Annotation of OpenXM_contrib/pari/doc/usersch4.tex, Revision 1.1
1.1 ! maekawa 1: \chapter{Programming PARI in Library Mode}
! 2:
! 3: \section{Introduction: initializations, universal objects}
! 4: \label{se:intro4}
! 5:
! 6: \noindent
! 7: To be able to use PARI in \idx{library mode}, you must write a C program and
! 8: link it to the PARI library. See the installation guide (in Appendix~A)
! 9: on how to create and install the library and include files. A sample Makefile
! 10: is presented in Appendix~B.
! 11:
! 12: Probably the best way to understand how programming is done is to work
! 13: through a complete example. We will write such a program in
! 14: \secref{se:prog}. Before doing this, a few explanations are in order.
! 15:
! 16: First, one must explain to the outside world what kind of objects and
! 17: routines we are going to use. This is done simply with the statement
! 18:
! 19: \bprog%
! 20: \#include <pari.h>
! 21: \eprog
! 22: \noindent
! 23: This file \tet{pari.h} imports all the necessary constants,
! 24: variables and functions, defines some important macros, and also defines the
! 25: fundamental type for all PARI objects: the type \teb{GEN}, which is
! 26: simply a pointer to \kbd{long}.
! 27:
! 28: \misctitle{Technical note}: we would have liked to define a type \kbd{GEN}
! 29: to be a pointer to itself. This unfortunately is not possible in C, except by
! 30: using structures, but then the names become unwieldy. The result of this is
! 31: that when we use a component of a PARI object, it will be a \kbd{long},
! 32: hence will need to be typecast to a \kbd{GEN} again if we want to avoid
! 33: warnings from the compiler. This will sometimes be quite tedious, but of
! 34: course is trivially done. See the discussion on typecasts in the next
! 35: section.
! 36:
! 37: After declaring the use of the file \kbd{pari.h}, the first executable
! 38: statement of a main program should be to initialize the PARI system, and in
! 39: particular the PARI stack which will be both a scratchboard and a repository
! 40: for computed objects. This is done with a call to the function%
! 41: \sidx{pari\string\_init}
! 42:
! 43: \kbd{void \key{pari\_init}(long size, long maxprime)}.
! 44:
! 45: \noindent The first argument is the number of bytes given to PARI to work
! 46: with (it should not reasonably be taken below 500000), and the second is the
! 47: upper limit on a precomputed prime number table. If you don't want prime
! 48: numbers, just put $\kbd{maxprime} = 2$. Be careful because lots a PARI
! 49: functions need this table (certainly all the ones of interest to number
! 50: theorists). If you wind up with the error message ``not enough precomputed
! 51: primes'', try to increase this value.
! 52:
! 53: \noindent We have now at our disposal:
! 54:
! 55: $\bullet$ a large PARI {\it \idx{stack}\/} containing nothing. It's a big
! 56: connected chunk of memory whose size you chose when invoking
! 57: \kbd{pari\_init}. All your computations are going to take place here.
! 58: When doing large computations, unwanted intermediate results clutter up
! 59: memory very fast so some kind of garbage collecting is needed. Most large
! 60: systems do garbage collecting when the memory is getting scarce, and this slows
! 61: down the performance. In PARI we have taken a different approach: you must do
! 62: your own cleaning up when the intermediate results are not needed anymore.
! 63: Special purpose routines have been written to do this; we will see later how
! 64: (and when, if at all) you should use them.
! 65:
! 66: $\bullet$ the following {\it universal objects\/} (by definition, objects
! 67: which do not belong on the stack): the integers 0, 1 and 2 (respectively
! 68: called \teb{gzero}, \teb{gun}, and \teb{gdeux}), the
! 69: fraction $\dfrac{1}{2}$ (\teb{ghalf}), the complex number $i$
! 70: (\teb{gi}). All of these are of type \kbd{GEN}.
! 71:
! 72: In addition, space is reserved for the polynomials $x_v$
! 73: \sidx{variable}
! 74: (\kbd{\teb{polx}[$v$]}), and the polynomials $1_v$ (\kbd{\teb{polun}[$v$]}).
! 75: Here, $x_v$ is the name of variable number $v$, where $0\le v\le
! 76: \kbd{\idx{MAXVARN}}$ (the exact value of which depends on your machine, at
! 77: least 16383 in any case). Both \kbd{polun} and \kbd{polx} are arrays of
! 78: \kbd{GEN}s, the index being the polynomial variable number.
! 79:
! 80: However, except for the ones corresponding to variables $0$ and \kbd{MAXVARN},
! 81: these polynomials are {\it not\/} created upon initialization. It
! 82: is the programmer's responsibility to fill them before use. We'll see how
! 83: this is done in \secref{se:vars} ({\it never\/} through direct assignment).
! 84:
! 85: $\bullet$ a {\it \idx{heap}\/} which is just a linked list of permanent
! 86: universal objects. For now, it contains exactly the ones listed above. You
! 87: will probably very rarely use the heap yourself; and if so, only as a
! 88: collection of individual copies of objects taken from the stack
! 89: (called \idx{clone}s in the sequel). Thus you need not bother with its
! 90: internal structure, which may change as PARI evolves. Some complex PARI
! 91: functions may create clones for special garbage collecting purposes, usually
! 92: destroying them when returning.
! 93:
! 94: $\bullet$ a table of primes (in fact of {\it differences\/} between
! 95: consecutive primes), called \teb{diffptr}, of type \kbd{byteptr}
! 96: (pointer to \kbd{unsigned char}). Its use is described in appendix~C.
! 97:
! 98: $\bullet$ access to all the built-in functions of the PARI library.
! 99: These are declared to the outside world when you include \kbd{pari.h}, but
! 100: need the above things to function properly. So if you forget the call to
! 101: \kbd{pari\_init}, you will immediately get a fatal error when running your
! 102: program (usually a segmentation fault).
! 103:
! 104: \section{Important technical notes}
! 105:
! 106: \subsec{Typecasts}.\label{se:typecast}\sidx{typecast}
! 107:
! 108: \noindent
! 109: We have seen that, due to the non-recursiveness of the PARI types from the
! 110: compiler's point of view, many typecasts will be necessary when programming
! 111: in PARI. To take an example, a vector $V$ of dimension 2 (two components)
! 112: will be represented by a chunk of memory pointed to by the \kbd{GEN}~\kbd{V}.
! 113: \kbd{V[0]} contains coded information, in particular about the type of the
! 114: object, its length, etc. \kbd{V[1]} and \kbd{V[2]} contain pointers to
! 115: the two components of \kbd{V}. Those coefficients \kbd{V[i]} themselves are in
! 116: chunks of memory whose complexity depends on their own types, and so on. This
! 117: is where typecasting will be necessary: a priori, \kbd{V[i]} (for
! 118: $\kbd{i}=1,2$) is a \kbd{long}, but we will want to use it as a \kbd{GEN}.
! 119: The following two constructions will be exceedingly frequent (\kbd{x} and
! 120: \kbd{V} are \kbd{GEN}s):
! 121: %
! 122: \bprog%
! 123: V[i] = (long) x;
! 124: x = (GEN) V[i];
! 125: \eprog
! 126: \noindent Note that a typecast is not a valid lvalue (cannot be put on the
! 127: left side of an assignment), so \kbd{(GEN)V[i] = x} would be incorrect, though
! 128: some compilers may accept it.
! 129:
! 130: Due to this annoyance, the PARI functions and variables that occur most
! 131: frequently have analogues which are macros including the typecast. The complete
! 132: list can be found in the file \kbd{paricast.h} (which is included by
! 133: \kbd{pari.h} and can be found at the same place). For instance you can
! 134: abbreviate:
! 135: %
! 136: \bprogtabs\+
! 137: -----------------&--------&
! 138: \cr\+
! 139: (long) gzero & -----> &\idx{zero}
! 140: \cr\+
! 141: (long) gun & -----> &\idx{un}
! 142: \cr\+
! 143: (long) polx[v]& -----> &\idx{lpolx}[v]
! 144: \cr\+
! 145: (long) gadd(x,y)& -----> &\idx{ladd(x,y)}
! 146: \cr\eprog
! 147: \noindent
! 148: In general, replacing a leading \kbd{g} by an \kbd{l} in the name of a PARI
! 149: function will typecast the result to \kbd{long}. Note that \kbd{ldiv} is an
! 150: ANSI C function which is is hidden in PARI by a macro of the same name
! 151: representing \kbd{(long)gdiv}.
! 152:
! 153: The macro $\teb{coeff}(x,m,n)$ exists with exactly the meaning of \kbd{x[m,n]}
! 154: under GP when \kbd{x} is a matrix. This is a purely syntactical trick
! 155: to reduce the number of typecasts and thus does not create a copy of the
! 156: coefficient (contrary to all the library {\it functions\/}). It can be put on
! 157: the left side of an assignment statement, and its value, of type \kbd{long}
! 158: integer, is a pointer to the desired coefficient object. The macro
! 159: \kbd{gcoeff} is a synonym for \kbd{(GEN) coeff}, hence cannot be put on
! 160: the left side of an assignment.
! 161:
! 162: To retrieve the values of elements of lists of \dots\ of
! 163: lists of vectors, without getting infuriated by gigantic lists of typecasts,
! 164: we have the \teb{mael} macros (for {\bf m}ultidimensional {\bf a}rray {\bf
! 165: el}ement). The syntax is $\key{mael}n(x,a_1,\dots,a_n)$, where $x$ is a
! 166: \kbd{GEN}, the $a_i$ are indexes, and $n$ is an integer between $2$ and $5$
! 167: (with a standalone \kbd{mael} as a synonym for \kbd{mael2}). This stands
! 168: for $x[a_1][a_2]\dots[a_n]$ (with all the necessary typecasts), and returns a
! 169: \kbd{long} (i.e.~they are valid lvalues). The $\kbd{gmael}n$ macros are
! 170: synonyms for $\kbd{(GEN) mael}n$. Note that due to the implementation of matrix
! 171: types in PARI (i.e.~as horizontal lists of vertical vectors), \kbd{coeff(x,y)}
! 172: is actually completely equivalent to \kbd{mael(y,x)}. It is suggested that
! 173: you use \kbd{coeff} in matrix context, and \kbd{mael} otherwise.
! 174:
! 175: \subsec{Variations on basic functions}.\label{se:low_level} In the library
! 176: syntax descriptions in Chapter~3, we have only given the basic names of the
! 177: functions. For example \kbd{gadd}$(x,y)$ assumes that $x$ and $y$ are PARI
! 178: objects (of type \kbd{GEN}), and {\it creates\/} the result $x+y$ on the PARI
! 179: stack. For most of the basic operators and functions, many other variants
! 180: are available. We give some examples for \kbd{gadd}, but the same is true for
! 181: all the basic operators, as well as for some simple common functions (the
! 182: complete list is given in Chapter~5):
! 183:
! 184: \kbd{GEN \teb{gaddgs}(GEN x, long y)}
! 185:
! 186: \kbd{GEN \teb{gaddsg}(long x, GEN y)}
! 187:
! 188: \noindent In the following three, \kbd{z} is a preexisting \kbd{GEN} and the
! 189: result of the corresponding operation is put into~\kbd{z}. The size of the PARI
! 190: stack does not change:
! 191:
! 192: \kbd{void \teb{gaddz}(GEN x, GEN y, GEN z)}
! 193:
! 194: \kbd{void \teb{gaddgsz}(GEN x, long y, GEN z)}
! 195:
! 196: \kbd{void \teb{gaddsgz}(GEN x, GEN y, GEN z)}
! 197:
! 198: \noindent There are also low level functions which are special cases of the
! 199: above:
! 200:
! 201: \kbd{GEN \teb{addii}(GEN x, GEN y)}: here $x$ and $y$ are \kbd{GEN}s of type
! 202: \typ{INT} (this is not checked).
! 203:
! 204: \kbd{GEN \teb{addrr}(GEN x, GEN y)}: here $x$ and $y$ are \kbd{GEN} reals
! 205: (type \typ{REAL}).
! 206:
! 207: \noindent
! 208: There also exist functions \teb{addir}, \teb{addri}, \teb{mpadd} (whose
! 209: two arguments can be of type integer or real), \teb{addis} (to add a \typ{INT}
! 210: and a \kbd{long}) and so on.
! 211:
! 212: All these functions can of course be called by the user but we feel that
! 213: the few microseconds lost in calling more general functions (in this case
! 214: \kbd{gadd}) are compensated by the fact that one needs to remember a much
! 215: smaller number of functions, and also because there is a hidden danger here:
! 216: the types of the objects that you use, if they are themselves results of a
! 217: previous computation, are not completely predetermined. For instance the
! 218: multiplication of a type real \typ{REAL} by a type integer \typ{INT}
! 219: {\it usually\/} gives a result of type real, except when the integer is~0, in
! 220: which case according to the PARI philosophy the result is the exact integer~0.
! 221: Hence if afterwards you call a function which specifically needs a real
! 222: type argument, you are going to be in trouble.
! 223:
! 224: If you really want to use these functions, their names are self-explanatory
! 225: once you know that {\bf i} stands for a PARI integer, {\bf r} for a PARI
! 226: real, {\bf mp} for i or r, {\bf s} for an ordinary signed long, whereas {\bf
! 227: z} (as a suffix) means that the result is not created on the PARI
! 228: stack but assigned to a preexisting GEN object passed as an extra argument.
! 229:
! 230: For completeness, Chapter 5 gives a description of all these
! 231: low-level functions.
! 232:
! 233: Please note that in the present version \vers{} the names of the functions
! 234: are not always consistent. This will be changed. Hence anyone programming in
! 235: PARI must be aware that the names of almost all functions that he uses might
! 236: be subject to change. If the need arises (i.e.~if there really are people out
! 237: there who delve into the innards of PARI), updated versions with no name
! 238: changes will be released.
! 239:
! 240: \subsec{Portability: 32-bit / 64-bit architectures}.
! 241:
! 242: \noindent
! 243: PARI supports both 32-bit and 64-bit based machines, but not simultaneously!
! 244: The library will have been compiled assuming a given architecture (a~priori
! 245: following a guess by the \kbd{Configure} program, see Appendix~A), and some
! 246: of the header files you include (through \kbd{pari.h}) will have been modified
! 247: to match the library.
! 248:
! 249: Portable macros are defined to bypass most machine dependencies. If you
! 250: want your programs to run identically on 32-bit and 64-bit machines, you will
! 251: have to use these, and not the corresponding numeric values, whenever the
! 252: precise size of your \kbd{long} integers might matter. Here are the most
! 253: important ones:
! 254:
! 255: \bprogtabs\+
! 256: ------------------&----------&------------&\cr
! 257: \+
! 258: & {\rm 64-bit} & {\rm 32-bit}
! 259: \cr\+
! 260: BITS\_IN\_LONG & 64 & 32
! 261: \cr\+
! 262: LONG\_IS\_64BIT & {\rm defined} & {\rm undefined}
! 263: \cr\+
! 264: DEFAULTPREC & 3 & 4 & {\rm ($\approx$ 19 decimal digits, %
! 265: see formula below)}
! 266: \cr\+
! 267: MEDDEFAULTPREC & 4 & 6 & {\rm ($\approx$ 38 decimal digits)}
! 268: \cr\+
! 269: BIGDEFAULTPREC & 5 & 8 & {\rm ($\approx$ 57 decimal digits)}
! 270: \cr
! 271: \eprog
! 272: \sidx{DEFAULTPREC}
! 273: \sidx{MEDDEFAULTPREC}
! 274: \sidx{BIGDEFAULTPREC}
! 275: \sidx{LONG\string\_IS\string\_64BIT}
! 276: \sidx{BITS\string\_IN\string\_LONG}
! 277:
! 278: \noindent
! 279: For instance, suppose you call a transcendental function, such as
! 280:
! 281: \kbd{GEN \key{gexp}(GEN x, long prec)}.
