@end iftex
@c %**end of header
+@direntry
+* CLN: (cln). Class Library for Numbers (C++).
+@end direntry
@c My own index.
@defindex my
This file documents @sc{cln}, a Class Library for Numbers.
Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
-Richard Kreckel, @code{<kreckel@@ginac.de>}.
+Richard B. Kreckel, @code{<kreckel@@ginac.de>}.
-Copyright (C) Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000.
+Copyright (C) Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005.
+Copyright (C) Richard B. Kreckel 2000, 2001, 2002, 2003, 2004, 2005.
Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
@author by Bruno Haible
@page
@vskip 0pt plus 1filll
-Copyright @copyright{} Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000.
+Copyright @copyright{} Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005.
+@sp 0
+Copyright @copyright{} Richard Kreckel 2000, 2001, 2002, 2003, 2004, 2005.
@sp 2
Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
@page
+@c Table of contents
+@contents
+
+
@node Top, Introduction, (dir), (dir)
@c @menu
The kernel of CLN has been written in assembly language for some CPUs
(@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
@item
-On all CPUs, CLN uses the superefficient low-level routines from GNU
-GMP version 2.
+@cindex GMP
+On all CPUs, CLN may be configured to use the superefficient low-level
+routines from GNU GMP version 3.
@item
It uses Karatsuba multiplication, which is significantly faster
for large numbers than the standard multiplication algorithm.
For very large numbers (more than 12000 decimal digits), it uses
@iftex
Sch{@"o}nhage-Strassen
+@cindex Sch{@"o}nhage-Strassen multiplication
@end iftex
@ifinfo
-Schönhage-Strassen
+Schnhage-Strassen
+@cindex Schnhage-Strassen multiplication
@end ifinfo
-multiplication, which is an asymptotically
-optimal multiplication algorithm, for multiplication, division and
-radix conversion.
+multiplication, which is an asymptotically optimal multiplication
+algorithm, for multiplication, division and radix conversion.
@end itemize
@noindent
The garbage collection imposes no burden on the main application.
@item
The library provides hooks for memory allocation and exceptions.
+@item
+@cindex namespace
+All non-macro identifiers are hidden in namespace @code{cln} in
+order to avoid name clashes.
@end itemize
@subsection C++ compiler
To build CLN, you need a C++ compiler.
-Actually, you need GNU @code{g++ 2.7.0} or newer.
-On HPPA, you need GNU @code{g++ 2.8.0} or newer.
-I recommend GNU @code{egcs 1.1} or newer.
+Actually, you need GNU @code{g++ 2.95} or newer.
The following C++ features are used:
-classes, member functions,
-overloading of functions and operators,
-constructors and destructors, inline, const,
-multiple inheritance, templates.
+classes, member functions, overloading of functions and operators,
+constructors and destructors, inline, const, multiple inheritance,
+templates and namespaces.
The following C++ features are not used:
-@code{new}, @code{delete}, virtual inheritance,
-exceptions.
-
-CLN relies on semi-automatic ordering of initializations
-of static and global variables, a feature which I could
-implement for GNU g++ only.
-
-@ignore
-@comment cl_modules.h requires g++
-Therefore nearly any C++ compiler will do.
-
-The following C++ compilers are known to compile CLN:
-@itemize @minus
-@item
-GNU @code{g++ 2.7.0}, @code{g++ 2.7.2}
-@item
-SGI @code{CC 4}
-@end itemize
+@code{new}, @code{delete}, virtual inheritance, exceptions.
-The following C++ compilers are known to be unusable for CLN:
-@itemize @minus
-@item
-On SunOS 4, @code{CC 2.1}, because it doesn't grok @code{//} comments
-in lines containing @code{#if} or @code{#elif} preprocessor commands.
-@item
-On AIX 3.2.5, @code{xlC}, because it doesn't grok the template syntax
-in @code{cl_SV.h} and @code{cl_GV.h}, because it forces most class types
-to have default constructors, and because it probably miscompiles the
-integer multiplication routines.
-@item
-On AIX 4.1.4.0, @code{xlC}, because when optimizing, it sometimes converts
-@code{short}s to @code{int}s by zero-extend.
-@item
-GNU @code{g++ 2.5.8}
-@item
-On HPPA, GNU @code{g++ 2.7.x}, because the semi-automatic ordering of
-initializations will not work.
-@end itemize
-@end ignore
+CLN relies on semi-automatic ordering of initializations of static and
+global variables, a feature which I could implement for GNU g++
+only. Also, it is not known whether this semi-automatic ordering works
+on all platforms when a non-GNU assembler is being used.
@subsection Make utility
+@cindex @code{make}
To build CLN, you also need to have GNU @code{make} installed.
+Only GNU @code{make} 3.77 is unusable for CLN; other versions work fine.
+
@subsection Sed utility
+@cindex @code{sed}
To build CLN on HP-UX, you also need to have GNU @code{sed} installed.
This is because the libtool script, which creates the CLN library, relies
The @code{configure} command checks out some features of your system and
C++ compiler and builds the @code{Makefile}s. The @code{make} command
-builds the library. This step may take 4 hours on an average workstation.
+builds the library. This step may take about an hour on an average workstation.
The @code{make check} runs some test to check that no important subroutine
has been miscompiled.
@example
$ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
-$ CC="gcc -V 2.7.2" CFLAGS="-O -g" \
- CXX="g++ -V 2.7.2" CXXFLAGS="-O -g" ./configure
-$ CC="gcc -V 2.8.1" CFLAGS="-O -fno-exceptions" \
- CXX="g++ -V 2.8.1" CXXFLAGS="-O -fno-exceptions" ./configure
-$ CC="gcc -V egcs-2.91.60" CFLAGS="-O2 -fno-exceptions" \
- CXX="g++ -V egcs-2.91.60" CFLAGS="-O2 -fno-exceptions" ./configure
-@end example
-@ignore
-@comment cl_modules.h requires g++
-You should not mix GNU and non-GNU compilers. So, if @code{CXX} is a non-GNU
-compiler, @code{CC} should be set to a non-GNU compiler as well. Examples:
-
-@example
-$ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O" ./configure
-$ CC="gcc -V 2.7.0" CFLAGS="-g" CXX="g++ -V 2.7.0" CXXFLAGS="-g" ./configure
-@end example
-
-On SGI Irix 5, if you wish not to use @code{g++}:
-
-@example
-$ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O -Olimit 16000" ./configure
-@end example
-
-On SGI Irix 6, if you wish not to use @code{g++}:
-
-@example
-$ CC="cc -32" CFLAGS="-O" CXX="CC -32" CXXFLAGS="-O -Olimit 34000" \
- ./configure --without-gmp
-$ CC="cc -n32" CFLAGS="-O" CXX="CC -n32" CXXFLAGS="-O \
- -OPT:const_copy_limit=32400 -OPT:global_limit=32400 -OPT:fprop_limit=4000" \
- ./configure --without-gmp
+$ CC="gcc -V egcs-2.91.60" CFLAGS="-O -g" \
+ CXX="g++ -V egcs-2.91.60" CXXFLAGS="-O -g" ./configure
+$ CC="gcc -V 2.95.2" CFLAGS="-O2 -fno-exceptions" \
+ CXX="g++ -V 2.95.2" CFLAGS="-O2 -fno-exceptions" ./configure
+$ CC="gcc -V 3.0.4" CFLAGS="-O2 -finline-limit=1000 -fno-exceptions" \
+ CXX="g++ -V 3.0.4" CFLAGS="-O2 -finline-limit=1000 -fno-exceptions" \
+ ./configure
@end example
-@end ignore
Note that for these environment variables to take effect, you have to set
them (assuming a Bourne-compatible shell) on the same line as the
calling @code{configure}. In a @code{csh} shell, you have to use the
@samp{setenv} command for setting each of the environment variables.
-On Linux, @code{g++} needs 15 MB to compile the tests. So you should better
-have 17 MB swap space and 1 MB room in $TMPDIR.
-
-If you use @code{g++} version 2.7.x, don't add @samp{-O2} to the CXXFLAGS,
-because @samp{g++ -O} generates better code for CLN than @samp{g++ -O2}.
-
-If you use @code{g++} version 2.8.x or egcs-2.91.x (a.k.a. egcs-1.1) or
-gcc-2.95.x, I recommend adding @samp{-fno-exceptions} to the CXXFLAGS.
-This will likely generate better code.
-
-If you use @code{g++} version egcs-2.91.x (egcs-1.1) or gcc-2.95.x on Sparc,
-add either @samp{-O} or @samp{-O2 -fno-schedule-insns} to the CXXFLAGS.
-With full @samp{-O2}, @code{g++} miscompiles the division routines. Also, for
---enable-shared to work, you need egcs-1.1.2 or newer.
+Currently CLN works only with the GNU @code{g++} compiler, and only in
+optimizing mode. So you should specify at least @code{-O} in the CXXFLAGS,
+or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use @code{-O}.)
+
+If you use @code{g++} 3.x, I recommend adding @samp{-finline-limit=1000}
+to the CXXFLAGS. This is essential for good code.
+
+If you use @code{g++} gcc-2.95.x or gcc-3.x , I recommend adding
+@samp{-fno-exceptions} to the CXXFLAGS. This will likely generate better code.
+
+If you use @code{g++} from gcc-3.0.4 or older on Sparc, add either
+@samp{-O}, @samp{-O1} or @samp{-O2 -fno-schedule-insns} to the
+CXXFLAGS. With full @samp{-O2}, @code{g++} miscompiles the division
+routines. If you use @code{g++} older than 2.95.3 on Sparc you should
+also specify @samp{--disable-shared} because of bad code produced in the
+shared library. Also, do not use gcc-3.0 on Sparc for compiling CLN, it
+won't work at all.
+
+If you use @code{g++} on OSF/1 or Tru64 using gcc-2.95.x, you should
+specify @samp{--disable-shared} because of linker problems with
+duplicate symbols in shared libraries. If you use @code{g++} from
+gcc-3.0.n, with n larger than 1, you should @emph{not} add
+@samp{-fno-exceptions} to the CXXFLAGS, since that will generate wrong
+code (gcc-3.1 is okay again, as is gcc-3.0).
+
+Also, please do not compile CLN with @code{g++} using the @code{-O3}
+optimization level. This leads to inferior code quality.
+
+If you use @code{g++} from gcc-3.1, it will need 235 MB of virtual memory.
+You might need some swap space if your machine doesn't have 512 MB of RAM.
+
+By default, both a shared and a static library are built. You can build
+CLN as a static (or shared) library only, by calling @code{configure} with
+the option @samp{--disable-shared} (or @samp{--disable-static}). While
+shared libraries are usually more convenient to use, they may not work
+on all architectures. Try disabling them if you run into linker
+problems. Also, they are generally somewhat slower than static
+libraries so runtime-critical applications should be linked statically.
+
+If you use @code{g++} from gcc-3.1 with option @samp{-g}, you will need
+some disk space: 335 MB for building as both a shared and a static library,
+or 130 MB when building as a shared library only.
+
+
+@subsection Using the GNU MP Library
+@cindex GMP
+
+Starting with version 1.1, CLN may be configured to make use of a
+preinstalled @code{gmp} library. Please make sure that you have at
+least @code{gmp} version 3.0 installed since earlier versions are
+unsupported and likely not to work. Enabling this feature by calling
+@code{configure} with the option @samp{--with-gmp} is known to be quite
+a boost for CLN's performance.
+
+If you have installed the @code{gmp} library and its header file in
+some place where your compiler cannot find it by default, you must help
+@code{configure} by setting @code{CPPFLAGS} and @code{LDFLAGS}. Here is
+an example:
-On MIPS (SGI Irix 6), pass option @code{--without-gmp} to configure. gmp does
-not work when compiled in @samp{n32} binary format on Irix.
-
-By default, only a static library is built. You can build CLN as a shared
-library too, by calling @code{configure} with the option @samp{--enable-shared}.
-To get it built as a shared library only, call @code{configure} with the options
-@samp{--enable-shared --disable-static}.
-
-If you use @code{g++} version egcs-2.91.x (egcs-1.1) on Sparc, you cannot
-use @samp{--enable-shared} because @code{g++} would miscompile parts of the
-library.
