2133 lines
49 KiB
Perl
2133 lines
49 KiB
Perl
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#
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# Complex numbers and associated mathematical functions
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# -- Raphael Manfredi Since Sep 1996
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# -- Jarkko Hietaniemi Since Mar 1997
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# -- Daniel S. Lewart Since Sep 1997
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#
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package Math::Complex;
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{ use 5.006; }
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use strict;
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our $VERSION = 1.59_01;
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use Config;
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our ($Inf, $ExpInf);
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our ($vax_float, $has_inf, $has_nan);
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BEGIN {
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$vax_float = (pack("d",1) =~ /^[\x80\x10]\x40/);
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$has_inf = !$vax_float;
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$has_nan = !$vax_float;
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unless ($has_inf) {
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# For example in vax, there is no Inf,
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# and just mentioning the DBL_MAX (1.70141183460469229e+38)
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# causes SIGFPE.
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# These are pretty useless without a real infinity,
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# but setting them makes for less warnings about their
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# undefined values.
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$Inf = "Inf";
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$ExpInf = "Inf";
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return;
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}
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my %DBL_MAX = # These are IEEE 754 maxima.
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(
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4 => '1.70141183460469229e+38',
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8 => '1.7976931348623157e+308',
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# AFAICT the 10, 12, and 16-byte long doubles
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# all have the same maximum.
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10 => '1.1897314953572317650857593266280070162E+4932',
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12 => '1.1897314953572317650857593266280070162E+4932',
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16 => '1.1897314953572317650857593266280070162E+4932',
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);
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my $nvsize = $Config{nvsize} ||
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($Config{uselongdouble} && $Config{longdblsize}) ||
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$Config{doublesize};
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die "Math::Complex: Could not figure out nvsize\n"
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unless defined $nvsize;
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die "Math::Complex: Cannot not figure out max nv (nvsize = $nvsize)\n"
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unless defined $DBL_MAX{$nvsize};
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my $DBL_MAX = eval $DBL_MAX{$nvsize};
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die "Math::Complex: Could not figure out max nv (nvsize = $nvsize)\n"
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unless defined $DBL_MAX;
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my $BIGGER_THAN_THIS = 1e30; # Must find something bigger than this.
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if ($^O eq 'unicosmk') {
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$Inf = $DBL_MAX;
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} else {
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local $SIG{FPE} = sub { };
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local $!;
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# We do want an arithmetic overflow, Inf INF inf Infinity.
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for my $t (
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'exp(99999)', # Enough even with 128-bit long doubles.
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'inf',
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'Inf',
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'INF',
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'infinity',
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'Infinity',
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'INFINITY',
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'1e99999',
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) {
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local $^W = 0;
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my $i = eval "$t+1.0";
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if (defined $i && $i > $BIGGER_THAN_THIS) {
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$Inf = $i;
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last;
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}
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}
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$Inf = $DBL_MAX unless defined $Inf; # Oh well, close enough.
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die "Math::Complex: Could not get Infinity"
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unless $Inf > $BIGGER_THAN_THIS;
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$ExpInf = eval 'exp(99999)';
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}
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# print "# On this machine, Inf = '$Inf'\n";
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}
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use Scalar::Util qw(set_prototype);
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use warnings;
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no warnings 'syntax'; # To avoid the (_) warnings.
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BEGIN {
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# For certain functions that we override, in 5.10 or better
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# we can set a smarter prototype that will handle the lexical $_
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# (also a 5.10+ feature).
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if ($] >= 5.010000) {
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set_prototype \&abs, '_';
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set_prototype \&cos, '_';
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set_prototype \&exp, '_';
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set_prototype \&log, '_';
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set_prototype \&sin, '_';
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set_prototype \&sqrt, '_';
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}
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}
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my $i;
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my %LOGN;
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# Regular expression for floating point numbers.
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# These days we could use Scalar::Util::lln(), I guess.
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my $gre = qr'\s*([\+\-]?(?:(?:(?:\d+(?:_\d+)*(?:\.\d*(?:_\d+)*)?|\.\d+(?:_\d+)*)(?:[eE][\+\-]?\d+(?:_\d+)*)?))|inf)'i;
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require Exporter;
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our @ISA = qw(Exporter);
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my @trig = qw(
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pi
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tan
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csc cosec sec cot cotan
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asin acos atan
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acsc acosec asec acot acotan
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sinh cosh tanh
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csch cosech sech coth cotanh
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asinh acosh atanh
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acsch acosech asech acoth acotanh
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);
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our @EXPORT = (qw(
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i Re Im rho theta arg
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sqrt log ln
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log10 logn cbrt root
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cplx cplxe
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atan2
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),
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@trig);
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my @pi = qw(pi pi2 pi4 pip2 pip4 Inf);
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our @EXPORT_OK = @pi;
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our %EXPORT_TAGS = (
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'trig' => [@trig],
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'pi' => [@pi],
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);
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use overload
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'=' => \&_copy,
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'+=' => \&_plus,
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'+' => \&_plus,
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'-=' => \&_minus,
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'-' => \&_minus,
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'*=' => \&_multiply,
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'*' => \&_multiply,
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'/=' => \&_divide,
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'/' => \&_divide,
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'**=' => \&_power,
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'**' => \&_power,
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'==' => \&_numeq,
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'<=>' => \&_spaceship,
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'neg' => \&_negate,
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'~' => \&_conjugate,
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'abs' => \&abs,
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'sqrt' => \&sqrt,
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'exp' => \&exp,
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'log' => \&log,
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'sin' => \&sin,
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'cos' => \&cos,
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'atan2' => \&atan2,
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'""' => \&_stringify;
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#
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# Package "privates"
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#
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my %DISPLAY_FORMAT = ('style' => 'cartesian',
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'polar_pretty_print' => 1);
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my $eps = 1e-14; # Epsilon
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#
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# Object attributes (internal):
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# cartesian [real, imaginary] -- cartesian form
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# polar [rho, theta] -- polar form
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# c_dirty cartesian form not up-to-date
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# p_dirty polar form not up-to-date
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# display display format (package's global when not set)
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#
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# Die on bad *make() arguments.
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sub _cannot_make {
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die "@{[(caller(1))[3]]}: Cannot take $_[0] of '$_[1]'.\n";
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}
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sub _make {
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my $arg = shift;
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my ($p, $q);
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if ($arg =~ /^$gre$/) {
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($p, $q) = ($1, 0);
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} elsif ($arg =~ /^(?:$gre)?$gre\s*i\s*$/) {
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($p, $q) = ($1 || 0, $2);
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} elsif ($arg =~ /^\s*\(\s*$gre\s*(?:,\s*$gre\s*)?\)\s*$/) {
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($p, $q) = ($1, $2 || 0);
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}
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if (defined $p) {
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$p =~ s/^\+//;
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$p =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
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$q =~ s/^\+//;
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$q =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
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}
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return ($p, $q);
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}
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sub _emake {
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my $arg = shift;
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my ($p, $q);
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if ($arg =~ /^\s*\[\s*$gre\s*(?:,\s*$gre\s*)?\]\s*$/) {
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($p, $q) = ($1, $2 || 0);
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} elsif ($arg =~ m!^\s*\[\s*$gre\s*(?:,\s*([-+]?\d*\s*)?pi(?:/\s*(\d+))?\s*)?\]\s*$!) {
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($p, $q) = ($1, ($2 eq '-' ? -1 : ($2 || 1)) * pi() / ($3 || 1));
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} elsif ($arg =~ /^\s*\[\s*$gre\s*\]\s*$/) {
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($p, $q) = ($1, 0);
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} elsif ($arg =~ /^\s*$gre\s*$/) {
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($p, $q) = ($1, 0);
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}
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if (defined $p) {
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$p =~ s/^\+//;
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$q =~ s/^\+//;
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$p =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
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$q =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
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}
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return ($p, $q);
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}
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sub _copy {
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my $self = shift;
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my $clone = {%$self};
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if ($self->{'cartesian'}) {
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$clone->{'cartesian'} = [@{$self->{'cartesian'}}];
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}
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if ($self->{'polar'}) {
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$clone->{'polar'} = [@{$self->{'polar'}}];
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}
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bless $clone,__PACKAGE__;
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return $clone;
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}
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#
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# ->make
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#
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# Create a new complex number (cartesian form)
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#
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sub make {
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my $self = bless {}, shift;
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my ($re, $im);
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if (@_ == 0) {
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($re, $im) = (0, 0);
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} elsif (@_ == 1) {
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return (ref $self)->emake($_[0])
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if ($_[0] =~ /^\s*\[/);
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($re, $im) = _make($_[0]);
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} elsif (@_ == 2) {
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($re, $im) = @_;
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}
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if (defined $re) {
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_cannot_make("real part", $re) unless $re =~ /^$gre$/;
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}
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$im ||= 0;
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_cannot_make("imaginary part", $im) unless $im =~ /^$gre$/;
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$self->_set_cartesian([$re, $im ]);
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$self->display_format('cartesian');
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return $self;
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}
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#
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# ->emake
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#
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# Create a new complex number (exponential form)
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#
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sub emake {
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my $self = bless {}, shift;
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my ($rho, $theta);
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if (@_ == 0) {
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($rho, $theta) = (0, 0);
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} elsif (@_ == 1) {
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return (ref $self)->make($_[0])
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if ($_[0] =~ /^\s*\(/ || $_[0] =~ /i\s*$/);
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($rho, $theta) = _emake($_[0]);
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} elsif (@_ == 2) {
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($rho, $theta) = @_;
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}
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if (defined $rho && defined $theta) {
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if ($rho < 0) {
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$rho = -$rho;
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$theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
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}
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}
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if (defined $rho) {
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_cannot_make("rho", $rho) unless $rho =~ /^$gre$/;
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}
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$theta ||= 0;
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_cannot_make("theta", $theta) unless $theta =~ /^$gre$/;
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$self->_set_polar([$rho, $theta]);
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$self->display_format('polar');
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return $self;
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}
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sub new { &make } # For backward compatibility only.