! 282:
! 283: \noindent The last argument \kbd{prec} is only used if \kbd{x} is an exact
! 284: object (otherwise the relative precision is determined by the precision
! 285: of~\kbd{x}). But since \kbd{prec} sets the size of the inexact result counted
! 286: in (\kbd{long}) {\it words\/} (including codewords), the same value of
! 287: \kbd{prec} will yield different results on 32-bit and 64-bit machines. Real
! 288: numbers have two codewords (see~\secref{se:impl}), so the formula for
! 289: computing the bit accuracy is\sidx{bit\string\_accuracy}
! 290: $$ \kbd{bit\_accuracy}(\kbd{prec}) = (\kbd{prec} - 2) * \kbd{BITS\_IN\_LONG}$$
! 291: (this is actually the definition of a macro). The corresponding accuracy
! 292: expressed in decimal digits would be
! 293: %
! 294: $$ \kbd{bit\_accuracy(prec)} * \log(2) / \log(10).$$
! 295: %
! 296: For example if the value of \kbd{prec} is 5, the corresponding accuracy for
! 297: 32-bit machines is $(5-2)*\log(2^{32})/\log(10)\approx 28$ decimal digits,
! 298: while for 64-bit machines it is $(5-2)*\log(2^{64})/\log(10)\approx 57$
! 299: decimal digits.
! 300:
! 301: Thus, you must take care to change the \kbd{prec} parameter you are supplying
! 302: according to the bit size, either using the default precisions given by the
! 303: various \kbd{DEFAULTPREC}s, or by using conditional constructs of the form:
! 304: %
! 305: \bprog%
! 306: \#ifndef LONG\_IS\_64BIT
! 307: \q prec = 4;
! 308: \#else
! 309: \q prec = 6;
! 310: \#endif
! 311: \eprog
! 312: \noindent which is in this case equivalent to the statement
! 313: \kbd{prec = MEDDEFAULTPREC;}.
! 314:
! 315: Note that for parity reasons, half the accuracies available on 32-bit
! 316: architectures (the odd ones) have no precise equivalents on 64-bit machines.
! 317:
! 318: \section{Creation of PARI objects, assignments, conversions}
! 319:
! 320: \subsec{Creation of PARI objects}.\sidx{creation}
! 321: The basic function which creates a PARI object is the function
! 322: \teb{cgetg} whose prototype is:
! 323:
! 324: \kbd{GEN \key{cgetg}(long length, long type)}.
! 325:
! 326: \noindent
! 327: Here \kbd{length} specifies the number of longwords to be allocated to the
! 328: object, and type is the type number of the object, preferably in symbolic
! 329: form (see \secref{se:impl} for the list of these). The precise effect of
! 330: this function is as follows: it first creates on the PARI {\it stack\/} a
! 331: chunk of memory of size \kbd{length} longwords, and saves the address of the
! 332: chunk which it will in the end return. If the stack has been used up, a
! 333: message to the effect that ``the PARI stack overflows'' will be printed,
! 334: and an error raised. Otherwise, it sets the type and length of the PARI object.
! 335: In effect, it fills correctly and completely its first codeword (\kbd{z[0]} or
! 336: \kbd{*z}). Many PARI objects also have a second codeword (types \typ{INT},
! 337: \typ{REAL}, \typ{PADIC}, \typ{POL}, and \typ{SER}). In case you want to
! 338: produce one of those from scratch (this should be exceedingly rare), {\it it
! 339: is your responsibility to fill this second codeword}, either explicitly (using
! 340: the macros described in \secref{se:impl}), or implicitly using an assignment
! 341: statement (using \kbd{gaffect}).
! 342:
! 343: Note that the argument \kbd{length} is predetermined for a number of types:
! 344: 3 for types \typ{INTMOD}, \typ{FRAC}, \typ{FRACN}, \typ{COMPLEX},
! 345: \typ{POLMOD}, \typ{RFRAC} and \typ{RFRACN}, 4 for type \typ{QUAD}
! 346: and \typ{QFI}, and 5 for type \typ{PADIC} and \typ{QFR}. However for the sake
! 347: of efficiency, no checking is done in the function \kbd{cgetg}, so
! 348: disasters will occur if you give an incorrect length.
! 349:
! 350: \misctitle{Notes}:
! 351: 1) The main use of this function is to prepare for later assigments
! 352: (see \secref{se:assign}). Most of the time you will use \kbd{GEN} objects
! 353: as they are created and returned by PARI functions. In this case you don't need
! 354: to use \kbd{cgetg} to create space to hold them.
! 355:
! 356: \noindent 2) For the creation of leaves, i.e.~integers or reals, which is
! 357: very common,
! 358:
! 359: \kbd{GEN \teb{cgeti}(long length)}
! 360:
! 361: \kbd{GEN \teb{cgetr}(long length)}
! 362:
! 363: \noindent should be used instead of \kbd{cgetg(length, t\_INT)} and
! 364: \kbd{cgetg(length, t\_REAL)} respectively.
! 365:
! 366: \noindent 3) The macros \teb{lgetg}, \teb{lgeti}, \teb{lgetr} are
! 367: predefined as \kbd{(long)cgetg}, \kbd{(long)cgeti}, \kbd{(long)cgetr},
! 368: respectively.
! 369:
! 370: \misctitle{Examples}: 1) \kbd{z = cgeti(DEFAULTPREC)} and
! 371: \kbd{cgetg(DEFAULTPREC, t\_INT)} create an integer object whose ``precision''
! 372: is \kbd{bit\_accuracy(DEFAULTPREC)} = 64. This means \kbd{z} can hold rational
! 373: integers of absolute value less than $2^{64}$. Note that in both cases, the
! 374: second codeword will {\it not\/} be filled. Of course we could use numerical
! 375: values, e.g.~\kbd{cgeti(4)}, but this would have different meanings on
! 376: different machines as \kbd{bit\_accuracy(4)} equals 64 on 32-bit machines,
! 377: but 128 on 64-bit machines.
! 378:
! 379: \noindent 2) The following creates a type ``complex'' object whose real and
! 380: imaginary parts can hold real numbers of precision
! 381: $\kbd{bit\_accuracy(MEDDEFAULTPREC)} = 96\hbox{ bits:}$
! 382: %
! 383: \bprog%
! 384: z = cgetg(3, t\_COMPLEX);
! 385: z[1] = lgetr(MEDDEFAULTPREC);
! 386: z[2] = lgetr(MEDDEFAULTPREC);
! 387: \eprog
! 388:
! 389: \noindent 3) To create a matrix object for $4\times 3$ matrices:
! 390: %
! 391: \bprog%
! 392: z = cgetg(4, t\_MAT);
! 393: for(i=1; i<4; i++) z[i] = lgetg(5, t\_COL);
! 394: \eprog
! 395: %
! 396: \noindent If one wishes to create space for the matrix elements themselves,
! 397: one has to follow this with a double loop to fill each column vector.
! 398:
! 399: These last two examples illustrate the fact that since PARI types are
! 400: recursive, all the branches of the tree must be created. The function
! 401: \teb{cgetg} creates only the ``root'', and other calls to \kbd{cgetg} must be
! 402: made to produce the whole tree. For matrices, a common mistake is to think
! 403: that \kbd{z = cgetg(4, t\_MAT)} (for example) will create the root of the
! 404: matrix: one needs also to create the column vectors of the matrix (obviously,
! 405: since we specified only one dimension in the first \kbd{cgetg}!). This is
! 406: because a matrix is really just a row vector of column vectors (hence a
! 407: priori not a basic type), but it has been given a special type number so that
! 408: operations with matrices become possible.
! 409:
! 410: \subsec{Assignments}.
! 411: Firstly, if \kbd{x} and \kbd{y} are both declared as \kbd{GEN} (i.e.~pointers
! 412: to something), the ordinary C assignment \kbd{y = x} makes perfect sense: we
! 413: are just moving a pointer around. However, physically modifying either
! 414: \kbd{x} or \kbd{y} (for instance, \kbd{x[1] = 0}) will also change the other
! 415: one, which is usually not desirable. \label{se:assign}
! 416:
! 417: \misctitle{Very important note}: Using the functions described in this
! 418: paragraph is very inefficient and often awkward: one of the \tet{gerepile}
! 419: functions (see~\secref{se:garbage}) should be preferred. See the paragraph
! 420: end for some exceptions to this rule.
! 421:
! 422: \noindent
! 423: The general PARI \idx{assignment} function is the function \teb{gaffect} with
! 424: the following syntax:
! 425:
! 426: \kbd{void \key{gaffect}(GEN x, GEN y)}
! 427:
! 428: \noindent
! 429: Its effect is to assign the PARI object \kbd{x} into the {\it preexisting\/}
! 430: object \kbd{y}. This copies the whole structure of \kbd{x} into \kbd{y} so
! 431: many conditions must be met for the assignment to be possible. For instance
! 432: it is allowed to assign an integer into a real number, but the converse is
! 433: forbidden. For that, you must use the truncation or rounding function of
! 434: your choice (see section 3.2). It can also happen that \kbd{y} is not large
! 435: enough or does not have the proper tree structure to receive the object
! 436: \kbd{x}. As an extreme example, assume \kbd{y} is the zero integer with length
! 437: equal to 2. Then all assignments of a non-zero integer into \kbd{y} will
! 438: result in an error message since \kbd{y} is not large enough to accommodate
! 439: a non-zero integer. In general common sense will tell you what is possible,
! 440: keeping in mind the PARI philosophy which says that if it makes sense it is
! 441: legal. For instance, the assignment of an imprecise object into a precise one
! 442: does {\it not\/} make sense. However, a change in precision of imprecise
! 443: objects is allowed.
! 444:
! 445: All functions ending in ``\kbd{z}'' such as \teb{gaddz}
! 446: (see~\secref{se:low_level}) implicitly use this function. In fact what they
! 447: exactly do is record {\teb{avma}} (see~\secref{se:garbage}), perform the
! 448: required operation, \teb{gaffect} the result to the last operand, then
! 449: restore the initial \kbd{avma}.
! 450:
! 451: You can assign ordinary C long integers into a PARI object (not necessarily
! 452: of type \typ{INT}). Use the function \teb{gaffsg} with the following
! 453: syntax:
! 454:
! 455: \kbd{void \key{gaffsg}(long s, GEN y)}
! 456:
! 457: \misctitle{Note}: due to the requirements mentionned above, it's usually
! 458: a bad idea to use \tet{gaffect} statements. Two exceptions:
! 459:
! 460: $\bullet$ for simple objects (e.g.~leaves) whose size is controlled, they can
! 461: be easier to use than gerepile, and about as efficient.
! 462:
! 463: $\bullet$ to coerce an inexact object to a given precision. For instance
! 464: \bprog%
! 465: gaffect(x, (tmp=cgetr(3))); x = tmp;
! 466: \eprog
! 467: \noindent at the beginning of a routine where precision can be kept to a
! 468: minimum (otherwise the precision of \kbd{x} will be used in all subsequent
! 469: computations, which will be a disaster if \kbd{x} is known to thousands of
! 470: digits).
! 471:
! 472: \subsec{Copy}. It is also very useful to \idx{copy} a PARI object, not
! 473: just by moving around a pointer as in the \kbd{y = x} example, but by
! 474: creating a copy of the whole tree structure, without pre-allocating a
! 475: possibly complicated \kbd{y} to use with \kbd{gaffect}. The function which
! 476: does this is called \teb{gcopy}, with the predefined macro
! 477: \teb{lcopy} as a synonym for \kbd{(long)gcopy}. Its syntax is:
! 478:
! 479: \kbd{GEN \key{gcopy}(GEN x)}
! 480:
! 481: \noindent and the effect is to create a new copy of x on the PARI stack.
! 482: Beware that universal objects which occur in specific components of certain
! 483: types (mainly moduli for types \typ{INTMOD} and \typ{PADIC}) are not copied,
! 484: as they are assumed to be permanent. In this case, \kbd{gcopy} only copies
! 485: the pointer. Use \kbd{GEN \teb{forcecopy}(GEN x)} if you want a
! 486: complete copy.
! 487:
! 488: Please be sure at this point that you really understand the difference between
! 489: \kbd{y = x}, \kbd{y = gcopy(x)}, and \kbd{gaffect(x,y)}: this will save you
! 490: from many ``obvious'' mistakes later on.
! 491:
! 492: \subsec{Clones}.\sidx{clone}
! 493: Sometimes, it may be more efficient to create a {\it permanent\/} copy of a
! 494: PARI object. This will not be created on the stack but on the heap. The
! 495: function which does this is called \teb{gclone}, with the predefined macro
! 496: \teb{lclone} as a synonym for \kbd{(long)gclone}. Its syntax is:
! 497:
! 498: \kbd{GEN \key{gclone}(GEN x)}
! 499:
! 500: A clone can be removed from the heap (thus destroyed) using
! 501:
! 502: \kbd{void \key{gunclone}(GEN x)}
! 503:
! 504: \noindent No PARI object should keep references to a clone which has been
! 505: destroyed. If you want to copy a clone back to the stack then delete it, use
! 506: \tet{forcecopy} and not \tet{gcopy}, otherwise some components might not be
! 507: copied (moduli of \typ{INTMOD}s and \typ{POLMOD}s for instance).
! 508:
! 509: \subsec{Conversions}.\sidx{conversions}
! 510: The following functions convert C objects to PARI objects (creating them on
! 511: the stack as usual):
! 512:
! 513: \kbd{GEN \teb{stoi}(long s)}: C \kbd{long} integer (``small'') to
! 514: PARI integer (\typ{INT})
! 515:
! 516: \kbd{GEN \teb{dbltor}(double s)}: C \kbd{double} to PARI real
! 517: (\typ{REAL}). The accuracy of the result is 19 decimal digits, i.e.~a type
! 518: \typ{REAL} of length \kbd{DEFAULTPREC}, although on 32-bit machines only 16
! 519: of them will be significant.
! 520:
! 521: \noindent We also have the converse functions:
! 522:
! 523: \kbd{long \teb{itos}(GEN x)}: \kbd{x} must be of type \typ{INT},
! 524:
! 525: \kbd{double \teb{rtodbl}(GEN x)}: \kbd{x} must be of type \typ{REAL},
! 526:
! 527: \noindent as well as the more general ones:
! 528:
! 529: \kbd{long \teb{gtolong}(GEN x)},
! 530:
! 531: \kbd{double \teb{gtodouble}(GEN x)}.
! 532:
! 533: \section{Garbage collection}\label{se:garbage}\sidx{garbage collecting}
! 534:
! 535: \subsec{Why and how}.
! 536:
! 537: \noindent
! 538: As we have seen, the \kbd{pari\_init} routine allocates a big range of
! 539: addresses (the {\it stack\/}) that are going to be used throughout. Recall that
! 540: all PARI objects are pointers. But for universal objects, they will all point
! 541: at some part of the stack.
! 542:
! 543: The stack starts at the address \kbd{bot} and ends just before \kbd{top}. This
! 544: means that the quantity
! 545: %
! 546: $$ (\kbd{top} - \kbd{bot})\,/\,\kbd{sizeof(long)} $$
! 547: %
! 548: is equal to the \kbd{size} argument of \kbd{pari\_init}. The PARI
! 549: stack also has a ``current stack pointer'' called \teb{avma}, which
! 550: stands for {\bf av}ailable {\bf m}emory {\bf a}ddress. These three variables
! 551: are global (declared for you by \kbd{pari.h}). For historical reasons they
! 552: are of type \kbd{long} and not of type \kbd{GEN} as would seem more natural.
! 553:
! 554: The stack is oriented upside-down: the more recent an object, the closer to
! 555: \kbd{bot}. Accordingly, initially \kbd{avma} = \kbd{top}, and \kbd{avma} gets
! 556: {\it decremented\/} as new objects are created. As its name indicates,
! 557: \kbd{avma} always points just {\it after\/} the first free address on the
! 558: stack, and \kbd{(GEN)avma} is always (a pointer to) the latest created object.
! 559: When \kbd{avma} reaches \kbd{bot}, the stack overflows, aborting all
! 560: computations, and an error message is issued. To avoid this {\it you\/} will
! 561: need to clean up the stack from time to time, when some bunch of intermediate
! 562: objects will not be needed anymore. This is called ``{\it garbage
! 563: collecting}.''
! 564:
! 565: We are now going to describe briefly how this is done. We will see many
! 566: concrete examples in the next subsection.
! 567:
! 568: \noindent$\bullet$
! 569: First, PARI routines will do their own garbage collecting, which
! 570: means that whenever a documented function from the library returns, only its
! 571: result(s) will have been added to the stack (non-documented ones may not do
! 572: this, for greater speed). In particular, a PARI function that does not return
! 573: a \kbd{GEN} does not clutter the stack. Thus, if your computation is small
! 574: enough (i.e.~you call few PARI routines, or most of them return \kbd{long}
! 575: integers), then you don't need to do any garbage collecting. This will probably
! 576: be the case in many of your subroutines. Of course the objects that were on
! 577: the stack {\it before\/} the function call are left alone. Except for the ones
! 578: listed below, PARI functions only collect their own garbage.
