infoimagwidthinthashrc OverflowError)rSrTrUhrVrVrWmpc_hashCs rocCs|\}}|t|||fSr_r(rSprecrbrTrUrVrVrW mpc_conjugatePsrscCs|tkSr_)mpc_zero)rSrVrVrWmpc_is_nonzeroTsrucCs,|\}}|\}}t||||t||||fSr_r)rSwrrrbabcdrVrVrWmpc_addWsr}cCs|\}}t|||||fSr_rv)rSxrrrbryrzrVrVrW mpc_add_mpf\srcCs,|\}}|\}}t||||t||||fSr_r*rwrVrVrWmpc_sub`srcCs|\}}t|||||fSr_r)rSprrrbryrzrVrVrW mpc_sub_mpfesrcCs |\}}t|||t|||fSr_)r'rSrrrbryrzrVrVrWmpc_posisrcCs |\}}t|||t|||fSr_rprrVrVrWmpc_negmsrcCs|\}}t||t||fSr_)r.)rSnryrzrVrVrW mpc_shiftqsrcCs|\}}t||||S)zEAbsolute value of a complex number, |a+bi|. Returns an mpf value.)r0rrVrVrWmpc_absusrcCs|\}}t||||S)z3Argument of a complex number. Returns an mpf value.)rHrrVrVrWmpc_arg{srcCs |\}}t|||t|||fSr_)r2rrVrVrW mpc_floorsrcCs |\}}t|||t|||fSr_)r3rrVrVrWmpc_ceilsrcCs |\}}t|||t|||fSr_)r4rrVrVrWmpc_nintsrcCs |\}}t|||t|||fSr_)r5rrVrVrWmpc_fracsrcCs\|\}}|\}}t||}t||} t||} t||} t|| ||} t| | ||} | | fS)z Complex multiplication. Returns the real and imaginary part of (a+bi)*(c+di), rounded to the specified precision. The rounding mode applies to the real and imaginary parts separately. )r+r*r))rSrxrrrbryrzr{r|rqrsrTrUrVrVrWmpc_muls    rc CsJ|\}}t||}t||}t||||}t||||}t|d} || fSNr)r+r*r.) rSrrrbryrzrrrrTrUrVrVrW mpc_squares   rcCs,|\}}t||||}t||||}||fSr_)r+rSrrrrbryrzrTrUrVrVrW mpc_mul_mpfsrcCs0|\}}tt||||}t||||}||fS)zB Multiply the mpc value z by I*x where x is an mpf value. )r(r+)rSr~rrrbryrzrTrUrVrVrWmpc_mul_imag_mpfsrcCs,|\}}t||||}t||||}||fSr_)r-)rSrrrrbryrzrTrUrVrVrW mpc_mul_intsrc Cs||\}}|\}}|d}tt||t|||} tt||t|||} tt||t|||} t| | ||t| | ||fSN )r)r+r*r,) rSrxrrrbryrzr{r|wpmagturVrVrWmpc_divsrcCs,|\}}t||||}t||||}||fS)zCalculate z/p where p is real)r,rrVrVrW mpc_div_mpfsrcCsL|\}}tt||t|||d}t||||}tt||||}||fS)zCalculate 1/z efficientlyrr)r+r,r()rSrrrbryrzmrTrUrVrVrWmpc_reciprocals rc CsX|\}}tt||t|||d}tt|||||}ttt|||||}||fS)z)Calculate p/z where p is real efficientlyrr) rrSrrrbryrzrrTrUrVrVrW mpc_mpf_divs rcCsnd}d}|rf|d@r>||||||||}}|d8}||||d||}}|d}q||fS)zgComplex integer power: computes (a+b*I)**n exactly for nonnegative n (a and b must be Python ints).rrrerV)ryrzrZwreZwimrVrVrWcomplex_int_pows" rcCs@|dtkrt||d||Sttt||d||d||S)Nrrr)r mpc_pow_mpfmpc_exprmpc_log)rSrxrrrbrVrVrWmpc_pows rc Cs||\}}}}|dkr.t|d|||>||S|dkrZt||d}t|d||||Sttt||d||d||S)Nrr) mpc_pow_intmpc_sqrtrrr) rSrrrrbZpsignZpmanZpexpZpbcZsqrtzrVrVrWrs