+@example
+$ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
+ CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
+@end example
@section Installing the library
+@cindex installation
As with any autoconfiguring GNU software, installation is as easy as this:
@example
Number
- cl_number
- <cl_number.h>
+ cl_number
+ <cln/number.h>
|
|
Real or complex number
cl_N
- <cl_complex.h>
+ <cln/complex.h>
|
|
Real number
cl_R
- <cl_real.h>
+ <cln/real.h>
|
+-------------------+-------------------+
| |
Rational number Floating-point number
cl_RA cl_F
-<cl_rational.h> <cl_float.h>
+<cln/rational.h> <cln/float.h>
| |
- | +-------------+-------------+-------------+
- Integer | | | |
- cl_I Short-Float Single-Float Double-Float Long-Float
- <cl_integer.h> cl_SF cl_FF cl_DF cl_LF
- <cl_sfloat.h> <cl_ffloat.h> <cl_dfloat.h> <cl_lfloat.h>
+ | +--------------+--------------+--------------+
+ Integer | | | |
+ cl_I Short-Float Single-Float Double-Float Long-Float
+<cln/integer.h> cl_SF cl_FF cl_DF cl_LF
+ <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
@end example
+@cindex @code{cl_number}
+@cindex abstract class
The base class @code{cl_number} is an abstract base class.
It is not useful to declare a variable of this type except if you want
to completely disable compile-time type checking and use run-time type
checking instead.
+@cindex @code{cl_N}
+@cindex real number
+@cindex complex number
The class @code{cl_N} comprises real and complex numbers. There is
no special class for complex numbers since complex numbers with imaginary
part @code{0} are automatically converted to real numbers.
+@cindex @code{cl_R}
The class @code{cl_R} comprises real numbers of different kinds. It is an
abstract class.
+@cindex @code{cl_RA}
+@cindex rational number
+@cindex integer
The class @code{cl_RA} comprises exact real numbers: rational numbers, including
integers. There is no special class for non-integral rational numbers
since rational numbers with denominator @code{1} are automatically converted
to integers.
+@cindex @code{cl_F}
The class @code{cl_F} implements floating-point approximations to real numbers.
It is an abstract class.
@section Exact numbers
+@cindex exact number
Some numbers are represented as exact numbers: there is no loss of information
when such a number is converted from its mathematical value to its internal
are coprime integers and the denominator is positive. If the resulting
denominator is @code{1}, the rational number is converted to an integer.
-Small integers (typically in the range @code{-2^30}@dots{}@code{2^30-1},
+@cindex immediate numbers
+Small integers (typically in the range @code{-2^29}@dots{}@code{2^29-1},
for 32-bit machines) are especially efficient, because they consume no heap
allocation. Otherwise the distinction between these immediate integers
(called ``fixnums'') and heap allocated integers (called ``bignums'')
@section Floating-point numbers
+@cindex floating-point number
Not all real numbers can be represented exactly. (There is an easy mathematical
proof for this: Only a countable set of numbers can be stored exactly in
are uncountably many real numbers.) So some approximation is needed.
CLN implements ordinary floating-point numbers, with mantissa and exponent.
+@cindex rounding error
The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{})
only return approximate results. For example, the value of the expression
@code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as
@itemize @bullet
@item
+@cindex @code{cl_SF}
Short floats, type @code{cl_SF}.
They have 1 sign bit, 8 exponent bits (including the exponent's sign),
and 17 mantissa bits (including the ``hidden'' bit).
They don't consume heap allocation.
@item
+@cindex @code{cl_FF}
Single floats, type @code{cl_FF}.
They have 1 sign bit, 8 exponent bits (including the exponent's sign),
and 24 mantissa bits (including the ``hidden'' bit).
This corresponds closely to the C/C++ type @samp{float}.
@item
+@cindex @code{cl_DF}
Double floats, type @code{cl_DF}.
They have 1 sign bit, 11 exponent bits (including the exponent's sign),
and 53 mantissa bits (including the ``hidden'' bit).
This corresponds closely to the C/C++ type @samp{double}.
@item
+@cindex @code{cl_LF}
Long floats, type @code{cl_LF}.
They have 1 sign bit, 32 exponent bits (including the exponent's sign),
and n mantissa bits (including the ``hidden'' bit), where n >= 64.
is too limited for your application, choose another floating-point type
with larger exponent range.
+@cindex @code{cl_F}
As a user of CLN, you can forget about the differences between the
four floating-point types and just declare all your floating-point
variables as being of type @code{cl_F}. This has the advantage that
@section Complex numbers
+@cindex complex number
Complex numbers, as implemented by the class @code{cl_N}, have a real
part and an imaginary part, both real numbers. A complex number whose
@section Conversions
+@cindex conversion
Conversions from any class to any its superclasses (``base classes'' in
C++ terminology) is done automatically.
@code{cl_R}, @code{cl_N}.
The easiest way to specify a value which is outside of the range of the
C++ built-in types is therefore to specify it as a string, like this:
+@cindex Rubik's cube
@example
cl_I order_of_rubiks_cube_group = "43252003274489856000";
@end example
@table @code
@item int cl_I_to_int (const cl_I& x)
+@cindex @code{cl_I_to_int ()}
@itemx unsigned int cl_I_to_uint (const cl_I& x)
+@cindex @code{cl_I_to_uint ()}
@itemx long cl_I_to_long (const cl_I& x)
+@cindex @code{cl_I_to_long ()}
@itemx unsigned long cl_I_to_ulong (const cl_I& x)
+@cindex @code{cl_I_to_ulong ()}
Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not
representable in the range of @var{ctype}, a runtime error occurs.
@end table
the functions
@table @code
-@item float cl_float_approx (const @var{type}& x)
-@itemx double cl_double_approx (const @var{type}& x)
+@item float float_approx (const @var{type}& x)
+@cindex @code{float_approx ()}
+@itemx double double_approx (const @var{type}& x)
+@cindex @code{double_approx ()}
Returns an approximation of @code{x} of C type @var{ctype}.
If @code{abs(x)} is too close to 0 (underflow), 0 is returned.
If @code{abs(x)} is too large (overflow), an IEEE infinity is returned.
C++ terminology) are not provided. Instead, you can assert and check
that a value belongs to a certain subclass, and return it as element of that
class, using the @samp{As} and @samp{The} macros.
+@cindex cast
+@cindex @code{As()()}
@code{As(@var{type})(@var{value})} checks that @var{value} belongs to
@var{type} and returns it as such.
+@cindex @code{The()()}
@code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
@var{type} and returns it as such. It is your responsibility to ensure
-that this assumption is valid.
+that this assumption is valid. Since macros and namespaces don't go
+together well, there is an equivalent to @samp{The}: the template
+@samp{the}.
+
Example:
@example
@group
cl_I x = @dots{};
if (!(x >= 0)) abort();
- cl_I ten_x = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
+ cl_I ten_x_a = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
// In general, it would be a rational number.
+ cl_I ten_x_b = the<cl_I>(expt(10,x)); // The same as above.
@end group
@end example
Each of the number classes declares its mathematical operations in the
corresponding include file. For example, if your code operates with
-objects of type @code{cl_I}, it should @code{#include <cl_integer.h>}.
+objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
@section Constructing numbers
@code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
and then apply the exponential function:
@example
- cl_float_format_t precision = cl_float_format(40);
+ float_format_t precision = float_format(40);
cl_F e = exp(cl_float(1,precision));
@end example
@table @code
@item @var{type} operator + (const @var{type}&, const @var{type}&)
+@cindex @code{operator + ()}
Addition.
@item @var{type} operator - (const @var{type}&, const @var{type}&)
+@cindex @code{operator - ()}
Subtraction.
@item @var{type} operator - (const @var{type}&)
Returns the negative of the argument.
@item @var{type} plus1 (const @var{type}& x)
+@cindex @code{plus1 ()}
Returns @code{x + 1}.
@item @var{type} minus1 (const @var{type}& x)
+@cindex @code{minus1 ()}
Returns @code{x - 1}.
@item @var{type} operator * (const @var{type}&, const @var{type}&)
+@cindex @code{operator * ()}
Multiplication.
@item @var{type} square (const @var{type}& x)
+@cindex @code{square ()}
Returns @code{x * x}.
@end table
@table @code
@item @var{type} operator / (const @var{type}&, const @var{type}&)
+@cindex @code{operator / ()}
Division.
@item @var{type} recip (const @var{type}&)
+@cindex @code{recip ()}
Returns the reciprocal of the argument.
@end table
The class @code{cl_I} doesn't define a @samp{/} operation because
in the C/C++ language this operator, applied to integral types,
denotes the @samp{floor} or @samp{truncate} operation (which one of these,
-is implementation dependent). (@xref{Rounding functions})
+is implementation dependent). (@xref{Rounding functions}.)
Instead, @code{cl_I} defines an ``exact quotient'' function:
@table @code
@item cl_I exquo (const cl_I& x, const cl_I& y)
+@cindex @code{exquo ()}
Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
@end table
@table @code
@item cl_I expt_pos (const cl_I& x, const cl_I& y)
+@cindex @code{expt_pos ()}
@itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
@code{y} must be > 0. Returns @code{x^y}.
@item cl_RA expt (const cl_RA& x, const cl_I& y)
+@cindex @code{expt ()}
@itemx cl_R expt (const cl_R& x, const cl_I& y)
@itemx cl_N expt (const cl_N& x, const cl_I& y)
Returns @code{x^y}.
@table @code
@item @var{type} abs (const @var{type}& x)
+@cindex @code{abs ()}
Returns the absolute value of @code{x}.
This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
@end table
@table @code
@item @var{type} signum (const @var{type}& x)
+@cindex @code{signum ()}
Returns the sign of @code{x}, in the same number format as @code{x}.
This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
@code{x} if @code{x} is zero. If @code{x} is real, the value is either
@table @code
@item cl_I numerator (const @var{type}& x)
+@cindex @code{numerator ()}
Returns the numerator of @code{x}.
@item cl_I denominator (const @var{type}& x)
+@cindex @code{denominator ()}
Returns the denominator of @code{x}.
@end table
@table @code
@item cl_N complex (const cl_R& a, const cl_R& b)
+@cindex @code{complex ()}
Returns the complex number @code{a+bi}, that is, the complex number with
real part @code{a} and imaginary part @code{b}.
@end table
@table @code
@item cl_R realpart (const @var{type}& x)
+@cindex @code{realpart ()}
Returns the real part of @code{x}.
@item cl_R imagpart (const @var{type}& x)
+@cindex @code{imagpart ()}
Returns the imaginary part of @code{x}.
@item @var{type} conjugate (const @var{type}& x)
+@cindex @code{conjugate ()}
Returns the complex conjugate of @code{x}.
@end table
@section Comparisons
+@cindex comparison
Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
@code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
@table @code
@item bool operator == (const @var{type}&, const @var{type}&)
+@cindex @code{operator == ()}
@itemx bool operator != (const @var{type}&, const @var{type}&)
+@cindex @code{operator != ()}
Comparison, as in C and C++.
-@item uint32 cl_equal_hashcode (const @var{type}&)
+@item uint32 equal_hashcode (const @var{type}&)
+@cindex @code{equal_hashcode ()}
Returns a 32-bit hash code that is the same for any two numbers which are
the same according to @code{==}. This hash code depends on the number's value,
not its type or precision.
@item cl_boolean zerop (const @var{type}& x)
+@cindex @code{zerop ()}
Compare against zero: @code{x == 0}
@end table
defines the following operations:
@table @code
-@item cl_signean cl_compare (const @var{type}& x, const @var{type}& y)
+@item cl_signean compare (const @var{type}& x, const @var{type}& y)
+@cindex @code{compare ()}
Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
-1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
@item bool operator <= (const @var{type}&, const @var{type}&)
+@cindex @code{operator <= ()}
@itemx bool operator < (const @var{type}&, const @var{type}&)
+@cindex @code{operator < ()}
@itemx bool operator >= (const @var{type}&, const @var{type}&)
+@cindex @code{operator >= ()}
@itemx bool operator > (const @var{type}&, const @var{type}&)
+@cindex @code{operator > ()}
Comparison, as in C and C++.