! 579:
! 580: \noindent$\bullet$
! 581: It may happen that you need to preserve {\it some\/} but not
! 582: {\it all\/} objects that were created after a certain point~--- for instance,
! 583: if the final result you need is not a \kbd{GEN}, or if some search proved
! 584: futile. Then, it is enough to record the value of \kbd{avma} just
! 585: {\it before\/} the first garbage is created, and restore it upon exit:
! 586:
! 587: \bprog%
! 588: long ltop = avma; /* record initial avma */
! 589: \h
! 590: garbage ...
! 591: avma = ltop; /* restore it */
! 592: \eprog
! 593: \noindent All objects created in the \kbd{garbage} zone will eventually
! 594: be overwritten: they should not be accessed anymore once \kbd{avma} has been
! 595: restored.
! 596:
! 597: \noindent$\bullet$
! 598: If you want to destroy (i.e.~give back the memory occupied by) the
! 599: latest PARI object on the stack (e.g.~the latest one obtained from a function
! 600: call), and the above method is not available (because the initial value of
! 601: \kbd{avma} has been lost), just use the function\sidx{destruction}%
! 602: \vadjust{\penalty500}%discourage page break
! 603:
! 604: \kbd{void \teb{cgiv}(GEN z)}
! 605:
! 606: \noindent where \kbd{z} is the object you want to give back.
! 607:
! 608: \noindent$\bullet$
! 609: Unfortunately life is not so simple, and sometimes you will want
! 610: to give back accumulated garbage {\it during\/} a computation without losing
! 611: recent data. For this you need the \kbd{gerepile} function (or one of its
! 612: variants described hereafter):
! 613:
! 614: \kbd{GEN \teb{gerepile}(long ltop, long lbot, GEN q)}
! 615:
! 616: \noindent
! 617: This function cleans up the stack between \kbd{ltop} and \kbd{lbot}, where
! 618: $\kbd{lbot} < \kbd{ltop}$, and returns the updated object \kbd{q}. This means:
! 619:
! 620: 1) we translate (copy) all the objects in the interval
! 621: $[\kbd{avma}, \kbd{lbot}[$, so that its right extremity abuts the address
! 622: \kbd{ltop}. Graphically
! 623:
! 624: \vbox{
! 625: \bprogtabs\+
! 626: {\rm end of stack} &|-&-----------&|++++++&|-/-/-/-/-/-/-&|++++++++&|
! 627: \cr\+
! 628: &bot & &avma &lbot <op &top
! 629: \cr\+
! 630: {\rm end of stack} |------------[++++++[-/-/-/-/-/-/-|++++++++|\q
! 631: {\rm beginning of stack}
! 632: \cr\+
! 633: & &{\rm free memory}
! 634: \cr\eprog
! 635: }\noindent becomes:
! 636:
! 637: \vbox{
! 638: \bprogtabs\+
! 639: {\rm end of stack} &|--------&------------------&|++++++&|++++++++&|
! 640: \cr\+
! 641: &bot & &avma <op &top
! 642: \cr\+
! 643: {\rm end of stack} |--------------------------[++++++[++++++++|\q
! 644: {\rm beginning of stack}
! 645: \cr\+
! 646: &&{\rm free memory}
! 647: \cr\eprog
! 648: }
! 649: \noindent where \kbd{++} denote significant objects, \kbd{--} the unused part
! 650: of the stack, and \kbd{-/-} the garbage we remove.
! 651:
! 652: 2) The function then inspects all the PARI objects between \kbd{avma} and
! 653: \kbd{lbot} (i.e.~the ones that we want to keep and that have been translated)
! 654: and looks at every component of such an object which is not a codeword. Each
! 655: such component is a pointer to an object whose address is either
! 656:
! 657: --- between \kbd{avma} and \kbd{lbot}, in which case it will be suitably
! 658: updated,
! 659:
! 660: --- larger than or equal to \kbd{ltop}, in which case it will not change, or
! 661:
! 662: --- between \kbd{lbot} and \kbd{ltop} in which case \kbd{gerepile} will
! 663: scream an error message at you (``significant pointers lost in gerepile'').
! 664:
! 665: 3) \key{avma} is updated (we add $\kbd{ltop} - \kbd{lbot}$ to the old value).
! 666:
! 667: 4) We return the (possibly updated) object \kbd{q}: if \kbd{q} initially
! 668: pointed between \kbd{avma} and \kbd{lbot}, we return the translated
! 669: address, as in~2). If not, the original address is still valid (and we return
! 670: it!).
! 671:
! 672: As stated above, no component of the remaining objects (in particular
! 673: \kbd{q}) should belong to the erased segment [\kbd{lbot}, \kbd{ltop}[, and
! 674: this is checked within \kbd{gerepile}. But beware as well that the addresses
! 675: of all the objects in the translated zone will have changed after a call to
! 676: \kbd{gerepile}: every pointer you may have kept around elsewhere, outside the
! 677: stack objects, which previously pointed into the zone below \kbd{ltop}
! 678: must be discarded. If you need to recover more than one object, use one of
! 679: the \kbd{gerepilemany} functions below.
! 680:
! 681: As a consequence of the preceding explanation, we must now state the most
! 682: important law about programming in PARI:
! 683:
! 684: {\bf If a given PARI object is to be relocated by \hbox{gerepile} then,
! 685: apart from universal objects, the chunks of memory used by its components
! 686: should be in consecutive memory locations}. All \kbd{GEN}s created by
! 687: documented PARI function are guaranteed to satisfy this.
! 688:
! 689: This is because the \kbd{gerepile} function knows only about {\it two
! 690: connected zones\/}: the garbage that will be erased (between \kbd{lbot} and
! 691: \kbd{ltop}) and the significant pointers that will be copied and updated.
! 692: If there is garbage interspersed with your objects, disasters will occur when
! 693: we try to update them and consider the corresponding ``pointers''. So be
! 694: {\it very\/} wary when you allow objects to become disconnected. Have a look
! 695: at the examples, it's not as complicated as it seems.
! 696:
! 697: \noindent In practice this is achieved by the following programming idiom:
! 698: \bprog%
! 699: ltop=avma; garbage(); lbot=avma; q=anything();
! 700: return gerepile(ltop, lbot, q); /* returns the updated q */
! 701: \eprog
! 702:
! 703: \noindent Beware that
! 704: \bprog%
! 705: ltop=avma; garbage();
! 706: return gerepile(ltop, avma, anything())
! 707: \eprog
! 708:
! 709: \noindent might work, but should be frowned upon. We can't predict whether
! 710: \kbd{avma} is going to be evaluated after or before the call to
! 711: \kbd{anything()}: it depends on the compiler. If we are out of luck, it will
! 712: be {\it after\/} the call, so the result will belong to the garbage zone and
! 713: the \kbd{gerepile} statement becomes equivalent to \kbd{avma = ltop}. Thus we
! 714: would return a pointer to random garbage.
! 715:
! 716: \noindent$\bullet$ A simple variant is
! 717:
! 718: \kbd{GEN \teb{gerepileupto}(long ltop, GEN q)}
! 719:
! 720: \noindent which cleans the stack between \kbd{ltop} and the {\it connected\/}
! 721: object \kbd{q} and returns \kbd{q} updated. For this to work, \kbd{q} must
! 722: have been created {\it before\/} all its components, otherwise they would
! 723: belong to the garbage zone! Documented PARI functions guarantee this. If you
! 724: stumble upon one that does not, consider it a bug worth reporting.
! 725:
! 726: \noindent$\bullet$
! 727: To cope with complicated cases where many objects have to be
! 728: preserved, you can use
! 729:
! 730: \kbd{void \teb{gerepilemany}(long ltop, GEN *gptr[],
! 731: \kbd{long} n)}
! 732:
! 733: \noindent which cleans up the most recent part of the stack (between
! 734: \kbd{ltop} and \kbd{avma}). All the \kbd{GEN}s pointed at by the elements of
! 735: the array \kbd{gptr} (of length \kbd{n}) are updated. A copy is done just
! 736: before the cleaning to preserve them, so they don't need to be connected
! 737: before the call. This is the most robust of the gerepile functions (the less
! 738: prone to user error), but also the slowest.
! 739:
! 740: \noindent$\bullet$ More efficient, but trickier to use is
! 741:
! 742: \kbd{void \teb{gerepilemanysp}(long ltop, long lbot, GEN *gptr[], long n)}
! 743:
! 744: \noindent which cleans the stack between \kbd{lbot} and \kbd{ltop} and
! 745: updates the \kbd{GEN}s pointed at by the elements of \kbd{gptr} without doing
! 746: any copying. This is subject to the same restrictions as \kbd{gerepile}, the
! 747: only difference being that more than one address gets updated.
! 748:
! 749: \subsec{Examples}.
! 750:
! 751: \noindent
! 752: Let \kbd{x} and \kbd{y} be two preexisting PARI objects and suppose that we
! 753: want to compute $\kbd{x}^2+ \kbd{y}^2$. This can trivially be done using the
! 754: following program (we skip the necessary declarations; everything in sight is
! 755: a \kbd{GEN}):
! 756:
! 757: \kbd{p1 = gsqr(x); p2 = gsqr(y); z = gadd(p1,p2);}
! 758:
! 759: \noindent
! 760: The \kbd{GEN} \kbd{z} indeed points at the desired quantity. However,
! 761: consider the stack: it contains as unnecessary garbage \kbd{p1} and \kbd{p2}.
! 762: More precisely it contains (in this order) \kbd{z}, \kbd{p2}, \kbd{p1}
! 763: (recall that, since the stack grows downward from the top, the most recent
! 764: object comes first). We need a way to get rid of this garbage (in this case
! 765: it causes no harm except that it occupies memory space, but in other cases
! 766: it could disconnect other PARI objects and this is dangerous).
! 767:
! 768: It would not have been possible to get rid of \kbd{p1}, \kbd{p2} before
! 769: \kbd{z} is computed, since they are used in the final operation. We cannot
! 770: record \kbd{avma} before \kbd{p1} is computed and restore it later, since
! 771: this would destroy \kbd{z} as well. It is not possible either to use the
! 772: function \kbd{cgiv} since \kbd{p1} and \kbd{p2} are not at the bottom of the
! 773: stack and we don't want to give back~\kbd{z}.
! 774:
! 775: But using \kbd{gerepile}, we can give back the memory locations corresponding
! 776: to \kbd{p1}, \kbd{p2}, and move the object \kbd{z} upwards so that no
! 777: space is lost. Specifically:
! 778: \bprog%
! 779: ltop = avma; /* remember the current address of the top of the stack */
! 780: p1 = gsqr(x); p2 = gsqr(y);
! 781: lbot = avma; /* keep the address of the bottom of the garbage pile */
! 782: z = gadd(p1, p2); /* z is now the last object on the stack */
! 783: z = gerepile(ltop, lbot, z); /* garbage collecting */
! 784: \eprog
! 785: \noindent Of course, the last two instructions could also have been
! 786: written more simply:
! 787:
! 788: \kbd{z = gerepile(ltop, lbot, gadd(p1,p2));}
! 789:
! 790: \noindent In fact \kbd{gerepileupto} is even simpler to use, because the
! 791: result of \kbd{gadd} will be the last object on the stack and \kbd{gadd} is
! 792: guaranteed to return an object suitable for \kbd{gerepileupto}:
! 793: \bprog%
! 794: ltop = avma;
! 795: z = gerepileupto(ltop, gadd(gsqr(x), gsqr(y)));
! 796: \eprog
! 797: \noindent
! 798: As you can see, in simple conditions the use of \kbd{gerepile} is not really
! 799: difficult. However make sure you understand exactly what has happened before
! 800: you go on (use the figure from the preceding section).
! 801:
! 802: \misctitle{Important remark}: as we will see presently it is often
! 803: necessary to do several \kbd{gerepile}s during a computation. However, the
! 804: fewer the better. The only condition for \kbd{gerepile} to work is that the
! 805: garbage be connected. If the computation can be arranged so that there is a
! 806: minimal number of connected pieces of garbage, then it should be done that
! 807: way.
! 808:
! 809: For example suppose we want to write a function of two \kbd{GEN} variables
! 810: \kbd{x} and \kbd{y} which creates the vector $\kbd{[x}^2+\kbd{y},
! 811: \kbd{y}^2+\kbd{x]}$. Without garbage collecting, one would write:
! 812: %
! 813: \bprog%
! 814: p1 = gsqr(x); p2 = gadd(p1, y);
! 815: p3 = gsqr(y); p4 = gadd(p3, x);
! 816: z = cgetg(3,t\_VEC); z[1] = (long)p2; z[2] = (long)p4;
! 817: \eprog
! 818: \noindent
! 819: This leaves a dirty stack containing (in this order) \kbd{z}, \kbd{p4},
! 820: \kbd{p3}, \kbd{p2}, \kbd{p1}. The garbage here consists of \kbd{p1} and
! 821: \kbd{p3}, which are separated by \kbd{p2}. But if we compute \kbd{p3}
! 822: {\it before\/} \kbd{p2} then the garbage becomes connected, and we get the
! 823: following program with garbage collecting:
! 824: %
! 825: \bprog%
! 826: ltop = avma; p1 = gsqr(x); p3 = gsqr(y); lbot = avma;
! 827: z = cgetg(3,t\_VEC); z[1] = ladd(p1,y); z[2] = ladd(p3,x);
! 828: z = gerepile(ltop,lbot,z);
! 829: \eprog
! 830:
! 831: \noindent Finishing by \kbd{z = gerepileupto(ltop, z)} would be ok as well.
! 832: But when you have the choice, it's usually clearer to brace the garbage
! 833: between \kbd{ltop}~/ \kbd{lbot} pairs.
! 834:
! 835: \noindent Beware that
! 836: \bprog%
! 837: ltop = avma; p1 = gadd(gsqr(x), y); p3 = gadd(gsqr(y), x);
! 838: z = cgetg(3,t\_VEC); z[1] = (long)p1; z[2] = (long)p3
! 839: z = gerepileupto(ltop,z); /* WRONG !!!~*/%
! 840: \eprog
! 841: \noindent would be a disaster since \kbd{p1} and \kbd{p3} would be created
! 842: before \kbd{z}, so the call to \kbd{gerepileupto} would overwrite them,
! 843: leaving \kbd{z[1]} and \kbd{z[2]} pointing at random data!
! 844:
! 845: We next want to write a program to compute the product of two complex numbers
! 846: $x$ and $y$, using a method which takes only 3 multiplications instead of 4.
! 847: Let $z = x*y$, and set $x = x_r + i*x_i$ and similarly for $y$ and $z$. The
! 848: well-known trick is to compute $p_1 = x_r*y_r$, $p_2=x_i*y_i$,
! 849: $p_3=(x_r+x_i)*(y_r+y_i)$, and then we have $z_r=p_1-p_2$,
! 850: $z_i=p_3-(p_1+p_2)$. The program is essentially as follows:
! 851: %
! 852: \bprog%
! 853: ltop = avma; p1 = gmul(x[1],y[1]); p2 = gmul(x[2],y[2]);
! 854: p3 = gmul(gadd(x[1],x[2]), gadd(y[1],y[2]));
! 855: p4 = gadd(p1,p2); lbot = avma;
! 856: z = cgetg(3,t\_COMPLEX); z[1] = lsub(p1,p2); z[2] = lsub(p3,p4);
! 857: z = gerepile(ltop,lbot,z);
! 858: \eprog
! 859: \noindent
! 860: ``Essentially,'' because for instance \kbd{x[1]} is a \kbd{long} and not a
! 861: \kbd{GEN}, so we need to insert many annoying typecasts:
! 862: \kbd{p1 = gmul((GEN)x[1], (GEN)y[1])} and so on.
! 863:
! 864: Let us now look at a less trivial example where more than one \kbd{gerepile}
! 865: is needed in practice (at the expense of efficiency, one can always use only
! 866: one using \kbd{gcopy}; see below). Suppose that we want to write a function
! 867: which multiplies a line vector by a matrix (such a function is of course
! 868: already part of \kbd{gmul}, but let's ignore this for a moment). Then the
! 869: most natural way is to do a \kbd{cgetg} of the result immediately, and then a
! 870: \kbd{gerepile} for each coefficient of the result vector to get rid of the
! 871: garbage which has accumulated while this particular coefficient was computed.
