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    1 Function: _header_operators
3 Section: operators
4 Doc:
6
7  \subsec{Boolean operators}\sidx{Boolean operators}
8
9  Any nonzero value is interpreted as \var{true} and any zero as \var{false}
10  (this includes empty vectors or matrices). The standard boolean operators
11  \kbd{||} (\idx{inclusive or}), \kbd{\&\&} (\idx{and})\sidx{or} and \kbd{!}
12  in prefix notation (\idx{not}) are available.
13  Their value is $1$ (true) or $0$ (false):
14  \bprog
15  ? a && b  \\ 1 iff a and b are nonzero
16  ? a || b  \\ 1 iff a or b is nonzero
17  ? !a      \\ 1 iff a is zero
18  @eprog
19
20  \subsec{Comparison}
21  The standard real \idx{comparison operators} \kbd{<=}, \kbd{<}, \kbd{>=},
22  \kbd{>}, are available in GP. The result is 1 if the comparison is true, 0
23  if it is false. These operators allow to compare integers (\typ{INT}),
24  rational (\typ{FRAC}) or real (\typ{REAL}) numbers,
26  (\kbd{oo}, \typ{INFINITY}).
27
28  By extension, two character strings (\typ{STR}) are compared using
29  the standard lexicographic order. Comparing a string to an object of a
31  comparison function.
32
33  \subsec{Equality}
34  Two operators allow to test for equality: \kbd{==} (equality up to type
35  coercion) and \kbd{===} (identity). The result is $1$ if equality is decided,
36  else $0$.
37
38  The operator \kbd{===} is strict: objects of different type or length are
39  never identical, polynomials in different variables are never identical,
40  even if constant. On the contrary, \kbd{==} is very liberal: $a~\kbd{==}~b$
41  decides whether there is a natural map sending $a$ to the domain of $b$
42  or sending $b$ to the domain of $a$, such that the comparison makes sense
43  and equality holds. For instance
44  \bprog
45  ? 4 == Mod(1,3) \\ equal
46  %1 = 1
47  ? 4 === Mod(1,3) \\ but not identical
48  %2 = 0
49
50  ? 'x == 'y   \\ not equal (nonconstant and different variables)
51  %3 = 0
52  ? Pol(0,'x) == Pol(0,'y)  \\ equal (constant: ignore variable)
53  %4 = 1
54  ? Pol(0,'x) == Pol(0,'y)  \\ not identical
55  %5 = 0
56
57  ? 0 == Pol(0) \\ equal
58  %6 = 1
59  ? [0] == 0     \\ equal
60  %7 = 1
61  ? [0, 0] == 0  \\ equal
62  %8 = 1
63  ? [0] == [0,0] \\ not equal
64  %9 = 1
65  @eprog\noindent In particular \kbd{==} is not transitive in general; it is
66  transitive when used to compare objects known to have the same type. The
67  operator \kbd{===} is transitive. The \kbd{==} operator allows two
68  equivalent negated forms: \kbd{!=} or \kbd{<>}; there is no negated form for
69  \kbd{===}.
70
71  Do not mistake \kbd{=} for \kbd{==}: it is the assignment statement.
72
73  \subseckbd{+$/$-} The expressions \kbd{+}$x$ and \kbd{-}$x$ refer
74  to monadic operators: the first does nothing, the second negates $x$.
75
76  The library syntax is \fun{GEN}{gneg}{GEN x} for \kbd{-}$x$.
77
78  \subseckbd{+} The expression $x$ \kbd{+} $y$ is the \idx{sum} of $x$ and $y$.
79  Addition between a scalar type $x$ and a \typ{COL} or \typ{MAT} $y$ returns
80  respectively $[y[1] + x, y[2],\dots]$ and $y + x \text{Id}$. Other additions
81  between a scalar type and a vector or a matrix, or between vector/matrices of
82  incompatible sizes are forbidden.
83
84  The library syntax is \fun{GEN}{gadd}{GEN x, GEN y}.
85
86  \subseckbd{-} The expression $x$ \kbd{-} $y$ is the \idx{difference} of $x$
87  and $y$. Subtraction between a scalar type $x$ and a \typ{COL} or \typ{MAT}
88  $y$ returns respectively $[y[1] - x, y[2],\dots]$ and $y - x \text{Id}$.
89  Other subtractions between a scalar type and a vector or a matrix, or
90  between vector/matrices of incompatible sizes are forbidden.
