NAME
Math::MPC - perl interface to the MPC (multi precision complex) library.
DEPENDENCIES
This module needs the MPC, MPFR and GMP C libraries. (Install GMP
first, followed by MPFR, followed by MPC.)
The GMP library is availble from http://gmplib.org
The MPFR library is available from http://www.mpfr.org/
The MPC library is available from
http://www.multiprecision.org/mpc/DESCRIPTION
A multiple precision complex number module utilising the MPC library.
Basically, this module simply wraps the 'mpc' complex number functions
provided by that library.
Operator overloading is also available.
The following documentation heavily plagiarises the mpc documentation.
use warnings;
use Math::MPFR qw(:mpfr); # needed by some Math::MPC functions
use Math::MPC qw(:mpc);
Rmpc_set_default_prec(500); # Set default precision to 500 bits
my $mpc1 = Math::MPC->new(12.5, 1125); # 12.5 + 1125*i
$mpc2 = sqrt($mpc1);
print "Square root of $mpc1 is $mpc2\n";
See also the Math::MPC test suite for some (simplistic) examples of
usage.ROUNDING MODE
A complex rounding mode is of the form MPC_RNDxy where "x" and "y"
are one of "N" (to nearest), "Z" (towards zero), "U" (towards plus
infinity), "D" (towards minus infinity). If building this module
against mpc-1.3.0 pr later, the additional option of "A" (away from
zero) is also available.
The first letter refers to the rounding mode for the real part, and
the second one for the imaginary part.
For example MPC_RNDZU indicates to round the real part towards
zero, and the imaginary part towards plus infinity.
The default rounding mode is MPC_RNDNN, but this can be changed
using the Rmpc_set_default_rounding_mode() function.MEMORY MANAGEMENT
Objects can be created with the Rmpc_init2 and Rmpc_init3 functions,
which return an object that has been blessed into the package
Math::MPC. Alternatively, blessed objects can also be created by
calling the new() function (either as a function or as a method).
These objects will be automatically cleaned up by the DESTROY()
function whenever they go out of scope.
Rmpc_init2_nobless and Rmpc_init3_nobless are the same as Rmpc_init2
and Rmpc_init3, except that they return an unblessed object.
If you create Math::MPC objects using the '_nobless' versions,
it will then be up to you to clean up the memory associated with
these objects by calling Rmpc_clear($op) for each object.
Alternatively such objects will be cleaned up when the script ends.
I don't know why you would want to create unblessed objects. The
point is that you can if you want to.MIXING MPC OBJECTS WITH MPFR & GMP OBJECTS
Some of the Math::MPC functions below take Math::MPFR, Math::GMP,
Math::GMPz, Math::GMPq, or Math::GMPf objects as arguments.
Obviously, to make use of these functions, you'll need to have
loaded the appropriate module.FUNCTIONS
Most of the following functions are simply wrappers around an mpc
function of the same name. eg. Rmpc_neg() is a wrapper around
mpc_neg().
"$rop", "$op1", "$op2", etc. are Math::MPC objects - the
return value of one of the Rmpc_init* functions. They are in fact
references to mpc structures. The "$op" variables are the operands
and "$rop" is the variable that stores the result of the operation.
Generally, $rop, $op1, $op2, etc. can be the same perl variable
referencing the same mpc structure, though often they will be
distinct perl variables referencing distinct mpc structures.
Eg something like Rmpc_add($r1, $r1, $r1, $rnd),
where $r1 *is* the same reference to the same mpc structure,
would add $r1 to itself and store the result in $r1. Alternatively,
you could (courtesy of operator overloading) simply code it
as $r1 += $r1. Otoh, Rmpc_add($r1, $r2, $r3, $rnd), where each of the
arguments is a different reference to a different mpc structure
would add $r2 to $r3 and store the result in $r1. Alternatively
it could be coded as $r1 = $r2 + $r3.
In the documentation that follows:
"$ui" means an integer that will fit into a C 'unsigned long int',
"$si" means an integer that will fit into a C 'signed long int'.
"$uj" means an integer that will fit into a C 'uintmax_t'. Don't
use the _uj functions unless your perl was compiled with 64
bit integer support.
"$sj" means an integer that will fit into a C 'intmax_t'. Don't
use the _sj functions unless your perl was compiled with 64
bit integer support.
"$double" is a C double.
"$ld" is a C long double. Don't use the _ld functions unless your
perl was compiled with long double support.
"$bool" means a value (usually a 'signed long int') in which
the only interest is whether it evaluates as false or true.
"$str" simply means a string of symbols that represent a number,
eg '1234567890987654321234567@7' which might be a base 10 number,
or 'zsa34760sdfgq123r5@11' which would have to represent at least
a base 36 number (because "z" is a valid digit only in bases 36
and above). Valid bases for MPC numbers are 2 to 36 (inclusive).
"$rnd" is simply one of the 16 rounding mode values (discussed above).
"$p" is the (unsigned long) value for precision.