rcCs|\}}|tkr"t||||tfS|tkrt||||}|d;}|dkrP|tfS|dkr`t|fS|dkrtt|tfS|dkrtt|fS|dkrtS|dkrt|||S|dkrt|||S|dkrt|||S|dkrtt|| |d||S|\}}} } |\} } } }|r| }| r"| } | | }t|}||t | |}|dkr|dkrf||K}| } n| | K} | } t || |\}}t |t || ||}t |t || ||}||fSt tt||d||d||S) Nrrrerdri'r)rr?r(mpc_onerrrrabsmaxrrrkrrr)rSrrrrbryrzvasignamanaexpabcbsignbmanbexpbbcdeZabs_deZ exact_sizerTrUrVrVrWrsX          rc Cs$|\}}|tkrT|tkr ||fS|dr@tt|||}t|fSt|||}|tfS|d}|dstt||f|||}t|d} t| ||}t|d} t| |} t|| ||}nhtt||f|||}t|d} t| ||}t|d} t| |} t|| ||}|drt|}t|}||fS)zComplex square root (principal branch). We have sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i where r = abs(a+bi), when a+bi is not a negative real number.rrr)rr/r(r)rr.r,r*) rSrrrbryrzrUrTrrrrrxrVrVrWr's6          rcCsd}tt||||}tt||||}z<|d|d|}|j}|j} tt|}tt| } Wndtk rt||}t||}t|} td| |} t||f| t f|\}} t |}t | } YnXd} |} |}t ||| D]}t || |d\}}t||d| ||}t||d| ||}||||||?}t|||}t|||}|||||?}| ||||?}||>|}||>|}||dt ||| |}||dt | || |} |} q|| fS)N2y?g?rr)rkrrealrirrmrr1rrrrrr)ryrzrrrstartZa1b1rrTrUfnnthextraZprevpZextra1rZre2Zim2Zr4apbpZrecZimcZrebZimbrVrVrWmpc_nthroot_fixedKsF      rcCs|\}}|ddkr2|tkr2t||||}|tfS|dkr|dkrFtS|dkr^t||f||S|dkrxtt||f||St||f| |dt|}tt|||S|dkrhtd|d}|\} } } } |\} }}}t||f|}|d |dd krh|d |d|krht ||}t ||}t ||||\}}d}t || |||}t || |||}||fSt |}|dd}t d||}t||f|tf||\}}t|d|d|d|d ||}t|d|d|d|d ||}||fS) zu Complex n-th root. Use Newton method as in the real case when it is faster, otherwise use z**(1/n) rrerrrg333333?rird)rrOrrr mpc_nthrootrrkrrrrrr1rr)rSrrrrbryrzrTZinverseZprec2rrrrrrrrpfafbfrUrrrrVrVrWrrsB   ,    ""rcCst|d||S)z Complex cubic root. rd)rrSrrrbrVrVrWmpc_cbrtsrc Cs||\}}|tkrt|||S|tkr4t|||tfSt||d|}t||d|\}}t||||}t||||} || fS)av Complex exponential function. We use the direct formula exp(a+bi) = exp(a) * (cos(b) + sin(b)*i) for the computation. This formula is very nice because it is pefectly stable; since we just do real multiplications, the only numerical errors that can creep in are single-ulp rounding errors. The formula is efficient since mpmath's real exp is quite fast and since we can compute cos and sin simultaneously. It is no problem if a and b are large; if the implementations of exp/cos/sin are accurate and efficient for all real numbers, then so is this function for all complex numbers. r)rr<r:r+) rSrrrbryrzrr{rrTrUrVrVrWrs