@item cl_boolean minusp (const @var{type}& x)
+@cindex @code{minusp ()}
Compare against zero: @code{x < 0}
@item cl_boolean plusp (const @var{type}& x)
+@cindex @code{plusp ()}
Compare against zero: @code{x > 0}
@item @var{type} max (const @var{type}& x, const @var{type}& y)
+@cindex @code{max ()}
Return the maximum of @code{x} and @code{y}.
@item @var{type} min (const @var{type}& x, const @var{type}& y)
+@cindex @code{min ()}
Return the minimum of @code{x} and @code{y}.
@end table
@section Rounding functions
+@cindex rounding
When a real number is to be converted to an integer, there is no ``best''
rounding. The desired rounding function depends on the application.
@table @code
@item cl_I floor1 (const @var{type}& x)
+@cindex @code{floor1 ()}
Returns @code{floor(x)}.
@item cl_I ceiling1 (const @var{type}& x)
+@cindex @code{ceiling1 ()}
Returns @code{ceiling(x)}.
@item cl_I truncate1 (const @var{type}& x)
+@cindex @code{truncate1 ()}
Returns @code{truncate(x)}.
@item cl_I round1 (const @var{type}& x)
+@cindex @code{round1 ()}
Returns @code{round(x)}.
@end table
@table @code
@item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
@itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
+@cindex @code{floor2 ()}
@itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
+@cindex @code{ceiling2 ()}
@itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
+@cindex @code{truncate2 ()}
@itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
+@cindex @code{round2 ()}
@end table
Sometimes, one wants the quotient as a floating-point number (of the
@table @code
@item @var{type} ffloor (const @var{type}& x)
+@cindex @code{ffloor ()}
@itemx @var{type} fceiling (const @var{type}& x)
+@cindex @code{fceiling ()}
@itemx @var{type} ftruncate (const @var{type}& x)
+@cindex @code{ftruncate ()}
@itemx @var{type} fround (const @var{type}& x)
+@cindex @code{fround ()}
@end table
and similarly for class @code{cl_R}, but with return type @code{cl_F}.
Each of the classes
@code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
defines the following operations:
+@cindex @code{cl_F_fdiv_t}
+@cindex @code{cl_SF_fdiv_t}
+@cindex @code{cl_FF_fdiv_t}
+@cindex @code{cl_DF_fdiv_t}
+@cindex @code{cl_LF_fdiv_t}
@table @code
@item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
@itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
+@cindex @code{ffloor2 ()}
@itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
+@cindex @code{fceiling2 ()}
@itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
+@cindex @code{ftruncate2 ()}
@itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
+@cindex @code{fround2 ()}
@end table
and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
+@cindex @code{cl_R_fdiv_t}
The class @code{cl_R} defines the following operations:
@table @code
@item @var{type} mod (const @var{type}& x, const @var{type}& y)
+@cindex @code{mod ()}
@itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
+@cindex @code{rem ()}
@end table
@table @code
@item @var{type} sqrt (const @var{type}& x)
+@cindex @code{sqrt ()}
@code{x} must be >= 0. This function returns the square root of @code{x},
normalized to be >= 0. If @code{x} is the square of a rational number,
@code{sqrt(x)} will be a rational number, else it will return a
@table @code
@item cl_boolean sqrtp (const @var{type}& x, @var{type}* root)
+@cindex @code{sqrtp ()}
This tests whether @code{x} is a perfect square. If so, it returns true
and the exact square root in @code{*root}, else it returns false.
@end table
@table @code
@item cl_boolean isqrt (const @var{type}& x, @var{type}* root)
+@cindex @code{isqrt ()}
@code{x} should be >= 0. This function sets @code{*root} to
@code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
the boolean value @code{(expt(*root,2) == x)}.
@table @code
@item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
+@cindex @code{rootp ()}
@code{x} must be >= 0. @code{n} must be > 0.
This tests whether @code{x} is an @code{n}th power of a rational number.
If so, it returns true and the exact root in @code{*root}, else it returns
@table @code
@item cl_N sqrt (const cl_N& z)
+@cindex @code{sqrt ()}
Returns the square root of @code{z}, as defined by the formula
@code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
or to a complex number are done if necessary. The range of the result is the
@section Transcendental functions
-
+@cindex transcendental functions
The transcendental functions return an exact result if the argument
is exact and the result is exact as well. Otherwise they must return
@table @code
@item cl_R exp (const cl_R& x)
+@cindex @code{exp ()}
@itemx cl_N exp (const cl_N& x)
Returns the exponential function of @code{x}. This is @code{e^x} where
@code{e} is the base of the natural logarithms. The range of the result
is the entire complex plane excluding 0.
@item cl_R ln (const cl_R& x)
+@cindex @code{ln ()}
@code{x} must be > 0. Returns the (natural) logarithm of x.
@item cl_N log (const cl_N& x)
+@cindex @code{log ()}
Returns the (natural) logarithm of x. If @code{x} is real and positive,
this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
The range of the result is the strip in the complex plane
@code{-pi < imagpart(log(x)) <= pi}.
@item cl_R phase (const cl_N& x)
+@cindex @code{phase ()}
Returns the angle part of @code{x} in its polar representation as a
complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
This is also the imaginary part of @code{log(x)}.
@code{log(a,b) = log(a)/log(b)}.
@item cl_N expt (const cl_N& x, const cl_N& y)
+@cindex @code{expt ()}
Exponentiation: Returns @code{x^y = exp(y*log(x))}.
@end table
The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
@table @code
-@item cl_F cl_exp1 (cl_float_format_t f)
+@item cl_F exp1 (float_format_t f)
+@cindex @code{exp1 ()}
Returns e as a float of format @code{f}.
-@item cl_F cl_exp1 (const cl_F& y)
+@item cl_F exp1 (const cl_F& y)
Returns e in the float format of @code{y}.
-@item cl_F cl_exp1 (void)
-Returns e as a float of format @code{cl_default_float_format}.
+@item cl_F exp1 (void)
+Returns e as a float of format @code{default_float_format}.
@end table
@table @code
@item cl_R sin (const cl_R& x)
+@cindex @code{sin ()}
Returns @code{sin(x)}. The range of the result is the interval
@code{-1 <= sin(x) <= 1}.
Returns @code{sin(z)}. The range of the result is the entire complex plane.
@item cl_R cos (const cl_R& x)
+@cindex @code{cos ()}
Returns @code{cos(x)}. The range of the result is the interval
@code{-1 <= cos(x) <= 1}.
@item cl_N cos (const cl_N& x)
Returns @code{cos(z)}. The range of the result is the entire complex plane.
-@item struct cl_cos_sin_t @{ cl_R cos; cl_R sin; @};
-@itemx cl_cos_sin_t cl_cos_sin (const cl_R& x)
+@item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
+@cindex @code{cos_sin_t}
+@itemx cos_sin_t cos_sin (const cl_R& x)
Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
+@cindex @code{cos_sin ()}
computing them separately. The relation @code{cos^2 + sin^2 = 1} will
hold only approximately.
@item cl_R tan (const cl_R& x)
+@cindex @code{tan ()}
@itemx cl_N tan (const cl_N& x)
Returns @code{tan(x) = sin(x)/cos(x)}.
@item cl_N cis (const cl_R& x)
+@cindex @code{cis ()}
@itemx cl_N cis (const cl_N& x)
Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
@code{e^(i*x) = cos(x) + i*sin(x)}.
+@cindex @code{asin}
+@cindex @code{asin ()}
@item cl_N asin (const cl_N& z)
Returns @code{arcsin(z)}. This is defined as
@code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
@end ignore
@item cl_N acos (const cl_N& z)
+@cindex @code{acos ()}
Returns @code{arccos(z)}. This is defined as
@code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
@ignore
Proof: This follows from the results about arcsin.
@end ignore
+@cindex @code{atan}
+@cindex @code{atan ()}
@item cl_R atan (const cl_R& x, const cl_R& y)
Returns the angle of the polar representation of the complex number
@code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
@end table
-The constant pi = 3.14@dots{} is returned by the following functions:
+@cindex pi
+@cindex Archimedes' constant
+Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
@table @code
-@item cl_F cl_pi (cl_float_format_t f)
+@item cl_F pi (float_format_t f)
+@cindex @code{pi ()}
Returns pi as a float of format @code{f}.
-@item cl_F cl_pi (const cl_F& y)
+@item cl_F pi (const cl_F& y)
Returns pi in the float format of @code{y}.
-@item cl_F cl_pi (void)
-Returns pi as a float of format @code{cl_default_float_format}.
+@item cl_F pi (void)
+Returns pi as a float of format @code{default_float_format}.
@end table
@table @code
@item cl_R sinh (const cl_R& x)
+@cindex @code{sinh ()}
Returns @code{sinh(x)}.
@item cl_N sinh (const cl_N& z)
Returns @code{sinh(z)}. The range of the result is the entire complex plane.
@item cl_R cosh (const cl_R& x)
+@cindex @code{cosh ()}
Returns @code{cosh(x)}. The range of the result is the interval
@code{cosh(x) >= 1}.
@item cl_N cosh (const cl_N& z)
Returns @code{cosh(z)}. The range of the result is the entire complex plane.
-@item struct cl_cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
-@itemx cl_cosh_sinh_t cl_cosh_sinh (const cl_R& x)
+@item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
+@cindex @code{cosh_sinh_t}
+@itemx cosh_sinh_t cosh_sinh (const cl_R& x)
+@cindex @code{cosh_sinh ()}
Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
hold only approximately.
@item cl_R tanh (const cl_R& x)
+@cindex @code{tanh ()}
@itemx cl_N tanh (const cl_N& x)
Returns @code{tanh(x) = sinh(x)/cosh(x)}.
@item cl_N asinh (const cl_N& z)
+@cindex @code{asinh ()}
Returns @code{arsinh(z)}. This is defined as
@code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
@code{arsinh(-z) = -arsinh(z)}.
@end ignore
@item cl_N acosh (const cl_N& z)
+@cindex @code{acosh ()}
Returns @code{arcosh(z)}. This is defined as
@code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
The range of the result is the half-strip in the complex domain
@end ignore
@item cl_N atanh (const cl_N& z)
+@cindex @code{atanh ()}
Returns @code{artanh(z)}. This is defined as
@code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
@code{artanh(-z) = -artanh(z)}. The range of the result is
@subsection Euler gamma
+@cindex Euler's constant
Euler's constant C = 0.577@dots{} is returned by the following functions:
@table @code
-@item cl_F cl_eulerconst (cl_float_format_t f)
+@item cl_F eulerconst (float_format_t f)
+@cindex @code{eulerconst ()}
Returns Euler's constant as a float of format @code{f}.
-@item cl_F cl_eulerconst (const cl_F& y)
+@item cl_F eulerconst (const cl_F& y)
Returns Euler's constant in the float format of @code{y}.
-@item cl_F cl_eulerconst (void)
-Returns Euler's constant as a float of format @code{cl_default_float_format}.
+@item cl_F eulerconst (void)
+Returns Euler's constant as a float of format @code{default_float_format}.
@end table
Catalan's constant G = 0.915@dots{} is returned by the following functions:
+@cindex Catalan's constant
@table @code
-@item cl_F cl_catalanconst (cl_float_format_t f)
+@item cl_F catalanconst (float_format_t f)
+@cindex @code{catalanconst ()}
Returns Catalan's constant as a float of format @code{f}.
-@item cl_F cl_catalanconst (const cl_F& y)
+@item cl_F catalanconst (const cl_F& y)
Returns Catalan's constant in the float format of @code{y}.
-@item cl_F cl_catalanconst (void)
-Returns Catalan's constant as a float of format @code{cl_default_float_format}.
+@item cl_F catalanconst (void)
+Returns Catalan's constant as a float of format @code{default_float_format}.
@end table
@subsection Riemann zeta
+@cindex Riemann's zeta
Riemann's zeta function at an integral point @code{s>1} is returned by the
following functions:
@table @code
-@item cl_F cl_zeta (int s, cl_float_format_t f)
+@item cl_F zeta (int s, float_format_t f)
+@cindex @code{zeta ()}
Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
-@item cl_F cl_zeta (int s, const cl_F& y)
+@item cl_F zeta (int s, const cl_F& y)
Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
-@item cl_F cl_zeta (int s)
+@item cl_F zeta (int s)
Returns Riemann's zeta function at @code{s} as a float of format
-@code{cl_default_float_format}.