"$mpf" is a Math::GMPf object (floating point). You'll need Math::GMPf
installed in order to create $mpf.
"$mpq" is a Math::GMPq object (rational). You'll need Nath::GMPq
installed in order to create $mpq.
"$mpz" is a Math::GMP or Math::GMPz object (integer). You'll need
Math::GMPz or Math::GMP installed in order to create $mpz.
"$mpfr" is a Math::MPFR object (floating point). You'll need to
'use Math::MPFR;' in order to create $mpfr. (Math::MPFR
a pre-requisite module for Math::MPC.)
"$cc" is a Math::Complex_C (double _Complex) object. You'll need to
'use Math::Complex_C' (or create your own double _Complex
object) in order to create $cc, and to use the functions that
take such an argument. (Math::Complex_C is *not* a
pre-requisite module for Math::MPC.)
"$lcc" is a Math::Complex_C::Long (long double _Complex) object.
You'll need to 'use Math::Complex_C' (or create your own
long double _Complex object in order to create $lcc, and
to use the functions that take such an argument.
######################
FUNCTION RETURN VALUES
Most MPC functions have a return value ($si) which is used to
indicate the position of the rounded real or imaginary parts with
respect to the exact (infinite precision) values. The functions
RMPC_INEX_RE($si) and RMPC_INEX_IM($si) return 0 if the corresponding
rounded value is exact, a negative value if the rounded value is less
than the exact one, and a positive value if it is greater than the
exact one. However, some functions do not completely fulfill this -
in some cases the sign is not guaranteed, and in some cases a
non-zero value is returned although the result is exact. In these
cases the function documentation explains the exact meaning of the
return value. However, the return value never wrongly indicates an
exact computation.
###########################
MANIPULATING ROUNDING MODES
Rmpc_set_default_rounding_mode($rnd);
Sets the default rounding mode to $rnd.
The default rounding mode is to nearest initially (MPC_RNDNN).
The default rounding mode is the rounding mode that is used in
overloaded operations.
$ui = Rmpc_get_default_rounding_mode();
Returns the numeric value of the current default rounding mode.
This will initially be 0 (MPC_RNDNN).
##########
INITIALIZATION
Normally, a variable should be initialized once only or at least
be cleared, using `Rmpc_clear', between initializations - but
don't explicitly call Rmpc_clear() on blessed objects. 'DESTROY'
(which calls 'Rmpc_clear') is automatically called on blessed
objects whenever they go out of scope.
First read the section 'MEMORY MANAGEMENT' (above).
Rmpc_set_default_prec($p);
Rmpc_set_default_prec2($p_re, $p_im);
Rmpc_set_default_prec sets the default precision to exactly $p
bits for both the real and imaginary parts. Rmpc_set_default_prec2
sets the default precision to be *exactly* $p_re bits for the real
part, and *exactly* $p_im bits for the imaginary part. The
precision of a variable means the number of bits used to store its
mantissa. All subsequent calls to `new' will use this precision,
but previously initialized variables are unaffected. This is also
the precision that will be used during some overloaded operations
(see OPERATOR OVERLOADING below).
The default precision is set to 53 bits initially (for both
real and imaginary components).
$ui = Rmpc_get_default_prec();
($ui_re, $ui_im) = Rmpc_get_default_prec2();
Rmpc_get_default_prec returns the current default real precision
iff the default real precision is the same as the current default
imaginary precision. Otherwise it returns zero.
Rmpc_get_default_prec2 returns both current default real precision
and current default imaginary precision (in bits).
$ui = Rmpc_get_prec($op);
If the real and imaginary part of $op have the same precision,
it is returned. Otherwise 0 is returned.
$ui = Rmpc_get_re_prec($op);
$ui = Rmpc_get_im_prec($op)
($re_prec, $im_prec) = Rmpc_get_prec2($op);
Get (respectively) the precision of the real part of $op, the
precision of the imaginary part of $op, or an array containing
precision of both real and imaginary parts of $op.
$rop = Math::MPC->new();
$rop = Math::MPC::new();
$rop = new Math::MPC();
Initialize $rop, and set its real and imaginary parts to NaN.
The precision of $rop is the default precision, which can be
changed by a call to `Rmpc_set_default_prec' or
`Rmpc_set_default_prec2' (documented above).
$rop = Rmpc_init2($p);
$rop = Rmpc_init2_nobless($p);
Initialize $rop, set the precision (of both real and imaginary
parts) to be *exactly* $p bits, and set its real and imaginary
parts to NaN.
$rop = Rmpc_init3($p_re, $p_im);
$rop = Rmpc_init3_nobless($p_r, $p_i);
Initialize $rop, set the precision of the real part to be
*exactly* $p_re bits, set the precision of the imaginary part to
be *exactly* $p_im bits, and set its real and imaginary parts to
NaN.
Rmpc_set_prec($op, $p);
Reset the precision of $op to be exactly $p bits, and set its
real/imaginary parts to NaN.