rcCs*t|d|d||}t|||}||fS)Nrr)r@rrqrVrVrWrs rc Cs|\}}|tkr t|||tfS|tkr8t|||tfS|d}t||\}}t||\}} t||||} t|| ||} | t| fS)aSComplex cosine. The formula used is cos(a+bi) = cos(a)*cosh(b) - sin(a)*sinh(b)*i. The same comments apply as for the complex exp: only real multiplications are pewrormed, so no cancellation errors are possible. The formula is also efficient since we can compute both pairs (cos, sin) and (cosh, sinh) in single stwps.)rrCrIr<r=r+r( rSrrrbryrzrr{rchshrTrUrVrVrWmpc_cossrc Cs|\}}|tkr t|||tfS|tkr8tt|||fS|d}t||\}}t||\}} t||||} t|| ||} | | fS)zComplex sine. We have sin(a+bi) = sin(a)*cosh(b) + cos(a)*sinh(b)*i. See the docstring for mpc_cos for additional comments.r)rrDrJr<r=r+rrVrVrWmpc_sinsrcCs|\}}|\}}}}|\} } } } |tkr8t|||tfS|tkrPtt|||fS|d} t|d}t|d}t|| \}}t|| \}}t||| }t||||}t||||}||fS)zcComplex tangent. Computed as tan(a+bi) = sin(2a)/M + sinh(2b)/M*i where M = cos(2a) + cosh(2b).r)rr>rKr.r<r=r)r,)rSrrrbryrzrrrrrrrrrr{rrrrrTrUrVrVrWmpc_tans      rc Cs|\}}|tkr t|||tfSt|t|d|d}|tkrPt|||tfS|d}t||\}}t||\}} t||||} t|| ||} | t| fSNrr)rrEr+r9rIrAr=r(rrVrVrW mpc_cos_pisrc Cs|\}}|tkr t|||tfSt|t|d|d}|tkrPtt|||fS|d}t||\}}t||\}} t||||} t|| ||} | | fSr)rrFr+r9rJrAr=rrVrVrW mpc_sin_pisrcCs|\}}|tkr0t|||\}}|tft|ffS|tkrXt|||\}}|tf|tffS|d} t|| \}}t|| \}}t||||} t||||} t||||} t||||} | t| f| | ffS)Nr)rr=r<r+r()rSrrrbryrzrrr{rrcrecimsresimrVrVrW mpc_cos_sinsrcCs|\}}|tkr0t|||\}}|tf|tffSt|t|d|d}|tkrpt|||\}}|tft|ffS|d} t|| \}}t|| \}}t||||} t||||} t||||} t||||} | t| f| | ffSr)rrAr+r9r=r()rSrrrbryrzr{rrrrrrrrrVrVrWmpc_cos_sin_pi%s rcCs|\}}t|t|f||S)z:Complex hyperbolic cosine. Computed as cosh(z) = cos(z*i).)rr(rrVrVrWmpc_cosh7srcCs$|\}}t||f||\}}||fS)z;Complex hyperbolic sine. Computed as sinh(z) = -i*sin(z*i).)rrrVrVrWmpc_sinh<srcCs$|\}}t||f||\}}||fS)z>Complex hyperbolic tangent. Computed as tanh(z) = -i*tan(z*i).)rrrVrVrWmpc_tanhBsrc Cs|\}}|d}tt||t|f}tt|||f}t||}t||} t|| ||\}}tt|dt|df} | dtkrt|r| dt f} | S)Nrrrr) r)rr(r*rrr.r$rXr) rSrrrbryrzrr~yl1l2rrVrVrWmpc_atanIs   rg:pΈ?g?cCs.