91
92  The library syntax is \fun{GEN}{gsub}{GEN x, GEN y} for $x$ \kbd{-} $y$.
93
94  \subseckbd{*} The expression $x$ \kbd{*} $y$ is the \idx{product} of $x$
95  and $y$. Among the prominent impossibilities are multiplication between
96  vector/matrices of incompatible sizes, between a \typ{INTMOD} or \typ{PADIC}
97  Restricted to scalars, \kbd{*} is commutative; because of vector and matrix
98  operations, it is not commutative in general.
99
100  Multiplication between two \typ{VEC}s or two \typ{COL}s is not
101  allowed; to take the \idx{scalar product} of two vectors of the same length,
102  transpose one of the vectors (using the operator \kbd{\til} or the function
103  \kbd{mattranspose}, see \secref{se:linear_algebra}) and multiply a line vector
104  by a column vector:
105  \bprog
106  ? a = [1,2,3];
107  ? a * a
108    ***   at top-level: a*a
109    ***                  ^--
110    *** _*_: forbidden multiplication t_VEC * t_VEC.
111  ? a * a~
112  %2 = 14
113  @eprog
114
115  If $x,y$ are binary quadratic forms, compose them; see also
116  \kbd{qfbnucomp} and \kbd{qfbnupow}. If $x,y$ are \typ{VECSMALL} of the same
117  length, understand them as permutations and compose them.
118
119  The library syntax is \fun{GEN}{gmul}{GEN x, GEN y} for $x$ \kbd{*} $y$.
120  Also available is \fun{GEN}{gsqr}{GEN x} for $x$ \kbd{*} $x$.
121
122  \subseckbd{/} The expression $x$ \kbd{/} $y$ is the \idx{quotient} of $x$
123  and $y$. In addition to the impossibilities for multiplication, note that if
124  the divisor is a matrix, it must be an invertible square matrix, and in that
125  case the result is $x*y^{-1}$. Furthermore note that the result is as exact
126  as possible: in particular, division of two integers always gives a rational
127  number (which may be an integer if the quotient is exact) and \emph{not} the
128  Euclidean quotient (see $x$ \kbd{\bs} $y$ for that), and similarly the
129  quotient of two polynomials is a rational function in general. To obtain the
130  approximate real value of the quotient of two integers, add \kbd{0.} to the
131  result; to obtain the approximate $p$-adic value of the quotient of two
132  integers, add \kbd{O(p\pow k)} to the result; finally, to obtain the
133  \idx{Taylor series} expansion of the quotient of two polynomials, add
134  \kbd{O(X\pow k)} to the result or use the \kbd{taylor} function
135  (see \secref{se:taylor}). \label{se:gdiv}
136
137  The library syntax is \fun{GEN}{gdiv}{GEN x, GEN y} for $x$ \kbd{/} $y$.
138
139  \subseckbd{\bs} The expression \kbd{$x$ \bs\ $y$} is the
140  \idx{Euclidean quotient} of $x$ and $y$. If $y$ is a real scalar, this is
141  defined as \kbd{floor($x$/$y$)} if $y > 0$, and \kbd{ceil($x$/$y$)} if
142  $y < 0$ and the division is not exact. Hence the remainder
143  \kbd{$x$ - ($x$\bs$y$)*$y$} is in $[0, |y|[$.
144
145  Note that when $y$ is an integer and $x$ a polynomial, $y$ is first promoted
146  to a polynomial of degree $0$. When $x$ is a vector or matrix, the operator
147  is applied componentwise.
148
149  The library syntax is \fun{GEN}{gdivent}{GEN x, GEN y}
150  for $x$ \kbd{\bs} $y$.
151
152  \subseckbd{\bs/} The expression $x$ \b{/} $y$ evaluates to the rounded
153  \idx{Euclidean quotient} of $x$ and $y$. This is the same as \kbd{$x$ \bs\ $y$}
154  except for scalar division: the quotient is such that the corresponding
155  remainder is smallest in absolute value and in case of a tie the quotient
156  closest to $+\infty$ is chosen (hence the remainder would belong to
157  $[{-}|y|/2, |y|/2[$).
158
159  When $x$ is a vector or matrix, the operator is applied componentwise.
160
161  The library syntax is \fun{GEN}{gdivround}{GEN x, GEN y}
162  for $x$ \b{/} $y$.