+@code{default_float_format}.
@end table
@table @code
@item cl_I lognot (const cl_I& x)
+@cindex @code{lognot ()}
@itemx cl_I operator ~ (const cl_I& x)
+@cindex @code{operator ~ ()}
Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
@item cl_I logand (const cl_I& x, const cl_I& y)
+@cindex @code{logand ()}
@itemx cl_I operator & (const cl_I& x, const cl_I& y)
+@cindex @code{operator & ()}
Logical and, like @code{x & y} in C.
@item cl_I logior (const cl_I& x, const cl_I& y)
+@cindex @code{logior ()}
@itemx cl_I operator | (const cl_I& x, const cl_I& y)
+@cindex @code{operator | ()}
Logical (inclusive) or, like @code{x | y} in C.
@item cl_I logxor (const cl_I& x, const cl_I& y)
+@cindex @code{logxor ()}
@itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
+@cindex @code{operator ^ ()}
Exclusive or, like @code{x ^ y} in C.
@item cl_I logeqv (const cl_I& x, const cl_I& y)
+@cindex @code{logeqv ()}
Bitwise equivalence, like @code{~(x ^ y)} in C.
@item cl_I lognand (const cl_I& x, const cl_I& y)
+@cindex @code{lognand ()}
Bitwise not and, like @code{~(x & y)} in C.
@item cl_I lognor (const cl_I& x, const cl_I& y)
+@cindex @code{lognor ()}
Bitwise not or, like @code{~(x | y)} in C.
@item cl_I logandc1 (const cl_I& x, const cl_I& y)
+@cindex @code{logandc1 ()}
Logical and, complementing the first argument, like @code{~x & y} in C.
@item cl_I logandc2 (const cl_I& x, const cl_I& y)
+@cindex @code{logandc2 ()}
Logical and, complementing the second argument, like @code{x & ~y} in C.
@item cl_I logorc1 (const cl_I& x, const cl_I& y)
+@cindex @code{logorc1 ()}
Logical or, complementing the first argument, like @code{~x | y} in C.
@item cl_I logorc2 (const cl_I& x, const cl_I& y)
+@cindex @code{logorc2 ()}
Logical or, complementing the second argument, like @code{x | ~y} in C.
@end table
These operations are all available though the function
@table @code
@item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
+@cindex @code{boole ()}
@end table
where @code{op} must have one of the 16 values (each one stands for a function
which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
@code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
@code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
@code{boole_orc1}, @code{boole_orc2}.
+@cindex @code{boole_clr}
+@cindex @code{boole_set}
+@cindex @code{boole_1}
+@cindex @code{boole_2}
+@cindex @code{boole_c1}
+@cindex @code{boole_c2}
+@cindex @code{boole_and}
+@cindex @code{boole_xor}
+@cindex @code{boole_eqv}
+@cindex @code{boole_nand}
+@cindex @code{boole_nor}
+@cindex @code{boole_andc1}
+@cindex @code{boole_andc2}
+@cindex @code{boole_orc1}
+@cindex @code{boole_orc2}
+
Other functions that view integers as bit strings:
@table @code
@item cl_boolean logtest (const cl_I& x, const cl_I& y)
+@cindex @code{logtest ()}
Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
@code{logand(x,y) != 0}.
@item cl_boolean logbitp (const cl_I& n, const cl_I& x)
+@cindex @code{logbitp ()}
Returns true if the @code{n}th bit (from the right) of @code{x} is set.
Bit 0 is the least significant bit.
-@item uintL logcount (const cl_I& x)
+@item uintC logcount (const cl_I& x)
+@cindex @code{logcount ()}
Returns the number of one bits in @code{x}, if @code{x} >= 0, or
the number of zero bits in @code{x}, if @code{x} < 0.
@end table
The following functions operate on intervals of bits in integers.
The type
@example
-struct cl_byte @{ uintL size; uintL position; @};
+struct cl_byte @{ uintC size; uintC position; @};
@end example
+@cindex @code{cl_byte}
represents the bit interval containing the bits
@code{position}@dots{}@code{position+size-1} of an integer.
The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
@table @code
@item cl_I ldb (const cl_I& n, const cl_byte& b)
+@cindex @code{ldb ()}
extracts the bits of @code{n} described by the bit interval @code{b}
and returns them as a nonnegative integer with @code{b.size} bits.
@item cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
+@cindex @code{ldb_test ()}
Returns true if some bit described by the bit interval @code{b} is set in
@code{n}.
@item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
+@cindex @code{dpb ()}
Returns @code{n}, with the bits described by the bit interval @code{b}
replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
@code{newbyte} are relevant.
@table @code
@item cl_I mask_field (const cl_I& n, const cl_byte& b)
+@cindex @code{mask_field ()}
returns an integer with the bits described by the bit interval @code{b}
copied from the corresponding bits in @code{n}, the other bits zero.
@item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
+@cindex @code{deposit_field ()}
returns an integer where the bits described by the bit interval @code{b}
come from @code{newbyte} and the other bits come from @code{n}.
@end table
@table @code
@item cl_boolean oddp (const cl_I& x)
+@cindex @code{oddp ()}
Returns true if the least significant bit of @code{x} is 1. Equivalent to
@code{mod(x,2) != 0}.
@item cl_boolean evenp (const cl_I& x)
+@cindex @code{evenp ()}
Returns true if the least significant bit of @code{x} is 0. Equivalent to
@code{mod(x,2) == 0}.
@item cl_I operator << (const cl_I& x, const cl_I& n)
+@cindex @code{operator << ()}
Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0.
Equivalent to @code{x * expt(2,n)}.
@item cl_I operator >> (const cl_I& x, const cl_I& n)
+@cindex @code{operator >> ()}
Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
Bits shifted out to the right are thrown away.
Equivalent to @code{floor(x / expt(2,n))}.
@item cl_I ash (const cl_I& x, const cl_I& y)
+@cindex @code{ash ()}
Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
by @code{-y} bits to the right (if @code{y}<=0). In other words, this
returns @code{floor(x * expt(2,y))}.
-@item uintL integer_length (const cl_I& x)
+@item uintC integer_length (const cl_I& x)
+@cindex @code{integer_length ()}
Returns the number of bits (excluding the sign bit) needed to represent @code{x}
in two's complement notation. This is the smallest n >= 0 such that
-2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
2^(n-1) <= x < 2^n.
-@item uintL ord2 (const cl_I& x)
+@item uintC ord2 (const cl_I& x)
+@cindex @code{ord2 ()}
@code{x} must be non-zero. This function returns the number of 0 bits at the
right of @code{x} in two's complement notation. This is the largest n >= 0
such that 2^n divides @code{x}.
-@item uintL power2p (const cl_I& x)
+@item uintC power2p (const cl_I& x)
+@cindex @code{power2p ()}
@code{x} must be > 0. This function checks whether @code{x} is a power of 2.
If @code{x} = 2^(n-1), it returns n. Else it returns 0.
(See also the function @code{logp}.)
@subsection Number theoretic functions
@table @code
-@item uint32 gcd (uint32 a, uint32 b)
+@item uint32 gcd (unsigned long a, unsigned long b)
+@cindex @code{gcd ()}
@itemx cl_I gcd (const cl_I& a, const cl_I& b)
This function returns the greatest common divisor of @code{a} and @code{b},
normalized to be >= 0.
@item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
+@cindex @code{xgcd ()}
This function (``extended gcd'') returns the greatest common divisor @code{g} of
@code{a} and @code{b} and at the same time the representation of @code{g}
as an integral linear combination of @code{a} and @code{b}:
@code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
@item cl_I lcm (const cl_I& a, const cl_I& b)
+@cindex @code{lcm ()}
This function returns the least common multiple of @code{a} and @code{b},
normalized to be >= 0.
@item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
+@cindex @code{logp ()}
@itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
@code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
rational number, this function returns true and sets *l = log(a,b), else
it returns false.
+
+@item int jacobi (signed long a, signed long b)
+@cindex @code{jacobi()}
+@itemx int jacobi (const cl_I& a, const cl_I& b)
+Returns the Jacobi symbol
+@tex
+$\left({a\over b}\right)$,
+@end tex
+@ifnottex
+(a/b),
+@end ifnottex
+@code{a,b} must be integers, @code{b>0} and odd. The result is 0
+iff gcd(a,b)>1.
+
+@item cl_boolean isprobprime (const cl_I& n)
+@cindex prime
+@cindex @code{isprobprime()}
+Returns true if @code{n} is a small prime or passes the Miller-Rabin
+primality test. The probability of a false positive is 1:10^30.
+
+@item cl_I nextprobprime (const cl_R& x)
+@cindex @code{nextprobprime()}
+Returns the smallest probable prime >=@code{x}.
@end table
@table @code
@item cl_I factorial (uintL n)
+@cindex @code{factorial ()}
@code{n} must be a small integer >= 0. This function returns the factorial
@code{n}! = @code{1*2*@dots{}*n}.
@item cl_I doublefactorial (uintL n)
+@cindex @code{doublefactorial ()}
@code{n} must be a small integer >= 0. This function returns the
doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
@code{n}!! = @code{2*4*@dots{}*n}, respectively.
@item cl_I binomial (uintL n, uintL k)
+@cindex @code{binomial ()}
@code{n} and @code{k} must be small integers >= 0. This function returns the
binomial coefficient
@tex
defines the following operations.
@table @code
-@item @var{type} scale_float (const @var{type}& x, sintL delta)
+@item @var{type} scale_float (const @var{type}& x, sintC delta)
+@cindex @code{scale_float ()}
@itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
because it copies @code{x} and modifies the exponent.
@table @code
@item sintL float_exponent (const @var{type}& x)
+@cindex @code{float_exponent ()}
Returns the exponent @code{e} of @code{x}.
For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
integer with @code{2^(e-1) <= abs(x) < 2^e}.
@item sintL float_radix (const @var{type}& x)
+@cindex @code{float_radix ()}
Returns the base of the floating-point representation. This is always @code{2}.
@item @var{type} float_sign (const @var{type}& x)
+@cindex @code{float_sign ()}
Returns the sign @code{s} of @code{x} as a float. The value is 1 for
@code{x} >= 0, -1 for @code{x} < 0.
-@item uintL float_digits (const @var{type}& x)
+@item uintC float_digits (const @var{type}& x)
+@cindex @code{float_digits ()}
Returns the number of mantissa bits in the floating-point representation
of @code{x}, including the hidden bit. The value only depends on the type
of @code{x}, not on its value.
-@item uintL float_precision (const @var{type}& x)
+@item uintC float_precision (const @var{type}& x)
+@cindex @code{float_precision ()}
Returns the number of significant mantissa bits in the floating-point
representation of @code{x}. Since denormalized numbers are not supported,
this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
@end table
The complete internal representation of a float is encoded in the type
-@code{cl_decoded_float} (or @code{cl_decoded_sfloat}, @code{cl_decoded_ffloat},
-@code{cl_decoded_dfloat}, @code{cl_decoded_lfloat}, respectively), defined by
+@cindex @code{decoded_float}
+@cindex @code{decoded_sfloat}
+@cindex @code{decoded_ffloat}
+@cindex @code{decoded_dfloat}
+@cindex @code{decoded_lfloat}
+@code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
+@code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
@example
-struct cl_decoded_@var{type}float @{
+struct decoded_@var{type}float @{
@var{type} mantissa; cl_I exponent; @var{type} sign;
@};
@end example
and returned by the function
@table @code
-@item cl_decoded_@var{type}float decode_float (const @var{type}& x)
+@item decoded_@var{type}float decode_float (const @var{type}& x)
+@cindex @code{decode_float ()}
For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
@code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
@end table
A complete decoding in terms of integers is provided as type
+@cindex @code{cl_idecoded_float}
@example
struct cl_idecoded_float @{
cl_I mantissa; cl_I exponent; cl_I sign;
@table @code
@item cl_idecoded_float integer_decode_float (const @var{type}& x)
+@cindex @code{integer_decode_float ()}
For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
@code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)}
bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
@table @code
@item cl_F float_sign (const cl_F& x, const cl_F& y)
+@cindex @code{float_sign ()}
This returns a floating point number whose precision and absolute value
is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
zero, it is treated as positive. Same for @code{y}.
