calc/LIBRARY

	USING THE ARBITRARY PRECISION ROUTINES IN A C PROGRAM

Part of the calc release consists of an arbitrary precision math link library.
This link library is used by the calc program to perform its own calculations.
If you wish, you can ignore the calc program entirely and call the arbitrary
precision math routines from your own C programs.

The link library is called libcalc.a, and provides routines to handle arbitrary
precision arithmetic with integers, rational numbers, or complex numbers.
There are also many numeric functions such as factorial and gcd, along
with some transcendental functions such as sin and exp.

Take a look at the sample sub-directory.  It contains a few simple
examples of how to use libcalc.a that might be helpful to look at
after you have read this file.

------------------
FIRST THINGS FIRST
------------------

...............................................................................
.									      .
. You MUST call libcalc_call_me_first() prior to using libcalc lib functions! .
.									      .
...............................................................................

The function libcalc_call_me_first() takes no args and returns void.  You
need call libcalc_call_me_first() only once.

-------------
INCLUDE FILES
-------------

To use any of these routines in your own programs, you need to include the
appropriate include file.  These include files are:

	zmath.h		(for integer arithmetic)
	qmath.h		(for rational arithmetic)
	cmath.h		(for complex number arithmetic)

You never need to include more than one of the above files, even if you wish
to use more than one type of arithmetic, since qmath.h automatically includes
zmath.h, and cmath.h automatically includes qmath.h.

The prototypes for the available routines are listed in the above include
files.	Some of these routines are meant for internal use, and so aren't
convenient for outside use.  So you should read the source for a routine
to see if it really does what you think it does.  I won't guarantee that
obscure internal routines won't change or disappear in future releases!

When calc is installed, all of the include files needed to build
libcalc.a along with the link library itself are installed into ${LIBDIR}.

External programs may want to compile with:

	-I${LIBDIR} -L${LIBDIR} -lcalc

--------------
ERROR HANDLING
--------------

Your program MUST provide a function called math_error.	 This is called by
the math routines on an error condition, such as malloc failures or a
division by zero.  The routine is called in the manner of printf, with a
format string and optional arguments.  (However, none of the low level math
routines currently uses formatting, so if you are lazy you can simply use
the first argument as a simple error string.)  For example, one of the
error calls you might expect to receive is:

	math_error("Division by zero");

Your program can handle errors in basically one of two ways.  Firstly, it
can simply print the error message and then exit.  Secondly, you can make
use of setjmp and longjmp in your program.  Use setjmp at some appropriate
level in your program, and use longjmp in the math_error routine to return
to that level and so recover from the error.  This is what the calc program
does.

For convenience, the link library libcalc.a contains a math_error routine.
By default, this routine simply prints a message to stderr and then exits.
By simply linking in this link library, any calc errors will result in a
error message on stderr followed by an exit.

External programs that wish to use this math_error may want to compile with:

	-I${LIBDIR} -L${LIBDIR} -lcalc

If one sets up calc_jmp_buf, and then sets calc_jmp to non-zero then
this routine will longjmp back (with the value of calc_jmp) instead.
In addition, the last calc error message will be found in calc_error;
this error is not printed to stderr.  The calc error message will
not have a trailing newline.

For example:

    #include <setjmp.h>

    extern jmp_buf calc_jmp_buf;
    extern int calc_jmp;
    extern char *calc_error;
    int error;

    ...

    if ((error = setjmp(calc_jmp_buf)) != 0) {

	    /* reinitialize calc after a longjmp */
	    reinitialize();

	    /* report the error */
	    printf("Ouch: %s\n", calc_error);
    }
    calc_jmp = 1;

---------------
OUTPUT ROUTINES
---------------

The output from the routines in the link library normally goes to stdout.  You
can divert that output to either another FILE handle, or else to a string.
Read the routines in zio.c to see what is available.  Diversions can be
nested.

You use math_setfp to divert output to another FILE handle.  Calling
math_setfp with stdout restores output to stdout.

Use math_divertio to begin diverting output into a string.  Calling
math_getdivertedio will then return a string containing the output, and
clears the diversion.  The string is reallocated as necessary, but since
it is in memory, there are obviously limits on the amount of data that can
be diverted into it.  The string needs freeing when you are done with it.

Calling math_cleardiversions will clear all the diversions to strings, and
is useful on an error condition to restore output to a known state.  You
should also call math_setfp on errors if you had changed that.

If you wish to mix your own output with numeric output from the math routines,
then you can call math_chr, math_str, math_fill, math_fmt, or math_flush.
These routines output single characters, output null-terminated strings,
output strings with space filling, output formatted strings like printf, and
flush the output.  Output from these routines is diverted as described above.