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#
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# cplx
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#
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# Creates a complex number from a (re, im) tuple.
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# This avoids the burden of writing Math::Complex->make(re, im).
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#
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sub cplx {
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return __PACKAGE__->make(@_);
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}
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#
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# cplxe
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#
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# Creates a complex number from a (rho, theta) tuple.
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# This avoids the burden of writing Math::Complex->emake(rho, theta).
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#
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sub cplxe {
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return __PACKAGE__->emake(@_);
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}
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|
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|
#
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|
# pi
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#
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# The number defined as pi = 180 degrees
|
||
|
#
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sub pi () { 4 * CORE::atan2(1, 1) }
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|
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#
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# pi2
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#
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# The full circle
|
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|
#
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sub pi2 () { 2 * pi }
|
||
|
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|
#
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||
|
# pi4
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|
#
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# The full circle twice.
|
||
|
#
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||
|
sub pi4 () { 4 * pi }
|
||
|
|
||
|
#
|
||
|
# pip2
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||
|
#
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||
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# The quarter circle
|
||
|
#
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||
|
sub pip2 () { pi / 2 }
|
||
|
|
||
|
#
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||
|
# pip4
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||
|
#
|
||
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# The eighth circle.
|
||
|
#
|
||
|
sub pip4 () { pi / 4 }
|
||
|
|
||
|
#
|
||
|
# _uplog10
|
||
|
#
|
||
|
# Used in log10().
|
||
|
#
|
||
|
sub _uplog10 () { 1 / CORE::log(10) }
|
||
|
|
||
|
#
|
||
|
# i
|
||
|
#
|
||
|
# The number defined as i*i = -1;
|
||
|
#
|
||
|
sub i () {
|
||
|
return $i if ($i);
|
||
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$i = bless {};
|
||
|
$i->{'cartesian'} = [0, 1];
|
||
|
$i->{'polar'} = [1, pip2];
|
||
|
$i->{c_dirty} = 0;
|
||
|
$i->{p_dirty} = 0;
|
||
|
return $i;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# _ip2
|
||
|
#
|
||
|
# Half of i.
|
||
|
#
|
||
|
sub _ip2 () { i / 2 }
|
||
|
|
||
|
#
|
||
|
# Attribute access/set routines
|
||
|
#
|
||
|
|
||
|
sub _cartesian {$_[0]->{c_dirty} ?
|
||
|
$_[0]->_update_cartesian : $_[0]->{'cartesian'}}
|
||
|
sub _polar {$_[0]->{p_dirty} ?
|
||
|
$_[0]->_update_polar : $_[0]->{'polar'}}
|
||
|
|
||
|
sub _set_cartesian { $_[0]->{p_dirty}++; $_[0]->{c_dirty} = 0;
|
||
|
$_[0]->{'cartesian'} = $_[1] }
|
||
|
sub _set_polar { $_[0]->{c_dirty}++; $_[0]->{p_dirty} = 0;
|
||
|
$_[0]->{'polar'} = $_[1] }
|
||
|
|
||
|
#
|
||
|
# ->_update_cartesian
|
||
|
#
|
||
|
# Recompute and return the cartesian form, given accurate polar form.
|
||
|
#
|
||
|
sub _update_cartesian {
|
||
|
my $self = shift;
|
||
|
my ($r, $t) = @{$self->{'polar'}};
|
||
|
$self->{c_dirty} = 0;
|
||
|
return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)];
|
||
|
}
|
||
|
|
||
|
#
|
||
|
#
|
||
|
# ->_update_polar
|
||
|
#
|
||
|
# Recompute and return the polar form, given accurate cartesian form.
|
||
|
#
|
||
|
sub _update_polar {
|
||
|
my $self = shift;
|
||
|
my ($x, $y) = @{$self->{'cartesian'}};
|
||
|
$self->{p_dirty} = 0;
|
||
|
return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0;
|
||
|
return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y),
|
||
|
CORE::atan2($y, $x)];
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_plus)
|
||
|
#
|
||
|
# Computes z1+z2.
|
||
|
#
|
||
|
sub _plus {
|
||
|
my ($z1, $z2, $regular) = @_;
|
||
|
my ($re1, $im1) = @{$z1->_cartesian};
|
||
|
$z2 = cplx($z2) unless ref $z2;
|
||
|
my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
|
||
|
unless (defined $regular) {
|
||
|
$z1->_set_cartesian([$re1 + $re2, $im1 + $im2]);
|
||
|
return $z1;
|
||
|
}
|
||
|
return (ref $z1)->make($re1 + $re2, $im1 + $im2);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_minus)
|
||
|
#
|
||
|
# Computes z1-z2.
|
||
|
#
|
||
|
sub _minus {
|
||
|
my ($z1, $z2, $inverted) = @_;
|
||
|
my ($re1, $im1) = @{$z1->_cartesian};
|
||
|
$z2 = cplx($z2) unless ref $z2;
|
||
|
my ($re2, $im2) = @{$z2->_cartesian};
|
||
|
unless (defined $inverted) {
|
||
|
$z1->_set_cartesian([$re1 - $re2, $im1 - $im2]);
|
||
|
return $z1;
|
||
|
}
|
||
|
return $inverted ?
|
||
|
(ref $z1)->make($re2 - $re1, $im2 - $im1) :
|
||
|
(ref $z1)->make($re1 - $re2, $im1 - $im2);
|
||
|
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_multiply)
|
||
|
#
|
||
|
# Computes z1*z2.
|
||
|
#
|
||
|
sub _multiply {
|
||
|
my ($z1, $z2, $regular) = @_;
|
||
|
if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
|
||
|
# if both polar better use polar to avoid rounding errors
|
||
|
my ($r1, $t1) = @{$z1->_polar};
|
||
|
my ($r2, $t2) = @{$z2->_polar};
|
||
|
my $t = $t1 + $t2;
|
||
|
if ($t > pi()) { $t -= pi2 }
|
||
|
elsif ($t <= -pi()) { $t += pi2 }
|
||
|
unless (defined $regular) {
|
||
|
$z1->_set_polar([$r1 * $r2, $t]);
|
||
|
return $z1;
|
||
|
}
|
||
|
return (ref $z1)->emake($r1 * $r2, $t);
|
||
|
} else {
|
||
|
my ($x1, $y1) = @{$z1->_cartesian};
|
||
|
if (ref $z2) {
|
||
|
my ($x2, $y2) = @{$z2->_cartesian};
|
||
|
return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2);
|
||
|
} else {
|
||
|
return (ref $z1)->make($x1*$z2, $y1*$z2);
|
||
|
}
|
||
|
}
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# _divbyzero
|
||
|
#
|
||
|
# Die on division by zero.
|
||
|
#
|
||
|
sub _divbyzero {
|
||
|
my $mess = "$_[0]: Division by zero.\n";
|
||
|
|
||
|
if (defined $_[1]) {
|
||
|
$mess .= "(Because in the definition of $_[0], the divisor ";
|
||
|
$mess .= "$_[1] " unless ("$_[1]" eq '0');
|
||
|
$mess .= "is 0)\n";
|
||
|
}
|
||
|
|
||
|
my @up = caller(1);
|
||
|
|
||
|
$mess .= "Died at $up[1] line $up[2].\n";
|
||
|
|
||
|
die $mess;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_divide)
|
||
|
#
|
||
|
# Computes z1/z2.
|
||
|
#
|
||
|
sub _divide {
|
||
|
my ($z1, $z2, $inverted) = @_;
|
||
|
if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
|
||
|
# if both polar better use polar to avoid rounding errors
|
||
|
my ($r1, $t1) = @{$z1->_polar};
|
||
|
my ($r2, $t2) = @{$z2->_polar};
|
||
|
my $t;
|
||
|
if ($inverted) {
|
||
|
_divbyzero "$z2/0" if ($r1 == 0);
|
||
|
$t = $t2 - $t1;
|
||
|
if ($t > pi()) { $t -= pi2 }
|
||
|
elsif ($t <= -pi()) { $t += pi2 }
|
||
|
return (ref $z1)->emake($r2 / $r1, $t);
|
||
|
} else {
|
||
|
_divbyzero "$z1/0" if ($r2 == 0);
|
||
|
$t = $t1 - $t2;
|
||
|
if ($t > pi()) { $t -= pi2 }
|
||
|
elsif ($t <= -pi()) { $t += pi2 }
|
||
|
return (ref $z1)->emake($r1 / $r2, $t);
|
||
|
}
|
||
|
} else {
|
||
|
my ($d, $x2, $y2);
|
||
|
if ($inverted) {
|
||
|
($x2, $y2) = @{$z1->_cartesian};
|
||
|
$d = $x2*$x2 + $y2*$y2;
|
||
|
_divbyzero "$z2/0" if $d == 0;
|
||
|
return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d);
|
||
|
} else {
|
||
|
my ($x1, $y1) = @{$z1->_cartesian};
|
||
|
if (ref $z2) {
|
||
|
($x2, $y2) = @{$z2->_cartesian};
|
||
|
$d = $x2*$x2 + $y2*$y2;
|
||
|
_divbyzero "$z1/0" if $d == 0;
|
||
|
my $u = ($x1*$x2 + $y1*$y2)/$d;
|
||
|
my $v = ($y1*$x2 - $x1*$y2)/$d;
|
||
|
return (ref $z1)->make($u, $v);
|
||
|
} else {
|
||
|
_divbyzero "$z1/0" if $z2 == 0;
|
||
|
return (ref $z1)->make($x1/$z2, $y1/$z2);
|
||
|
}
|
||
|
}
|
||
|
}
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_power)