@section Conversion functions
+@cindex conversion
@subsection Conversion to floating-point numbers
-The type @code{cl_float_format_t} describes a floating-point format.
+The type @code{float_format_t} describes a floating-point format.
+@cindex @code{float_format_t}
@table @code
-@item cl_float_format_t cl_float_format (uintL n)
+@item float_format_t float_format (uintL n)
+@cindex @code{float_format ()}
Returns the smallest float format which guarantees at least @code{n}
decimal digits in the mantissa (after the decimal point).
-@item cl_float_format_t cl_float_format (const cl_F& x)
+@item float_format_t float_format (const cl_F& x)
Returns the floating point format of @code{x}.
-@item cl_float_format_t cl_default_float_format
+@item float_format_t default_float_format
+@cindex @code{default_float_format}
Global variable: the default float format used when converting rational numbers
to floats.
@end table
defines the following operations:
@table @code
-@item cl_F cl_float (const @var{type}&x, cl_float_format_t f)
+@item cl_F cl_float (const @var{type}&x, float_format_t f)
+@cindex @code{cl_float ()}
Returns @code{x} as a float of format @code{f}.
@item cl_F cl_float (const @var{type}&x, const cl_F& y)
Returns @code{x} in the float format of @code{y}.
@item cl_F cl_float (const @var{type}&x)
-Returns @code{x} as a float of format @code{cl_default_float_format} if
+Returns @code{x} as a float of format @code{default_float_format} if
it is an exact number, or @code{x} itself if it is already a float.
@end table
Every floating-point format has some characteristic numbers:
@table @code
-@item cl_F most_positive_float (cl_float_format_t f)
+@item cl_F most_positive_float (float_format_t f)
+@cindex @code{most_positive_float ()}
Returns the largest (most positive) floating point number in float format @code{f}.
-@item cl_F most_negative_float (cl_float_format_t f)
+@item cl_F most_negative_float (float_format_t f)
+@cindex @code{most_negative_float ()}
Returns the smallest (most negative) floating point number in float format @code{f}.
-@item cl_F least_positive_float (cl_float_format_t f)
+@item cl_F least_positive_float (float_format_t f)
+@cindex @code{least_positive_float ()}
Returns the least positive floating point number (i.e. > 0 but closest to 0)
in float format @code{f}.
-@item cl_F least_negative_float (cl_float_format_t f)
+@item cl_F least_negative_float (float_format_t f)
+@cindex @code{least_negative_float ()}
Returns the least negative floating point number (i.e. < 0 but closest to 0)
in float format @code{f}.
-@item cl_F float_epsilon (cl_float_format_t f)
+@item cl_F float_epsilon (float_format_t f)
+@cindex @code{float_epsilon ()}
Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
-@item cl_F float_negative_epsilon (cl_float_format_t f)
+@item cl_F float_negative_epsilon (float_format_t f)
+@cindex @code{float_negative_epsilon ()}
Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
@end table
@table @code
@item cl_RA rational (const @var{type}& x)
+@cindex @code{rational ()}
Returns the value of @code{x} as an exact number. If @code{x} is already
an exact number, this is @code{x}. If @code{x} is a floating-point number,
the value is a rational number whose denominator is a power of 2.
@table @code
@item cl_RA rationalize (const cl_R& x)
+@cindex @code{rationalize ()}
If @code{x} is a floating-point number, it actually represents an interval
of real numbers, and this function returns the rational number with
smallest denominator (and smallest numerator, in magnitude)
A random generator is a machine which produces (pseudo-)random numbers.
-The include file @code{<cl_random.h>} defines a class @code{cl_random_state}
+The include file @code{<cln/random.h>} defines a class @code{random_state}
which contains the state of a random generator. If you make a copy
of the random number generator, the original one and the copy will produce
the same sequence of random numbers.
a complicated but deterministic way.
The global variable
+@cindex @code{random_state}
+@cindex @code{default_random_state}
@example
-cl_random_state cl_default_random_state
+random_state default_random_state
@end example
contains a default random number generator. It is used when the functions
-below are called without @code{cl_random_state} argument.
+below are called without @code{random_state} argument.
@table @code
-@item uint32 random32 (cl_random_state& randomstate)
+@item uint32 random32 (random_state& randomstate)
@itemx uint32 random32 ()
+@cindex @code{random32 ()}
Returns a random unsigned 32-bit number. All bits are equally random.
-@item cl_I random_I (cl_random_state& randomstate, const cl_I& n)
+@item cl_I random_I (random_state& randomstate, const cl_I& n)
@itemx cl_I random_I (const cl_I& n)
+@cindex @code{random_I ()}
@code{n} must be an integer > 0. This function returns a random integer @code{x}
in the range @code{0 <= x < n}.
-@item cl_F random_F (cl_random_state& randomstate, const cl_F& n)
+@item cl_F random_F (random_state& randomstate, const cl_F& n)
@itemx cl_F random_F (const cl_F& n)
+@cindex @code{random_F ()}
@code{n} must be a float > 0. This function returns a random floating-point
number of the same format as @code{n} in the range @code{0 <= x < n}.
-@item cl_R random_R (cl_random_state& randomstate, const cl_R& n)
+@item cl_R random_R (random_state& randomstate, const cl_R& n)
@itemx cl_R random_R (const cl_R& n)
+@cindex @code{random_R ()}
Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
if @code{n} is a float.
@end table
@section Obfuscating operators
+@cindex modifying operators
The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
@code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
@example
#define WANT_OBFUSCATING_OPERATORS
@end example
+@cindex @code{WANT_OBFUSCATING_OPERATORS}
to the beginning of your source files, before the inclusion of any CLN
include files. This flag will enable the following operators:
@table @code
@item @var{type}& operator += (@var{type}&, const @var{type}&)
+@cindex @code{operator += ()}
@itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
+@cindex @code{operator -= ()}
@itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
+@cindex @code{operator *= ()}
@itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
+@cindex @code{operator /= ()}
@end table
For the class @code{cl_I}:
@itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
@itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
@itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
+@cindex @code{operator &= ()}
@itemx @var{type}& operator |= (@var{type}&, const @var{type}&)
+@cindex @code{operator |= ()}
@itemx @var{type}& operator ^= (@var{type}&, const @var{type}&)
+@cindex @code{operator ^= ()}
@itemx @var{type}& operator <<= (@var{type}&, const @var{type}&)
+@cindex @code{operator <<= ()}
@itemx @var{type}& operator >>= (@var{type}&, const @var{type}&)
+@cindex @code{operator >>= ()}
@end table
For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
@table @code
@item @var{type}& operator ++ (@var{type}& x)
+@cindex @code{operator ++ ()}
The prefix operator @code{++x}.
@item void operator ++ (@var{type}& x, int)
The postfix operator @code{x++}.
@item @var{type}& operator -- (@var{type}& x)
+@cindex @code{operator -- ()}
The prefix operator @code{--x}.
@item void operator -- (@var{type}& x, int)
@chapter Input/Output
+@cindex Input/Output
@section Internal and printed representation
+@cindex representation
All computations deal with the internal representations of the numbers.
"20.0" and "20.000".
Converting an internal to an external representation is called ``printing'',
+@cindex printing
converting an external to an internal representation is called ``reading''.
-In CLN, is it always true that conversion of an internal to an external
+@cindex reading
+In CLN, it is always true that conversion of an internal to an external
representation and then back to an internal representation will yield the
same internal representation. Symbolically: @code{read(print(x)) == x}.
This is called ``print-read consistency''.
or @samp{e}, which denotes a default float format. The precision specifying
suffix has the syntax _@var{prec} where @var{prec} denotes the number of
valid mantissa digits (in decimal, excluding leading zeroes), cf. also
-function @samp{cl_float_format}.
+function @samp{float_format}.
@item Complex numbers
External representation:
@section Input functions
-Including @code{<cl_io.h>} defines a type @code{cl_istream}, which is
-the type of the first argument to all input functions. Unless you build
-and use CLN with the macro CL_IO_STDIO being defined, @code{cl_istream}
-is the same as @code{istream&}.
-
-The variable
-@itemize @asis
-@item
-@code{cl_istream cl_stdin}
-@end itemize
-contains the standard input stream.
-
-These are the simple input functions:
+Including @code{<cln/io.h>} defines a number of simple input functions
+that read from @code{std::istream&}:
@table @code
-@item int freadchar (cl_istream stream)
+@item int freadchar (std::istream& stream)
Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
if the end of stream was encountered or an error occurred.
-@item int funreadchar (cl_istream stream, int c)
+@item int funreadchar (std::istream& stream, int c)
Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
last @code{freadchar} operation on @code{stream}.
@end table
Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
@code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
-defines, in @code{<cl_@var{type}_io.h>}, the following input function:
+defines, in @code{<cln/@var{type}_io.h>}, the following input function:
@table @code
-@item cl_istream operator>> (cl_istream stream, @var{type}& result)
+@item std::istream& operator>> (std::istream& stream, @var{type}& result)
Reads a number from @code{stream} and stores it in the @code{result}.
@end table
-The most flexible input functions, defined in @code{<cl_@var{type}_io.h>},
+The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
are the following:
@table @code
-@item cl_N read_complex (cl_istream stream, const cl_read_flags& flags)
-@itemx cl_R read_real (cl_istream stream, const cl_read_flags& flags)
-@itemx cl_F read_float (cl_istream stream, const cl_read_flags& flags)
-@itemx cl_RA read_rational (cl_istream stream, const cl_read_flags& flags)
-@itemx cl_I read_integer (cl_istream stream, const cl_read_flags& flags)
+@item cl_N read_complex (std::istream& stream, const cl_read_flags& flags)
+@itemx cl_R read_real (std::istream& stream, const cl_read_flags& flags)
+@itemx cl_F read_float (std::istream& stream, const cl_read_flags& flags)
+@itemx cl_RA read_rational (std::istream& stream, const cl_read_flags& flags)
+@itemx cl_I read_integer (std::istream& stream, const cl_read_flags& flags)
Reads a number from @code{stream}. The @code{flags} are parameters which
affect the input syntax. Whitespace before the number is silently skipped.
@item unsigned int rational_base
The base in which rational numbers are read.
-@item cl_float_format_t float_flags.default_float_format
+@item float_format_t float_flags.default_float_format
The float format used when reading floats with exponent marker @samp{e}.
-@item cl_float_format_t float_flags.default_lfloat_format
+@item float_format_t float_flags.default_lfloat_format
The float format used when reading floats with exponent marker @samp{l}.
@item cl_boolean float_flags.mantissa_dependent_float_format
@section Output functions
-Including @code{<cl_io.h>} defines a type @code{cl_ostream}, which is
-the type of the first argument to all output functions. Unless you build
-and use CLN with the macro CL_IO_STDIO being defined, @code{cl_ostream}
-is the same as @code{ostream&}.
-
-The variable
-@itemize @asis
-@item
-@code{cl_ostream cl_stdout}
-@end itemize
-contains the standard output stream.
-
-The variable
-@itemize @asis
-@item
-@code{cl_ostream cl_stderr}
-@end itemize
-contains the standard error output stream.
-
-These are the simple output functions:
+Including @code{<cln/io.h>} defines a number of simple output functions
+that write to @code{std::ostream&}:
@table @code
-@item void fprintchar (cl_ostream stream, char c)
+@item void fprintchar (std::ostream& stream, char c)
Prints the character @code{x} literally on the @code{stream}.
-@item void fprint (cl_ostream stream, const char * string)
+@item void fprint (std::ostream& stream, const char * string)
Prints the @code{string} literally on the @code{stream}.
-@item void fprintdecimal (cl_ostream stream, int x)
-@itemx void fprintdecimal (cl_ostream stream, const cl_I& x)
+@item void fprintdecimal (std::ostream& stream, int x)
+@itemx void fprintdecimal (std::ostream& stream, const cl_I& x)
Prints the integer @code{x} in decimal on the @code{stream}.
-@item void fprintbinary (cl_ostream stream, const cl_I& x)
+@item void fprintbinary (std::ostream& stream, const cl_I& x)
Prints the integer @code{x} in binary (base 2, without prefix)
on the @code{stream}.
-@item void fprintoctal (cl_ostream stream, const cl_I& x)
+@item void fprintoctal (std::ostream& stream, const cl_I& x)
Prints the integer @code{x} in octal (base 8, without prefix)
on the @code{stream}.