You can change the default output mode by calling math_setmode, and you can
change the default number of digits printed by calling math_setdigits.	These
routines return the previous values.  The possible modes are described in
zmath.h.

--------------
USING INTEGERS
--------------

The arbitrary precision integer routines define a structure called a ZVALUE.
This is defined in zmath.h.  A ZVALUE contains a pointer to an array of
integers, the length of the array, and a sign flag.  The array is allocated
using malloc, so you need to free this array when you are done with a
ZVALUE.	 To do this, you should call zfree with the ZVALUE as an argument
(or call freeh with the pointer as an argument) and never try to free the
array yourself using free.  The reason for this is that sometimes the pointer
points to one of two statically allocated arrays which should NOT be freed.

The ZVALUE structures are passed to routines by value, and are returned
through pointers.  For example, to multiply two small integers together,
you could do the following:

	ZVALUE	z1, z2, z3;

	itoz(3L, &z1);
	itoz(4L, &z2);
	zmul(z1, z2, &z3);

Use zcopy to copy one ZVALUE to another.  There is no sharing of arrays
between different ZVALUEs even if they have the same value, so you MUST
use this routine.  Simply assigning one value into another will cause
problems when one of the copies is freed.  However, the special ZVALUE
values _zero_ and _one_ CAN be assigned to variables directly, since their
values of 0 and 1 are so common that special checks are made for them.

For initial values besides 0 or 1, you need to call itoz to convert a long
value into a ZVALUE, as shown in the above example.  Or alternatively,
for larger numbers you can use the atoz routine to convert a string which
represents a number into a ZVALUE.  The string can be in decimal, octal,
hex, or binary according to the leading digits.

Always make sure you free a ZVALUE when you are done with it or when you
are about to overwrite an old ZVALUE with another value by passing its
address to a routine as a destination value, otherwise memory will be
lost.  The following shows an example of the correct way to free memory
over a long sequence of operations.

	ZVALUE z1, z2, z3;

	z1 = _one_;
	atoz("12345678987654321", &z2);
	zadd(z1, z2, &z3);
	zfree(z1);
	zfree(z2);
	zsquare(z3, &z1);
	zfree(z3);
	itoz(17L, &z2);
	zsub(z1, z2, &z3);
	zfree(z1);
	zfree(z2);
	zfree(z3);

There are some quick checks you can make on integers.  For example, whether
or not they are zero, negative, even, and so on.  These are all macros
defined in zmath.h, and should be used instead of checking the parts of the
ZVALUE yourself.  Examples of such checks are:

	ziseven(z)	(number is even)
	zisodd(z)	(number is odd)
	ziszero(z)	(number is zero)
	zisneg(z)	(number is negative)
	zispos(z)	(number is positive)
	zisunit(z)	(number is 1 or -1)
	zisone(z)	(number is 1)
	zisnegone(z)	(number is -1)
	zistwo(z)	(number is 2)
	zisabstwo(z)	(number is 2 or -2)
	zisabsleone(z)	(number is -1, 0 or 1)
	zislezero(z)	(number is <= 0)
	zisleone(z)	(number is <= 1)
	zge16b(z)	(number is >= 2^16)
	zge24b(z)	(number is >= 2^24)
	zge31b(z)	(number is >= 2^31)
	zge32b(z)	(number is >= 2^32)
	zge64b(z)	(number is >= 2^64)

Typically the largest unsigned long is typedefed to FULL.  The following
macros are useful in dealing with this data type:

	MAXFULL		(largest positive FULL value)
	MAXUFULL	(largest unsigned FULL value)
	zgtmaxfull(z)	(number is > MAXFULL)
	zgtmaxufull(z)	(number is > MAXUFULL)
	zgtmaxlong(z)	(number is > MAXLONG, largest long value)
	zgtmaxulong(z)	(number is > MAXULONG, largest unsigned long value)

If zgtmaxufull(z) is false, then one may quickly convert the absolute
value of number into a full with the macro:

	ztofull(z)	(convert abs(number) to FULL)
	ztoulong(z)	(convert abs(number) to an unsigned long)
	ztolong(z)	(convert abs(number) to a long)

If the value is too large for ztofull(), ztoulong() or ztolong(), only
the low order bits converted.

There are two types of comparisons you can make on ZVALUEs.  This is whether
or not they are equal, or the ordering on size of the numbers.	The zcmp
function tests whether two ZVALUEs are equal, returning TRUE if they differ.
The zrel function tests the relative sizes of two ZVALUEs, returning -1 if
the first one is smaller, 0 if they are the same, and 1 if the first one
is larger.