163
164  \subseckbd{\%} The expression \kbd{$x$ \% $y$} evaluates to the modular
165  \idx{Euclidean remainder} of $x$ and $y$, which we now define. When $x$ or $y$
166  is a nonintegral real number, \kbd{$x$\%$y$} is defined as
167  \kbd{$x$ - ($x$\bs$y$)*$y$}. Otherwise, if $y$ is an integer, this is
168  the smallest
169  nonnegative integer congruent to $x$ modulo $y$. (This actually coincides
170  with the previous definition if and only if $x$ is an integer.) If $y$ is a
171  polynomial, this is the polynomial of smallest degree congruent to
172  $x$ modulo $y$. For instance:
173  \bprog
174  ? (1/2) % 3
175  %1 = 2
176  ? 0.5 % 3
177  %2 = 0.5000000000000000000000000000
178  ? (1/2) % 3.0
179  %3 = 1/2
180  @eprog
181  Note that when $y$ is an integer and $x$ a polynomial, $y$ is first promoted
182  to a polynomial of degree $0$. When $x$ is a vector or matrix, the operator
183  is applied componentwise.
184
185  The library syntax is \fun{GEN}{gmod}{GEN x, GEN y}
186  for $x$ \kbd{\%} $y$.
187
188  \subseckbd{\var{op}=} When \var{op} is a binary arithmetic operator among
189  \kbd{+}, \kbd{-}, \kbd{\%}, \kbd{/}, \kbd{\bs} or \kbd{\bs/}, the construct
190  $x \var{op}= y$ is a shortcut for $x = x \var{op} y$.
191  \bprog
192  ? v[1] += 10  \\ increment v[1] by 10
193  ? a /= 2 \\ divide a by 2
194  @eprog
195
196  \subseckbd{++} \kbd{$x$++} is a shortcut for \kbd{$x$ = $x$ + 1}.
197
198  \subseckbd{--} \kbd{$x$--} is a shortcut for \kbd{$x$ = $x$ - 1}.
199
200  \subseckbd{\pow} The expression $x\hbox{\kbd{\pow}}n$ is \idx{powering}.
201
202  \item If the exponent $n$ is an integer, then exact operations are performed
203  using binary (left-shift) powering techniques. By definition, $x^0$ is
204  (an empty product interpreted as) an exact $1$ in the underlying prime
205  ring:
206  \bprog
207  ? 0.0 ^ 0
208  %1 = 1
209  ? (1 + O(2^3)) ^ 0
210  %2 = 1
211  ? (1 + O(x)) ^ 0
212  %3 = 1
213  ? Mod(2,4)^0
214  %4 = Mod(1,4)
215  ? Mod(x,x^2)^0
216  %5 = Mod(1, x^2)
217  @eprog\noindent
218  If $x$ is a $p$-adic number, its precision will increase if $v_p(n) > 0$ and
219  $n \neq 0$. Powering a binary quadratic form (types \typ{QFI} and
220  \typ{QFR}) returns a representative of the class, which is reduced if the
221  input was. (In particular, \kbd{x \pow 1} returns $x$ itself, whether it is
222  reduced or not.)
223
224  PARI rewrites the multiplication $x * x$ of two \emph{identical}
225  objects as $x^2$. Here, identical means the operands are reference the same
226  chunk of memory; no equality test is performed. This is no longer true when
227  more than two arguments are involved.
228  \bprog
229  ? a = 1 + O(2); b = a;
230  ? a * a  \\ = a^2, precision increases
231  %2 = 1 + O(2^3)
232  ? a * b \\ not rewritten as a^2
233  %3 = 1 + O(2)
234  ? a*a*a \\ not rewritten as a^3
235  %4 = 1 + O(2)
236  @eprog
237
238  \item If the exponent is a rational number $p/q$ the behaviour depends
239  on~$x$. If $x$ is a complex number, return $\exp(n \log x)$ (principal
240  branch), in an exact form if possible:
241  \bprog
242  ? 4^(1/2)  \\ 4 being a square, this is exact
243  %1 = 2
244  ? 2^(1/2)  \\ now inexact
245  %2 = 1.4142135623730950488016887242096980786
246  ? (-1/4)^(1/2) \\ exact again
247  %3 = 1/2*I
248  ? (-1)^(1/3)
249  %4 = 0.500...+ 0.866...*I
250  @eprog\noindent Note that even though $-1$ is an exact cube root of $-1$,
251  it is not $\exp(\log(-1)/3)$; the latter is returned.
252
253  Otherwise return a solution $y$ of $y^q = x^p$ if it exists; beware that
254  this is defined up to $q$-th roots of 1 in the base field. Intmods modulo
255  composite numbers are not supported.
256  \bprog
257  ? Mod(7,19)^(1/2)
258  %1 = Mod(11, 19) \\ is any square root
259  ? sqrt(Mod(7,19))
260  %2 = Mod(8, 19)  \\ is the smallest square root
261  ? Mod(1,4)^(1/2)
262   ***   at top-level: Mod(1,4)^(1/2)
263   ***                         ^------
264   *** _^_: not a prime number in gpow: 4.