-@item void fprinthexadecimal (cl_ostream stream, const cl_I& x)
+@item void fprinthexadecimal (std::ostream& stream, const cl_I& x)
Prints the integer @code{x} in hexadecimal (base 16, without prefix)
on the @code{stream}.
@end table
Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
@code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
-defines, in @code{<cl_@var{type}_io.h>}, the following output functions:
+defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
@table @code
-@item void fprint (cl_ostream stream, const @var{type}& x)
-@itemx cl_ostream operator<< (cl_ostream stream, const @var{type}& x)
+@item void fprint (std::ostream& stream, const @var{type}& x)
+@itemx std::ostream& operator<< (std::ostream& stream, const @var{type}& x)
Prints the number @code{x} on the @code{stream}. The output may depend
-on the global printer settings in the variable @code{cl_default_print_flags}.
+on the global printer settings in the variable @code{default_print_flags}.
The @code{ostream} flags and settings (flags, width and locale) are
ignored.
@end table
-The most flexible output function, defined in @code{<cl_@var{type}_io.h>},
+The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
are the following:
@example
-void print_complex (cl_ostream stream, const cl_print_flags& flags,
+void print_complex (std::ostream& stream, const cl_print_flags& flags,
const cl_N& z);
-void print_real (cl_ostream stream, const cl_print_flags& flags,
+void print_real (std::ostream& stream, const cl_print_flags& flags,
const cl_R& z);
-void print_float (cl_ostream stream, const cl_print_flags& flags,
+void print_float (std::ostream& stream, const cl_print_flags& flags,
const cl_F& z);
-void print_rational (cl_ostream stream, const cl_print_flags& flags,
+void print_rational (std::ostream& stream, const cl_print_flags& flags,
const cl_RA& z);
-void print_integer (cl_ostream stream, const cl_print_flags& flags,
+void print_integer (std::ostream& stream, const cl_print_flags& flags,
const cl_I& z);
@end example
Prints the number @code{x} on the @code{stream}. The @code{flags} are
If this flag is true, type specific exponent markers have precedence over 'E'.
Default is false.
-@item cl_float_format_t default_float_format
+@item float_format_t default_float_format
Floating point numbers of this format will be printed using the 'E' exponent
-marker. Default is @code{cl_float_format_ffloat}.
+marker. Default is @code{float_format_ffloat}.
@item cl_boolean complex_readably
If this flag is true, complex numbers will be printed using the Common Lisp
using this variable name. Default is @code{"x"}.
@end table
-The global variable @code{cl_default_print_flags} contains the default values,
-used by the function @code{fprint},
+The global variable @code{default_print_flags} contains the default values,
+used by the function @code{fprint}.
@chapter Rings
@example
Ring
cl_ring
- <cl_ring.h>
+ <cln/ring.h>
@end example
Rings can be compared for equality:
Given a ring @code{R}, the following members can be used.
@table @code
-@item void R->fprint (cl_ostream stream, const cl_ring_element& x)
+@item void R->fprint (std::ostream& stream, const cl_ring_element& x)
+@cindex @code{fprint ()}
@itemx cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
+@cindex @code{equal ()}
@itemx cl_ring_element R->zero ()
+@cindex @code{zero ()}
@itemx cl_boolean R->zerop (const cl_ring_element& x)
+@cindex @code{zerop ()}
@itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
+@cindex @code{plus ()}
@itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
+@cindex @code{minus ()}
@itemx cl_ring_element R->uminus (const cl_ring_element& x)
+@cindex @code{uminus ()}
@itemx cl_ring_element R->one ()
+@cindex @code{one ()}
@itemx cl_ring_element R->canonhom (const cl_I& x)
+@cindex @code{canonhom ()}
@itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
+@cindex @code{mul ()}
@itemx cl_ring_element R->square (const cl_ring_element& x)
+@cindex @code{square ()}
@itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
+@cindex @code{expt_pos ()}
@end table
The following rings are built-in.
@table @code
@item cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
+@cindex @code{instanceof ()}
Tests whether the given number is an element of the number ring R.
@end table
@chapter Modular integers
+@cindex modular integer
@section Modular integer rings
+@cindex ring
CLN implements modular integers, i.e. integers modulo a fixed integer N.
The modulus is explicitly part of every modular integer. CLN doesn't
@example
Ring
cl_ring
- <cl_ring.h>
+ <cln/ring.h>
|
|
Modular integer ring
cl_modint_ring
- <cl_modinteger.h>
+ <cln/modinteger.h>
@end example
+@cindex @code{cl_modint_ring}
and the class of all modular integers (elements of modular integer rings) is
@example
Modular integer
cl_MI
- <cl_modinteger.h>
+ <cln/modinteger.h>
@end example
Modular integer rings are constructed using the function
@table @code
-@item cl_modint_ring cl_find_modint_ring (const cl_I& N)
+@item cl_modint_ring find_modint_ring (const cl_I& N)
+@cindex @code{find_modint_ring ()}
This function returns the modular ring @samp{Z/NZ}. It takes care
of finding out about special cases of @code{N}, like powers of two
and odd numbers for which Montgomery multiplication will be a win,
+@cindex Montgomery multiplication
and precomputes any necessary auxiliary data for computing modulo @code{N}.
There is a cache table of rings, indexed by @code{N} (or, more precisely,
by @code{abs(N)}). This ensures that the precomputation costs are reduced
@table @code
@item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
+@cindex @code{operator == ()}
@itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
+@cindex @code{operator != ()}
These compare two modular integer rings for equality. Two different calls
-to @code{cl_find_modint_ring} with the same argument necessarily return the
+to @code{find_modint_ring} with the same argument necessarily return the
same ring because it is memoized in the cache table.
@end table
@table @code
@item cl_I R->modulus
+@cindex @code{modulus}
This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
@item cl_MI R->zero()
+@cindex @code{zero ()}
This returns @code{0 mod N}.
@item cl_MI R->one()
+@cindex @code{one ()}
This returns @code{1 mod N}.
@item cl_MI R->canonhom (const cl_I& x)
+@cindex @code{canonhom ()}
This returns @code{x mod N}.
@item cl_I R->retract (const cl_MI& x)
+@cindex @code{retract ()}
This is a partial inverse function to @code{R->canonhom}. It returns the
standard representative (@code{>=0}, @code{<N}) of @code{x}.
-@item cl_MI R->random(cl_random_state& randomstate)
+@item cl_MI R->random(random_state& randomstate)
@itemx cl_MI R->random()
+@cindex @code{random ()}
This returns a random integer modulo @code{N}.
@end table
@table @code
@item cl_modint_ring x.ring ()
+@cindex @code{ring ()}
Returns the ring to which the modular integer @code{x} belongs.
@item cl_MI operator+ (const cl_MI&, const cl_MI&)
-Returns the sum of two modular integers. One of the arguments may also be
-a plain integer.
+@cindex @code{operator + ()}
+Returns the sum of two modular integers. One of the arguments may also
+be a plain integer.
@item cl_MI operator- (const cl_MI&, const cl_MI&)
-Returns the difference of two modular integers. One of the arguments may also be
-a plain integer.
+@cindex @code{operator - ()}
+Returns the difference of two modular integers. One of the arguments may also
+be a plain integer.
@item cl_MI operator- (const cl_MI&)
Returns the negative of a modular integer.
@item cl_MI operator* (const cl_MI&, const cl_MI&)
-Returns the product of two modular integers. One of the arguments may also be
-a plain integer.
+@cindex @code{operator * ()}
+Returns the product of two modular integers. One of the arguments may also
+be a plain integer.
@item cl_MI square (const cl_MI&)
+@cindex @code{square ()}
Returns the square of a modular integer.
@item cl_MI recip (const cl_MI& x)
+@cindex @code{recip ()}
Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
must be coprime to the modulus, otherwise an error message is issued.
@item cl_MI div (const cl_MI& x, const cl_MI& y)
+@cindex @code{div ()}
Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
@code{y} must be coprime to the modulus, otherwise an error message is issued.
@item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
+@cindex @code{expt_pos ()}
@code{y} must be > 0. Returns @code{x^y}.
@item cl_MI expt (const cl_MI& x, const cl_I& y)
+@cindex @code{expt ()}
Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
modulus, else an error message is issued.
@item cl_MI operator<< (const cl_MI& x, const cl_I& y)
+@cindex @code{operator << ()}
Returns @code{x*2^y}.
@item cl_MI operator>> (const cl_MI& x, const cl_I& y)
+@cindex @code{operator >> ()}
Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
or an error message is issued.
@item bool operator== (const cl_MI&, const cl_MI&)
+@cindex @code{operator == ()}
@itemx bool operator!= (const cl_MI&, const cl_MI&)
+@cindex @code{operator != ()}
Compares two modular integers, belonging to the same modular integer ring,
for equality.
@item cl_boolean zerop (const cl_MI& x)
+@cindex @code{zerop ()}
Returns true if @code{x} is @code{0 mod N}.
@end table
input/output).
@table @code
-@item void fprint (cl_ostream stream, const cl_MI& x)
-@itemx cl_ostream operator<< (cl_ostream stream, const cl_MI& x)
+@item void fprint (std::ostream& stream, const cl_MI& x)
+@cindex @code{fprint ()}
+@itemx std::ostream& operator<< (std::ostream& stream, const cl_MI& x)
+@cindex @code{operator << ()}
Prints the modular integer @code{x} on the @code{stream}. The output may depend
-on the global printer settings in the variable @code{cl_default_print_flags}.
+on the global printer settings in the variable @code{default_print_flags}.
@end table
@chapter Symbolic data types
+@cindex symbolic type
CLN implements two symbolic (non-numeric) data types: strings and symbols.
@section Strings
+@cindex string
+@cindex @code{cl_string}
The class
@example
String
cl_string
- <cl_string.h>
+ <cln/string.h>
@end example
implements immutable strings.
Assignment from @code{cl_string} and @code{const char *}.
@item s.length()
+@cindex @code{length ()}
@itemx strlen(s)
+@cindex @code{strlen ()}
Returns the length of the string @code{s}.
@item s[i]
+@cindex @code{operator [] ()}
Returns the @code{i}th character of the string @code{s}.
@code{i} must be in the range @code{0 <= i < s.length()}.
@item bool equal (const cl_string& s1, const cl_string& s2)
+@cindex @code{equal ()}
Compares two strings for equality. One of the arguments may also be a
plain @code{const char *}.
@end table
@section Symbols
+@cindex symbol
+@cindex @code{cl_symbol}
Symbols are uniquified strings: all symbols with the same name are shared.
This means that comparison of two symbols is fast (effectively just a pointer
@code{sym}.
@item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
+@cindex @code{equal ()}
Compares two symbols for equality. This is very fast.
@end table
@chapter Univariate polynomials
+@cindex polynomial
+@cindex univariate polynomial
@section Univariate polynomial rings
CLN implements univariate polynomials (polynomials in one variable) over an
arbitrary ring. The indeterminate variable may be either unnamed (and will be
-printed according to @code{cl_default_print_flags.univpoly_varname}, which
+printed according to @code{default_print_flags.univpoly_varname}, which
defaults to @samp{x}) or carry a given name. The base ring and the
indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
(accidentally) mix elements of different polynomial rings, e.g.