|
||
|
#
|
||
|
# Computes z1**z2 = exp(z2 * log z1)).
|
||
|
#
|
||
|
sub _power {
|
||
|
my ($z1, $z2, $inverted) = @_;
|
||
|
if ($inverted) {
|
||
|
return 1 if $z1 == 0 || $z2 == 1;
|
||
|
return 0 if $z2 == 0 && Re($z1) > 0;
|
||
|
} else {
|
||
|
return 1 if $z2 == 0 || $z1 == 1;
|
||
|
return 0 if $z1 == 0 && Re($z2) > 0;
|
||
|
}
|
||
|
my $w = $inverted ? &exp($z1 * &log($z2))
|
||
|
: &exp($z2 * &log($z1));
|
||
|
# If both arguments cartesian, return cartesian, else polar.
|
||
|
return $z1->{c_dirty} == 0 &&
|
||
|
(not ref $z2 or $z2->{c_dirty} == 0) ?
|
||
|
cplx(@{$w->_cartesian}) : $w;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_spaceship)
|
||
|
#
|
||
|
# Computes z1 <=> z2.
|
||
|
# Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i.
|
||
|
#
|
||
|
sub _spaceship {
|
||
|
my ($z1, $z2, $inverted) = @_;
|
||
|
my ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
|
||
|
my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
|
||
|
my $sgn = $inverted ? -1 : 1;
|
||
|
return $sgn * ($re1 <=> $re2) if $re1 != $re2;
|
||
|
return $sgn * ($im1 <=> $im2);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_numeq)
|
||
|
#
|
||
|
# Computes z1 == z2.
|
||
|
#
|
||
|
# (Required in addition to _spaceship() because of NaNs.)
|
||
|
sub _numeq {
|
||
|
my ($z1, $z2, $inverted) = @_;
|
||
|
my ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
|
||
|
my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
|
||
|
return $re1 == $re2 && $im1 == $im2 ? 1 : 0;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_negate)
|
||
|
#
|
||
|
# Computes -z.
|
||
|
#
|
||
|
sub _negate {
|
||
|
my ($z) = @_;
|
||
|
if ($z->{c_dirty}) {
|
||
|
my ($r, $t) = @{$z->_polar};
|
||
|
$t = ($t <= 0) ? $t + pi : $t - pi;
|
||
|
return (ref $z)->emake($r, $t);
|
||
|
}
|
||
|
my ($re, $im) = @{$z->_cartesian};
|
||
|
return (ref $z)->make(-$re, -$im);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_conjugate)
|
||
|
#
|
||
|
# Compute complex's _conjugate.
|
||
|
#
|
||
|
sub _conjugate {
|
||
|
my ($z) = @_;
|
||
|
if ($z->{c_dirty}) {
|
||
|
my ($r, $t) = @{$z->_polar};
|
||
|
return (ref $z)->emake($r, -$t);
|
||
|
}
|
||
|
my ($re, $im) = @{$z->_cartesian};
|
||
|
return (ref $z)->make($re, -$im);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (abs)
|
||
|
#
|
||
|
# Compute or set complex's norm (rho).
|
||
|
#
|
||
|
sub abs {
|
||
|
my ($z, $rho) = @_ ? @_ : $_;
|
||
|
unless (ref $z) {
|
||
|
if (@_ == 2) {
|
||
|
$_[0] = $_[1];
|
||
|
} else {
|
||
|
return CORE::abs($z);
|
||
|
}
|
||
|
}
|
||
|
if (defined $rho) {
|
||
|
$z->{'polar'} = [ $rho, ${$z->_polar}[1] ];
|
||
|
$z->{p_dirty} = 0;
|
||
|
$z->{c_dirty} = 1;
|
||
|
return $rho;
|
||
|
} else {
|
||
|
return ${$z->_polar}[0];
|
||
|
}
|
||
|
}
|
||
|
|
||
|
sub _theta {
|
||
|
my $theta = $_[0];
|
||
|
|
||
|
if ($$theta > pi()) { $$theta -= pi2 }
|
||
|
elsif ($$theta <= -pi()) { $$theta += pi2 }
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# arg
|
||
|
#
|
||
|
# Compute or set complex's argument (theta).
|
||
|
#
|
||
|
sub arg {
|
||
|
my ($z, $theta) = @_;
|
||
|
return $z unless ref $z;
|
||
|
if (defined $theta) {
|
||
|
_theta(\$theta);
|
||
|
$z->{'polar'} = [ ${$z->_polar}[0], $theta ];
|
||
|
$z->{p_dirty} = 0;
|
||
|
$z->{c_dirty} = 1;
|
||
|
} else {
|
||
|
$theta = ${$z->_polar}[1];
|
||
|
_theta(\$theta);
|
||
|
}
|
||
|
return $theta;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (sqrt)
|
||
|
#
|
||
|
# Compute sqrt(z).
|
||
|
#
|
||
|
# It is quite tempting to use wantarray here so that in list context
|
||
|
# sqrt() would return the two solutions. This, however, would
|
||
|
# break things like
|
||
|
#
|
||
|
# print "sqrt(z) = ", sqrt($z), "\n";
|
||
|
#
|
||
|
# The two values would be printed side by side without no intervening
|
||
|
# whitespace, quite confusing.
|
||
|
# Therefore if you want the two solutions use the root().
|
||
|
#
|
||
|
sub sqrt {
|
||
|
my ($z) = @_ ? $_[0] : $_;
|
||
|
my ($re, $im) = ref $z ? @{$z->_cartesian} : ($z, 0);
|
||
|
return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re)
|
||
|
if $im == 0;
|
||
|
my ($r, $t) = @{$z->_polar};
|
||
|
return (ref $z)->emake(CORE::sqrt($r), $t/2);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# cbrt
|
||
|
#
|
||
|
# Compute cbrt(z) (cubic root).
|
||
|
#
|
||
|
# Why are we not returning three values? The same answer as for sqrt().
|
||
|
#
|
||
|
sub cbrt {
|
||
|
my ($z) = @_;
|
||
|
return $z < 0 ?
|
||
|
-CORE::exp(CORE::log(-$z)/3) :
|
||
|
($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
|
||
|
unless ref $z;
|
||
|
my ($r, $t) = @{$z->_polar};
|
||
|
return 0 if $r == 0;
|
||
|
return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# _rootbad
|
||
|
#
|
||
|
# Die on bad root.
|
||
|
#
|
||
|
sub _rootbad {
|
||
|
my $mess = "Root '$_[0]' illegal, root rank must be positive integer.\n";
|
||
|
|
||
|
my @up = caller(1);
|
||
|
|
||
|
$mess .= "Died at $up[1] line $up[2].\n";
|
||
|
|
||
|
die $mess;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# root
|
||
|
#
|
||
|
# Computes all nth root for z, returning an array whose size is n.
|
||
|
# `n' must be a positive integer.
|
||
|
#
|
||
|
# The roots are given by (for k = 0..n-1):
|
||
|
#
|
||
|
# z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n))
|
||
|
#
|
||
|
sub root {
|
||
|
my ($z, $n, $k) = @_;
|
||
|
_rootbad($n) if ($n < 1 or int($n) != $n);
|
||
|
my ($r, $t) = ref $z ?
|
||
|
@{$z->_polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
|
||
|
my $theta_inc = pi2 / $n;
|
||
|
my $rho = $r ** (1/$n);
|
||
|
my $cartesian = ref $z && $z->{c_dirty} == 0;
|
||
|
if (@_ == 2) {
|
||
|
my @root;
|
||
|
for (my $i = 0, my $theta = $t / $n;
|
||
|
$i < $n;
|
||
|
$i++, $theta += $theta_inc) {
|
||
|
my $w = cplxe($rho, $theta);
|
||
|
# Yes, $cartesian is loop invariant.
|
||
|
push @root, $cartesian ? cplx(@{$w->_cartesian}) : $w;
|
||
|
}
|
||
|
return @root;
|
||
|
} elsif (@_ == 3) {
|
||
|
my $w = cplxe($rho, $t / $n + $k * $theta_inc);
|
||
|
return $cartesian ? cplx(@{$w->_cartesian}) : $w;
|
||
|
}
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# Re
|
||
|
#
|
||
|
# Return or set Re(z).
|
||
|
#
|
||
|
sub Re {
|
||
|
my ($z, $Re) = @_;
|
||
|
return $z unless ref $z;
|
||
|
if (defined $Re) {
|
||
|
$z->{'cartesian'} = [ $Re, ${$z->_cartesian}[1] ];
|
||
|
$z->{c_dirty} = 0;
|
||
|
$z->{p_dirty} = 1;
|
||
|
} else {
|
||
|
return ${$z->_cartesian}[0];