265  @eprog
266
267  \item If the exponent is a negative integer or rational number,
268  an \idx{inverse} must be computed. For noninvertible \typ{INTMOD} $x$, this
269  will fail and (for $n$ an integer) implicitly exhibit a factor of the modulus:
270  \bprog
271  ? Mod(4,6)^(-1)
272    ***   at top-level: Mod(4,6)^(-1)
273    ***                         ^-----
274    *** _^_: impossible inverse modulo: Mod(2, 6).
275  @eprog\noindent
276  Here, a factor 2 is obtained directly. In general, take the gcd of the
277  representative and the modulus. This is most useful when performing
278  complicated operations modulo an integer $N$ whose factorization is
279  unknown. Either the computation succeeds and all is well, or a factor $d$
280  is discovered and the computation may be restarted modulo $d$ or $N/d$.
281
282  For noninvertible \typ{POLMOD} $x$, the behavior is the same:
283  \bprog
284  ? Mod(x^2, x^3-x)^(-1)
285    ***   at top-level: Mod(x^2,x^3-x)^(-1)
286    ***                               ^-----
287    *** _^_: impossible inverse in RgXQ_inv: Mod(x^2, x^3 - x).
288  @eprog\noindent Note that the underlying algorihm (subresultant) assumes
289  that the base ring is a domain:
290  \bprog
291  ? a = Mod(3*y^3+1, 4); b = y^6+y^5+y^4+y^3+y^2+y+1; c = Mod(a,b);
292  ? c^(-1)
293    ***   at top-level: Mod(a,b)^(-1)
294    ***                         ^-----
295    *** _^_: impossible inverse modulo: Mod(2, 4).
296  @eprog\noindent
297  In fact $c$ is invertible, but $\Z/4\Z$ is not a domain and the algorithm
298  fails. It is possible for the algorithm to succeed in such situations
299  and any returned result will be correct, but chances are that an error
300  will occur first. In this specific case, one should work with $2$-adics.
301  In general, one can also try the following approach
302  \bprog
303  ? inversemod(a, b) =
304  { my(m, v = variable(b));
305    m = polsylvestermatrix(polrecip(a), polrecip(b));
306    m = matinverseimage(m, matid(#m)[,1]);
307    Polrev(m[1..poldegree(b)], v);
308  }
309  ? inversemod(a,b)
310  %2 = Mod(2,4)*y^5 + Mod(3,4)*y^3 + Mod(1,4)*y^2 + Mod(3,4)*y + Mod(2,4)
311  @eprog\noindent
312  This is not guaranteed to work either since \kbd{matinverseimage} must also
313  invert pivots. See \secref{se:linear_algebra}.
314
315  For a \typ{MAT} $x$, the matrix is expected to be square and invertible, except
316  in the special case \kbd{x\pow(-1)} which returns a left inverse if one exists
317  (rectangular $x$ with full column rank).
318  \bprog
319  ? x = Mat([1;2])
320  %1 =
321  [1]
322
323  [2]
324
325  ? x^(-1)
326  %2 =
327  [1 0]
328  @eprog
329
330  \item Finally, if the exponent $n$ is not an rational number, powering is
331  treated as the transcendental function $\exp(n\log x)$, although it will be
332  more precise than the latter when $n$ and $x$ are exact:
333  \bprog
334  ? s = 1/2 + 10^14 * I
335  ? localprec(200); z = 2^s  \\ for reference
336  ? exponent(2^s - z)
337  %3 = -127  \\ perfect
338  ? exponent(exp(s * log(2)) - z)
339  %4 = -84 \\ not so good
340  @eprog\noindent The second computation is less precise because $\log(2)$ is
341  first computed to $38$ decimal digits, then multiplied by $s$, which has a
342  huge imaginary part amplifying the error.
343
344  In this case, $x \mapsto x^n$ is treated as a transcendental function and
345  and in particular acts
346  componentwise on vector or matrices, even square matrices ! (See
347  \secref{se:trans}.) If $x$ is $0$ and $n$ is an inexact $0$, this will raise
348  an exception:
349  \bprog
350  ? 4 ^ 1.0
351  %1 = 4.0000000000000000000000000000000000000
352  ? 0^ 0.0
353   ***   at top-level: 0^0.0
354   ***                  ^----
355   *** _^_: domain error in gpow(0,n): n <= 0
356  @eprog
357
358  The library syntax is \fun{GEN}{gpow}{GEN x, GEN n, long prec}
359  for $x\hbox{\kbd{\pow}}n$.