@example
Ring
cl_ring
- <cl_ring.h>
+ <cln/ring.h>
|
|
Univariate polynomial ring
cl_univpoly_ring
- <cl_univpoly.h>
+ <cln/univpoly.h>
|
+----------------+-------------------+
| | |
Complex polynomial ring | Modular integer polynomial ring
cl_univpoly_complex_ring | cl_univpoly_modint_ring
- <cl_univpoly_complex.h> | <cl_univpoly_modint.h>
+ <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
|
+----------------+
| |
Real polynomial ring |
cl_univpoly_real_ring |
- <cl_univpoly_real.h> |
+ <cln/univpoly_real.h> |
|
+----------------+
| |
Rational polynomial ring |
cl_univpoly_rational_ring |
- <cl_univpoly_rational.h> |
+ <cln/univpoly_rational.h> |
|
+----------------+
|
Integer polynomial ring
cl_univpoly_integer_ring
- <cl_univpoly_integer.h>
+ <cln/univpoly_integer.h>
@end example
and the corresponding classes of univariate polynomials are
@example
Univariate polynomial
cl_UP
- <cl_univpoly.h>
+ <cln/univpoly.h>
|
+----------------+-------------------+
| | |
Complex polynomial | Modular integer polynomial
cl_UP_N | cl_UP_MI
- <cl_univpoly_complex.h> | <cl_univpoly_modint.h>
+ <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
|
+----------------+
| |
Real polynomial |
cl_UP_R |
- <cl_univpoly_real.h> |
+ <cln/univpoly_real.h> |
|
+----------------+
| |
Rational polynomial |
cl_UP_RA |
- <cl_univpoly_rational.h> |
+ <cln/univpoly_rational.h> |
|
+----------------+
|
Integer polynomial
cl_UP_I
- <cl_univpoly_integer.h>
+ <cln/univpoly_integer.h>
@end example
Univariate polynomial rings are constructed using the functions
@table @code
-@item cl_univpoly_ring cl_find_univpoly_ring (const cl_ring& R)
-@itemx cl_univpoly_ring cl_find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
+@item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
+@itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
This function returns the polynomial ring @samp{R[X]}, unnamed or named.
@code{R} may be an arbitrary ring. This function takes care of finding out
about special cases of @code{R}, such as the rings of complex numbers,
This ensures that two calls of this function with the same arguments will
return the same polynomial ring.
-@item cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R)
-@itemx cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
-@itemx cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R)
-@itemx cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
-@itemx cl_univpoly_rational_ring cl_find_univpoly_ring (const cl_rational_ring& R)
-@itemx cl_univpoly_rational_ring cl_find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
-@itemx cl_univpoly_integer_ring cl_find_univpoly_ring (const cl_integer_ring& R)
-@itemx cl_univpoly_integer_ring cl_find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
-@itemx cl_univpoly_modint_ring cl_find_univpoly_ring (const cl_modint_ring& R)
-@itemx cl_univpoly_modint_ring cl_find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
-These functions are equivalent to the general @code{cl_find_univpoly_ring},
+@itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
+@cindex @code{find_univpoly_ring ()}
+@itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
+@itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
+@itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
+@itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
+@itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
+@itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
+@itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
+@itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
+@itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
+These functions are equivalent to the general @code{find_univpoly_ring},
only the return type is more specific, according to the base ring's type.
@end table
@table @code
@item cl_ring R->basering()
-This returns the base ring, as passed to @samp{cl_find_univpoly_ring}.
+@cindex @code{basering ()}
+This returns the base ring, as passed to @samp{find_univpoly_ring}.
@item cl_UP R->zero()
+@cindex @code{zero ()}
This returns @code{0 in R}, a polynomial of degree -1.
@item cl_UP R->one()
-This returns @code{1 in R}, a polynomial of degree <= 0.
+@cindex @code{one ()}
+This returns @code{1 in R}, a polynomial of degree == 0.
@item cl_UP R->canonhom (const cl_I& x)
+@cindex @code{canonhom ()}
This returns @code{x in R}, a polynomial of degree <= 0.
@item cl_UP R->monomial (const cl_ring_element& x, uintL e)
+@cindex @code{monomial ()}
This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
indeterminate.
@item cl_UP R->create (sintL degree)
+@cindex @code{create ()}
Creates a new polynomial with a given degree. The zero polynomial has degree
@code{-1}. After creating the polynomial, you should put in the coefficients,
using the @code{set_coeff} member function, and then call the @code{finalize}
@table @code
@item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
+@cindex @code{set_coeff ()}
This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
After changing a polynomial and before applying any "normal" operation on it,
you should call its @code{finalize} member function.
@item void finalize (cl_UP& x)
+@cindex @code{finalize ()}
This function marks the endpoint of destructive modifications of a polynomial.
It normalizes the internal representation so that subsequent computations have
less overhead. Doing normal computations on unnormalized polynomials may
@table @code
@item cl_univpoly_ring x.ring ()
+@cindex @code{ring ()}
Returns the ring to which the univariate polynomial @code{x} belongs.
@item cl_UP operator+ (const cl_UP&, const cl_UP&)
+@cindex @code{operator + ()}
Returns the sum of two univariate polynomials.
@item cl_UP operator- (const cl_UP&, const cl_UP&)
+@cindex @code{operator - ()}
Returns the difference of two univariate polynomials.
@item cl_UP operator- (const cl_UP&)
Returns the negative of a univariate polynomial.
@item cl_UP operator* (const cl_UP&, const cl_UP&)
+@cindex @code{operator * ()}
Returns the product of two univariate polynomials. One of the arguments may
also be a plain integer or an element of the base ring.
@item cl_UP square (const cl_UP&)
+@cindex @code{square ()}
Returns the square of a univariate polynomial.
@item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
+@cindex @code{expt_pos ()}
@code{y} must be > 0. Returns @code{x^y}.
@item bool operator== (const cl_UP&, const cl_UP&)
+@cindex @code{operator == ()}
@itemx bool operator!= (const cl_UP&, const cl_UP&)
+@cindex @code{operator != ()}
Compares two univariate polynomials, belonging to the same univariate
polynomial ring, for equality.
@item cl_boolean zerop (const cl_UP& x)
+@cindex @code{zerop ()}
Returns true if @code{x} is @code{0 in R}.
@item sintL degree (const cl_UP& x)
+@cindex @code{degree ()}
Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
+@item sintL ldegree (const cl_UP& x)
+@cindex @code{degree ()}
+Returns the low degree of the polynomial. This is the degree of the first
+non-vanishing polynomial coefficient. The zero polynomial has ldegree @code{-1}.
+
@item cl_ring_element coeff (const cl_UP& x, uintL index)
+@cindex @code{coeff ()}
Returns the coefficient of @code{X^index} in the polynomial @code{x}.
@item cl_ring_element x (const cl_ring_element& y)
+@cindex @code{operator () ()}
Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
then @samp{x(y)} returns the value of the substitution of @code{y} into
@code{x}.
@item cl_UP deriv (const cl_UP& x)
+@cindex @code{deriv ()}
Returns the derivative of the polynomial @code{x} with respect to the
indeterminate @code{X}.
@end table
input/output).
@table @code
-@item void fprint (cl_ostream stream, const cl_UP& x)
-@itemx cl_ostream operator<< (cl_ostream stream, const cl_UP& x)
+@item void fprint (std::ostream& stream, const cl_UP& x)
+@cindex @code{fprint ()}
+@itemx std::ostream& operator<< (std::ostream& stream, const cl_UP& x)
+@cindex @code{operator << ()}
Prints the univariate polynomial @code{x} on the @code{stream}. The output may
depend on the global printer settings in the variable
-@code{cl_default_print_flags}.
+@code{default_print_flags}.
@end table
@section Special polynomials
The following functions return special polynomials.
@table @code
-@item cl_UP_I cl_tschebychev (sintL n)
-Returns the n-th Tchebychev polynomial (n >= 0).
+@item cl_UP_I tschebychev (sintL n)
+@cindex @code{tschebychev ()}
+@cindex Chebyshev polynomial
+Returns the n-th Chebyshev polynomial (n >= 0).
-@item cl_UP_I cl_hermite (sintL n)
+@item cl_UP_I hermite (sintL n)
+@cindex @code{hermite ()}
+@cindex Hermite polynomial
Returns the n-th Hermite polynomial (n >= 0).
-@item cl_UP_RA cl_legendre (sintL n)
+@item cl_UP_RA legendre (sintL n)
+@cindex @code{legendre ()}
+@cindex Legende polynomial
Returns the n-th Legendre polynomial (n >= 0).
-@item cl_UP_I cl_laguerre (sintL n)
+@item cl_UP_I laguerre (sintL n)
+@cindex @code{laguerre ()}
+@cindex Laguerre polynomial
Returns the n-th Laguerre polynomial (n >= 0).
@end table
@chapter Internals
@section Why C++ ?
+@cindex advocacy
Using C++ as an implementation language provides
Efficiency: It compiles to machine code.
@item
+@cindex portability
Portability: It runs on all platforms supporting a C++ compiler. Because
of the availability of GNU C++, this includes all currently used 32-bit and
64-bit platforms, independently of the quality of the vendor's C++ compiler.
Type safety: The C++ compilers knows about the number types and complains if,
for example, you try to assign a float to an integer variable. However,
a drawback is that C++ doesn't know about generic types, hence a restriction
-like that @code{operation+ (const cl_MI&, const cl_MI&)} requires that both
+like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
arguments belong to the same modular ring cannot be expressed as a compile-time
information.
Object sharing: An operation like @code{x+0} returns @code{x} without copying
it.
@item
+@cindex garbage collection
+@cindex reference counting
Garbage collection: A reference counting mechanism makes sure that any
number object's storage is freed immediately when the last reference to the
object is gone.
@item
+@cindex immediate numbers
Small integers are represented as immediate values instead of pointers
to heap allocated storage. This means that integers @code{> -2^29},
@code{< 2^29} don't consume heap memory, unless they were explicitly allocated
The kernel of CLN has been written in assembly language for some CPUs
(@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
@item
-On all CPUs, CLN uses the superefficient low-level routines from GNU
-GMP version 2.
+On all CPUs, CLN may be configured to use the superefficient low-level
+routines from GNU GMP version 3.
@item
For large numbers, CLN uses, instead of the standard @code{O(N^2)}
algorithm, the Karatsuba multiplication, which is an
For very large numbers (more than 12000 decimal digits), CLN uses
@iftex
Sch{@"o}nhage-Strassen
+@cindex Sch{@"o}nhage-Strassen multiplication
@end iftex
@ifinfo
-Schönhage-Strassen
+Schnhage-Strassen
+@cindex Schnhage-Strassen multiplication
@end ifinfo
-multiplication, which is an asymptotically
-optimal multiplication algorithm.
+multiplication, which is an asymptotically optimal multiplication
+algorithm.
@item
These fast multiplication algorithms also give improvements in the speed
of division and radix conversion.
@section Garbage collection
+@cindex garbage collection
All the number classes are reference count classes: They only contain a pointer
to an object in the heap. Upon construction, assignment and destruction of
@section Compiler options
+@cindex compiler options
Until you have installed CLN in a public place, the following options are
needed:
public directory as well (normally @code{/usr/local/lib}), hence when
linking a CLN application it is sufficient to give the flag @code{-lcln}.
+Since CLN version 1.1, there are two tools to make the creation of
+software packages that use CLN easier:
+@itemize @bullet
+@item
+@cindex @code{cln-config}
+@code{cln-config} is a shell script that you can use to determine the
+compiler and linker command line options required to compile and link a
+program with CLN. Start it with @code{--help} to learn about its options
+or consult the manpage that comes with it.
+@item
+@cindex @code{AC_PATH_CLN}
+@code{AC_PATH_CLN} is for packages configured using GNU automake.
+The synopsis is:
+@example
+@code{AC_PATH_CLN([@var{MIN-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])}
+@end example
+This macro determines the location of CLN using @code{cln-config}, which
+is either found in the user's path, or from the environment variable
+@code{CLN_CONFIG}. It tests the installed libraries to make sure that
+their version is not earlier than @var{MIN-VERSION} (a default version
+will be used if not specified). If the required version was found, sets
+the @env{CLN_CPPFLAGS} and the @env{CLN_LIBS} variables. This
+macro is in the file @file{cln.m4} which is installed in
+@file{$datadir/aclocal}. Note that if automake was installed with a
+different @samp{--prefix} than CLN, you will either have to manually
+move @file{cln.m4} to automake's @file{$datadir/aclocal}, or give
+aclocal the @samp{-I} option when running it. Here is a possible example
+to be included in your package's @file{configure.ac}:
+@example
+AC_PATH_CLN(1.1.0, [
+ LIBS="$LIBS $CLN_LIBS"
+ CPPFLAGS="$CPPFLAGS $CLN_CPPFLAGS"
+], AC_MSG_ERROR([No suitable installed version of CLN could be found.]))