Rmpc_set_re_prec($op, $p);
Rmpc_set_im_prec($op, $p);
Set (respectively) the precision of the real part of $op to be
exactly $p bits and the precision of the imaginary part of $op
to be exactly $p bits. In both cases the value is set to NaN.
(There are currently no corresponding MPC library functions.)
##########
ASSIGNMENT
$si = Rmpc_set($rop, $op, $rnd);
$si = Rmpc_set_ui($rop, $ui, $rnd);
$si2 = Rmpc_set_si($rop, $si1, $rnd);
$si = Rmpc_set_d($rop, $double, $rnd);
$si = Rmpc_set_uj($rop, $uj, $rnd);
$si = Rmpc_set_sj($rop, $sj, $rnd);
$si = Rmpc_set_ld($rop, $ld, $rnd);
$si = Rmpc_set_f($rop, $mpf, $rnd);
$si = Rmpc_set_q($rop, $mpq, $rnd);
$si = Rmpc_set_z($rop, $mpz, $rnd);
$si = Rmpc_set_fr($rop, $mpfr, $rnd);
$si = Rmpc_set_dc($rop, $cc, $rnd); # deprecated - use Rmpc_set_DC instead
$si = Rmpc_set_ldc($rop, $lcc, $rnd); # deprecated - use Rmpc_set_LDC instead
$si = Rmpc_set_DC($rop, $cc, $rnd); # $cc is a Math:Complex_C object
$si = Rmpc_set_LDC($rop, $lcc, $rnd); # $lcc is a Math::Complex_C::L object
Rmpc_set_F128C($rop, $f128c, $rnd); # $f128c is a Math::Complex_C::Q object
Set the value of $rop from 2nd arg, rounded to the precision of
$rop towards the given direction $rnd.
Don't use Rmpc_set_ld unless perl has been built with long
double support. Don't use Rmpc_set_uj or Rmpc_set_sj unless
perl has been built with long long int support.
For Rmpc_set_DC and Rmpc_set_LDC, requires an mpc library (version 0.9
or later) that has been built with support for 'double _Complex' and 'long
double _Complex' data types (resp.). For Rmpc_set_F128C, the underlying
mpfr library needs to have been built with --enable-float128 option.
$si = Rmpc_set_str($rop, $string, $base, $rnd);
$si = Rmpc_strtoc($rop, $string, $base, $rnd);
Set $rop to the value represented in $string (in base $base), rounded
in accordance with $rnd. See the mpc documentation for details.
$si = Rmpc_set_ui_ui($rop, $ui1, $ui2, $rnd);
$si3 = Rmpc_set_si_si($rop, $si1, $si2, $rnd);
$si = Rmpc_set_d_d($rop, $double1, $double2, $rnd);
$si = Rmpc_set_f_f($rop, $mpf1, $mpf2, $rnd);
$si = Rmpc_set_q_q($rop, $mpq1, $mpq2, $rnd);
$si = Rmpc_set_z_z($rop, $mpz1, $mpz2, $rnd);
$si = Rmpc_set_fr_fr($rop, $mpfr1, $mpfr2, $rnd);
Sets the real part of $rop from 2nd arg, and the imaginary part
of $rop from 3rd arg, according to the rounding mode $rnd.
$si = Rmpc_set_uj_uj($rop, $uj1, $uj2, $rnd);
$si = Rmpc_set_sj_sj($rop, $sj1, $sj2, $rnd);
$si = Rmpc_set_ld_ld($rop, $ld1, $ld2, $rnd);
Don't use the first 2 functions unless Math::MPC::_has_longlong()
returns a true value. Don't use the 3rd function unless
Math::MPC::_has_longdouble() returns true.
Sets the real part of $rop from 2nd arg, and the imaginary part
of $rop from 3rd arg, according to the rounding mode $rnd.
$si = Rmpc_set_x_y($rop, $op1, $op2, $rnd);
You need to replace the 'x' and the 'y' with any one of 'ui',
'si', 'd', 'uj', 'sj', 'ld', 'f', 'q', 'z' and 'fr' - eg:
Rmpc_set_ui_d($rop, $ui, $double, $rnd);
Don't use the 'uj' or 'sj' variants if Math::MPC::_has_longlong()
doesn't return a true value. And don't use the 'ld' variants if
Math::MPC_haslongdouble() doesn't return a true value.
Sets the real part of $rop from 2nd arg, and the imaginary part
of $rop from 3rd arg, according to the rounding mode $rnd.
################################################
COMBINED INITIALIZATION AND ASSIGNMENT
NOTE: Do NOT use these functions if $rop has already been initialised
or created by calling new(). Instead use the Rmpc_set* functions in
the section 'ASSIGNMENT' (above).
First read the section 'MEMORY MANAGEMENT' (above).