|
||
|
}
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# Im
|
||
|
#
|
||
|
# Return or set Im(z).
|
||
|
#
|
||
|
sub Im {
|
||
|
my ($z, $Im) = @_;
|
||
|
return 0 unless ref $z;
|
||
|
if (defined $Im) {
|
||
|
$z->{'cartesian'} = [ ${$z->_cartesian}[0], $Im ];
|
||
|
$z->{c_dirty} = 0;
|
||
|
$z->{p_dirty} = 1;
|
||
|
} else {
|
||
|
return ${$z->_cartesian}[1];
|
||
|
}
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# rho
|
||
|
#
|
||
|
# Return or set rho(w).
|
||
|
#
|
||
|
sub rho {
|
||
|
Math::Complex::abs(@_);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# theta
|
||
|
#
|
||
|
# Return or set theta(w).
|
||
|
#
|
||
|
sub theta {
|
||
|
Math::Complex::arg(@_);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (exp)
|
||
|
#
|
||
|
# Computes exp(z).
|
||
|
#
|
||
|
sub exp {
|
||
|
my ($z) = @_ ? @_ : $_;
|
||
|
return CORE::exp($z) unless ref $z;
|
||
|
my ($x, $y) = @{$z->_cartesian};
|
||
|
return (ref $z)->emake(CORE::exp($x), $y);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# _logofzero
|
||
|
#
|
||
|
# Die on logarithm of zero.
|
||
|
#
|
||
|
sub _logofzero {
|
||
|
my $mess = "$_[0]: Logarithm of zero.\n";
|
||
|
|
||
|
if (defined $_[1]) {
|
||
|
$mess .= "(Because in the definition of $_[0], the argument ";
|
||
|
$mess .= "$_[1] " unless ($_[1] eq '0');
|
||
|
$mess .= "is 0)\n";
|
||
|
}
|
||
|
|
||
|
my @up = caller(1);
|
||
|
|
||
|
$mess .= "Died at $up[1] line $up[2].\n";
|
||
|
|
||
|
die $mess;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (log)
|
||
|
#
|
||
|
# Compute log(z).
|
||
|
#
|
||
|
sub log {
|
||
|
my ($z) = @_ ? @_ : $_;
|
||
|
unless (ref $z) {
|
||
|
_logofzero("log") if $z == 0;
|
||
|
return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi);
|
||
|
}
|
||
|
my ($r, $t) = @{$z->_polar};
|
||
|
_logofzero("log") if $r == 0;
|
||
|
if ($t > pi()) { $t -= pi2 }
|
||
|
elsif ($t <= -pi()) { $t += pi2 }
|
||
|
return (ref $z)->make(CORE::log($r), $t);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# ln
|
||
|
#
|
||
|
# Alias for log().
|
||
|
#
|
||
|
sub ln { Math::Complex::log(@_) }
|
||
|
|
||
|
#
|
||
|
# log10
|
||
|
#
|
||
|
# Compute log10(z).
|
||
|
#
|
||
|
|
||
|
sub log10 {
|
||
|
return Math::Complex::log($_[0]) * _uplog10;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# logn
|
||
|
#
|
||
|
# Compute logn(z,n) = log(z) / log(n)
|
||
|
#
|
||
|
sub logn {
|
||
|
my ($z, $n) = @_;
|
||
|
$z = cplx($z, 0) unless ref $z;
|
||
|
my $logn = $LOGN{$n};
|
||
|
$logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n)
|
||
|
return &log($z) / $logn;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (cos)
|
||
|
#
|
||
|
# Compute cos(z) = (exp(iz) + exp(-iz))/2.
|
||
|
#
|
||
|
sub cos {
|
||
|
my ($z) = @_ ? @_ : $_;
|
||
|
return CORE::cos($z) unless ref $z;
|
||
|
my ($x, $y) = @{$z->_cartesian};
|
||
|
my $ey = CORE::exp($y);
|
||
|
my $sx = CORE::sin($x);
|
||
|
my $cx = CORE::cos($x);
|
||
|
my $ey_1 = $ey ? 1 / $ey : Inf();
|
||
|
return (ref $z)->make($cx * ($ey + $ey_1)/2,
|
||
|
$sx * ($ey_1 - $ey)/2);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (sin)
|
||
|
#
|
||
|
# Compute sin(z) = (exp(iz) - exp(-iz))/2.
|
||
|
#
|
||
|
sub sin {
|
||
|
my ($z) = @_ ? @_ : $_;
|
||
|
return CORE::sin($z) unless ref $z;
|
||
|
my ($x, $y) = @{$z->_cartesian};
|
||
|
my $ey = CORE::exp($y);
|
||
|
my $sx = CORE::sin($x);
|
||
|
my $cx = CORE::cos($x);
|
||
|
my $ey_1 = $ey ? 1 / $ey : Inf();
|
||
|
return (ref $z)->make($sx * ($ey + $ey_1)/2,
|
||
|
$cx * ($ey - $ey_1)/2);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# tan
|
||
|
#
|
||
|
# Compute tan(z) = sin(z) / cos(z).
|
||
|
#
|
||
|
sub tan {
|
||
|
my ($z) = @_;
|
||
|
my $cz = &cos($z);
|
||
|
_divbyzero "tan($z)", "cos($z)" if $cz == 0;
|
||
|
return &sin($z) / $cz;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# sec
|
||
|
#
|
||
|
# Computes the secant sec(z) = 1 / cos(z).
|
||
|
#
|
||
|
sub sec {
|
||
|
my ($z) = @_;
|
||
|
my $cz = &cos($z);
|
||
|
_divbyzero "sec($z)", "cos($z)" if ($cz == 0);
|
||
|
return 1 / $cz;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# csc
|
||
|
#
|
||
|
# Computes the cosecant csc(z) = 1 / sin(z).
|
||
|
#
|
||
|
sub csc {
|
||
|
my ($z) = @_;
|
||
|
my $sz = &sin($z);
|
||
|
_divbyzero "csc($z)", "sin($z)" if ($sz == 0);
|
||
|
return 1 / $sz;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# cosec
|
||
|
#
|
||
|
# Alias for csc().
|
||
|
#
|
||
|
sub cosec { Math::Complex::csc(@_) }
|
||
|
|
||
|
#
|
||
|
# cot
|
||
|
#
|
||
|
# Computes cot(z) = cos(z) / sin(z).
|
||
|
#
|
||
|
sub cot {
|
||
|
my ($z) = @_;
|
||
|
my $sz = &sin($z);
|
||
|
_divbyzero "cot($z)", "sin($z)" if ($sz == 0);
|
||
|
return &cos($z) / $sz;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# cotan
|
||
|
#
|
||
|
# Alias for cot().
|
||
|
#
|
||
|
sub cotan { Math::Complex::cot(@_) }
|
||
|
|
||
|
#
|
||
|
# acos
|
||
|
#
|
||
|
# Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
|
||
|
#
|
||
|
sub acos {
|
||
|
my $z = $_[0];
|
||
|
return CORE::atan2(CORE::sqrt(1-$z*$z), $z)
|
||
|
if (! ref $z) && CORE::abs($z) <= 1;
|
||
|
$z = cplx($z, 0) unless ref $z;
|
||
|
my ($x, $y) = @{$z->_cartesian};
|
||
|
return 0 if $x == 1 && $y == 0;
|
||
|
my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
|
||
|
my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
|
||
|
my $alpha = ($t1 + $t2)/2;
|
||
|
my $beta = ($t1 - $t2)/2;
|
||
|
$alpha = 1 if $alpha < 1;
|
||
|
if ($beta > 1) { $beta = 1 }
|
||
|
elsif ($beta < -1) { $beta = -1 }
|
||
|
my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta);
|
||
|
my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
|
||
|
$v = -$v if $y > 0 || ($y == 0 && $x < -1);
|
||
|
return (ref $z)->make($u, $v);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# asin
|
||
|
#
|
||
|
# Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
|
||
|
#
|
||
|
sub asin {
|
||
|
my $z = $_[0];
|
||
|
return CORE::atan2($z, CORE::sqrt(1-$z*$z))
|
||
|
if (! ref $z) && CORE::abs($z) <= 1;
|
||
|
$z = cplx($z, 0) unless ref $z;
|
||
|
my ($x, $y) = @{$z->_cartesian};
|
||
|
return 0 if $x == 0 && $y == 0;
|
||
|
my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
|
||
|
my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
|
||
|
my $alpha = ($t1 + $t2)/2;
|
||
|
my $beta = ($t1 - $t2)/2;
|
||
|
$alpha = 1 if $alpha < 1;
|
||
|
if ($beta > 1) { $beta = 1 }
|
||
|
elsif ($beta < -1) { $beta = -1 }
|
||
|
my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta));
|
||
|
my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
|
||
|
$v = -$v if $y > 0 || ($y == 0 && $x < -1);
|
||
|
return (ref $z)->make($u, $v);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# atan
|
||
|
#
|
||
|
# Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)).
|
||
|
#
|
||
|
sub atan {
|
||
|
my ($z) = @_;
|
||
|
return CORE::atan2($z, 1) unless ref $z;
|
||
|
my ($x, $y) = ref $z ? @{$z->_cartesian} : ($z, 0);
|
||
|
return 0 if $x == 0 && $y == 0;
|
||
|
_divbyzero "atan(i)" if ( $z == i);
|
||
|
_logofzero "atan(-i)" if (-$z == i); # -i is a bad file test...
