HEADER (pari-2.13.0) | : | HEADER (pari-2.13.1) | ||
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skipping to change at line 238 | skipping to change at line 238 | |||

\bprog | \bprog | |||

? a = 1 + O(2); b = a; | ? a = 1 + O(2); b = a; | |||

? a * a \\ = a^2, precision increases | ? a * a \\ = a^2, precision increases | |||

%2 = 1 + O(2^3) | %2 = 1 + O(2^3) | |||

? a * b \\ not rewritten as a^2 | ? a * b \\ not rewritten as a^2 | |||

%3 = 1 + O(2) | %3 = 1 + O(2) | |||

? a*a*a \\ not rewritten as a^3 | ? a*a*a \\ not rewritten as a^3 | |||

%4 = 1 + O(2) | %4 = 1 + O(2) | |||

@eprog | @eprog | |||

\item If the exponent is a rational number $p/q$, return a solution $y$ of | \item If the exponent is a rational number $p/q$ the behaviour depends | |||

$y^q = x^p$ if it exists. Beware that this is defined up to $q$-th roots of 1 | on~$x$. If $x$ is a complex number, return $\exp(n \log x)$ (principal | |||

in the base field. Intmods modulo composite numbers are not supported. | branch), in an exact form if possible: | |||

\bprog | \bprog | |||

? 4^(1/2) \\ 4 being a square, this is exact | ? 4^(1/2) \\ 4 being a square, this is exact | |||

%1 = 2 | %1 = 2 | |||

? 2^(1/2) \\ now inexact | ? 2^(1/2) \\ now inexact | |||

%2 = 1.4142135623730950488016887242096980786 | %2 = 1.4142135623730950488016887242096980786 | |||

? (-1/4)^(1/2) \\ exact again | ||||

%3 = 1/2*I | ||||

? (-1)^(1/3) | ||||

%4 = 0.500...+ 0.866...*I | ||||

@eprog\noindent Note that even though $-1$ is an exact cube root of $-1$, | ||||

it is not $\exp(\log(-1)/3)$; the latter is returned. | ||||

Otherwise return a solution $y$ of $y^q = x^p$ if it exists; beware that | ||||

this is defined up to $q$-th roots of 1 in the base field. Intmods modulo | ||||

composite numbers are not supported. | ||||

\bprog | ||||

? Mod(7,19)^(1/2) | ? Mod(7,19)^(1/2) | |||

%3 = Mod(11, 19) \\ is any square root | %1 = Mod(11, 19) \\ is any square root | |||

? sqrt(Mod(7,19)) | ? sqrt(Mod(7,19)) | |||

%4 = Mod(8, 19) \\ is the smallest square root | %2 = Mod(8, 19) \\ is the smallest square root | |||

? Mod(1,4)^(1/2) | ? Mod(1,4)^(1/2) | |||

*** at top-level: Mod(1,4)^(1/2) | *** at top-level: Mod(1,4)^(1/2) | |||

*** ^------ | *** ^------ | |||

*** _^_: not a prime number in gpow: 4. | *** _^_: not a prime number in gpow: 4. | |||

@eprog | @eprog | |||

\item If the exponent is a negative integer or rational number, | \item If the exponent is a negative integer or rational number, | |||

an \idx{inverse} must be computed. For noninvertible \typ{INTMOD} $x$, this | an \idx{inverse} must be computed. For noninvertible \typ{INTMOD} $x$, this | |||

will fail and (for $n$ an integer) implicitly exhibit a factor of the modulus: | will fail and (for $n$ an integer) implicitly exhibit a factor of the modulus: | |||

\bprog | \bprog | |||

End of changes. 4 change blocks. | ||||

6 lines changed or deleted | | 16 lines changed or added |