$rop = Math::MPC->new($arg1 [, $arg2]);
$rop = Math::MPC::new($arg1 [, $arg2]);
$rop = new Math::MPC($arg1, [, $arg2]);
Returns a Math::MPC object whose real component has a value of $arg1,
rounded in the default rounding direction, with default precision.
If $arg2 is supplied, the imaginary component of the returned
Math::MPC object is set to $arg2, rounded in the default rounding
direction, with default precision. Otherwise the imaginary component
of the returned Math::MPC object is set to zero. $arg1 & $arg2 can be
either a number (signed integer, unsigned integer, signed fraction or
unsigned fraction), a string that represents a numeric value, a
Math::MPFR object, a Math::GMP object, a Math::GMPz object, a
Math::GMPq object or a Math::GMPf object.
If a string argument begins with "0b" or "0B", then the string is
treated as a base 2 string. Elsif it begins with "0x" or "0X" it is
treated as a base 16 string. Else it is treated as a base 10 string.
##########
ARITHMETIC
$si = Rmpc_add($rop, $op1, $op2, $rnd);
$si = Rmpc_add_ui($rop, $op, $ui, $rnd);
$si = Rmpc_add_fr($rop, $op, $mpfr, $rnd);
Set $rop to 2nd arg + 3rd arg rounded in the direction $rnd.
$si = Rmpc_sub($rop, $op1, $op2, $rnd);
$si = Rmpc_sub_ui($rop, $op, $ui, $rnd);
$si = Rmpc_ui_sub($rop, $ui, $op, $rnd);
Set $rop to 2nd arg - 3rd arg rounded in the direction $rnd.
$si = Rmpc_ui_ui_sub($rop, $ui_r, $ui_i, $op, $rnd);
The real part of $rop is set to $ui_r minus the real part of $op
(rounded in the direction $rnd) - and the imaginary part of $rop
is set to $ui_r minus the imaginary part of $op (rounded in the
direction $rnd)
$si = Rmpc_mul($rop, $op1, $op2, $rnd);
$si = Rmpc_mul_ui($rop, $op, $ui, $rnd);
$si = Rmpc_mul_si($rop, $op, $si1, $rnd);
$si = Rmpc_mul_sj($rop, $op, $sj, $rnd); # Math::MPC XSub
$si = Rmpc_mul_d($rop, $op, $double, $rnd);# Math::MPC XSub
$si = Rmpc_mul_ld($rop, $op, $ld, $rnd); # Math::MPC XSub
$si = Rmpc_mul_fr($rop, $op, $mpfr, $rnd);
Set $rop to 2nd arg * 3rd arg rounded in the direction $rnd.
The 'sj'/'ld' versions are available only on perls built with
'64 bit int'/'long double' support.
$si = Rmpc_mul_i($rop, $op, $si1, $rnd);
If $si1 >= 0 (non-negative), set $rop to $op times the
imaginary unit i - else set $rop to $op times -i.
$si = Rmpc_div($rop, $op1, $op2, $rnd);
$si = Rmpc_div_ui($rop, $op, $ui, $rnd);
$si = Rmpc_ui_div($rop, $ui, $op, $rnd);
$si = Rmpc_div_d($rop, $op, $double, $rnd); # Math::MPC XSub
$si = Rmpc_d_div($rop, $double, $op, $rnd); # Math::MPC XSub
$si = Rmpc_div_sj($rop, $op, $sj, $rnd); # Math::MPC XSub
$si = Rmpc_sj_div($rop, $sj, $op, $rnd); # Math::MPC XSub
$si = Rmpc_div_ld($rop, $op, $ld, $rnd); # Math::MPC XSub
$si = Rmpc_ld_div($rop, $ld, $op, $rnd); # Math::MPC XSub
$si = Rmpc_div_fr($rop, $op, $mpfr, $rnd);
Set $rop to 2nd arg / 3rd arg rounded in the direction $rnd.
The 'sj'/'ld' versions are available only on perls built with
'64 bit int'/'long double' support.
$si = Rmpc_sqr($rop, $op, $rnd);
Set $rop to the square of $op, rounded in direction $rnd.
$si = Rmpc_fma ($rop, $op1, $op2, $op3, $rnd);
Set $rop to $op1 * $op2 + $op3, rounded according to $rnd, with
only one final rounding.
$si = Rmpc_dot ($rop, \@op1, \@op2, $ui, $rnd); # mpc-1.1.1 &
# later only
Set $rop to the dot product of elements of @op1 by those of
@op2, whose common size is $ui (== scalar @op1), correctly
rounded in the direction $rnd.
$si = Rmpc_sum ($rop, \@op, $ui, $rnd); # mpc-1.1.1 &
# later only
Set $rop to the sum of the first $ui elements of @op,
correctly rounded in the direction $rnd.
$si = Rmpc_sqrt($rop, $op, $rnd);
Set $rop to the square root of the 2nd arg rounded in the
direction $rnd. When the return value is 0, it means the result
is exact. Else it's unknown whether the result is exact or not.