|
||
|
my $log = &log((i + $z) / (i - $z));
|
||
|
return _ip2 * $log;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# asec
|
||
|
#
|
||
|
# Computes the arc secant asec(z) = acos(1 / z).
|
||
|
#
|
||
|
sub asec {
|
||
|
my ($z) = @_;
|
||
|
_divbyzero "asec($z)", $z if ($z == 0);
|
||
|
return acos(1 / $z);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# acsc
|
||
|
#
|
||
|
# Computes the arc cosecant acsc(z) = asin(1 / z).
|
||
|
#
|
||
|
sub acsc {
|
||
|
my ($z) = @_;
|
||
|
_divbyzero "acsc($z)", $z if ($z == 0);
|
||
|
return asin(1 / $z);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# acosec
|
||
|
#
|
||
|
# Alias for acsc().
|
||
|
#
|
||
|
sub acosec { Math::Complex::acsc(@_) }
|
||
|
|
||
|
#
|
||
|
# acot
|
||
|
#
|
||
|
# Computes the arc cotangent acot(z) = atan(1 / z)
|
||
|
#
|
||
|
sub acot {
|
||
|
my ($z) = @_;
|
||
|
_divbyzero "acot(0)" if $z == 0;
|
||
|
return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z)
|
||
|
unless ref $z;
|
||
|
_divbyzero "acot(i)" if ($z - i == 0);
|
||
|
_logofzero "acot(-i)" if ($z + i == 0);
|
||
|
return atan(1 / $z);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# acotan
|
||
|
#
|
||
|
# Alias for acot().
|
||
|
#
|
||
|
sub acotan { Math::Complex::acot(@_) }
|
||
|
|
||
|
#
|
||
|
# cosh
|
||
|
#
|
||
|
# Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2.
|
||
|
#
|
||
|
sub cosh {
|
||
|
my ($z) = @_;
|
||
|
my $ex;
|
||
|
unless (ref $z) {
|
||
|
$ex = CORE::exp($z);
|
||
|
return $ex ? ($ex == $ExpInf ? Inf() : ($ex + 1/$ex)/2) : Inf();
|
||
|
}
|
||
|
my ($x, $y) = @{$z->_cartesian};
|
||
|
$ex = CORE::exp($x);
|
||
|
my $ex_1 = $ex ? 1 / $ex : Inf();
|
||
|
return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
|
||
|
CORE::sin($y) * ($ex - $ex_1)/2);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# sinh
|
||
|
#
|
||
|
# Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2.
|
||
|
#
|
||
|
sub sinh {
|
||
|
my ($z) = @_;
|
||
|
my $ex;
|
||
|
unless (ref $z) {
|
||
|
return 0 if $z == 0;
|
||
|
$ex = CORE::exp($z);
|
||
|
return $ex ? ($ex == $ExpInf ? Inf() : ($ex - 1/$ex)/2) : -Inf();
|
||
|
}
|
||
|
my ($x, $y) = @{$z->_cartesian};
|
||
|
my $cy = CORE::cos($y);
|
||
|
my $sy = CORE::sin($y);
|
||
|
$ex = CORE::exp($x);
|
||
|
my $ex_1 = $ex ? 1 / $ex : Inf();
|
||
|
return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2,
|
||
|
CORE::sin($y) * ($ex + $ex_1)/2);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# tanh
|
||
|
#
|
||
|
# Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
|
||
|
#
|
||
|
sub tanh {
|
||
|
my ($z) = @_;
|
||
|
my $cz = cosh($z);
|
||
|
_divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
|
||
|
my $sz = sinh($z);
|
||
|
return 1 if $cz == $sz;
|
||
|
return -1 if $cz == -$sz;
|
||
|
return $sz / $cz;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# sech
|
||
|
#
|
||
|
# Computes the hyperbolic secant sech(z) = 1 / cosh(z).
|
||
|
#
|
||
|
sub sech {
|
||
|
my ($z) = @_;
|
||
|
my $cz = cosh($z);
|
||
|
_divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
|
||
|
return 1 / $cz;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# csch
|
||
|
#
|
||
|
# Computes the hyperbolic cosecant csch(z) = 1 / sinh(z).
|
||
|
#
|
||
|
sub csch {
|
||
|
my ($z) = @_;
|
||
|
my $sz = sinh($z);
|
||
|
_divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
|
||
|
return 1 / $sz;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# cosech
|
||
|
#
|
||
|
# Alias for csch().
|
||
|
#
|
||
|
sub cosech { Math::Complex::csch(@_) }
|
||
|
|
||
|
#
|
||
|
# coth
|
||
|
#
|
||
|
# Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z).
|
||
|
#
|
||
|
sub coth {
|
||
|
my ($z) = @_;
|
||
|
my $sz = sinh($z);
|
||
|
_divbyzero "coth($z)", "sinh($z)" if $sz == 0;
|
||
|
my $cz = cosh($z);
|
||
|
return 1 if $cz == $sz;
|
||
|
return -1 if $cz == -$sz;
|
||
|
return $cz / $sz;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# cotanh
|
||
|
#
|
||
|
# Alias for coth().
|
||
|
#
|
||
|
sub cotanh { Math::Complex::coth(@_) }
|
||
|
|
||
|
#
|
||
|
# acosh
|
||
|
#
|
||
|
# Computes the area/inverse hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
|
||
|
#
|
||
|
sub acosh {
|
||
|
my ($z) = @_;
|
||
|
unless (ref $z) {
|
||
|
$z = cplx($z, 0);
|
||
|
}
|
||
|
my ($re, $im) = @{$z->_cartesian};
|
||
|
if ($im == 0) {
|
||
|
return CORE::log($re + CORE::sqrt($re*$re - 1))
|
||
|
if $re >= 1;
|
||
|
return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re))
|
||
|
if CORE::abs($re) < 1;
|
||
|
}
|
||
|
my $t = &sqrt($z * $z - 1) + $z;
|
||
|
# Try Taylor if looking bad (this usually means that
|
||
|
# $z was large negative, therefore the sqrt is really
|
||
|
# close to abs(z), summing that with z...)
|
||
|
$t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
|
||
|
if $t == 0;
|
||
|
my $u = &log($t);
|
||
|
$u->Im(-$u->Im) if $re < 0 && $im == 0;
|
||
|
return $re < 0 ? -$u : $u;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# asinh
|
||
|
#
|
||
|
# Computes the area/inverse hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
|
||
|
#
|
||
|
sub asinh {
|
||
|
my ($z) = @_;
|
||
|
unless (ref $z) {
|
||
|
my $t = $z + CORE::sqrt($z*$z + 1);
|
||
|
return CORE::log($t) if $t;
|
||
|
}
|
||
|
my $t = &sqrt($z * $z + 1) + $z;
|
||
|
# Try Taylor if looking bad (this usually means that
|
||
|
# $z was large negative, therefore the sqrt is really
|
||
|
# close to abs(z), summing that with z...)
|
||
|
$t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
|
||
|
if $t == 0;
|
||
|
return &log($t);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# atanh
|
||
|
#
|
||
|
# Computes the area/inverse hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)).
|
||
|
#
|
||
|
sub atanh {
|
||
|
my ($z) = @_;
|
||
|
unless (ref $z) {
|
||
|
return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1;
|
||
|
$z = cplx($z, 0);
|
||
|
}
|
||
|
_divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0);
|
||
|
_logofzero 'atanh(-1)' if (1 + $z == 0);
|
||
|
return 0.5 * &log((1 + $z) / (1 - $z));
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# asech
|
||
|
#
|
||
|
# Computes the area/inverse hyperbolic secant asech(z) = acosh(1 / z).
|
||
|
#
|
||
|
sub asech {
|
||
|
my ($z) = @_;
|
||
|
_divbyzero 'asech(0)', "$z" if ($z == 0);
|
||
|
return acosh(1 / $z);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# acsch
|
||
|
#
|
||
|
# Computes the area/inverse hyperbolic cosecant acsch(z) = asinh(1 / z).
|
||
|
#
|
||
|
sub acsch {
|
||
|
my ($z) = @_;
|
||
|
_divbyzero 'acsch(0)', $z if ($z == 0);
|
||
|
return asinh(1 / $z);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# acosech
|
||
|
#
|
||
|
# Alias for acosh().
|
||
|
#
|
||
|
sub acosech { Math::Complex::acsch(@_) }
|
||
|
|
||
|
#
|
||
|
# acoth
|
||
|
#
|
||
|
# Computes the area/inverse hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
|
||
|
#
|
||
|
sub acoth {
|
||
|
my ($z) = @_;
|
||
|
_divbyzero 'acoth(0)' if ($z == 0);
|
||
|
unless (ref $z) {
|
||
|
return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
|
||
|
$z = cplx($z, 0);
|
||
|
}
|
||
|
_divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0);
|
||
|
_logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0);
|
||
|
return &log((1 + $z) / ($z - 1)) / 2;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# acotanh
|
||
|
#
|
||
|
# Alias for acot().
|
||
|
#
|
||
|
sub acotanh { Math::Complex::acoth(@_) }
|
||
|
|
||
|
#
|
||
|
# (atan2)
|
||
|
#
|
||
|
# Compute atan(z1/z2), minding the right quadrant.
|
||
|
#
|
||
|
sub atan2 {
|
||
|
my ($z1, $z2, $inverted) = @_;
|
||
|
my ($re1, $im1, $re2, $im2);