+@end example
+@end itemize
+
+
+@section Compatibility to old CLN versions
+@cindex namespace
+@cindex compatibility
+
+As of CLN version 1.1 all non-macro identifiers were hidden in namespace
+@code{cln} in order to avoid potential name clashes with other C++
+libraries. If you have an old application, you will have to manually
+port it to the new scheme. The following principles will help during
+the transition:
+@itemize @bullet
+@item
+All headers are now in a separate subdirectory. Instead of including
+@code{cl_}@var{something}@code{.h}, include
+@code{cln/}@var{something}@code{.h} now.
+@item
+All public identifiers (typenames and functions) have lost their
+@code{cl_} prefix. Exceptions are all the typenames of number types,
+(cl_N, cl_I, cl_MI, @dots{}), rings, symbolic types (cl_string,
+cl_symbol) and polynomials (cl_UP_@var{type}). (This is because their
+names would not be mnemonic enough once the namespace @code{cln} is
+imported. Even in a namespace we favor @code{cl_N} over @code{N}.)
+@item
+All public @emph{functions} that had by a @code{cl_} in their name still
+carry that @code{cl_} if it is intrinsic part of a typename (as in
+@code{cl_I_to_int ()}).
+@end itemize
+When developing other libraries, please keep in mind not to import the
+namespace @code{cln} in one of your public header files by saying
+@code{using namespace cln;}. This would propagate to other applications
+and can cause name clashes there.
+
@section Include files
+@cindex include files
+@cindex header files
Here is a summary of the include files and their contents.
@table @code
-@item <cl_object.h>
+@item <cln/object.h>
General definitions, reference counting, garbage collection.
-@item <cl_number.h>
+@item <cln/number.h>
The class cl_number.
-@item <cl_complex.h>
+@item <cln/complex.h>
Functions for class cl_N, the complex numbers.
-@item <cl_real.h>
+@item <cln/real.h>
Functions for class cl_R, the real numbers.
-@item <cl_float.h>
+@item <cln/float.h>
Functions for class cl_F, the floats.
-@item <cl_sfloat.h>
+@item <cln/sfloat.h>
Functions for class cl_SF, the short-floats.
-@item <cl_ffloat.h>
+@item <cln/ffloat.h>
Functions for class cl_FF, the single-floats.
-@item <cl_dfloat.h>
+@item <cln/dfloat.h>
Functions for class cl_DF, the double-floats.
-@item <cl_lfloat.h>
+@item <cln/lfloat.h>
Functions for class cl_LF, the long-floats.
-@item <cl_rational.h>
+@item <cln/rational.h>
Functions for class cl_RA, the rational numbers.
-@item <cl_integer.h>
+@item <cln/integer.h>
Functions for class cl_I, the integers.
-@item <cl_io.h>
+@item <cln/io.h>
Input/Output.
-@item <cl_complex_io.h>
+@item <cln/complex_io.h>
Input/Output for class cl_N, the complex numbers.
-@item <cl_real_io.h>
+@item <cln/real_io.h>
Input/Output for class cl_R, the real numbers.
-@item <cl_float_io.h>
+@item <cln/float_io.h>
Input/Output for class cl_F, the floats.
-@item <cl_sfloat_io.h>
+@item <cln/sfloat_io.h>
Input/Output for class cl_SF, the short-floats.
-@item <cl_ffloat_io.h>
+@item <cln/ffloat_io.h>
Input/Output for class cl_FF, the single-floats.
-@item <cl_dfloat_io.h>
+@item <cln/dfloat_io.h>
Input/Output for class cl_DF, the double-floats.
-@item <cl_lfloat_io.h>
+@item <cln/lfloat_io.h>
Input/Output for class cl_LF, the long-floats.
-@item <cl_rational_io.h>
+@item <cln/rational_io.h>
Input/Output for class cl_RA, the rational numbers.
-@item <cl_integer_io.h>
+@item <cln/integer_io.h>
Input/Output for class cl_I, the integers.
-@item <cl_input.h>
+@item <cln/input.h>
Flags for customizing input operations.
-@item <cl_output.h>
+@item <cln/output.h>
Flags for customizing output operations.
-@item <cl_malloc.h>
-@code{cl_malloc_hook}, @code{cl_free_hook}.
-@item <cl_abort.h>
+@item <cln/malloc.h>
+@code{malloc_hook}, @code{free_hook}.
+@item <cln/abort.h>
@code{cl_abort}.
-@item <cl_condition.h>
+@item <cln/condition.h>
Conditions/exceptions.
-@item <cl_string.h>
+@item <cln/string.h>
Strings.
-@item <cl_symbol.h>
+@item <cln/symbol.h>
Symbols.
-@item <cl_proplist.h>
+@item <cln/proplist.h>
Property lists.
-@item <cl_ring.h>
+@item <cln/ring.h>
General rings.
-@item <cl_null_ring.h>
+@item <cln/null_ring.h>
The null ring.
-@item <cl_complex_ring.h>
+@item <cln/complex_ring.h>
The ring of complex numbers.
-@item <cl_real_ring.h>
+@item <cln/real_ring.h>
The ring of real numbers.
-@item <cl_rational_ring.h>
+@item <cln/rational_ring.h>
The ring of rational numbers.
-@item <cl_integer_ring.h>
+@item <cln/integer_ring.h>
The ring of integers.
-@item <cl_numtheory.h>
+@item <cln/numtheory.h>
Number threory functions.
-@item <cl_modinteger.h>
+@item <cln/modinteger.h>
Modular integers.
-@item <cl_V.h>
+@item <cln/V.h>
Vectors.
-@item <cl_GV.h>
+@item <cln/GV.h>
General vectors.
-@item <cl_GV_number.h>
+@item <cln/GV_number.h>
General vectors over cl_number.
-@item <cl_GV_complex.h>
+@item <cln/GV_complex.h>
General vectors over cl_N.
-@item <cl_GV_real.h>
+@item <cln/GV_real.h>
General vectors over cl_R.
-@item <cl_GV_rational.h>
+@item <cln/GV_rational.h>
General vectors over cl_RA.
-@item <cl_GV_integer.h>
+@item <cln/GV_integer.h>
General vectors over cl_I.
-@item <cl_GV_modinteger.h>
+@item <cln/GV_modinteger.h>
General vectors of modular integers.
-@item <cl_SV.h>
+@item <cln/SV.h>
Simple vectors.
-@item <cl_SV_number.h>
+@item <cln/SV_number.h>
Simple vectors over cl_number.
-@item <cl_SV_complex.h>
+@item <cln/SV_complex.h>
Simple vectors over cl_N.
-@item <cl_SV_real.h>
+@item <cln/SV_real.h>
Simple vectors over cl_R.
-@item <cl_SV_rational.h>
+@item <cln/SV_rational.h>
Simple vectors over cl_RA.
-@item <cl_SV_integer.h>
+@item <cln/SV_integer.h>
Simple vectors over cl_I.
-@item <cl_SV_ringelt.h>
+@item <cln/SV_ringelt.h>
Simple vectors of general ring elements.
-@item <cl_univpoly.h>
+@item <cln/univpoly.h>
Univariate polynomials.
-@item <cl_univpoly_integer.h>
+@item <cln/univpoly_integer.h>
Univariate polynomials over the integers.
-@item <cl_univpoly_rational.h>
+@item <cln/univpoly_rational.h>
Univariate polynomials over the rational numbers.
-@item <cl_univpoly_real.h>
+@item <cln/univpoly_real.h>
Univariate polynomials over the real numbers.
-@item <cl_univpoly_complex.h>
+@item <cln/univpoly_complex.h>
Univariate polynomials over the complex numbers.
-@item <cl_univpoly_modint.h>
+@item <cln/univpoly_modint.h>
Univariate polynomials over modular integer rings.
-@item <cl_timing.h>
+@item <cln/timing.h>
Timing facilities.
-@item <cln.h>
+@item <cln/cln.h>
Includes all of the above.
@end table
@section An Example
A function which computes the nth Fibonacci number can be written as follows.
+@cindex Fibonacci number
@example
-#include <cl_integer.h>
-#include <cl_real.h>
+#include <cln/integer.h>
+#include <cln/real.h>
+using namespace cln;
// Returns F_n, computed as the nearest integer to
// ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
const cl_I fibonacci (int n)
@{
// Need a precision of ((1+sqrt(5))/2)^-n.
- cl_float_format_t prec = cl_float_format((int)(0.208987641*n+5));
+ float_format_t prec = float_format((int)(0.208987641*n+5));
cl_R sqrt5 = sqrt(cl_float(5,prec));
cl_R phi = (1+sqrt5)/2;
return round1( expt(phi,n)/sqrt5 );
Let's explain what is going on in detail.
-The include file @code{<cl_integer.h>} is necessary because the type
-@code{cl_I} is used in the function, and the include file @code{<cl_real.h>}
+The include file @code{<cln/integer.h>} is necessary because the type
+@code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
is needed for the type @code{cl_R} and the floating point number functions.
-The order of the include files does not matter.
+The order of the include files does not matter. In order not to write
+out @code{cln::}@var{foo} in this simple example we can safely import
+the whole namespace @code{cln}.
Then comes the function declaration. The argument is an @code{int}, the
result an integer. The return type is defined as @samp{const cl_I}, not
automatically reclaimed (garbage collected). Only the result survives and
gets passed to the caller.
+The file @code{fibonacci.cc} in the subdirectory @code{examples}
+contains this implementation together with an even faster algorithm.
@section Debugging support
+@cindex debugging
When debugging a CLN application with GNU @code{gdb}, two facilities are
available from the library:
for printing number objects. In order to get this function, you have
to define the macro @samp{CL_DEBUG} and then include all the header files
for which you want @code{cl_print} debugging support. For example:
+@cindex @code{CL_DEBUG}
@example
#define CL_DEBUG
-#include <cl_string.h>
+#include <cln/string.h>
@end example
Now, if you have in your program a variable @code{cl_string s}, and
inspect it under @code{gdb}, the output may look like this:
@code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
is needed for this member function to be implemented. Under @code{gdb},
you call it like this:
+@cindex @code{debug_print ()}
@example
(gdb) print s
$7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
@chapter Customizing
+@cindex customizing
@section Error handling
To catch such a fatal error, you need to define the function @code{cl_abort}
yourself, with the prototype
@example
-#include <cl_abort.h>
+#include <cln/abort.h>
void cl_abort (void);
@end example
+@cindex @code{cl_abort ()}
This function must not return control to its caller.
@section Floating-point underflow
+@cindex underflow
Floating point underflow denotes the situation when a floating-point number
is to be created which is so close to @code{0} that its exponent is too
cl_boolean cl_inhibit_floating_point_underflow
@end example
to @code{cl_true}, the error will be inhibited, and a floating-point zero
-will be generated instead.
-The default value of @code{cl_inhibit_floating_point_underflow} is
-@code{cl_false}.
+will be generated instead. The default value of
+@code{cl_inhibit_floating_point_underflow} is @code{cl_false}.
@section Customizing I/O
The output of the function @code{fprint} may be customized by changing the
-value of the global variable @code{cl_default_print_flags}.
+value of the global variable @code{default_print_flags}.
+@cindex @code{default_print_flags}
@section Customizing the memory allocator
Every memory allocation of CLN is done through the function pointer
-@code{cl_malloc_hook}. Freeing of this memory is done through the function
-pointer @code{cl_free_hook}. The default versions of these functions,
+@code{malloc_hook}. Freeing of this memory is done through the function
+pointer @code{free_hook}. The default versions of these functions,
provided in the library, call @code{malloc} and @code{free} and check
the @code{malloc} result against @code{NULL}.
If you want to provide another memory allocator, you need to define
-the variables @code{cl_malloc_hook} and @code{cl_free_hook} yourself,
+the variables @code{malloc_hook} and @code{free_hook} yourself,
like this:
@example
-#include <cl_malloc.h>
-void* (*cl_malloc_hook) (size_t size) = @dots{};
-void (*cl_free_hook) (void* ptr) = @dots{};
+#include <cln/malloc.h>
+namespace cln @{
+ void* (*malloc_hook) (size_t size) = @dots{};
+ void (*free_hook) (void* ptr) = @dots{};
+@}
@end example
+@cindex @code{malloc_hook ()}
+@cindex @code{free_hook ()}
The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
It is not possible to change the memory allocator at runtime, because
@printindex my
-@c Table of contents
-@contents
-
-
@bye
git://www.ginac.de / cln.git / blobdiff