! 872: We leave the details to the reader, who can look at the answer in the file
! 873: \kbd{basemath/gen1.c}, in the function \kbd{gmul}, case \typ{VEC} times case
! 874: \typ{MAT}. It would theoretically be possible to have a single connected
! 875: piece of garbage, but it would be a much less natural and unnecessarily
! 876: complicated program.
! 877:
! 878: Let us now take an example which is probably the least trivial way of using
! 879: \kbd{gerepile}, but is unfortunately sometimes necessary. Although it is not
! 880: an infrequent occurrence, we will not give a specific example but a general
! 881: one: suppose that we want to do a computation (usually inside a larger
! 882: function) producing more than one PARI object as a result, say two for
! 883: instance. Then even if we set up the work properly, before cleaning up we
! 884: will have a stack which has the desired results \kbd{z1}, \kbd{z2} (say),
! 885: and then connected garbage from lbot to ltop. If we write
! 886:
! 887: \kbd{z1 = gerepile(ltop,lbot,z1);}
! 888:
! 889: \noindent
! 890: then the stack will be cleaned, the pointers fixed up, but we will have lost
! 891: the address of \kbd{z2}. This is where we need one of the \idx{gerepilemany}
! 892: functions: we declare
! 893: \bprog%
! 894: GEN *gptr[2]; /* Array of pointers to GENs */
! 895: gptr[0] = \&z1; gptr[1] = \&z2;
! 896: \eprog
! 897: \noindent and now the call \kbd{gerepilemany(ltop, gptr, 2)} copies \kbd{z1}
! 898: and \kbd{z2} to new locations, cleans the stack from \kbd{ltop} to the old
! 899: \kbd{avma}, and updates the pointers \kbd{z1} and \kbd{z2}. Here we don't
! 900: assume anything about the stack: the garbage can be disconnected and
! 901: \kbd{z1}, \kbd{z2} need not be at the bottom of the stack. If all of these
! 902: assumptions are in fact satisfied, then we can call \kbd{gerepilemanysp}
! 903: instead, which will be faster since we don't need the initial copy.
! 904:
! 905: Another important usage is ``random'' garbage collection during loops
! 906: whose size requirements we cannot control in advance:
! 907: \bprog%
! 908: \q long ltop = avma, limit = (avma+bot)/2;
! 909: \q GEN x, y;
! 910: \h
! 911: \q while (...)
! 912: \q $\{$
! 913: \q\q garbage(); x = anything();
! 914: \q\q garbage(); y = anything()
! 915: \q\q garbage();
! 916: \q\q if (avma < limit) /* memory is running low (half spent since entry) */
! 917: \q\q $\{$
! 918: \q\q\q GEN *gptr[2];
! 919: \q\q\q gptr[0] = \&x; gptr[1] = \&y;
! 920: \q\q\q gerepilemany(ltop, gptr, 2);
! 921: \q\q $\}$
! 922: \q $\}$
! 923: \eprog
! 924: \noindent Here we assume that only \kbd{x} and \kbd{y} are needed from one
! 925: iteration to the next. As it would be too costly to call gerepile once for
! 926: each iteration, we only do it when it seems to have become necessary. Of
! 927: course, when the need arises, you can use bigger \kbd{gptr} arrays: in the
! 928: PARI library source code, we once needed to preserve up to 10 objects at a
! 929: time (in a variant of the LLL algorithm)!
! 930:
! 931: \misctitle{Technical note:} the statement \kbd{limit = (avma+bot)/2} is
! 932: dangerous since the addition can overflow, which would result in
! 933: \kbd{limit} being negative. This will prevent garbage collection in the
! 934: loop. To avoid this problem, we provide a robust macro
! 935: \kbd{stack\_lim(avma,$n$)}\sidx{stack\string\_lim}, which denotes an
! 936: address where $2^{n-1} / (2^{n-1}+1)$ of the total stack space is
! 937: exhausted ($1/2$ for $n=1$, $2/3$ for $n=2$). Hence, the above snippet should be written as
! 938:
! 939: \bprog%
! 940: \q long ltop = avma, limit = stack\_lim(avma,1);
! 941: \q \dots
! 942: \eprog
! 943:
! 944: \subsec{Some hints and tricks}. Even for the authors, the use of
! 945: \kbd{gerepile} was not evident at first. Hence we give some indications on
! 946: how to avoid most problems connected with garbage collecting:
! 947:
! 948: First, although it looks complicated, \kbd{gerepile} has turned out to be a
! 949: very flexible and fast garbage collector, which compares very favorably
! 950: with much more sophisticated methods used in other systems. A few tests that
! 951: we have done indicate that the price paid for using \kbd{gerepile}, when
! 952: properly used, is usually around 1 or 2 percents of the total time, which is
! 953: quite acceptable.
! 954:
! 955: Secondly, in many cases, in particular when the tree structure and the size of
! 956: the PARI objects which will appear in a computation are under control, one
! 957: can avoid \kbd{gerepile} altogether by creating sufficiently large objects at
! 958: the beginning (using \teb{cgetg}), and then using assignment statements and
! 959: operations ending with z (such as \teb{gaddz}). Coming back to our first
! 960: example, note that if we know that x and y are of type real and of length
! 961: less than or equal to 5, we can program without using \kbd{gerepile} at all:
! 962:
! 963: \bprog%
! 964: z = cgetr(5); ltop = avma;
! 965: p1 = gsqr(x); p2 = gsqr(y); gaddz(p1,p2,z);
! 966: avma = ltop;%
! 967: \eprog
! 968: \noindent This practice will usually be {\it slower\/} than a craftily used
! 969: \kbd{gerepile} though, and is certainly more cumbersome to use. As a rule,
! 970: assignment statements should generally be avoided.
! 971: \smallskip
! 972:
! 973: Thirdly, the philosophy of \kbd{gerepile} is the following: keep the value of
! 974: the stack pointer \kbd{avma} at the beginning, and just {\it before\/} the
! 975: last operation. Afterwards, it would be too late since the lower end address
! 976: of the garbage zone would have been lost. Of course you can always use
! 977: \kbd{gerepileupto}, but you will have to assume that the object was created
! 978: {\it before\/} its components.
! 979:
! 980: Finally, if everything seems hopeless, at the expense of speed you can do the
! 981: following: after saving the value of \kbd{avma} in \kbd{ltop}, perform your
! 982: computation as you wish, in any order, leaving a messy stack. Let \kbd{z} be
! 983: your result. Then write the following:
! 984:
! 985: \kbd{z = gerepileupto(ltop, gcopy(z));}
! 986:
! 987: \noindent
! 988: The trick is to force a copy of \kbd{z} to be created at the bottom of the
! 989: stack, hence all the rest including the initial \kbd{z} becomes connected
! 990: garbage. If you need to keep more than one result, use \kbd{gerepilemany}
! 991: (of which the above is just a special case).
! 992:
! 993: \smallskip If you followed us this far, congratulations, and rejoice: the
! 994: rest is much easier.
! 995:
! 996: \section{Implementation of the PARI types}
! 997: \label{se:impl}
! 998:
! 999: \noindent
! 1000: Although it is a little tedious, we now go through each type and explain its
! 1001: implementation. Let \kbd{z} be a \kbd{GEN}, pointing at a PARI object. In
! 1002: the following paragraphs, we will constantly mix two points of view: on the
! 1003: one hand, \kbd{z} will be treated as the C pointer it is (in the context of
! 1004: program fragments like \kbd{z[1]}), on the other, as PARI's handle on (the
! 1005: internal representation of) some mathematical entity, so we will shamelessly
! 1006: write $\kbd{z} \ne 0$ to indicate that the {\it value\/} thus represented
! 1007: is nonzero (in which case the {\it pointer\/}~\kbd{z} certainly will be
! 1008: non-\kbd{NULL}). We offer no apologies for this style. In fact, you had
! 1009: better feel comfortable juggling both views simultaneously in your mind if
! 1010: you want to write correct PARI programs.
! 1011:
! 1012: Common to all the types is the
! 1013: first codeword \kbd{z[0]}, which we don't have to worry about since this is
! 1014: taken care of by \kbd{cgetg}. Its precise structure will depend on the
! 1015: machine you are using, but it always contain the following data: the
! 1016: {\it internal \idx{type number}\/} associated to the symbolic type name, the
! 1017: {\it\idx{length}\/} of the root in longwords, and a technical bit which
! 1018: indicates whether the object is a clone (see below) or not. This last one is
! 1019: used by GP for internal garbage collecting, you won't have to worry about it.
! 1020:
! 1021: \noindent These data can be handled through the following {\it macros\/}:
! 1022:
! 1023: \kbd{long \teb{typ}(GEN z)} returns the type number of \kbd{z}.
! 1024:
! 1025: \kbd{void \teb{settyp}(GEN z, long n)} sets the type number of \kbd{z}
! 1026: to \kbd{n} (you should not have to use this function if you
! 1027: use \kbd{cgetg}).
! 1028:
! 1029: \kbd{long \teb{lg}(GEN z)} returns the length (in longwords) of the
! 1030: root of \kbd{z}.
! 1031:
! 1032: \kbd{long \teb{setlg}(GEN z, long l)} sets the length of \kbd{z}
! 1033: to \kbd{l} (you should not have to use this function if you use
! 1034: \kbd{cgetg}; however, see an advanced example in \secref{se:prog}).
! 1035:
! 1036: \noindent
! 1037: (If you know enough about PARI to need to access the ``clone'' bit, then you'll
! 1038: be able to find usage hints in the code (esp. \kbd{killbloc()} and
! 1039: \kbd{matrix\_block()}). It {\it is\/} technical after all.)
! 1040:
! 1041: These macros are written in such a way that you don't need to worry about
! 1042: type casts when using them: i.e.~if \kbd{z} is a \kbd{GEN}, \kbd{typ(z[2])}
! 1043: will be accepted by your compiler, without your having to explicitly type
! 1044: \kbd{typ((GEN)z[2])}. Note that for the sake of efficiency, none of the
! 1045: codeword-handling macros check the types of their arguments even when there
! 1046: are stringent restrictions on their use.
! 1047:
! 1048: The possible second codeword is used differently by the different types, and
! 1049: we will describe it as we now consider each of them in turn:
! 1050:
! 1051: \def\sectype#1#2{ \subsec{Type \typ{#1} (#2s):}\sidx{#2} }
! 1052: \def\sectypes#1#2#3{ \subsec{Types \typ{#1} and \typ{#2} (#3s):}\sidx{#3} }
! 1053: \def\sectypeindex#1#2#3{ \subsec{Type \typ{#1} (#2):}\sidx{#3} }
! 1054:
! 1055: \sectype{INT}{integer} this type has a second codeword \kbd{z[1]} which
! 1056: contains the following information:
! 1057:
! 1058: the sign of \kbd{z}: coded as $1$, $0$ or $-1$ if $\kbd{z} > 0$, $\kbd{z} = 0$,
! 1059: $\kbd{z} < 0$ respectively.
! 1060:
! 1061: the {\it effective length\/} of \kbd{z}, i.e.~the total number of significant
! 1062: longwords. This means the following: apart from the integer 0, every integer
! 1063: is ``normalized'', meaning that the first mantissa longword (i.e.~\kbd{z[2]})
! 1064: is non-zero. However, the integer may have been created with a longer length.
! 1065: Hence the ``length'' which is in \kbd{z[0]} can be larger than the
! 1066: ``effective length'' which is in \kbd{z[1]}. Accessing \kbd{z[i]}
! 1067: for \kbd{i} larger than or equal to the effective length will yield random
! 1068: results.
! 1069:
! 1070: \noindent This information is handled using the following macros:
! 1071:
! 1072: \kbd{long \teb{signe}(GEN z)} returns the sign of \kbd{z}.
! 1073:
! 1074: \kbd{void \teb{setsigne}(GEN z, long s)} sets the sign of
! 1075: \kbd{z} to \kbd{s}.
! 1076:
! 1077: \kbd{long \teb{lgefint}(GEN z)} returns the \idx{effective length}
! 1078: of \kbd{z}.
! 1079:
! 1080: \kbd{void \teb{setlgefint}(GEN z, long l)} sets the effective length
! 1081: of \kbd{z} to \kbd{l}.
! 1082:
! 1083: The integer 0 can be recognized either by its sign being~0, or by its
! 1084: effective length being equal to~2. When $\kbd{z} \ne 0$, the word
! 1085: \kbd{z[2]} exists and is non-zero, and the absolute value of \kbd{z}
! 1086: is (\kbd{z[2]},\kbd{z[3]},\dots,\kbd{z[lgefint(z)-1]}) in base
! 1087: \kbd{2\pow BITS\_IN\_LONG}, where as usual in this notation \kbd{z[2]} is
! 1088: the highest order longword.
! 1089:
! 1090: \noindent The following further macros are available:
! 1091:
! 1092: \kbd{long \teb{mpodd}(GEN x)} which is 1 if \kbd{x} is odd, and 0
! 1093: otherwise.
! 1094:
! 1095: \kbd{long \teb{mod2}(GEN x)}, \kbd{\key{mod4}(x)}, and so on up
! 1096: to \kbd{\key{mod64}(x)}, which give the residue class of \kbd{x} modulo the
! 1097: corresponding power of 2, {\it for positive\/}~\kbd{x} (you will obtain weird
! 1098: results if you use these on the integer 0 or on negative numbers).
! 1099:
! 1100: These macros directly access the binary data and are thus much faster than
! 1101: the generic modulo functions. Besides, they return long integers instead of
! 1102: \kbd{GEN}s, so they don't clutter up the stack.
! 1103:
! 1104: \sectype{REAL}{real number} this type has a second codeword z[1] which
! 1105: also encodes its sign (obtained or set using the same functions as for the
! 1106: integers), and a biased binary exponent (i.e.~the actual exponent value plus
! 1107: some constant bias, actually a power of~2, whose value is given by
! 1108: \kbd{HIGHEXPOBIT}). This exponent can be handled using the following macros:
! 1109:
! 1110: \kbd{long \teb{expo}(GEN z)} returns the true (unbiased) exponent
! 1111: of \kbd{z}. This is defined even when \kbd{z} is equal to zero, see
! 1112: \secref{se:whatzero}.
! 1113:
! 1114: \kbd{void \teb{setexpo}(GEN z, long e)} sets the exponent of \kbd{z}
! 1115: to \kbd{e}, of course after adding the bias.
! 1116:
! 1117: \noindent Note the functions:
! 1118:
! 1119: \kbd{long \teb{gexpo}(GEN z)} which tries to return an exponent
! 1120: for \kbd{z}, even if \kbd{z} is not a real number.
! 1121:
! 1122: \kbd{long \teb{gsigne}(GEN z)} which returns a sign for \kbd{z},
! 1123: even when \kbd{z} is neither real nor integer (a rational number for
! 1124: instance).
! 1125:
! 1126: The real zero is characterized by having its sign equal to 0. However,
! 1127: usually the first mantissa word \kbd{z[2]} is defined and equal to~0. This
! 1128: fact must {\it never\/} be used to recognize a real~0. If \kbd{z} is not equal
! 1129: to~0, the first mantissa word \kbd{z[2]} is normalized, i.e.~its most
! 1130: significant bit is~1. The mantissa is (\kbd{z[2]},\kbd{z[3]},\dots,%
! 1131: \kbd{z[lg(z]-1]}) in base \kbd{2\pow BITS\_IN\_LONG}. Here, \kbd{z[2]} is
! 1132: the most significant longword, and the mantissa takes values between
! 1133: 1 (included) and 2 (excluded). Thus, assume that \kbd{sizeof(long)} is 32 and
! 1134: the length is 4, the real number $3.5$ is represented as \kbd{z[0]} (encoding
! 1135: $\kbd{type} = \typ{REAL}$, $\kbd{lg} = 4$), \kbd{z[1]} (encoding $\kbd{sign} =
! 1136: 1$, $\kbd{expo} = 1$), $\kbd{z[2]} = \kbd{0xe0000000}$,
! 1137: $\kbd{z[3]} = \kbd{0x0}$.