$si = Rmpc_rootofunity ($rop, $ui1, $ui2, $rnd); # mpc-1.1.0 &
# later only
Set $rop to the standard primitive $ui1-th root of unity raised
to the power $ui2, that is, exp (2 * pi * i * $ui2 / $ui1),
rounded according to $rnd with the precision of $rop.
$si = Rmpc_agm ($rop, $op1, $op2, $rnd); # mpc-1.3.0 &
# later only
Set $rop to the arithmetic-geometric mean (AGM) of $op1 and $op2,
rounded according to $rnd with the precision of $rop. Let's call
the complex number 1 + (0 * i) "$one". Concerning the branch cut,
the function is computed by homogeneity either as:
1) $op1 * AGM($one,b0) with b0=$op2/$op1 if |$op1|>=|$op2|
or, otherwise, as:
2) $op2 * AGM($one,b0) with b0=$op1/$op2.
Then when b0 is real and negative, AGM(1,b0) is chosen to have
positive imaginary part.
$si = Rmpc_pow($rop, $op1, $op2, $rnd);
$si = Rmpc_pow_d($rop, $op1, $double, $rnd);
$si = Rmpc_pow_ld($rop, $op1, $ld, $rnd);
$si2 = Rmpc_pow_si($rop, $op1, $si, $rnd);
$si = Rmpc_pow_ui($rop, $op1, $ui, $rnd);
$si = Rmpc_pow_z($rop, $op1, $mpz, $rnd);
$si = Rmpc_pow_fr($rop, $op1, $mpfr, $rnd);
Set $op to ($op1 ** 3rd arg) rounded in the direction $rnd.
Rmpc_pow_ld is available only on perls that have "long double"
support.
$si = Rmpc_neg($rop, $op, $rnd);
Set $rop to -$op rounded in the direction $rnd. Just
changes the sign if $rop and $op are the same variable.
$si = Rmpc_conj($rop, $op, $rnd);
Set $rop to the conjugate of $op rounded in the direction $rnd.
Just changes the sign of the imaginary part if $rop and $op are
the same variable.
$si = Rmpc_abs($mpfr, $op, $rnd);
Set the floating-point number $mpfr to the absolute value of $op,
rounded in the direction $rnd. Return 0 iff the result is exact.
$si = Rmpc_norm($mpfr, $op, $rnd);
Set the floating-point number $mpfr to the norm of $op (ie the
square of its absolute value), rounded in the direction $rnd.
Return 0 iff the result is exact.
$si = Rmpc_mul_2exp($rop, $op, $ui, $rnd);
$si = Rmpc_mul_2ui ($rop, $op, $ui, $rnd);#same as Rmpc_mul_2exp
$si1 = Rmpc_mul_2si ($rop, $op, $si2, $rnd);
Set $rop to $op times 2 raised to 3rd arg rounded according to
$rnd. Just increases the exponents of the real and imaginary
parts by value of 3rd arg when $rop and $op are identical.
$si = Rmpc_div_2exp($rop, $op, $ui, $rnd);
$si = Rmpc_div_2ui ($rop, $op, $ui, $rnd);#same as Rmpc_div_2exp
$si1 = Rmpc_div_2si ($rop, $op, $si2, $rnd);
Set $rop to $op divided by 2 raised to 3rd arg rounded according
to $rnd. Just decreases the exponents of the real and imaginary
parts by value of 3rd arg when $rop and $op are identical.
Rmpc_swap($op1, $op2);
Swap the values (and precisions) of op1 and op2 efficiently.
##########
COMPARISON
$si = Rmpc_cmp($op1, $op2);
$si = Rmpc_cmp_si($op, $si1);
Compare 1st and 2nd args. The return value $si can be decomposed
into $x = RMPC_INEX_RE($si) and $y = RMPC_INEX_IM($si), such that $x
is positive if the real part of the 1st arg is greater than that of
the 2nd arg, zero if both real parts are equal, and negative if the
real part of the 1st arg is less than that of the 2nd arg.
Likewise for $y.
Both 1st and 2nd args are considered to their full own precision,
which may differ.
It is not allowed that one of the operands has a NaN (Not-a-Number)
part.
The storage of the return value is such that equality can be simply
checked with Rmpc_cmp($first_arg, $second_arg) == 0.
$si = Rmpc_cmp_si_si($op, $si1, $si2);
As for the above comparison functions - except that $op is being
compared with $si1 + ($si2 * i).
$si = Rmpc_cmp_abs ($op1, $op2); # requires mpc-1.1.0 and later
Compare the absolute values of $op1 and $op2. The return value is 0
if both are the same (including infinity), positive if the absolute
value of OP1 is greater than that of OP2, and negative if it is
smaller. If $op1 or $op2 has a real or imaginary part which is NaN,
the function behaves like 'mpfr_cmp' on two real numbers of which
at least one is NaN.
#######
SPECIAL
$si = Rmpc_exp($rop, $op, $rnd);
Set $rop to the exponential of $op, rounded according to $rnd
with the precision of $rop.