|
||
|
if ($inverted) {
|
||
|
($re1, $im1) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
|
||
|
($re2, $im2) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
|
||
|
} else {
|
||
|
($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
|
||
|
($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
|
||
|
}
|
||
|
if ($im1 || $im2) {
|
||
|
# In MATLAB the imaginary parts are ignored.
|
||
|
# warn "atan2: Imaginary parts ignored";
|
||
|
# http://documents.wolfram.com/mathematica/functions/ArcTan
|
||
|
# NOTE: Mathematica ArcTan[x,y] while atan2(y,x)
|
||
|
my $s = $z1 * $z1 + $z2 * $z2;
|
||
|
_divbyzero("atan2") if $s == 0;
|
||
|
my $i = &i;
|
||
|
my $r = $z2 + $z1 * $i;
|
||
|
return -$i * &log($r / &sqrt( $s ));
|
||
|
}
|
||
|
return CORE::atan2($re1, $re2);
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# display_format
|
||
|
# ->display_format
|
||
|
#
|
||
|
# Set (get if no argument) the display format for all complex numbers that
|
||
|
# don't happen to have overridden it via ->display_format
|
||
|
#
|
||
|
# When called as an object method, this actually sets the display format for
|
||
|
# the current object.
|
||
|
#
|
||
|
# Valid object formats are 'c' and 'p' for cartesian and polar. The first
|
||
|
# letter is used actually, so the type can be fully spelled out for clarity.
|
||
|
#
|
||
|
sub display_format {
|
||
|
my $self = shift;
|
||
|
my %display_format = %DISPLAY_FORMAT;
|
||
|
|
||
|
if (ref $self) { # Called as an object method
|
||
|
if (exists $self->{display_format}) {
|
||
|
my %obj = %{$self->{display_format}};
|
||
|
@display_format{keys %obj} = values %obj;
|
||
|
}
|
||
|
}
|
||
|
if (@_ == 1) {
|
||
|
$display_format{style} = shift;
|
||
|
} else {
|
||
|
my %new = @_;
|
||
|
@display_format{keys %new} = values %new;
|
||
|
}
|
||
|
|
||
|
if (ref $self) { # Called as an object method
|
||
|
$self->{display_format} = { %display_format };
|
||
|
return
|
||
|
wantarray ?
|
||
|
%{$self->{display_format}} :
|
||
|
$self->{display_format}->{style};
|
||
|
}
|
||
|
|
||
|
# Called as a class method
|
||
|
%DISPLAY_FORMAT = %display_format;
|
||
|
return
|
||
|
wantarray ?
|
||
|
%DISPLAY_FORMAT :
|
||
|
$DISPLAY_FORMAT{style};
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# (_stringify)
|
||
|
#
|
||
|
# Show nicely formatted complex number under its cartesian or polar form,
|
||
|
# depending on the current display format:
|
||
|
#
|
||
|
# . If a specific display format has been recorded for this object, use it.
|
||
|
# . Otherwise, use the generic current default for all complex numbers,
|
||
|
# which is a package global variable.
|
||
|
#
|
||
|
sub _stringify {
|
||
|
my ($z) = shift;
|
||
|
|
||
|
my $style = $z->display_format;
|
||
|
|
||
|
$style = $DISPLAY_FORMAT{style} unless defined $style;
|
||
|
|
||
|
return $z->_stringify_polar if $style =~ /^p/i;
|
||
|
return $z->_stringify_cartesian;
|
||
|
}
|
||
|
|
||
|
#
|
||
|
# ->_stringify_cartesian
|
||
|
#
|
||
|
# Stringify as a cartesian representation 'a+bi'.
|
||
|
#
|
||
|
sub _stringify_cartesian {
|
||
|
my $z = shift;
|
||
|
my ($x, $y) = @{$z->_cartesian};
|
||
|
my ($re, $im);
|
||
|
|
||
|
my %format = $z->display_format;
|
||
|
my $format = $format{format};
|
||
|
|
||
|
if ($x) {
|
||
|
if ($x =~ /^NaN[QS]?$/i) {
|
||
|
$re = $x;
|
||
|
} else {
|
||
|
if ($x =~ /^-?\Q$Inf\E$/oi) {
|
||
|
$re = $x;
|
||
|
} else {
|
||
|
$re = defined $format ? sprintf($format, $x) : $x;
|
||
|
}
|
||
|
}
|
||
|
} else {
|
||
|
undef $re;
|
||
|
}
|
||
|
|
||
|
if ($y) {
|
||
|
if ($y =~ /^(NaN[QS]?)$/i) {
|
||
|
$im = $y;
|
||
|
} else {
|
||
|
if ($y =~ /^-?\Q$Inf\E$/oi) {
|
||
|
$im = $y;
|
||
|
} else {
|
||
|
$im =
|
||
|
defined $format ?
|
||
|
sprintf($format, $y) :
|
||
|
($y == 1 ? "" : ($y == -1 ? "-" : $y));
|
||
|
}
|
||
|
}
|
||
|
$im .= "i";
|
||
|
} else {
|
||
|
undef $im;
|
||
|
}
|
||
|
|
||
|
my $str = $re;
|
||
|
|
||
|
if (defined $im) {
|
||
|
if ($y < 0) {
|
||
|
$str .= $im;
|
||
|
} elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) {
|
||
|
$str .= "+" if defined $re;
|
||
|
$str .= $im;
|
||
|
}
|
||
|
} elsif (!defined $re) {
|
||
|
$str = "0";
|
||
|
}
|
||
|
|
||
|
return $str;
|
||
|
}
|
||
|
|
||
|
|
||
|
#
|
||
|
# ->_stringify_polar
|
||
|
#
|
||
|
# Stringify as a polar representation '[r,t]'.
|
||
|
#
|
||
|
sub _stringify_polar {
|
||
|
my $z = shift;
|
||
|
my ($r, $t) = @{$z->_polar};
|
||
|
my $theta;
|
||
|
|
||
|
my %format = $z->display_format;
|
||
|
my $format = $format{format};
|
||
|
|
||
|
if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?\Q$Inf\E$/oi) {
|
||
|
$theta = $t;
|
||
|
} elsif ($t == pi) {
|
||
|
$theta = "pi";
|
||
|
} elsif ($r == 0 || $t == 0) {
|
||
|
$theta = defined $format ? sprintf($format, $t) : $t;
|
||
|
}
|
||
|
|
||
|
return "[$r,$theta]" if defined $theta;
|
||
|
|
||
|
#
|
||
|
# Try to identify pi/n and friends.
|
||
|
#
|
||
|
|
||
|
$t -= int(CORE::abs($t) / pi2) * pi2;
|
||
|
|
||
|
if ($format{polar_pretty_print} && $t) {
|
||
|
my ($a, $b);
|
||
|
for $a (2..9) {
|
||
|
$b = $t * $a / pi;
|
||
|
if ($b =~ /^-?\d+$/) {
|
||
|
$b = $b < 0 ? "-" : "" if CORE::abs($b) == 1;
|
||
|
$theta = "${b}pi/$a";
|
||
|
last;
|
||
|
}
|
||
|
}
|
||
|
}
|
||
|
|
||
|
if (defined $format) {
|
||
|
$r = sprintf($format, $r);
|
||
|
$theta = sprintf($format, $t) unless defined $theta;
|
||
|
} else {
|
||
|
$theta = $t unless defined $theta;
|
||
|
}
|
||
|
|
||
|
return "[$r,$theta]";
|
||
|
}
|
||
|
|
||
|
sub Inf {
|
||
|
return $Inf;
|
||
|
}
|
||
|
|
||
|
1;
|
||
|
__END__
|
||
|
|
||
|
=pod
|
||
|
|
||
|
=head1 NAME
|
||
|
|
||
|
Math::Complex - complex numbers and associated mathematical functions
|
||
|
|
||
|
=head1 SYNOPSIS
|
||
|
|
||
|
use Math::Complex;
|
||
|
|
||
|
$z = Math::Complex->make(5, 6);
|
||
|
$t = 4 - 3*i + $z;
|
||
|
$j = cplxe(1, 2*pi/3);
|
||
|
|
||
|
=head1 DESCRIPTION
|
||
|
|
||
|
This package lets you create and manipulate complex numbers. By default,
|
||
|
I<Perl> limits itself to real numbers, but an extra C<use> statement brings
|
||
|
full complex support, along with a full set of mathematical functions
|
||
|
typically associated with and/or extended to complex numbers.
|
||
|
|
||
|
If you wonder what complex numbers are, they were invented to be able to solve
|
||
|
the following equation:
|
||
|
|
||
|
x*x = -1
|
||
|
|
||
|
and by definition, the solution is noted I<i> (engineers use I<j> instead since
|
||
|
I<i> usually denotes an intensity, but the name does not matter). The number
|
||
|
I<i> is a pure I<imaginary> number.
|
||
|
|
||
|
The arithmetics with pure imaginary numbers works just like you would expect
|
||
|
it with real numbers... you just have to remember that
|
||
|
|
||
|
i*i = -1
|
||
|
|
||
|
so you have:
|
||
|
|
||
|
5i + 7i = i * (5 + 7) = 12i
|
||
|
4i - 3i = i * (4 - 3) = i
|
||
|
4i * 2i = -8
|
||
|
6i / 2i = 3
|
||
|
1 / i = -i
|
||
|
|
||
|
Complex numbers are numbers that have both a real part and an imaginary
|
||
|
part, and are usually noted:
|
||
|
|
||
|
a + bi
|
||
|
|
||
|
where C<a> is the I<real> part and C<b> is the I<imaginary> part. The
|
||
|
arithmetic with complex numbers is straightforward. You have to
|
||
|
keep track of the real and the imaginary parts, but otherwise the
|
||
|
rules used for real numbers just apply:
|
||
|
|
||
|
(4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
|
||
|
(2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
|
||
|
|
||
|
A graphical representation of complex numbers is possible in a plane
|
||
|
(also called the I<complex plane>, but it's really a 2D plane).