|\}}|d}|tkrttt||}|dsZ|dkrHt|||tfSt|||tfSn|drt||}tt|||} |dkr|t| fStt |d| fSn8t|||} |dkrt| fSt||}t |dt| fSd} } |drt|}d} |drt|}d} tt||}t t||} t | ||} t |||}t t | ||d}t |||}t |||}tt||ds|dkrt||}n t||}n t |||}|ds,t |t | | ||} t |||}t t |t | |||d}|dkrtt t|||||}ntt |t||||}n|t |t | | ||} t |t||||}t t | ||d}t |t|||}|dkrtt ||||}ntt ||||}tt||ds\t |t | | ||}t|drt |||}t |||}t t |||d}nt|||}t t |||d}t |t |t||}tt tt |t|||||}n,ttt |||t||}tt ||||}| r|dkrtt|||}nt|}| s|dkrt|}| r|dkrt|}t|d|d|d|d||}t|d|d|d|d||}||fS)a& complex acos for n = 0, asin for n = 1 The algorithm is described in T.E. Hull, T.F. Fairgrieve and P.T.P. Tang 'Implementing the Complex Arcsine and Arcosine Functions using Exception Handling', ACM Trans. on Math. Software Vol. 23 (1997), p299 The complex acos and asin can be defined as acos(z) = acos(beta) - I*sign(a)* log(alpha + sqrt(alpha**2 -1)) asin(z) = asin(beta) + I*sign(a)* log(alpha + sqrt(alpha**2 -1)) where z = a + I*b alpha = (1/2)*(r + s); beta = (1/2)*(r - s) = a/alpha r = sqrt((a+1)**2 + y**2); s = sqrt((a-1)**2 + y**2) These expressions are rewritten in different ways in different regions, delimited by two crossovers alpha_crossover and beta_crossover, and by abs(a) <= 1, in order to improve the numerical accuracy. rrrrrerd)rr*rr&rMrLr9rNr(r.r)r0r,r+beta_crossoverrGr/alpha_crossoverr;r)rSrrrbrryrzrampir{rrrrralphabetab2rTZAxr|c1c2ZAm1rUrVrVrW acos_asin_s                     " ""rcCst|||dSNrrrrVrVrWmpc_acossrcCst|||dSrrrrVrVrWmpc_asinsrcCs,|\}}t|t|f||\}}t||fSr_)rr(rrVrVrW mpc_asinhsrcCs<t|||\}}|ds |tkr,t||fS|t|fSdSr)rrr(rrVrVrW mpc_acoshs rcCsj|d}t|t|}tt||}t||}t||}tt|||d}|dtkrft|rft|df}|S)Nrrrr)r}rrrrr$rXr)rSrrrbrryrzrrVrVrW mpc_atanhs     rc Cs|\}}|tkr t|||tfStt|d|dt|d|d}||d}t|}tt|dt|}t|tf||} t ||} t | | |} t | | |} t | |||} | S)Nrerdrr) rrPrrrBr)r.r%rrrrr) rSrrrbrTrUsizerryrzrrrVrVrW mpc_fibonacci s*    rfcCstdSr_r8r~rrrbrVrVrWmpf_expjsrcCs|\}}|tkrt|||S|tkr8tt|||tfStt||d}t||d\}}t||||}t||||}||fSr)rr<r:r(r+)rSrrrbrTrUeyr{rrVrVrWmpc_expjs rcCstdSr_rrrVrVrW mpf_expjpi(src Cs|\}}|tkrt|||S|\}}}}|d} |rF| td||7} ttt| || }|tkrrt|||tfSt||d} t||d\} } t| | ||}t| | ||}||fS)Nrr)rrArr(r+r9r:) rSrrrbrTrUsignmanexpbcrrr{rrVrVrW mpc_expjpi+s  rsagez&Warning: Sage imports in libmpc failed)r)r)r)r)__doc__rfbackendrrrrrZlibmpfrr r r r r rrrrrrrrrrrrrrrrrrr r!r"r#r$r%r&r'r(r)r*r+r,r-r.r/r0r1r2r3r4r5r6r7r8Z libelefunr9r:r;r<r=r>r?r@rArBrCrDrErFrGrHrIrJrKrLrMrNrOrPrrtZmpc_twoZmpc_halfrQrYrXrZr^rcrorsrur}rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrZsage.libs.mpmath.ext_libmpZlibsZmpmathZ ext_libmpZ_lbmp ImportErrorAttributeErrorprintrVrVrVrWsh                    ( $' )