! 1138:
! 1139: \sectype{INTMOD}{integermod} \kbd{z[1]} points to the modulus, and \kbd{z[2]}
! 1140: at the number representing the class \kbd{z}. Both are separate \kbd{GEN}
! 1141: objects, and both must be of type integer.
! 1142: In principle \kbd{z[1]} $>$ 0 and 0 $\le$ \kbd{z[2]} $<$ \kbd{z[1]}, but this
! 1143: rule does not have to be strictly obeyed by the user. Any integermod obtained
! 1144: as the result of a PARI function call will satisfy these conditions.
! 1145:
! 1146: It is good practice to keep the modulus object on the heap, so that new
! 1147: integermods resulting from operations can point at this common object,
! 1148: instead of carrying along their own copies of it on the stack. The library
! 1149: functions implement this practice almost by default.
! 1150:
! 1151: \sectypes{FRAC}{FRACN}{rational number} \kbd{z[1]} points to the numerator,
! 1152: and \kbd{z[2]} to the denominator. Both must be of type integer. In principle
! 1153: $\kbd{z[2]} > 0$, but this rule does not have to be strictly obeyed. Note
! 1154: that a type \typ{FRACN} rational number can be converted to irreducible
! 1155: form using the function \kbd{GEN \teb{gred}(GEN x)}.
! 1156:
! 1157: \sectype{COMPLEX}{complex number} \kbd{z[1]} points to the real part, and
! 1158: \kbd{z[2]} to the imaginary part. A priori \kbd{z[1]} and \kbd{z[2]} can be
! 1159: of any type, but only certain types are useful and make sense.
! 1160:
! 1161: \sectypeindex{PADIC}{$p$-adic numbers}{p-adic number} this type has a second
! 1162: codeword \kbd{[1]} which contains the following information: the $p$-adic
! 1163: precision (the exponent of $p$ modulo which the $p$-adic unit corresponding
! 1164: to \kbd{z} is defined if \kbd{z} is not~0), i.e.~one less than the number
! 1165: of significant $p$-adic digits, and the biased exponent of \kbd{z} (the
! 1166: bias being equal to \kbd{HIGHVALPBIT} here). This information can be
! 1167: handled using the following functions:
! 1168:
! 1169: \kbd{long \teb{precp}(GEN z)} returns the $p$-adic precision of
! 1170: \kbd{z}.
! 1171:
! 1172: \kbd{void \teb{setprecp}(GEN z, long l)} sets the $p$-adic precision
! 1173: of \kbd{z} to \kbd{l}.
! 1174:
! 1175: \kbd{long \teb{valp}(GEN z)} returns the $p$-adic valuation of
! 1176: \kbd{z} (i.e. the unbiased exponent). This is defined even if \kbd{z} is
! 1177: equal to~0, see \secref{se:whatzero}.
! 1178:
! 1179: \kbd{void \teb{setvalp}(GEN z, long e)} sets the $p$-adic valuation
! 1180: of \kbd{z} to \kbd{e}.
! 1181:
! 1182: In addition to this codeword, \kbd{z[2]} points to the prime $p$, \kbd{z[3]}
! 1183: points to $p^{\text{precp(z)}}$, and \kbd{z[4]} points to an integer
! 1184: representing the $p$-adic unit associated to \kbd{z} modulo \kbd{z[3]} (or
! 1185: points to zero if \kbd{z} is zero). To summarize, we have the equality:
! 1186: $$ \kbd{z} = p^{\text{valp(z)}} * (\kbd{z[4]} + O(\kbd{z[3]})) =
! 1187: p^{\text{valp(z)}} * (\kbd{z[4]} + O(p^{\text{precp(z)}})) $$
! 1188:
! 1189: \sectype{QUAD}{quadratic number} \kbd{z[1]} points to the polynomial defining
! 1190: the quadratic field, \kbd{z[2]} to the ``real part'' and \kbd{z[3]} to the
! 1191: ``imaginary part'', which are to be taken as the coefficients of \kbd{z} with
! 1192: respect to the ``canonical'' basis $(1,w)$, see~\secref{se:compquad}. Complex
! 1193: numbers are a particular case of quadratics but deserve a separate type.
! 1194:
! 1195: \sectype{POLMOD}{polmod} exactly as for integermods, \kbd{z[1]} points to
! 1196: the modulus, and \kbd{z[2]} to a polynomial representing the class of~\kbd{z}.
! 1197: Both must be of type polynomial. However, one must obey the rules
! 1198: explained in Chapter 2 concerning the hierarchical structure of the variables
! 1199: of a polymod.
! 1200:
! 1201: \sectype{POL}{polynomial} this type has a second codeword which is analogous
! 1202: to the one for integers. It contains a ``{\it sign\/}'': 0 if the polynomial
! 1203: is equal to~0, and 1 if not (see however the important remark below), a {\it
! 1204: variable number\/} (e.g.~0 for $x$, 1 for $y$, etc\dots), and an {\it
! 1205: effective length}.
! 1206:
! 1207: \noindent These data can be handled with the following macros:
! 1208:
! 1209: \teb{signe} and \teb{setsigne} as for reals and integers.
! 1210:
! 1211: \kbd{long \teb{lgef}(GEN z)} returns the \idx{effective length} of \kbd{z}.
! 1212:
! 1213: \kbd{void \teb{setlgef}(GEN z, long l)} sets the effective length
! 1214: of \kbd{z} to \kbd{l}.
! 1215:
! 1216: \kbd{long \teb{varn}(GEN z)} returns the variable number of the object \kbd{z}.
! 1217:
! 1218: \kbd{void \teb{setvarn}(GEN z, long v)} sets the variable number
! 1219: of \kbd{z} to \kbd{v}.
! 1220:
! 1221: Note also the function \kbd{long \teb{gvar}(GEN z)} which tries
! 1222: to return a \idx{variable number} for \kbd{z}, even if \kbd{z} is not a
! 1223: polynomial or power series. The variable number of a scalar type is set by
! 1224: definition equal to \tet{BIGINT}.
! 1225:
! 1226: The components \kbd{z[2]}, \kbd{z[3]},\dots \kbd{z[lgef(z)-1]} point to the
! 1227: coefficients of the polynomial {\it in ascending order}, with \kbd{z[2]}
! 1228: being the constant term and so on. Note that the {\it \idx{degree}\/} of the
! 1229: polynomial is equal to its effective length minus three. The function
! 1230:
! 1231: \kbd{long \teb{degree}(GEN x)} returns the degree of \kbd{x} with
! 1232: respect to its main variable even when \kbd{x} is not a polynomial (a
! 1233: rational function for instance). By convention, the degree of $0$ is~$-1$.
! 1234:
! 1235: \misctitle{Important remark}. A zero polynomial can be characterized by the
! 1236: fact that its sign is~0. However, its effective length may be equal to 2, or
! 1237: greater than 2. If it is greater than 2, this means that all the coefficients
! 1238: of the polynomial are equal to zero (as they should for a zero polynomial),
! 1239: but not all of these zeros are exact zeros, and more precisely the leading
! 1240: term \kbd{z[lgef(z)-1]} is not an exact zero.
! 1241:
! 1242: \sectypeindex{SER}{power series}{power series} This type also has a second
! 1243: codeword, which encodes a ``{\it sign\/}'', i.e.~0 if the power series is 0,
! 1244: and 1 if not, a {\it variable number\/} as for polynomials, and a {\it biased
! 1245: exponent\/} with a bias of \kbd{HIGHVALPBIT}. This information can be handled
! 1246: with the following functions: \teb{signe}, \teb{setsigne}, \teb{varn},
! 1247: \teb{setvarn} as for polynomials, and \teb{valp}, \teb{setvalp} for the
! 1248: exponent as for $p$-adic numbers. Beware: do {\it not\/} use \teb{expo} and
! 1249: \teb{setexpo} on power series.
! 1250:
! 1251: If the power series is non-zero, \kbd{z[2]}, \kbd{z[3]},\dots
! 1252: \kbd{z[lg(z)-1]} point to the coefficients of \kbd{z} in ascending order,
! 1253: \kbd{z[2]} being the first non-zero coefficient. Note that the exponent of a
! 1254: power series can be negative, i.e.~we are then dealing with a Laurent series
! 1255: (with a finite number of negative terms).
! 1256:
! 1257: \sectypes{RFRAC}{RFRACN}{rational function} \kbd{z[1]} points to the
! 1258: numerator, and \kbd{z[2]} on the denominator. Both must be of type
! 1259: polynomial. Note that a type \typ{RFRACN} rational function can be
! 1260: converted to irreducible form using the function \teb{gred}.
! 1261:
! 1262: \sectype{QFR}{indefinite binary quadratic form} \kbd{z[1]}, \kbd{z[2]},
! 1263: \kbd{z[3]} point to the three coefficients of the form and should be of type
! 1264: integer. \kbd{z[4]} is Shanks's distance function, and should be of type
! 1265: real.
! 1266:
! 1267: \sectype{QFI}{definite binary quadratic form} \kbd{z[1]}, \kbd{z[2]},
! 1268: \kbd{z[3]} point to the three coefficients of the form. All three should be of
! 1269: type integer.
! 1270:
! 1271: \sectypes{VEC}{COL}{vector}\sidx{row vector}\sidx{column vector}
! 1272: \kbd{z[1]}, \kbd{z[2]},\dots \kbd{z[lg(z)-1]}
! 1273: point to the components of the vector.
! 1274:
! 1275: \sectypeindex{MAT}{matrices}{matrix} \kbd{z[1]}, \kbd{z[2]},\dots
! 1276: \kbd{z[lg(z)-1]} point to the column vectors of \kbd{z}, i.e.~they must be of
! 1277: type \typ{COL} and of the same length.
! 1278:
! 1279: \noindent The last two were introduced for specific GP use, and you'll be
! 1280: much better off using the standard malloc'ed C constructs when programming
! 1281: in library mode. We quote them just for completeness (advising you not to
! 1282: use them):
! 1283:
! 1284: \sectype{LIST}{list} This one has a second codeword which contains an
! 1285: effective length (handled through \teb{lgef}~/ \teb{setlgef}).
! 1286: \kbd{z[2]},\dots, \kbd{z[lgef(z)-1]} contain the components of the list.
! 1287:
! 1288: \sectype{STR}{character string} \kbd{\teb{GSTR}(z)} (= \kbd{(z+1)}) points to
! 1289: the first character of the (\kbd{NULL}-terminated) string.
! 1290:
! 1291: \section{PARI variables}\label{se:vars}
! 1292: \subsec{Multivariate objects}
! 1293:
! 1294: \noindent
! 1295: We now consider variables and formal computations. As we have seen in
! 1296: \secref{se:impl}, the codewords for types \typ{POL} and \typ{SER}
! 1297: encode a ``variable number''. This is an integer, ranging from $0$ to
! 1298: \kbd{MAXVARN}. The lower it is, the higher the variable priority. PARI does
! 1299: not know anything about intelligent ``sparse'' representation of polynomials.
! 1300: So a multivariate polynomial in PARI is just a polynomial (in one variable),
! 1301: whose coefficients are themselves (arbitrary) polynomials. All computations
! 1302: are then just done formally on the coefficients as if the polynomial was
! 1303: univariate.
! 1304:
! 1305: In fact, the way an object will be considered in formal computations depends
! 1306: entirely on its ``principal variable number'' which is given by the function
! 1307:
! 1308: \kbd{long \teb{gvar}(GEN z)}
! 1309:
! 1310: \noindent which returns a \idx{variable number} for \kbd{z}, even if \kbd{z}
! 1311: is not a polynomial or power series. The variable number of a scalar type is
! 1312: set by definition equal to \tet{BIGINT} which is bigger than any legal
! 1313: variable number. The variable number of a recursive type which is not a
! 1314: polynomial or power series is the minimal variable number of its components.
! 1315: But for polynomials and power series only the ``outermost'' number counts:
! 1316: the representation is not symmetrical at all.
! 1317:
! 1318: Under GP, one need not worry too much since the interpreter will define
! 1319: the variables as it sees them and do the right thing with the polynomials
! 1320: produced (however, have a look at the remark in \secref{se:rempolmod}). But
! 1321: in library mode, they are tricky objects if you intend to build polynomials
! 1322: yourself (and not just let PARI functions produce them, which is usually less
! 1323: efficient). For instance, it does not make sense to have a variable number
! 1324: occur in the components of a polynomial whose main variable has a higher
! 1325: number (lower priority), even though there's nothing PARI can do to prevent
! 1326: you from doing it.
! 1327:
! 1328: \subsec{Creating variables}
! 1329: A basic difficulty is to ``create'' a variable. As we have seen in
! 1330: \secref{se:intro4}, a plethora of objects is associated to variable
! 1331: number~$v$. Here is the complete list: \teb{polun}$[v]$ and
! 1332: \teb{polx}$[v]$, which you can use in library mode and which represent,
! 1333: respectively, the monic monomials of degrees 0 and 1 in~$v$;
! 1334: \teb{varentries}$[v]$, and \teb{polvar}$[v]$. The latter two are only
! 1335: meaningful to GP, but they have to be set nevertheless. All of them must be
! 1336: properly defined before you can use a given integer as a variable number.
! 1337:
! 1338: Initially, this is done for $0$ (the variable \kbd{x} under GP), and
! 1339: \tet{MAXVARN}, which is there to address the need for a ``temporary'' new
! 1340: variable, which would not be used in regular objects (created by the
! 1341: library). We call the latter type a ``temporary variable''. The regular
! 1342: variables meant to be used in regular objects, are called ``user
! 1343: variables\sidx{variable (user)}''.
! 1344:
! 1345: \subsubsec{User variables}\sidx{variable (user)}:
! 1346: When the program starts, \kbd{x} is the only user variable (number~$0$). To
! 1347: define new ones, use
! 1348:
! 1349: \kbd{long \key{fetch\_user\_var}(char *$s$)}%
! 1350: \sidx{fetch\string\_user\string\_var}
! 1351:
! 1352: \noindent which inspects the user variable named $s$ (creating it if
! 1353: needed), and returns its variable number.
! 1354: \bprog%
! 1355: long v = fetch\_user\_var("y");
! 1356: GEN gy = polx[v];
! 1357: \eprog
! 1358: This function raises an error if $s$ is already known as a function name to
! 1359: the interpreter.
! 1360:
! 1361: \misctitle{Caveat:} it is possible to use \teb{flissexpr}
! 1362: (see~\secref{se:flisexpr}) to execute a GP command and create GP variables
! 1363: on the fly as needed:
! 1364:
! 1365: \bprog%
! 1366: GEN gy = flissexpr("y"); /* supposedly returns polx[$v$], for some $v$ */
! 1367: long v = gvar(gy);
! 1368: \eprog
! 1369:
! 1370: \noindent This is dangerous, especially when programming functions that
! 1371: will be used under GP. The code above reads the value of \kbd{y}, as it is
! 1372: currently known by the GP interpreter (possibly creating it in the
! 1373: process). All is well and good if \kbd{y} hasn't been tampered with in
! 1374: previous GP commands. But if \kbd{y} has been modified (e.g \kbd {y = 1}),
! 1375: then the value of \kbd{gy} is not what you expected it to be and corresponds
! 1376: instead to the current value of the GP variable (e.g \kbd{gun}).
! 1377:
! 1378: \subsubsec{Temporary variables}\sidx{variable (temporary)}:
! 1379: \kbd{MAXVARN} should be enough for most cases, but you can create more
! 1380: temporary variables using
! 1381:
! 1382: \kbd{long \key{fetch\_var}()}\sidx{fetch\string\_var}\label{se:fetch_var}
! 1383:
! 1384: \noindent
! 1385: This returns a variable number which is guaranteed to be unused by the
! 1386: library at the time you get it and as long as you do not delete it (we'll see
! 1387: how to do that shortly). This has {\it lower\/} number (i.e.~{\it higher\/}
! 1388: priority) than any temporary variable produced so far (\kbd{MAXVARN} is
! 1389: assumed to be the first such). This call updates all the aforementioned
! 1390: internal arrays. In particular, after the statement \kbd{v = fetch\_var()},
! 1391: you can use \kbd{polun[v]} and \kbd{polx[v]}. The variables created in this
! 1392: way have no identifier assigned to them though, and they will be printed as
! 1393: \kbd{\#<\text{number}>}, except for \kbd{MAXVARN} which will be printed
! 1394: as~\kbd{\#}. You can assign a name to a temporary variable, after creating
! 1395: it, by calling the function
! 1396:
! 1397: \kbd{void \key{name\_var}(long n, char *s)}\sidx{name\string\_var}
! 1398:
! 1399: \noindent after which the output machinery will use the name \kbd{s} to
! 1400: represent the variable number~\kbd{n}. The GP parser will {\it not\/}
! 1401: recognize it by that name, however, and calling this on a variable known
! 1402: to~GP will raise an error. Temporary variables are meant to be used as free
! 1403: variables, and you should never assign values or functions to them as you
! 1404: would do with variables under~GP. For that, you need a user variable.