---------------
USING FRACTIONS
---------------

The arbitrary precision fractional routines define a structure called NUMBER.
This is defined in qmath.h.  A NUMBER contains two ZVALUEs for the numerator
and denominator of a fraction, and a count of the number of uses there are
for this NUMBER.  The numerator and denominator are always in lowest terms,
and the sign of the number is contained in the numerator.  The denominator
is always positive.  If the NUMBER is an integer, the denominator has the
value 1.

Unlike ZVALUEs, NUMBERs are passed using pointers, and pointers to them are
returned by functions.	So the basic type for using fractions is not really
(NUMBER), but is (NUMBER *).  NUMBERs are allocated using the qalloc routine.
This returns a pointer to a number which has the value 1.  Because of the
special property of a ZVALUE of 1, the numerator and denominator of this
returned value can simply be overwritten with new ZVALUEs without needing
to free them first.  The following illustrates this:

	NUMBER *q;

	q = qalloc();
	itoz(55L, &q->num);

A better way to create NUMBERs with particular values is to use the itoq,
iitoq, or atoq functions.  Using itoq makes a long value into a NUMBER,
using iitoq makes a pair of longs into the numerator and denominator of a
NUMBER (reducing them first if needed), and atoq converts a string representing
a number into the corresponding NUMBER.	 The atoq function accepts input in
integral, fractional, real, or exponential formats.  Examples of allocating
numbers are:

	NUMBER *q1, *q2, *q3;

	q1 = itoq(66L);
	q2 = iitoq(2L, 3L);
	q3 = atoq("456.78");

Also unlike ZVALUEs, NUMBERs are quickly copied.  This is because they contain
a link count, which is the number of pointers there are to the NUMBER.	The
qlink macro is used to copy a pointer to a NUMBER, and simply increments
the link count and returns the same pointer.  Since it is a macro, the
argument should not be a function call, but a real pointer variable.  The
qcopy routine will actually make a new copy of a NUMBER, with a new link
count of 1.  This is not usually needed.

NUMBERs are deleted using the qfree routine.  This decrements the link count
in the NUMBER, and if it reaches zero, then it will deallocate both of
the ZVALUEs contained within the NUMBER, and then puts the NUMBER structure
onto a free list for quick reuse.  The following is an example of allocating
NUMBERs, copying them, adding them, and finally deleting them again.

	NUMBER *q1, *q2, *q3;

	q1 = itoq(111L);
	q2 = qlink(q1);
	q3 = qqadd(q1, q2);
	qfree(q1);
	qfree(q2);
	qfree(q3);

Because of the passing of pointers and the ability to copy numbers easily,
you might wish to use the rational number routines even for integral
calculations.  They might be slightly slower than the raw integral routines,
but are more convenient to program with.

The prototypes for the fractional routines are defined in qmath.h.
Many of the definitions for integer functions parallel the ones defined
in zmath.h.  But there are also functions used only for fractions.
Examples of these are qnum to return the numerator, qden to return the
denominator, qint to return the integer part of, qfrac to return the
fractional part of, and qinv to invert a fraction.

There are some transcendental functions in the link library, such as sin
and cos.  These cannot be evaluated exactly as fractions.  Therefore,
they accept another argument which tells how accurate you want the result.
This is an "epsilon" value, and the returned value will be within that
quantity of the correct value.	This is usually an absolute difference,
but for some functions (such as exp), this is a relative difference.
For example, to calculate sin(0.5) to 100 decimal places, you could do:

	NUMBER *q, *ans, *epsilon;

	q = atoq("0.5");
	epsilon = atoq("1e-100");
	ans = qsin(q, epsilon);

There are many convenience macros similar to the ones for ZVALUEs which can
give quick information about NUMBERs.  In addition, there are some new ones
applicable to fractions.  These are all defined in qmath.h.  Some of these
macros are:

	qiszero(q)	(number is zero)
	qisneg(q)	(number is negative)
	qispos(q)	(number is positive)
	qisint(q)	(number is an integer)
	qisfrac(q)	(number is fractional)
	qisunit(q)	(number is 1 or -1)
	qisone(q)	(number is 1)
	qisnegone(q)	(number is -1)
	qistwo(q)	(number is 2)
	qiseven(q)	(number is an even integer)
	qisodd(q)	(number is an odd integer)
	qistwopower(q)	(number is a power of 2 >= 1)

The comparisons for NUMBERs are similar to the ones for ZVALUEs.  You use the
qcmp and qrel functions.