|
||
|
The number
|
||
|
|
||
|
z = a + bi
|
||
|
|
||
|
is the point whose coordinates are (a, b). Actually, it would
|
||
|
be the vector originating from (0, 0) to (a, b). It follows that the addition
|
||
|
of two complex numbers is a vectorial addition.
|
||
|
|
||
|
Since there is a bijection between a point in the 2D plane and a complex
|
||
|
number (i.e. the mapping is unique and reciprocal), a complex number
|
||
|
can also be uniquely identified with polar coordinates:
|
||
|
|
||
|
[rho, theta]
|
||
|
|
||
|
where C<rho> is the distance to the origin, and C<theta> the angle between
|
||
|
the vector and the I<x> axis. There is a notation for this using the
|
||
|
exponential form, which is:
|
||
|
|
||
|
rho * exp(i * theta)
|
||
|
|
||
|
where I<i> is the famous imaginary number introduced above. Conversion
|
||
|
between this form and the cartesian form C<a + bi> is immediate:
|
||
|
|
||
|
a = rho * cos(theta)
|
||
|
b = rho * sin(theta)
|
||
|
|
||
|
which is also expressed by this formula:
|
||
|
|
||
|
z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
|
||
|
|
||
|
In other words, it's the projection of the vector onto the I<x> and I<y>
|
||
|
axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta>
|
||
|
the I<argument> of the complex number. The I<norm> of C<z> is
|
||
|
marked here as C<abs(z)>.
|
||
|
|
||
|
The polar notation (also known as the trigonometric representation) is
|
||
|
much more handy for performing multiplications and divisions of
|
||
|
complex numbers, whilst the cartesian notation is better suited for
|
||
|
additions and subtractions. Real numbers are on the I<x> axis, and
|
||
|
therefore I<y> or I<theta> is zero or I<pi>.
|
||
|
|
||
|
All the common operations that can be performed on a real number have
|
||
|
been defined to work on complex numbers as well, and are merely
|
||
|
I<extensions> of the operations defined on real numbers. This means
|
||
|
they keep their natural meaning when there is no imaginary part, provided
|
||
|
the number is within their definition set.
|
||
|
|
||
|
For instance, the C<sqrt> routine which computes the square root of
|
||
|
its argument is only defined for non-negative real numbers and yields a
|
||
|
non-negative real number (it is an application from B<R+> to B<R+>).
|
||
|
If we allow it to return a complex number, then it can be extended to
|
||
|
negative real numbers to become an application from B<R> to B<C> (the
|
||
|
set of complex numbers):
|
||
|
|
||
|
sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
|
||
|
|
||
|
It can also be extended to be an application from B<C> to B<C>,
|
||
|
whilst its restriction to B<R> behaves as defined above by using
|
||
|
the following definition:
|
||
|
|
||
|
sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
|
||
|
|
||
|
Indeed, a negative real number can be noted C<[x,pi]> (the modulus
|
||
|
I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative
|
||
|
number) and the above definition states that
|
||
|
|
||
|
sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
|
||
|
|
||
|
which is exactly what we had defined for negative real numbers above.
|
||
|
The C<sqrt> returns only one of the solutions: if you want the both,
|
||
|
use the C<root> function.
|
||
|
|
||
|
All the common mathematical functions defined on real numbers that
|
||
|
are extended to complex numbers share that same property of working
|
||
|
I<as usual> when the imaginary part is zero (otherwise, it would not
|
||
|
be called an extension, would it?).
|
||
|
|
||
|
A I<new> operation possible on a complex number that is
|
||
|
the identity for real numbers is called the I<conjugate>, and is noted
|
||
|
with a horizontal bar above the number, or C<~z> here.
|
||
|
|
||
|
z = a + bi
|
||
|
~z = a - bi
|
||
|
|
||
|
Simple... Now look:
|
||
|
|
||
|
z * ~z = (a + bi) * (a - bi) = a*a + b*b
|
||
|
|
||
|
We saw that the norm of C<z> was noted C<abs(z)> and was defined as the
|
||
|
distance to the origin, also known as:
|
||
|
|
||
|
rho = abs(z) = sqrt(a*a + b*b)
|
||
|
|
||
|
so
|
||
|
|
||
|
z * ~z = abs(z) ** 2
|
||
|
|
||
|
If z is a pure real number (i.e. C<b == 0>), then the above yields:
|
||
|
|
||
|
a * a = abs(a) ** 2
|
||
|
|
||
|
which is true (C<abs> has the regular meaning for real number, i.e. stands
|
||
|
for the absolute value). This example explains why the norm of C<z> is
|
||
|
noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet
|
||
|
is the regular C<abs> we know when the complex number actually has no
|
||
|
imaginary part... This justifies I<a posteriori> our use of the C<abs>
|
||
|
notation for the norm.
|
||
|
|
||
|
=head1 OPERATIONS
|
||
|
|
||
|
Given the following notations:
|
||
|
|
||
|
z1 = a + bi = r1 * exp(i * t1)
|
||
|
z2 = c + di = r2 * exp(i * t2)
|
||
|
z = <any complex or real number>
|
||
|
|
||
|
the following (overloaded) operations are supported on complex numbers:
|
||
|
|
||
|
z1 + z2 = (a + c) + i(b + d)
|
||
|
z1 - z2 = (a - c) + i(b - d)
|
||
|
z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
|
||
|
z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
|
||
|
z1 ** z2 = exp(z2 * log z1)
|
||
|
~z = a - bi
|
||
|
abs(z) = r1 = sqrt(a*a + b*b)
|
||
|
sqrt(z) = sqrt(r1) * exp(i * t/2)
|
||
|
exp(z) = exp(a) * exp(i * b)
|
||
|
log(z) = log(r1) + i*t
|
||
|
sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
|
||
|
cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
|
||
|
atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.
|
||
|
|
||
|
The definition used for complex arguments of atan2() is
|
||
|
|
||
|
-i log((x + iy)/sqrt(x*x+y*y))
|
||
|
|
||
|
Note that atan2(0, 0) is not well-defined.
|
||
|
|
||
|
The following extra operations are supported on both real and complex
|
||
|
numbers:
|
||
|
|
||
|
Re(z) = a
|
||
|
Im(z) = b
|
||
|
arg(z) = t
|
||
|
abs(z) = r
|
||
|
|
||
|
cbrt(z) = z ** (1/3)
|
||
|
log10(z) = log(z) / log(10)
|
||
|
logn(z, n) = log(z) / log(n)
|
||
|
|
||
|
tan(z) = sin(z) / cos(z)
|
||
|
|
||
|
csc(z) = 1 / sin(z)
|
||
|
sec(z) = 1 / cos(z)
|
||
|
cot(z) = 1 / tan(z)
|
||
|
|
||
|
asin(z) = -i * log(i*z + sqrt(1-z*z))
|
||
|
acos(z) = -i * log(z + i*sqrt(1-z*z))
|
||
|
atan(z) = i/2 * log((i+z) / (i-z))
|
||
|
|
||
|
acsc(z) = asin(1 / z)
|
||
|
asec(z) = acos(1 / z)
|
||
|
acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))
|
||
|
|
||
|
sinh(z) = 1/2 (exp(z) - exp(-z))
|
||
|
cosh(z) = 1/2 (exp(z) + exp(-z))
|
||
|
tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))
|
||
|
|
||
|
csch(z) = 1 / sinh(z)
|
||
|
sech(z) = 1 / cosh(z)
|
||
|
coth(z) = 1 / tanh(z)
|
||
|
|
||
|
asinh(z) = log(z + sqrt(z*z+1))
|
||
|
acosh(z) = log(z + sqrt(z*z-1))
|
||
|
atanh(z) = 1/2 * log((1+z) / (1-z))
|
||
|
|
||
|
acsch(z) = asinh(1 / z)
|
||
|
asech(z) = acosh(1 / z)
|
||
|
acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
|
||
|
|
||
|
I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>,
|
||
|
I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>,
|
||
|
I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>,
|
||
|
I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>,
|
||
|
C<rho>, and C<theta> can be used also as mutators. The C<cbrt>
|
||
|
returns only one of the solutions: if you want all three, use the
|
||
|
C<root> function.
|
||
|
|
||
|
The I<root> function is available to compute all the I<n>
|
||
|
roots of some complex, where I<n> is a strictly positive integer.
|
||
|
There are exactly I<n> such roots, returned as a list. Getting the
|
||
|
number mathematicians call C<j> such that:
|
||
|
|
||
|
1 + j + j*j = 0;
|
||
|
|
||
|
is a simple matter of writing:
|
||
|
|
||
|
$j = ((root(1, 3))[1];
|
||
|
|
||
|
The I<k>th root for C<z = [r,t]> is given by:
|
||
|
|
||
|
(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
|
||
|
|
||
|
You can return the I<k>th root directly by C<root(z, n, k)>,
|
||
|
indexing starting from I<zero> and ending at I<n - 1>.
|
||
|
|
||
|
The I<spaceship> numeric comparison operator, E<lt>=E<gt>, is also
|
||
|
defined. In order to ensure its restriction to real numbers is conform
|
||
|
to what you would expect, the comparison is run on the real part of
|
||
|
the complex number first, and imaginary parts are compared only when
|
||
|
the real parts match.