! 1405:
! 1406: All objects created by \kbd{fetch\_var} are on the heap and not on the stack,
! 1407: thus they are not subject to standard garbage collecting (they won't be
! 1408: destroyed by a \kbd{gerepile} or \kbd{avma = ltop} statement). When you don't
! 1409: need a variable number anymore, you can delete it using
! 1410:
! 1411: \kbd{long \key{delete\_var}()}\sidx{delete\string\_var}
! 1412:
! 1413: \noindent which deletes the {\it latest\/} temporary variable created and
! 1414: returns the variable number of the previous one (or simply returns 0 if you
! 1415: try, in vain, to delete \kbd{MAXVARN}). Of course you should make sure that
! 1416: the deleted variable does not appear anywhere in the objects you use later
! 1417: on. Here is an example:
! 1418:
! 1419: \bprog\obr
! 1420: \q long first = fetch\_var();
! 1421: \q long n1 = fetch\_var();
! 1422: \q long n2 = fetch\_var(); /* prepare three variables for internal use */
! 1423: \q ...
! 1424: \q /* delete all variables before leaving */
! 1425: \q do \obr\ num = delete\_var(); \cbr\ while (num \&\& num <= first);
! 1426: \cbr\eprog
! 1427:
! 1428: \noindent
! 1429: The (dangerous) statement
! 1430:
! 1431: \bprog%
! 1432: while (delete\_var()) /* empty */;
! 1433: \eprog
! 1434:
! 1435: \noindent removes all temporary variables that were in use, except
! 1436: \kbd{MAXVARN} which cannot be deleted.
! 1437:
! 1438: \section{Input and output}
! 1439:
! 1440: \noindent
! 1441: Two important aspects have not yet been explained which are specific to
! 1442: library mode: input and output of PARI objects.
! 1443:
! 1444: \subsec{Input}.
! 1445:
! 1446: \noindent
! 1447: For \idx{input}, PARI provides you with two powerful high level functions
! 1448: which enables you to input your objects as if you were under GP. In fact,
! 1449: the second one {\it is\/} essentially the GP syntactical parser, hence you
! 1450: can use it not only for input but for (most) computations that you can do
! 1451: under GP. These functions are called \teb{flisexpr} and \teb{flisseq}. The
! 1452: first one has the following syntax:\label{se:flisexpr}
! 1453:
! 1454: \kbd{GEN \teb{flisexpr}(char *s)}
! 1455:
! 1456: \noindent
! 1457: Its effect is to analyze the input string s and to compute the result as in
! 1458: GP. However it is limited to one expression. If you want to read and
! 1459: evaluate a sequence of expressions, use
! 1460:
! 1461: \kbd{GEN \teb{flisseq}(char *s)}
! 1462:
! 1463: \noindent\sidx{filter}
! 1464: In fact these two functions start by {\it filtering\/} out all spaces and
! 1465: comments in the input string (that's what the initial \kbd{f} stands for).
! 1466: They then call the underlying basic functions, the GP parser proper: \kbd{GEN
! 1467: \teb{lisexpr}(char *s)} and \kbd{GEN \teb{lisseq}(char *s)}, which are
! 1468: slightly faster but which you probably don't need.
! 1469:
! 1470: To read a \kbd{GEN} from a file, you can use the simpler interface
! 1471:
! 1472: \kbd{GEN \teb{lisGEN}(FILE *file)}
! 1473:
! 1474: which reads a character string of arbitrary length from the stream \kbd{file}
! 1475: (up to the first newline character), applies \kbd{flisexpr} to it, and
! 1476: returns the resulting \kbd{GEN}. This way, you won't have to worry about
! 1477: allocating buffers to hold the string. To interactively input an expression,
! 1478: use \kbd{lisGEN(stdin)}.
! 1479:
! 1480: Once in a while, it may be necessary to evaluate a GP expression sequence
! 1481: involving a call to a function you have defined in~C. This is easy using
! 1482: \teb{install} which allows you to manipulate quite an arbitrary function (GP
! 1483: knows about pointers!). The syntax is
! 1484:
! 1485: \kbd{void \teb{install}(void *f, char *name, char *code)}
! 1486:
! 1487: \noindent where \kbd{f} is the (address of) the function (cast to the C type
! 1488: \kbd{void*}), \kbd{name} is the name by which you want to access your
! 1489: function from within your GP expressions, and \kbd{code} is a character
! 1490: string describing the function call prototype (see~\secref{se:gp.interface}
! 1491: for the precise description of prototype strings). In case the function
! 1492: returns a \kbd{GEN}, it should satisfy \kbd{gerepileupto} assumptions (see
! 1493: \secref{se:garbage}).
! 1494:
! 1495: \subsec{Output}.
! 1496:
! 1497: \noindent
! 1498: For \idx{output}, there exist essentially three different functions (with
! 1499: variants), corresponding to the three main GP output formats (as described in
! 1500: \secref{se:output}), plus three extra ones, respectively devoted to
! 1501: \TeX\ output, string output, and (advanced) debugging.
! 1502:
! 1503: \noindent $\bullet$ ``raw'' format, obtained by using the function
! 1504: \teb{brute} with the following syntax:
! 1505:
! 1506: \kbd{void \teb{brute}(GEN obj, char x, long n);}
! 1507:
! 1508: \noindent
! 1509: This prints the PARI object \kbd{obj} in \idx{format} \kbd{x0.n}, using the
! 1510: notations from \secref{se:format}. Recall that here \kbd{x} is either
! 1511: \kbd{'e'}, \kbd{'f'} or \kbd{'g'} corresponding to the three numerical output
! 1512: formats, and \kbd{n} is the number of printed significant digits, and should
! 1513: be set to $-1$ if all of them are wanted (these arguments only affect the
! 1514: printing of real numbers). Usually you won't need that much flexibility, so
! 1515: most of the time you will get by with the function
! 1516:
! 1517: \kbd{void \teb{outbrute}(GEN obj)}, which is equivalent to
! 1518: \kbd{brute(x,'g',-1)},
! 1519:
! 1520: \noindent or even better, with
! 1521:
! 1522: \kbd{void \teb{output}(GEN obj)} which is equivalent to \kbd{outbrute(obj)}
! 1523: followed by a newline and a buffer flush. This is especially nice during
! 1524: debugging. For instance using \kbd{dbx} or \kbd{gdb}, if \kbd{obj} is a
! 1525: \kbd{GEN}, typing \kbd{print output(obj)} will enable you to see the content
! 1526: of \kbd{obj} (provided the optimizer has not put it into a register, but it's
! 1527: rarely a good idea to debug optimized code).
! 1528:
! 1529: \noindent $\bullet$ ``prettymatrix'' format: this format is identical to the
! 1530: preceding one except for matrices. The relevant functions are:
! 1531:
! 1532: \kbd{void \teb{matbrute}(GEN obj, char x, long n)}
! 1533:
! 1534: \kbd{void \teb{outmat}(GEN obj)}, which is followed by a newline
! 1535: and a buffer flush.
! 1536:
! 1537: \noindent $\bullet$ ``prettyprint'' format: the basic function has an
! 1538: additional parameter \kbd{m}, corresponding to the (minimum) field width
! 1539: used for printing integers:
! 1540:
! 1541: \kbd{void \teb{sor}(GEN obj, char x, long n, long m)}
! 1542:
! 1543: \noindent The simplified version is
! 1544:
! 1545: \kbd{void \teb{outbeaut}(GEN obj)} which is equivalent to
! 1546: \kbd{sor(obj,'g',-1,0)} followed by a newline and a buffer flush.
! 1547:
! 1548: \noindent $\bullet$ The first extra format corresponds to the \teb{texprint}
! 1549: function of GP, and gives a \TeX{} output of the result. It is obtained by
! 1550: using:
! 1551:
! 1552: \kbd{void \teb{texe}(GEN obj, char x, long n)}
! 1553:
! 1554: \noindent $\bullet$ The second one is the function \teb{GENtostr} which
! 1555: converts a PARI \kbd{GEN} to an ASCII string. The syntax is
! 1556:
! 1557: \kbd{char* \teb{GENtostr}(GEN obj)}, wich returns a
! 1558: \kbd{malloc}'ed character string (which you should \kbd{free} after use).
! 1559:
! 1560: \noindent $\bullet$ The third and final one outputs the \idx{hexadecimal tree}
! 1561: corresponding to the GP command \kbd{\b x} using the function
! 1562:
! 1563: \kbd{void \teb{voir}(GEN obj, long nb)}, which will only output the first
! 1564: \kbd{nb} words corresponding to leaves (very handy when you have a look at
! 1565: big recursive structures). If you set this parameter to $-1$ all significant
! 1566: words will be printed. Usually this last type of output would only be used for
! 1567: debugging purposes.
! 1568:
! 1569: \misctitle{Remark}. Apart from \teb{GENtostr}, all PARI output is done on
! 1570: the stream \teb{outfile}, which by default is initialized to \teb{stdout}. If
! 1571: you want that your output be directed to another file, you should use the
! 1572: function \kbd{void \teb{switchout}(char *name)} where \kbd{name} is a
! 1573: character string giving the name of the file you are going to use. The
! 1574: output will be {\it appended\/} at the end of the file. In order to close
! 1575: the file, simply call \kbd{switchout(NULL)}.
! 1576:
! 1577: Similarly, errors are sent to the stream \teb{errfile} (\teb{stderr}
! 1578: by default), and input is done on the stream \teb{infile}, which you can change
! 1579: using the function \teb{switchin} which is analogous to \teb{switchout}.
! 1580:
! 1581: \misctitle{(Advanced) Remark}. All output is done according to the values
! 1582: of the \teb{pariOut}~/ \teb{pariErr} global variables which are pointers to
! 1583: structs of pointer to functions. If you really intend to use these, this
! 1584: probably means you are rewriting GP. In that case, have a look at the code in
! 1585: \kbd{language/es.c} (\kbd{init80()} or \kbd{GENtostr()} for instance).
! 1586:
! 1587: \subsec{Errors}.\sidx{error}\sidx{talker}
! 1588:
! 1589: \noindent
! 1590: If you want your functions to issue error messages, you can use the general
! 1591: error handling routine \teb{err}. The basic syntax is
! 1592: %
! 1593: \bprog%
! 1594: err(talker, "error message");
! 1595: \eprog
! 1596:
! 1597: \noindent
! 1598: This will print the corresponding error message and exit the program (in
! 1599: library mode; go back to the GP prompt otherwise).\label{se:err} You can
! 1600: also use it in the more versatile guise
! 1601: \bprog%
! 1602: err(talker, format, ...);
! 1603: \eprog\noindent
! 1604: where \kbd{format} describes the format to use to write the remaining
! 1605: operands, as in the \teb{printf} function (however, see the next section).
! 1606: The simple syntax above is just a special case with a constant format and no
! 1607: remaining arguments.
! 1608:
! 1609: \noindent
! 1610: The general syntax is
! 1611:
! 1612: \kbd{void \teb{err}(numerr,...)}
! 1613:
! 1614: \noindent where \kbd{numerr} is a codeword which indicates what to do with
! 1615: the remaining arguments and what message to print. The list of valid keywords
! 1616: is in \kbd{language/errmessages.c} together with the basic corresponding
! 1617: message. For instance, \kbd{err(typeer,"matexp")} will print the message:
! 1618:
! 1619: \kbd{ *** incorrect type in matexp.}
! 1620:
! 1621: \noindent
! 1622: Among the codewords are {\it warning\/} keywords (all those which start with
! 1623: the prefix \kbd{warn}). In that case, \teb{err} does {\it not\/} abort the
! 1624: computation, just print the requested message and go on. The basic example is
! 1625:
! 1626: \kbd{err(warner, "Strategy 1 failed. Trying strategy 2")}
! 1627:
! 1628: \noindent which is the exact equivalent of \kbd{err(talker,...)} except that
! 1629: you certainly don't want to stop the program at this point, just inform the
! 1630: user that something important has occured (in particular, this output would be
! 1631: suitably highlighted under GP, whereas a simple \kbd{printf} would not).
! 1632:
! 1633: \subsec{Debugging output}.\sidx{debugging}\sidx{format}\label{se:dbg_output}
! 1634:
! 1635: \noindent
! 1636: The global variables \teb{DEBUGLEVEL} and \teb{DEBUGMEM} (corresponding
! 1637: to the default \teb{debug} and \teb{debugmem}, see \secref{se:defaults})
! 1638: are used throughout the PARI code to govern the amount of diagnostic and
! 1639: debugging output, depending on their values. You can use them to debug your
! 1640: own functions, especially after having made them accessible under GP through
! 1641: the command \teb{install} (see \secref{se:install}).
! 1642:
! 1643: For debugging output, you can use \kbd{printf} and the standard output
! 1644: functions (\teb{brute} or \teb{output} mainly), but also some special purpose
! 1645: functions which embody both concepts, the main one being
! 1646:
! 1647: \kbd{void \teb{fprintferr}(char *pariformat, ...)}
! 1648:
! 1649: \noindent
! 1650: Now let's define what a PARI format is. It is a character string, similar
! 1651: to the one \kbd{printf} uses, where \kbd{\%} characters have a special
! 1652: meaning. It describes the format to use when printing the remaining operands.
! 1653: But, in addition to the standard format types, you can use \kbd{\%Z} to
! 1654: denote a \kbd{GEN} object (we would have liked to pick \kbd{\%G} but it was
! 1655: already in use!). For instance you could write:
! 1656:
! 1657: \kbd{err(talker, "x[\%d] = \%Z is not invertible!", i, x[i])},
! 1658:
! 1659: \noindent since the \teb{err} function accepts PARI formats. Here \kbd{i} is an
! 1660: \kbd{int}, \kbd{x} a \kbd{\idx{GEN}} which is not a leaf and this would
! 1661: insert in raw format the value of the \kbd{GEN} \kbd{x[i]}.
! 1662:
! 1663: \subsec{Timers and timing output}.
! 1664:
! 1665: \noindent
! 1666: To profile your functions, you can use the PARI timer. The functions
! 1667: \kbd{long \teb{timer}()} and \kbd{long \teb{timer2}()} return the
! 1668: elapsed time since the last call of the same function (in milliseconds). Two
! 1669: different functions (identical except for their independent time-of-last-call
! 1670: memories!) are provided so you can have both global timing and fine tuned
! 1671: profiling.
! 1672:
! 1673: You can also use \kbd{void \teb{msgtimer}(char *format,...)},
! 1674: which prints prints \kbd{Time}, then the remaining arguments as specified by
! 1675: \kbd{format} (which is a PARI format), then the output of \kbd{timer2}.
! 1676:
! 1677: \section{A complete program}
! 1678: \label{se:prog}
! 1679:
! 1680: \noindent
! 1681: Now that the preliminaries are out of the way, the best way to learn how to
! 1682: use the library mode is to work through a detailed non-trivial example of a
! 1683: main program. We will write a program which computes the exponential of a
! 1684: square matrix~$x$. The complete listing is given in Appendix~B, but each
! 1685: part of the program will be produced and explained here. We will use an
! 1686: algorithm which is not optimal but is not far from the one used for the PARI
! 1687: function \teb{gexp} (in fact embodied in the function \kbd{mpexp1}). This
! 1688: consists in calculating the sum of the series:
! 1689: $$e^{x/(2^n)}=\sum_{k=0}^\infty \dfrac{(x/(2^n))^k}{k!}$$
! 1690: for a suitable positive integer $n$, and then computing $e^x$ by repeated
! 1691: squarings. First, we will need to compute the $L^2$-norm of the matrix~$x$,
! 1692: i.e.~the quantity:
! 1693: $$z=\|x\|_2=\sqrt{\sum_{i,j}x_{i,j}^2}.$$
! 1694: We will then choose the integer $n$ such that the $L^2$-norm of $x/(2^n)$ is
! 1695: less than or equal to~1, i.e.