There are four predefined values for fractions.	 You should qlink them when
you want to use them.  These are _qzero_, _qone_, _qnegone_, and _qonehalf_.
These have the values 0, 1, -1, and 1/2.  An example of using them is:

	NUMBER *q1, *q2;

	q1 = qlink(&_qonehalf_);
	q2 = qlink(&_qone_);

---------------------
USING COMPLEX NUMBERS
---------------------

The arbitrary precision complex arithmetic routines define a structure
called COMPLEX.	 This is defined in cmath.h.  This contains two NUMBERs
for the real and imaginary parts of a complex number, and a count of the
number of links there are to this COMPLEX number.

The complex number routines work similarly to the fractional routines.
You can allocate a COMPLEX structure using comalloc (NOT calloc!).
You can construct a COMPLEX number with desired real and imaginary
fractional parts using qqtoc.  You can copy COMPLEX values using clink
which increments the link count.  And you free a COMPLEX value using cfree.
The following example illustrates this:

	NUMBER *q1, *q2;
	COMPLEX *c1, *c2, *c3;

	q1 = itoq(3L);
	q2 = itoq(4L);
	c1 = qqtoc(q1, q2);
	qfree(q1);
	qfree(q2);
	c2 = clink(c1);
	c3 = cmul(c1, c2);
	cfree(c1);
	cfree(c2);
	cfree(c3);

As a shortcut, when you want to manipulate a COMPLEX value by a real value,
you can use the caddq, csubq, cmulq, and cdivq routines.  These accept one
COMPLEX value and one NUMBER value, and produce a COMPLEX value.

There is no direct routine to convert a string value into a COMPLEX value.
But you can do this yourself by converting two strings into two NUMBERS,
and then using the qqtoc routine.

COMPLEX values are always returned from these routines.	 To split out the
real and imaginary parts into normal NUMBERs, you can simply qlink the
two components, as shown in the following example:

	COMPLEX *c;
	NUMBER *rp, *ip;

	c = calloc();
	rp = qlink(c->real);
	ip = qlink(c->imag);

There are many macros for checking quick things about complex numbers,
similar to the ZVALUE and NUMBER macros.  In addition, there are some
only used for complex numbers.	Examples of macros are:

	cisreal(c)	(number is real)
	cisimag(c)	(number is pure imaginary)
	ciszero(c)	(number is zero)
	cisnegone(c)	(number is -1)
	cisone(c)	(number is 1)
	cisrunit(c)	(number is 1 or -1)
	cisiunit(c)	(number is i or -i)
	cisunit(c)	(number is 1, -1, i, or -i)
	cistwo(c)	(number is 2)
	cisint(c)	(number is has integer real and imaginary parts)
	ciseven(c)	(number is has even real and imaginary parts)
	cisodd(c)	(number is has odd real and imaginary parts)

There is only one comparison you can make for COMPLEX values, and that is
for equality.  The ccmp function returns TRUE if two complex numbers differ.

There are three predefined values for complex numbers.	You should clink
them when you want to use them.	 They are _czero_, _cone_, and _conei_.
These have the values 0, 1, and i.

----------------
LAST THINGS LAST
----------------

If you wish, when you are all doen you can call libcalc_call_me_last()
to free a small amount of storage associated with the libcalc_call_me_first()
call.  This is not required, but is does bring things to a closure.

The function libcalc_call_me_last() takes no args and returns void.  You
need call libcalc_call_me_last() only once.

## Copyright (C) 1999  David I. Bell and Landon Curt Noll
##
## Calc is open software; you can redistribute it and/or modify it under
## the terms of the version 2.1 of the GNU Lesser General Public License
## as published by the Free Software Foundation.
##
## Calc is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
## Public License for more details.
##
## A copy of version 2.1 of the GNU Lesser General Public License is
## distributed with calc under the filename COPYING-LGPL.  You should have
## received a copy with calc; if not, write to Free Software Foundation, Inc.
## 59 Temple Place, Suite 330, Boston, MA  02111-1307, USA.
##
## @(#) $Revision: 29.1 $
## @(#) $Id: LIBRARY,v 29.1 1999/12/14 09:15:29 chongo Exp $
## @(#) $Source: /usr/local/src/cmd/calc/RCS/LIBRARY,v $
##
## Under source code control:	1993/07/30 19:44:49
## File existed as early as:	1993
##
## chongo <was here> /\oo/\	http://reality.sgi.com/chongo/
## Share and enjoy!  :-)	http://reality.sgi.com/chongo/tech/comp/calc/