$si = Rmpc_log($rop, $op, $rnd);
Set $rop to the log of $op, rounded according to $rnd with the
precision of $rop.
$si = Rmpc_log10($rop, $op, $rnd);
Set $rop to the base 10 log of $op, rounded according to $rnd with
the precision of $rop.
$si = Rmpc_arg ($mpfr, $op, $rnd);
Set $mpfr to the argument of $op, with a branch cut along the
negative real axis. ($mpfr is a Math::MPFR object.)
$si = Rmpc_proj ($rop, $op, $rnd);
Compute a projection of $op onto the Riemann sphere. Set $rop
to $op, rounded in the direction $rnd, except when at least one
part of $op is infinite (even if the other part is a NaN) in
which case the real part of $rop is set to plus infinity and its
imaginary part to a signed zero with the same sign as the
imaginary part of $op.
Rmpc_set_nan($op);
Set $op to 'NaN +I*NaN'.
##########
TRIGONOMETRIC
$si = Rmpc_sin($rop, $op, $rnd);
Set $rop to the sine of $op, rounded according to $rnd with the
precision of $rop.
$si = Rmpc_cos($rop, $op, $rnd);
Set $rop to the cosine of $op, rounded according to $rnd with
the precision of $rop.
$si = Rmpc_sin_cos($r_sin, $r_cos, $op, $rnd_sin, $rnd_cos);
Needs version 0.9 or later of the mpc C library.
Set $r_sin/$r_cos to the sin/cos of $op, rounded according to
$rnd_sin/$rnd_cos.
(If the mpc C library is pre version 0.9, calling this
function will cause the program to die with an appropriate
error message.)
$si = Rmpc_tan($rop, $op, $rnd);
Set $rop to the tangent of $op, rounded according to $rnd with
the precision of $rop.
$si = Rmpc_sinh($rop, $op, $rnd);
Set $rop to the hyperbolic sine of $op, rounded according to
$rnd with the precision of $rop.
$si = Rmpc_cosh($rop, $op, $rnd);
Set $rop to the hyperbolic cosine of $op, rounded according to
$rnd with the precision of $rop.
$si = Rmpc_tanh($rop, $op, $rnd);
Set $rop to the hyperbolic tangent of $op, rounded according to
$rnd with the precision of $rop.
$si = Rmpc_asin ($rop, $op, $rnd);
Set $rop to the inverse sine of $op, rounded according to
$rnd with the precision of $rop.
$si = Rmpc_acos ($rop, $op, $rnd);
Set $rop to the inverse cosine of $op, rounded according to
$rnd with the precision of $rop.
$si = Rmpc_atan ($rop, $op, $rnd);
Set $rop to the inverse tangent of $op, rounded according to
$rnd with the precision of $rop.
$si = Rmpc_asinh ($rop, $op, $rnd);
Set $rop to the inverse hyperbolic sine of $op, rounded
according to $rnd with the precision of $rop.
$si = Rmpc_acosh ($rop, $op, $rnd);
Set $rop to the inverse hyperbolic cosine of $op, rounded
according to$rnd with the precision of $rop.
$si = Rmpc_atanh ($rop, $op, $rnd);
Set $rop to the inverse hyperbolic tangent of $op, rounded
according to$rnd with the precision of $rop.
##########
MODULAR
$si = Rmpc_eta_fund ($rop, $op, $rnd); # mpc-1.3.0 &
# later only
Assuming that the argument $op lies in the fundamental domain for
Sl_2(Z), that is, it has real part not below -1/2 and not above
+1/2 and absolute value at least 1, return the value of the
Dedekind eta-function in $rop. For arguments outside the
fundamental domain the function is expected to loop indefinitely.
Use the in_fund_dom function (see next) if you first want to verfiy
that the value is, indeed, inside the fundamental domain.
$bool = in_fund_dom($rop);
Returns TRUE if $rop is insided the fundamental domain (as
described above in the Rmpc_eta_fund documentation).
Else returns FALSE.
##########
CONVERSION
($real, $im) = c_string($op, $base, $digits, $rnd);
$real = r_string($op, $base, $digits, $rnd);
$im = i_string($op, $base, $digits, $rnd);
$real is a string containing the value of the real part of $op.
$im is a string containing the value of the imaginary part of $op.
$real and $im will be of the form XeY (X@Y for bases greater than 10)
- where X is the mantissa (in base $base) and Y is the exponent (in
base 10).
For example, -31.4132' would be returned as -3.14132e1. $digits is the
number of digits that will be written in the mantissa. If $digits is
zero, the mantissa will contain the maximum number of digits
accurately representable. The mantissa will be rounded in the
direction specified by $rnd.
@vals = Rmpc_deref4($op, $base, $digits, $rnd);
@vals contains (in order) the real mantissa, the real exponent, the
imaginary mantissa, and the imaginary exponent of $op.The mantissas,
expressed in base $base and rounded according to $rnd), contain an
implicit radix point to the left of the first (ie leftmost) digit.