! 1696: $$ n = \left\lceil{\ln(z)}\big/{\ln(2)}\right\rceil $$
! 1697: if $z\ge1$, and~$n=0$ otherwise. Then the series will converge at least as
! 1698: fast as the usual one for $e^1$, and the cutoff error will be easy to
! 1699: estimate. In fact a larger value of $n$ would be preferable, but this is
! 1700: slightly machine dependent and more complicated, and will be left to the
! 1701: reader.
! 1702:
! 1703: Let us start writing our program. So as to be able to use it in other
! 1704: contexts, we will structure it in the following way: a main program which
! 1705: will do the input and output, and a function which we shall call \kbd{matexp}
! 1706: which does the real work. The main program is easy to write. It can be
! 1707: something like this:
! 1708:
! 1709: \bprog%
! 1710: \#include <pari.h>
! 1711: \h
! 1712: GEN matexp(GEN x, long prec);
! 1713: \h
! 1714: int
! 1715: main()
! 1716: \obr
! 1717: \q long d, prec = 3;
! 1718: \q GEN x;
! 1719: \h
! 1720: \q /* take a stack of a 10\pow 6 bytes, no prime table */
! 1721: \q pari\_init(1000000,2);
! 1722: \q printf("precision of the computation in decimal digits:\bs n");
! 1723: \q d = itos(lisGEN(stdin));
! 1724: \q if (d > 0) prec = (long) (d*pariK1+3);
! 1725: \h
! 1726: \q printf("input your matrix in GP format:\bs n");
! 1727: \q x = matexp(lisGEN(stdin), prec);
! 1728: \h
! 1729: \q sor(x,'g',d,0);
! 1730: \q exit(0);
! 1731: \cbr\eprog
! 1732: \noindent
! 1733: The variable \kbd{prec} represents the length in longwords of the real
! 1734: numbers used. \teb{pariK1} is a constant (defined in \kbd{paricom.h}) equal
! 1735: to $\ln(10) / (\ln(2) * \kbd{BITS\_IN\_LONG})$, which allows us to convert
! 1736: from a number of decimal digits to a number of longwords, independently of
! 1737: the actual bit size of your long integers. The function \teb{lisGEN} reads an
! 1738: expression (here from standard input) and converts it to a \kbd{GEN}, like
! 1739: the GP parser itself would. This means it takes care of whitespace etc.\ in
! 1740: the input, and can do computations (e.g.~\kbd{matid(2)} or \kbd{[1,0; 0,1]}
! 1741: are equally valid inputs).
! 1742:
! 1743: Finally, \kbd{sor} is the general output routine. We have chosen to give
! 1744: \kbd{d} significant digits since this is what was asked for. Note that there
! 1745: is a trick hidden here: if a negative \kbd{d} was input, then the computation
! 1746: will be done in precision 3 (i.e.~about 9.7 decimal digits for 32-bit
! 1747: machines and 19.4 for 64-bit machines) and in the function \kbd{sor}, giving
! 1748: a negative third argument outputs all the significant digits, which is entirely
! 1749: appropriate. Now let us attack the main course, the function \kbd{matexp}:
! 1750: %
! 1751: \bprog%
! 1752: GEN
! 1753: matexp(GEN x, long prec)
! 1754: \obr
! 1755: \q long lx=lg(x),i,k,n,lbot, ltop = avma;
! 1756: \q GEN y,r,s,p1,p2;
! 1757: \h
! 1758: \q /* check that x is a square matrix */
! 1759: \q if (typ(x) != t\_MAT) err(talker,"this expression is not a matrix");
! 1760: \q if (lx == 1) return cgetg(1, t\_MAT);
! 1761: \q if (lx != lg(x[1])) err(talker,"not a square matrix");
! 1762: \h
! 1763: \q /* compute the L2 norm of x */
! 1764: \q s = gzero;
! 1765: \q for (i=1; i<lx; i++)
! 1766: \q\q s = gadd(s, gnorml2((GEN)x[i]));
! 1767: \q if (typ(s) == t\_REAL) setlg(s,3);
! 1768: \q s = gsqrt(s,3); /* we do not need much precision on s */
! 1769: \h
! 1770: \q /* if s<1, we are happy */
! 1771: \q k = expo(s);
! 1772: \q if (k < 0) \obr\ n = 0; p1 = x; \cbr
! 1773: \q else \obr\ n = k+1; p1 = gmul2n(x,-n); setexpo(s,-1); \cbr\eprog
! 1774: \noindent
! 1775: Before continuing, several remarks are in order.
! 1776:
! 1777: First, before starting this computation which will produce garbage on the
! 1778: stack, we have carefully saved the value of the stack pointer \kbd{avma} in
! 1779: \kbd{ltop}. Note that we are going to assume throughout that the garbage does
! 1780: not overflow the currently available stack. If it ever did, we would
! 1781: have several options~--- allocate a larger stack in the main program (for
! 1782: instance change 1000000 into 2000000), do some \kbd{gerepile}ing along the
! 1783: way, or (if you know what you are doing) use \tet{allocatemoremem}.
! 1784:
! 1785: Secondly, the \teb{err} function is the general error handler for the
! 1786: PARI library. This will abort the program after printing the required
! 1787: message.
! 1788:
! 1789: Thirdly, notice how we handle the special case $\kbd{lx} = 1$ (empty matrix)
! 1790: {\it before\/} accessing \kbd{lx(x[1])}. Doing it the other way round could
! 1791: produce a fatal error (a segmentation fault or a bus error, most probably).
! 1792: Indeed, if \kbd{x} is of length 1, then \kbd{x[1]} is not a component of
! 1793: \kbd{x}. It is just the contents of the memory cell which happens to follow
! 1794: the one pointed to by \kbd{x}, and thus has no reason to be a valid \kbd{GEN}.
! 1795: Now recall that none of the codeword handling macros do any kind of type
! 1796: checking (see \secref{se:impl}), thus \teb{lg} would consider \kbd{x[1]}
! 1797: as a valid address, and try to access \kbd{*((GEN)x[1])} (the first codeword)
! 1798: which is unlikely to be a legal memory address.
! 1799:
! 1800: In the fourth place, to compute the square of the $L^2$-norm of \kbd{x} we
! 1801: just add the squares of the $L^2$-norms of the column vectors which we obtain
! 1802: using the library function \teb{gnorml2}. Had this function not existed, the
! 1803: norm computation would of course have been just as easy to write, but we would
! 1804: have needed a double loop.
! 1805:
! 1806: We then take the square root of \kbd{s}, in precision~3 (the smallest
! 1807: possible). The \kbd{prec} argument of transcendental functions (here~$3$) is
! 1808: only taken into account when the arguments are {\it exact\/} objects, and thus
! 1809: no a priori precision can be determined from the objects themselves. To cater
! 1810: for this possibility, if \kbd{s} is of type \typ{REAL}, we use the function
! 1811: \teb{setlg} which effectively sets the precision of \kbd{s} to the required
! 1812: value. Note that here, since we are using a numeric value for a \kbd{cget}
! 1813: function, the program will run slightly differently on 32-bit and 64-bit
! 1814: machines: we want to use the smallest possible bit accuracy, and this is equal
! 1815: to \kbd{BITS\_IN\_LONG}.
! 1816:
! 1817: Note that the matrix \kbd{x} is allowed to have complex entries, but the
! 1818: function \kbd{gnorml2} guarantees that \kbd{s} is a non-negative real number
! 1819: (not necessarily of type \typ{REAL} of course). If we had not known this fact,
! 1820: we would simply have added the instruction \kbd{s = greal(s);} just after the
! 1821: \kbd{for} loop.
! 1822:
! 1823: Note also that the function \kbd{gnorml2} works as desired on matrices, so we
! 1824: really did not need this loop at all (\kbd{s = gnorml2(x)} would have been
! 1825: enough), but we wanted to give examples of function usage. Similarly, it is of
! 1826: course not necessary to take the square root for testing whether the norm
! 1827: exceeds~$1$.
! 1828:
! 1829: In the fifth place, note that we initialized the sum \kbd{s} to \teb{gzero},
! 1830: which is an exact zero. This is logical, but has some disadvantages: if all
! 1831: the entries of the matrix are integers (or rational numbers), the computation
! 1832: will take rather long, about twice as long as with real numbers of the same
! 1833: length. It would be better to initialize \kbd{s} to a real zero, using for
! 1834: instance the instructions:
! 1835:
! 1836: \kbd{s = cgetr(prec+1); gaffsg(0,s);}
! 1837:
! 1838: \noindent
! 1839: This raises the question: which real zero does this produce (have a look
! 1840: at~\secref{se:whatzero})? In fact, the following choice has been made: it will
! 1841: give you the zero with exponent equal to $-\kbd{BITS\_IN\_LONG}$ times the
! 1842: number of longwords in the mantissa, i.e.~$-\kbd{bit\_accuracy(lg(s))}$.
! 1843: Instead of the above idiom, you can also use the function
! 1844: \kbd{GEN \teb{realzero}(long prec)}, which simply returns a real zero to
! 1845: accuracy \kbd{-bit\_accuracy(prec)}.
! 1846:
! 1847: The sixth remark here is about how to determine the approximate size
! 1848: of a real number. The fastest way to do this is to look at its binary
! 1849: exponent. Hence we need to have \kbd{s} actually represented as a real number,
! 1850: and not as an integer or a rational number. The result of transcendental
! 1851: functions is guaranteed to be of type \typ{REAL}, or complex with \typ{REAL}
! 1852: components, thus this is indeed the case after the call to \kbd{gsqrt} since
! 1853: its argument is a nonnegative (real) number.
! 1854:
! 1855: Finally, note the use of the function \teb{gmul2n}. It has the following
! 1856: syntax:
! 1857:
! 1858: \kbd{GEN \teb{gmul2n}(GEN x, long n);}
! 1859:
! 1860: \noindent and the effect is simply to multiply \kbd{x} by $2^{\text n}$,
! 1861: where \kbd{n} can be positive or negative. This is much faster than
! 1862: \kbd{gmul} or \kbd{gmulgs}.
! 1863:
! 1864: There is another function \teb{gshift} with exactly the same syntax.
! 1865: When \kbd{n} is non-negative, the effects of these two functions are the same.
! 1866: However, when \kbd{n} is negative, \kbd{gshift} acts like a right shift of
! 1867: \kbd{-n}, hence does not noramlly perform an exact division on integers. The
! 1868: function \kbd{gshift} is the PARI analogue of the C or GP operators \kbd{<<}
! 1869: and~\kbd{>>}.
! 1870:
! 1871: We now come to the heart of the function. We have a \kbd{GEN} \kbd{p1} which
! 1872: points to a certain matrix of which we want to take the exponential. We will
! 1873: want to transform this matrix into a matrix with real (or complex of real)
! 1874: entries before starting the computation. To do this, we simply multiply by
! 1875: the real number 1 in precision $\kbd{prec}+1$ (to be on the side of safety).
! 1876: To sum the series, we will use three variables: a variable \kbd{p2} which at
! 1877: stage $k$ will contain $\kbd{p1}^k/k!$, a variable \kbd{y} which will contain
! 1878: $\sum_{i=0}^k \kbd{p1}^i/i!$, and a variable \kbd{r} which will contain the
! 1879: size estimate $\kbd{s}^k/k!$. Note that we do not use Horner's rule. This is
! 1880: simply because we are lazy and do not want to compute in advance the number
! 1881: of terms that we need. We leave this modification (and many other
! 1882: improvements!) to the reader. The program continues as follows:
! 1883:
! 1884: \bprog%
! 1885: \q /* initializations before the loop */
! 1886: \q r = cgetr(prec+1); gaffsg(1,r); p1 = gmul(r,p1);
! 1887: \q y = gscalmat(r,lx-1); /* creates scalar matrix with r on diagonal */
! 1888: \q p2 = p1; r = s; k = 1;
! 1889: \q y = gadd(y,p2);
! 1890: \h
! 1891: \q /* now the main loop */
! 1892: \q while (expo(r) >= -BITS\_IN\_LONG*(prec-1))
! 1893: \q \obr
! 1894: \q\q k++; p2 = gdivgs(gmul(p2,p1),k);
! 1895: \q\q r = gdivgs(gmul(s,r),k); y = gadd(y,p2);
! 1896: \q \cbr
! 1897: \h
! 1898: \q /* now square back n times if necessary */
! 1899: \q if (!n) \obr\ lbot = avma; y = gcopy(y); \cbr
! 1900: \q else
! 1901: \q \obr
! 1902: \q\q for (i=0; i<n; i++) \obr\ lbot = avma; y = gsqr(y); \cbr
! 1903: \q \cbr
! 1904: \q return gerepile(ltop,lbot,y);
! 1905: \cbr\eprog
! 1906: \noindent
! 1907: A few remarks once again. First note the use of the function
! 1908: \teb{gscalmat} with the following syntax:
! 1909:
! 1910: \kbd{GEN \teb{gscalmat}(GEN x, long m);}
! 1911:
! 1912: \noindent
! 1913: The effect of this function is to create the $\kbd{m}\times\kbd{m}$ scalar
! 1914: matrix whose diagonal entries are~\kbd{x}. Hence the length of the matrix
! 1915: including the codeword will in fact be \kbd{m+1}. There is a corresponding
! 1916: function \teb{gscalsmat} which takes a long as a first argument.
! 1917:
! 1918: If we refer to what has been said above, the main loop should be self-evident.
! 1919:
! 1920: When we do the final squarings, according to the fundamental dogma on the
! 1921: use of \teb{gerepile}, we keep the value of \kbd{avma} in \kbd{lbot} just {\it
! 1922: before\/} the squaring, so that if it is the last one, \kbd{lbot} will indeed
! 1923: be the bottom address of the garbage pile, and \kbd{gerepile} will work. Note
! 1924: that it takes a completely negligible time to do this in each loop compared
! 1925: to a matrix squaring. However, when \kbd{n} is initially equal to 0, no
! 1926: squaring has to be done, and we have our final result ready but we lost the
! 1927: address of the bottom of the garbage pile. Hence we use the trick of copying
! 1928: \kbd{y} again to the top of the stack. This is inefficient, but does the trick.
! 1929: If we wanted to avoid this using only \kbd{gerepile}, the best thing to do
! 1930: would be to put the instruction \kbd{lbot=avma} just before both occurrences
! 1931: of the instruction \kbd{y=gadd(p2,y)}. Of course, we could also rewrite the
! 1932: last block as follows:
! 1933:
! 1934: \bprog%
! 1935: \q /* now square back n times */
! 1936: \q for (i=0; i<n; i++) y = gsqr(y);
! 1937: \q return gerepileupto(ltop,y);
! 1938: \cbr\eprog
! 1939: \noindent
! 1940: because it does not matter to \teb{gerepileupto} that we have lost the address
! 1941: just before the final result (note that the loop is not executed if \kbd{n}
! 1942: is~0). It is safe to use \kbd{gerepileupto} here as \kbd{y} will have been
! 1943: created by either \kbd{gsqr} or \kbd{gadd}, both of which are guaranteed to
! 1944: return suitable objects.
! 1945:
! 1946: \misctitle{Remarks}. As such, the program should work most of the time if
! 1947: \kbd{x} is a square matrix with real or complex entries. Indeed, since
! 1948: essentially the first thing that we do is to multiply by the real number~1,
! 1949: the program should work for integer, real, rational, complex or quadratic
! 1950: entries. This is in accordance with the behavior of transcendental functions.
! 1951:
! 1952: Furthermore, since this program is intended to be only an illustrative
! 1953: example, it has been written a little sloppily. In particular many error
! 1954: checks have been omitted, and the efficiency is far from optimal. An evident
! 1955: improvement would be the use of \kbd{gerepileupto} mentioned above. Another
! 1956: improvement is to multiply the matrix x by the real number 1 right at the
! 1957: beginning, speeding up the computation of the $L^2$-norm in many cases. These
! 1958: improvements are included in the version given in Appendix~B. Still another
! 1959: improvement would come from a better choice of~\kbd{n}. If the reader takes a
! 1960: look at the implementation of the function \kbd{mpexp1} in the file
! 1961: \kbd{basemath/trans1.c}, he can make the necessary changes himself. Finally,
! 1962: there exist other algorithms of a different nature to compute the exponential
! 1963: of a matrix.
! 1964:
! 1965: \section{Adding functions to PARI}
! 1966: \subsec{Nota Bene}.
! 1967: %
! 1968: As already mentioned, modified versions of the PARI package should NOT be
! 1969: spread without our prior approval. If you do modify PARI, however, it is
! 1970: certainly for a good reason, hence we would like to know about it, so that
! 1971: everyone can benefit from it. There is then a good chance that the
! 1972: modifications that you have made will be incorporated into the next release.
! 1973:
! 1974: (Recall the e-mail address: \kbd{pari@math.u-bordeaux.fr}, or use the mailing
! 1975: lists).
! 1976:
! 1977: Roughly four types of modifications can be made. The first type includes all
! 1978: improvements to the documentation, in a broad sense. This includes correcting
! 1979: typos or inacurracies of course, but also items which are not really covered
! 1980: in this document, e.g.~if you happen to write a tutorial, or pieces of code
! 1981: exemplifying some fine points that you think were unduly omitted.
! 1982:
! 1983: The second type is to expand or modify the configuration routines and skeleton
! 1984: files (the \kbd{Configure} script and anything in the \kbd{config/}
! 1985: subdirectory) so that compilation is possible (or easier, or more efficient)
! 1986: on an operating system previously not catered for. This includes discovering
! 1987: and removing idiosyncrasies in the code that would hinder its portability.
! 1988:
! 1989: The third type is to modify existing (mathematical) code, either to correct
! 1990: bugs, to add new functionalities to existing functions, or to improve their
! 1991: efficiency.
! 1992:
! 1993: Finally the last type is to add new functions to PARI. We explain here how
! 1994: to do this, so that in particular the new function can be called from GP.
! 1995:
! 1996: \subsec{The calling interface from GP, parser codes}.
! 1997: \label{se:gp.interface}
! 1998: A \idx{parser code} is a character string describing all the GP parser
! 1999: needs to know about the function prototype. It contains a sequence of the
! 2000: following atoms:
! 2001:
! 2002: \settabs\+\indent&\kbd{Dxxx}\quad&\cr
! 2003: \noindent $\bullet$ Syntax requirements, used by functions like
! 2004: \kbd{for}, \kbd{sum}, etc.:
! 2005: %
! 2006: \+& \kbd{=} & separator \kbd{=} required at this point (between two
! 2007: arguments)\cr
! 2008:
! 2009: \noindent$\bullet$ Mandatory arguments, appearing in the same order as the
! 2010: input arguments they describe:
! 2011: %
! 2012: \+& \kbd{G} & \kbd{GEN}\cr
! 2013: \+& \kbd{\&}& \kbd{*GEN}\cr
! 2014: \+& \kbd{L} & long {\rm (we implicitly identify \kbd{int} with \kbd{long})}\cr
! 2015: \+& \kbd{S} & symbol (i.e.~GP identifier name). Function expects a
! 2016: \kbd{*entree}\cr
! 2017: \+& \kbd{V} & variable (as \kbd{S}, but rejects symbols associated to
! 2018: functions)\cr
! 2019: \+& \kbd{n} & variable, expects a \idx{variable number} (a \kbd{long}, not an
! 2020: \kbd{*entree})\cr
! 2021: \+& \kbd{I} & string containing GP code (useful for control statements, %
! 2022: to be processed with \kbd{lisexpr}\cr
! 2023: \+&&\quad or \kbd{lisseq})\cr
! 2024: \+& \kbd{r} & raw input (treated as a string without quotes). Quoted %
! 2025: args are copied as strings.\cr
! 2026: \+&&\quad Stops at first unquoted \kbd{')'} or \kbd{','}. Special chars can
! 2027: be quoted using '\kbd{\bs{}}'.\cr
! 2028: \+&&\quad Example: \kbd{aa"b\bs n)"c} yields the string \kbd{"aab\bs{n})c"}\cr
! 2029: \+& \kbd{s} & expanded string. Example: \kbd{Pi"x"2} yields \kbd{"3.142x2"}.\cr
! 2030: \+&& Unquoted components can be of any PARI type (converted following current
! 2031: output format)\cr
! 2032:
! 2033: \noindent$\bullet$ Optional arguments:
! 2034: %
! 2035: \+& \kbd{s*} & any number of strings, possibly 0 (see \kbd{s})\cr
! 2036: \+& \kbd{s*p} & idem, setting $\kbd{prettyp} = 1$ (i.e.~in beautified
! 2037: format)\cr
! 2038: \+& \kbd{s*t} & idem, in \TeX\ format\cr
! 2039: \+& \kbd{D\var{xxx}} & argument has a default value.\cr
! 2040:
! 2041: The format to indicate a default value (atom starts with a \kbd{D}) is
! 2042: ``\kbd{D\var{value},\var{type},}'', where \var{type} is the code for any
! 2043: mandatory atom (previous group), \var{value} is any valide GP expression
! 2044: which is converted according to \var{type}, and the ending comma is
! 2045: mandatory. For instance \kbd{D0,L,} stands for ``this optional argument will
! 2046: be converted to a \kbd{long}, and is \kbd{0} by default''. So if the
! 2047: user-given argument reads \kbd{1 + 3} at this point, \kbd{(long)4} is sent to
! 2048: the function (via \tet{itos}$()$); and \kbd{(long)0} if the argument is
! 2049: ommitted. The following special syntaxes are available:
! 2050:
! 2051: \begingroup
! 2052: \settabs\+\indent\indent&\kbd{Dxxx}\quad& optional \kbd{*GEN},&\cr
! 2053: \+&\kbd{DG}& optional \kbd{GEN}, & send {\tt NULL} if argument omitted.\cr
! 2054:
! 2055: \+&\kbd{D\&}& optional \kbd{*GEN}, send {\tt NULL} if argument omitted.\cr
! 2056:
! 2057: \+&\kbd{DV}& optional \kbd{*entree}, send {\tt NULL} if argument omitted.\cr
! 2058:
! 2059: \+&\kbd{DI}& optional \kbd{*char}, send {\tt NULL} if argument omitted.\cr
! 2060:
! 2061: \+&\kbd{Dn}& optional variable number, $-1$ if omitted.\cr
! 2062: \endgroup
! 2063:
! 2064: \noindent$\bullet$ Automatic arguments:
! 2065: %
! 2066: \+& \kbd{f} & Fake \kbd{*long}. C function requires a pointer but we
! 2067: don't use the resulting \kbd{long}\cr
! 2068: \+& \kbd{p} & real precision (default \kbd{realprecision})\cr
! 2069: \+& \kbd{P} & series precision (default \kbd{seriesprecision},
! 2070: global variable \kbd{precdl} for the library)\cr
! 2071:
! 2072: \noindent $\bullet$ Return type: \kbd{GEN} by default, otherwise the
! 2073: following can appear anywhere in the code string:
! 2074: %
! 2075: \+& \kbd{l} & return \kbd{long}\cr
! 2076: \+& \kbd{v} & return \kbd{void}\cr
! 2077:
! 2078: No more than 8 arguments can be given (syntax requirements and return types
! 2079: are not considered as arguments). This is currently hardcoded but can
! 2080: trivially be changed by modifying the definition of \kbd{argvec} in
! 2081: \kbd{anal.c:identifier()}. This limitation should disappear in future
! 2082: versions.
! 2083:
! 2084: When the function is called under GP, the prototype is scanned and each time
! 2085: an atom corresponding to a mandatory argument is met, a user-given argument
! 2086: is read (GP outputs an error message it the argument was missing). Each time
! 2087: an optional atom is met, a default value is inserted if the user omits the
! 2088: argument. The ``automatic'' atoms fill in the argument list transparently,
! 2089: supplying the current value of the corresponding variable (or a dummy
! 2090: pointer).
! 2091:
! 2092: For instance, here is how you would code the following prototypes (which
! 2093: don't involve default values):
! 2094: \bprogtabs\+\indent& void name(GEN x, GEN y, long prec)\quad& ---->\quad&\cr
! 2095: \+& GEN name(GEN x, GEN y, long prec) & ----> & "GGp"\cr
! 2096: \+& void name(GEN x, long y, long prec) & ----> & "vGLp" {\rm (or %
! 2097: \kbd{"GvLp"})}\cr
! 2098: \+& long name(GEN x) & ----> & "Gl"\cr
! 2099: \eprog
! 2100:
! 2101: If you want more examples, GP gives you easy access to the parser codes
! 2102: associated to all GP functions: just type \kbd{\b{h} \var{function}}. You
! 2103: can then compare with the C prototypes as they stand in the code.
! 2104:
! 2105: \misctitle{Remark}: If you need to implement complicated control statements
! 2106: (probably for some improved summation functions), you'll need to know about
! 2107: the \teb{entree} type, which is not documented. Check the comment before
! 2108: the function list at the end of \kbd{language/init.c} and the source code
! 2109: in \kbd{language/sumiter.c}. You should be able to make something of it.
! 2110: \smallskip
! 2111:
! 2112: \subsec{Coding guidelines}.
! 2113: \noindent
! 2114: Code your function in a file of its own, using as a guide other functions
! 2115: in the PARI sources. One important thing to remember is to clean the stack
! 2116: before exiting your main function (usually using \kbd{gerepile}), since
! 2117: otherwise successive calls to the function will clutter the stack with
! 2118: unnecessary garbage, and stack overflow will occur sooner. Also, if it
! 2119: returns a \kbd{GEN} and you want it to be accessible to GP, you have to
! 2120: make sure this \kbd{GEN} is suitable for \kbd{gerepileupto} (see
! 2121: \secref{se:garbage}).
! 2122:
! 2123: If error messages are to be generated in your function, use the general
! 2124: error handling routine \kbd{err} (see \secref{se:err}). Recall that, apart
! 2125: from the \kbd{warn} variants, this function does not return but ends with
! 2126: a \kbd{longjmp} statement. As well, instead of explicit \kbd{printf}~/
! 2127: \kbd{fprintf} statements, use the following encapsulated variants:
! 2128:
! 2129: \kbd{void \teb{pariputs}(char *s)}: write \kbd{s} to the GP output stream.
! 2130:
! 2131: \kbd{void \teb{fprintferr}(char *s)}: write \kbd{s} to the GP error
! 2132: stream (this function is in fact much more versatile, see
! 2133: \secref{se:dbg_output}).
! 2134:
! 2135: Declare all public functions in \kbd{paridecl.h} (you want the outside
! 2136: world to know about them). The other ones should be declared \kbd{static} in
! 2137: your file.
! 2138:
! 2139: Your function is now ready to be used in library mode after compilation and
! 2140: creation of the library. If possible, compile it as a shared library (see
! 2141: the \kbd{Makefile} coming with the \kbd{matexp} example in the
! 2142: distribution). It is however still inaccessible from GP.\smallskip
! 2143:
! 2144: \subsec{Integration with GP as a shared module}
! 2145:
! 2146: To tell GP about your function, you must do the following. First, find a
! 2147: name for it. It does not have to match the one used in library mode, but
! 2148: consistency is nice. It has to be a valid GP identifier, i.e.~use only
! 2149: alphabetic characters, digits and the underscore character (\kbd{\_}), the
! 2150: first character being alphabetic.
! 2151:
! 2152: Then you have to figure out the correct \idx{parser code} corresponding to
! 2153: the function prototype. This has been explained above
! 2154: (\secref{se:gp.interface}).
! 2155:
! 2156: Now, assuming your Operating System is supported by \tet{install}, simply
! 2157: write a GP script like the following:
! 2158:
! 2159: \bprog%
! 2160: install(\var{name}, \var{code}, \var{gpname}, \var{library})
! 2161: addhelp(\var{gpname}, \var{some help text})
! 2162: \eprog
! 2163: \noindent(see \secref{se:addhelp} and~\ref{se:install}). The \idx{addhelp}
! 2164: part is not mandatory, but very useful if you want others to use your
! 2165: module.
! 2166:
! 2167: Read that file from your GP session (from your \idx{preferences file} for
! 2168: instance, see \secref{se:gprc}), and that's it, you can use the new
! 2169: function \var{gpname} under GP (and we would very much like to hear about
! 2170: it!).
! 2171:
! 2172: \subsec{Integration the hard way}
! 2173:
! 2174: If \tet{install} is not available for your Operating System, it's more
! 2175: complicated: you have to hardcode your function in the GP binary (or
! 2176: install \idx{Linux}). Here's what needs to be done:
! 2177:
! 2178: In the definition of \kbd{functions\_basic} (file \kbd{language/init.c}),
! 2179: add your entry in exact alphabetical order by its GP name (note that digits
! 2180: come before letters), in a line of the form:
! 2181:
! 2182: \kbd{\obr"gpname",V,(void*)libname,secno,"code"\cbr,}
! 2183:
! 2184: \noindent where
! 2185:
! 2186: \kbd{libname} is the name of your function in library mode,
! 2187:
! 2188: \kbd{gpname} the name that you have chosen to call it under GP,
! 2189:
! 2190: \kbd{secno} is the section number of Chapter~3 in which this function would
! 2191: belong (type \kbd{?} in GP to see the list),
! 2192:
! 2193: \kbd{V} is a number between 0 and 99. Right now, there are only two
! 2194: significant values: zero means that it's possible to call the function
! 2195: without argument, and non-zero means it needs at least one argument.
! 2196:
! 2197: \kbd{code} is the parser code.
! 2198:
! 2199: Once this has been done, in the file \kbd{language/helpmessages.c} add in
! 2200: exact alphabetical order a short message describing the effect of your
! 2201: function:
! 2202: \kbd{"name(x,y,...)=short descriptive message",}
! 2203:
! 2204: The message must be a single line, of arbitrary length. Do not use
! 2205: \kbd{\bs{n}}; the necessary newlines will be inserted by GP's online help
! 2206: functions. Optional arguments should be shown between braces (see the other
! 2207: messages for comparison).\smallskip
! 2208:
! 2209: Now, you can recompile GP.
! 2210:
! 2211: \subsec{Example}.
! 2212: %
! 2213: A complete description could look like this:
! 2214:
! 2215: \bprog%
! 2216: \obr
! 2217: \q install(bnfinit0, GD0,L,DGp, ClassGroupInit, "libpari.so")
! 2218: \q addhelp(ClassGroupInit, "ClassGroupInit(P,\obr flag=0\cbr,\obr data=[]\cbr):
! 2219: \q\q compute the necessary data for ...")
! 2220: \cbr
! 2221: \eprog
! 2222: which means we have a function \kbd{ClassGroupInit} under GP, which calls the
! 2223: library function \kbd{bnfinit0} . The function has one mandatory
! 2224: argument (\kbd{V} is non-zero), and possibly two more (two \kbd{'D'} in the
! 2225: code), plus the current real precision. More precisely, the first argument
! 2226: is a \kbd{GEN}, the second one is converted to a \kbd{long} using
! 2227: \kbd{itos} (\kbd{0} is passed if it is omitted), and the third one is also
! 2228: a \kbd{GEN}, but we pass \kbd{NULL} if no argument was supplied by the
! 2229: user. This matches the C prototype (from \kbd{paridecl.h}):
! 2230: %
! 2231: \bprog%
! 2232: GEN bnfinit0(GEN P, long flag, GEN data, long prec)
! 2233: \eprog
! 2234:
! 2235: This function is in fact coded in \kbd{basemath/buch2.c}, and will in this
! 2236: case be completely identical to the GP function \kbd{bnfinit} but GP does
! 2237: not need to know about this, only that it can be found somewhere in the
! 2238: shared library \kbd{libpari.so}.
! 2239:
! 2240: \misctitle{Important note} You see in this example that it is the
! 2241: function's responsibility to correctly interpret its operands: \kbd{data =
! 2242: NULL} is interpreted {\it by the function\/} as an empty vector. Note that
! 2243: since \kbd{NULL} is never a valid \kbd{GEN} pointer, this trick always
! 2244: enables you to distinguish between a default value and actual input: the
! 2245: user could explicitly supply an empty vector!
! 2246:
! 2247: \misctitle{Note} If \kbd{install} were not available we would have to
! 2248: modify \kbd{language/helpmessages.c}, and \kbd{language/init.c} and
! 2249: recompile GP. The entry in \kbd{functions\_basic} corresponding to the
! 2250: function above is actually
! 2251: \bprog\obr
! 2252: "bnfinit", 91, (void*)bnfinit0, 6, "GD0,L,DGp"
! 2253: \cbr\eprog
! 2254: \vfill\eject
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