The exponents are always expressed in base 10. $digits is the number
of digits that will be written in the mantissa. If $digits is zero
the mantissa will contain the maximum number of digits accurately
representable.
RMPC_RE($mpfr, $op);
RMPC_IM($mpfr, $op);
Set $mpfr to the value of the real (respectively imaginary) component
of $op. $mpfr will be an exact copy of the real/imaginary component
of op - ie the precision of $mpfr will be set to the precision of the
real/imaginary component of $op before the copy is made. Hence no need
for a rounding arg to be supplied.
$si = Rmpc_real($mpfr, $op, $rnd);
$si = Rmpc_imag($mpfr, $op, $rnd);
Set $mpfr to the value of the real (respectively imaginary) part of
$op, rounded in the direction $rnd. ($mpfr is a Math::MPFR object.)
#############
I-O FUNCTIONS
$ul = Rmpc_inp_str($rop, $stream, $base, $rnd);
Input a string in base $base from $stream, rounded according to $rnd,
and put the read complex in $rop. Each of the real and imaginary
parts should be of the form X@Ym or, if the base is 10 or less,
alternatively XeY or XEY. (X is the mantissa, Y is the exponent.
The mantissa is always in the specified base. The exponent is always
read in decimal. This function first reads the real part, followed by
the imaginary part. The argument $base may be in the range 0,2..36.
Return the number of bytes read, or if an error occurred, return 0.
$ul =
Rmpc_out_str([$pre,] $stream, $base, $digits, $op, $rnd [, $suf]);
This function changed from 1st release (version 0.45) of Math::MPC.
Output $op to $stream, in base $base, rounded according to $rnd. First
the real part is printed, followed by the imaginary part. The base may
be 0,2..36. Print at most $digits significant digits for each
part, or if $digits is 0, the maximum number of digits accurately
representable by $op. In addition to the significant digits, a decimal
point at the right of the first digit and a trailing exponent, in the
form eYYY , are printed. (If $base is greater than 10, "@" will be
used as exponent delimiter.) $pre and $suf are optional arguments
containing a string that will be prepended/appended to the output.
Return the number of bytes written. (The contents of $pre and $suf
are not included in the count.)
$string = Rmpc_get_str($base, $how_many, $op, $rnd);
Convert $op to a string containing the real and imaginary parts of
$op. The number of significant digits for both real and imaginary
parts is specified by $how_many. It is also possible to let
$how_many be zero, in which case the number of digits is chosen large
enough so that re-reading the printed value with the same precision,
assuming both output and input use rounding to nearest, will recover
the original value of $op. See the mpc documentation for details.
$si = Rmpc_get_dc($cc, $op, $rnd); # deprecated, use Rmpc_get_DC instead
$si = Rmpc_get_ldc($lcc, $op $rnd);# deprecated, use Rmpc_get_LDC instead
$si = Rmpc_get_DC($cc, $op, $rnd); # $cc is a Math:Complex_C object
$si = Rmpc_get_LDC($lcc, $op, $rnd); # $lcc is a Math::Complex_C::L object
Rmpc_get_F128C($f128c, $op, $rnd);# $f128c is Math::Complex_C::Q object
Set the 'double _Complex'/'long double _Complex'/'__float128 _Complex'
object to the value of $op, rounded according to $rnd. For Rmpc_get_DC
and Rmpc_get_LDC, requires an mpc library (version 0.9 or later) that
has been built with support for 'double _Complex' and 'long double
_Complex' data types (resp.). For Rmpc_get_F128C, the underlying mpfr
library needs to have been built with --enable-float128 option.
####################
OPERATOR OVERLOADING
Overloading works with numbers, strings (bases 2, 10, and 16
only - see step '4.' below) and Math::MPC objects.
Overloaded operations are performed using the current
"default rounding mode" (which you can determine using the
'Rmpc_get_default_rounding_mode' function, and change using
the 'Rmpc_set_default_rounding_mode' function).
Be aware that when you use overloading with a string operand,
the overload subroutine converts that string operand to a
Math::MPC object with *current default precision*, and using
the *current default rounding mode*.
Be aware also, that the sign of zero may not always be handled
correctly by the overload subroutines. If it's important to you
that the sign of zero be handled correctly, don't use the
overloaded operators. (For multiplication, division, addition
and subtraction the sign of zero will be handled correctly by the
overloaded operators if both operands are Math::MPC objects.)
For the purposes of the overloaded 'not', '!' and 'bool'
operators, a "false" Math::MPC object is one with real and
imaginary parts that are both "false" - where "false" currently
means either 0 or NaN.
(A "true" Math::MPC object is, of course, simply one that is not
"false".)
The following operators are overloaded:
+ - * / ** sqrt (Return object has default precision)
+= -= *= /= **= (Precision remains unchanged)
== !=
! bool
abs (Returns an MPFR object, blessed into package Math::MPFR)
exp log (Return object has default precision)
sin cos (Return object has default precision)
""
= (The copy has the same precision as the copied object.)
NOTE: Making use of the '=' overloading is not recommended unless
you understand its caveats. See 'perldoc overload' and
read it thoroughly, including the documentation regarding
'copy constructors'.
Attempting to use the overloaded operators with objects that
have been blessed into some package other than 'Math::MPC'
will not work. The workaround is to convert this "foreign"
object to a Math::MPC object - thus allowing it to work with
the overloaded operator.
In those situations where the overload subroutine operates on 2
perl variables, then obviously one of those perl variables is
a Math::MPC object. To determine the value of the other variable
the subroutine works through the following steps (in order),
using the first value it finds, or croaking if it gets
to step 6:
1. If the variable is a UV then that value is used. The variable
is considered to be a UV if the IOK and IsUV flags are set.
2. If the variable is an IV, then that value is used.
The variable is considered to be an IV if the IOK flag is set.
3. If the variable is a string (ie the POK flag is set) then the
value of that string is used. If the POK flag is set, but the
string is not a valid number, the subroutine croaks with an
appropriate error message. If the string starts with '0b' or
'0B' it is regarded as a base 2 number. If it starts with '0x'
or '0X' it is regarded as a base 16 number. Otherwise it is
regarded as a base 10 number.
4. If the variable is an NV (floating point value) then that
value is used. The variable is considered to be an NV if the
NOK flag is set.
5. If the variable is a Math::MPC object then the value of that
object is used.
6. If none of the above is true, then the second variable is
deemed to be of an invalid type. The subroutine croaks with
an appropriate error message.
#####################
MISCELLANEOUS
$ui = MPC_VERSION_MAJOR;
Returns the 'x' in the 'x.y.z' of the MPC library version.
Value is as specified by the header file (mpc.h) that was
used to build Math::MPC.
$ui =MPC_VERSION_MINOR;
Returns the 'y' in the 'x.y.z' of the MPC library version.
Value is as specified by the header file (mpc.h) that was
used to build Math::MPC.
$ui = MPC_VERSION_PATCHLEVEL;
Returns the 'z' in the 'x.y.z' of the MPC library version.
Value is as specified by the header file (mpc.h) that was
used to build Math::MPC.
$ui = MPC_VERSION();
An integer value derived from the library's major, minor and
patchlevel values. Value is as specified by the header file
(mpc.h) that was used to build Math::MPC.
$ui = MPC_VERSION_NUM($major, $minor, $patchlevel);
Returns an integer in the same format as used by MPC_VERSION,
using the given $major, $minor and $patchlevel. Value is as
specified by the header file (mpc.h) that was used to build
Math::MPC.
$string = MPC_VERSION_STRING;
$string contains the MPC library version ('x.y.z'), as defined
by the header file (mpc.h) that was used to build Math::MPC
$string = Rmpc_get_version();
$string contains the MPC library version ('x.y.z'), as defined
by the mpc library being used by Math::MPC.
$MPFR_version = Math::MPC::mpfr_v();
$MPFR_version is set to the version of the mpfr library
being used by the mpc library that Math::MPC uses.
(The function is not exportable.)
$GMP_version = Math::MPC::gmp_v();
$GMP_version is set to the version of the gmp library being
used by the mpc library that Math::MPC uses.
(The function is not exportable.)
$iv = Math::MPC::nok_pokflag(); # not exported
Returns the value of the nok_pok flag. This flag is
initialized to zero, but incemented by 1 whenever a
scalar that is both a float (NOK) and string (POK) is passed
to new() or to an overloaded operator. The value of the flag
therefore tells us how many times such events occurred . The
flag can be reset to 0 by running clear_nok_pok().
Math::MPC::set_nok_pok($iv); # not exported
Resets the nok_pok flag to the value specified by $iv.
Math::MPC::clear_nok_pok(); # not exported
Resets the nok_pok flag to 0.(Essentially the same as
running set_nok_pok(0).)
####################
RADIUS FUNCTIONS
See the Math::MPC::Radius documentation:
perldoc Math::MPC::Radius
####################
BALL FUNCTIONS
See the Math::MPC::Ball documentation:
perldoc Math::MPC::Ball
####################TODO
For completeness, we probably should wrap mpc_realref and
mpc_imagref - though I don't think there's much to be
achieved by doing this in a *perl* context.BUGS
You can get segfaults if you pass the wrong type of
argument to the functions - so if you get a segfault, the
first thing to do is to check that the argument types
you have supplied are appropriate.
Also, as mentioned above in the "OPERATOR OVERLOADING" section,
the overloaded operators are not guaranteed to always handle the
sign of zero correctly.LICENSE
This program is free software; you may redistribute it and/or
modify it under the same terms as Perl itself.
Copyright 2006-2011, 2014, 2016, 2018, 2020, 2022 SisyphusAUTHOR
Sisyphus <sisyphus at(@) cpan dot (.) org>