|
||
|
|
||
|
=head1 CREATION
|
||
|
|
||
|
To create a complex number, use either:
|
||
|
|
||
|
$z = Math::Complex->make(3, 4);
|
||
|
$z = cplx(3, 4);
|
||
|
|
||
|
if you know the cartesian form of the number, or
|
||
|
|
||
|
$z = 3 + 4*i;
|
||
|
|
||
|
if you like. To create a number using the polar form, use either:
|
||
|
|
||
|
$z = Math::Complex->emake(5, pi/3);
|
||
|
$x = cplxe(5, pi/3);
|
||
|
|
||
|
instead. The first argument is the modulus, the second is the angle
|
||
|
(in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a
|
||
|
notation for complex numbers in the polar form).
|
||
|
|
||
|
It is possible to write:
|
||
|
|
||
|
$x = cplxe(-3, pi/4);
|
||
|
|
||
|
but that will be silently converted into C<[3,-3pi/4]>, since the
|
||
|
modulus must be non-negative (it represents the distance to the origin
|
||
|
in the complex plane).
|
||
|
|
||
|
It is also possible to have a complex number as either argument of the
|
||
|
C<make>, C<emake>, C<cplx>, and C<cplxe>: the appropriate component of
|
||
|
the argument will be used.
|
||
|
|
||
|
$z1 = cplx(-2, 1);
|
||
|
$z2 = cplx($z1, 4);
|
||
|
|
||
|
The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
|
||
|
understand a single (string) argument of the forms
|
||
|
|
||
|
2-3i
|
||
|
-3i
|
||
|
[2,3]
|
||
|
[2,-3pi/4]
|
||
|
[2]
|
||
|
|
||
|
in which case the appropriate cartesian and exponential components
|
||
|
will be parsed from the string and used to create new complex numbers.
|
||
|
The imaginary component and the theta, respectively, will default to zero.
|
||
|
|
||
|
The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
|
||
|
understand the case of no arguments: this means plain zero or (0, 0).
|
||
|
|
||
|
=head1 DISPLAYING
|
||
|
|
||
|
When printed, a complex number is usually shown under its cartesian
|
||
|
style I<a+bi>, but there are legitimate cases where the polar style
|
||
|
I<[r,t]> is more appropriate. The process of converting the complex
|
||
|
number into a string that can be displayed is known as I<stringification>.
|
||
|
|
||
|
By calling the class method C<Math::Complex::display_format> and
|
||
|
supplying either C<"polar"> or C<"cartesian"> as an argument, you
|
||
|
override the default display style, which is C<"cartesian">. Not
|
||
|
supplying any argument returns the current settings.
|
||
|
|
||
|
This default can be overridden on a per-number basis by calling the
|
||
|
C<display_format> method instead. As before, not supplying any argument
|
||
|
returns the current display style for this number. Otherwise whatever you
|
||
|
specify will be the new display style for I<this> particular number.
|
||
|
|
||
|
For instance:
|
||
|
|
||
|
use Math::Complex;
|
||
|
|
||
|
Math::Complex::display_format('polar');
|
||
|
$j = (root(1, 3))[1];
|
||
|
print "j = $j\n"; # Prints "j = [1,2pi/3]"
|
||
|
$j->display_format('cartesian');
|
||
|
print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
|
||
|
|
||
|
The polar style attempts to emphasize arguments like I<k*pi/n>
|
||
|
(where I<n> is a positive integer and I<k> an integer within [-9, +9]),
|
||
|
this is called I<polar pretty-printing>.
|
||
|
|
||
|
For the reverse of stringifying, see the C<make> and C<emake>.
|
||
|
|
||
|
=head2 CHANGED IN PERL 5.6
|
||
|
|
||
|
The C<display_format> class method and the corresponding
|
||
|
C<display_format> object method can now be called using
|
||
|
a parameter hash instead of just a one parameter.
|
||
|
|
||
|
The old display format style, which can have values C<"cartesian"> or
|
||
|
C<"polar">, can be changed using the C<"style"> parameter.
|
||
|
|
||
|
$j->display_format(style => "polar");
|
||
|
|
||
|
The one parameter calling convention also still works.
|
||
|
|
||
|
$j->display_format("polar");
|
||
|
|
||
|
There are two new display parameters.
|
||
|
|
||
|
The first one is C<"format">, which is a sprintf()-style format string
|
||
|
to be used for both numeric parts of the complex number(s). The is
|
||
|
somewhat system-dependent but most often it corresponds to C<"%.15g">.
|
||
|
You can revert to the default by setting the C<format> to C<undef>.
|
||
|
|
||
|
# the $j from the above example
|
||
|
|
||
|
$j->display_format('format' => '%.5f');
|
||
|
print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
|
||
|
$j->display_format('format' => undef);
|
||
|
print "j = $j\n"; # Prints "j = -0.5+0.86603i"
|
||
|
|
||
|
Notice that this affects also the return values of the
|
||
|
C<display_format> methods: in list context the whole parameter hash
|
||
|
will be returned, as opposed to only the style parameter value.
|
||
|
This is a potential incompatibility with earlier versions if you
|
||
|
have been calling the C<display_format> method in list context.
|
||
|
|
||
|
The second new display parameter is C<"polar_pretty_print">, which can
|
||
|
be set to true or false, the default being true. See the previous
|
||
|
section for what this means.
|
||
|
|
||
|
=head1 USAGE
|
||
|
|
||
|
Thanks to overloading, the handling of arithmetics with complex numbers
|
||
|
is simple and almost transparent.
|
||
|
|
||
|
Here are some examples:
|
||
|
|
||
|
use Math::Complex;
|
||
|
|
||
|
$j = cplxe(1, 2*pi/3); # $j ** 3 == 1
|
||
|
print "j = $j, j**3 = ", $j ** 3, "\n";
|
||
|
print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
|
||
|
|
||
|
$z = -16 + 0*i; # Force it to be a complex
|
||
|
print "sqrt($z) = ", sqrt($z), "\n";
|
||
|
|
||
|
$k = exp(i * 2*pi/3);
|
||
|
print "$j - $k = ", $j - $k, "\n";
|
||
|
|
||
|
$z->Re(3); # Re, Im, arg, abs,
|
||
|
$j->arg(2); # (the last two aka rho, theta)
|
||
|
# can be used also as mutators.
|
||
|
|
||
|
=head1 CONSTANTS
|
||
|
|
||
|
=head2 PI
|
||
|
|
||
|
The constant C<pi> and some handy multiples of it (pi2, pi4,
|
||
|
and pip2 (pi/2) and pip4 (pi/4)) are also available if separately
|
||
|
exported:
|
||
|
|
||
|
use Math::Complex ':pi';
|
||
|
$third_of_circle = pi2 / 3;
|
||
|
|
||
|
=head2 Inf
|
||
|
|
||
|
The floating point infinity can be exported as a subroutine Inf():
|
||
|
|
||
|
use Math::Complex qw(Inf sinh);
|
||
|
my $AlsoInf = Inf() + 42;
|
||
|
my $AnotherInf = sinh(1e42);
|
||
|
print "$AlsoInf is $AnotherInf\n" if $AlsoInf == $AnotherInf;
|
||
|
|
||
|
Note that the stringified form of infinity varies between platforms:
|
||
|
it can be for example any of
|
||
|
|
||
|
inf
|
||
|
infinity
|
||
|
INF
|
||
|
1.#INF
|
||
|
|
||
|
or it can be something else.
|
||
|
|
||
|
Also note that in some platforms trying to use the infinity in
|
||
|
arithmetic operations may result in Perl crashing because using
|
||
|
an infinity causes SIGFPE or its moral equivalent to be sent.
|
||
|
The way to ignore this is
|
||
|
|
||
|
local $SIG{FPE} = sub { };
|
||
|
|
||
|
=head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
|
||
|
|
||
|
The division (/) and the following functions
|
||
|
|
||
|
log ln log10 logn
|
||
|
tan sec csc cot
|
||
|
atan asec acsc acot
|
||
|
tanh sech csch coth
|
||
|
atanh asech acsch acoth
|
||
|
|
||
|
cannot be computed for all arguments because that would mean dividing
|
||
|
by zero or taking logarithm of zero. These situations cause fatal
|
||
|
runtime errors looking like this
|
||
|
|
||
|
cot(0): Division by zero.
|
||
|
(Because in the definition of cot(0), the divisor sin(0) is 0)
|
||
|
Died at ...
|
||
|
|
||
|
or
|
||
|
|
||
|
atanh(-1): Logarithm of zero.
|
||
|
Died at...
|
||
|
|
||
|
For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
|
||
|
C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the
|
||
|
logarithmic functions and the C<atanh>, C<acoth>, the argument cannot
|
||
|
be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be
|
||
|
C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be
|
||
|
C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument
|
||
|
cannot be C<-i> (the negative imaginary unit). For the C<tan>,
|
||
|
C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k>
|
||
|
is any integer. atan2(0, 0) is undefined, and if the complex arguments
|
||
|
are used for atan2(), a division by zero will happen if z1**2+z2**2 == 0.
|
||
|
|
||
|
Note that because we are operating on approximations of real numbers,
|
||
|
these errors can happen when merely `too close' to the singularities
|
||
|
listed above.
|
||
|
|
||
|
=head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS
|
||
|
|
||
|
The C<make> and C<emake> accept both real and complex arguments.
|
||
|
When they cannot recognize the arguments they will die with error
|
||
|
messages like the following
|
||
|
|
||
|
Math::Complex::make: Cannot take real part of ...
|
||
|
Math::Complex::make: Cannot take real part of ...
|
||
|
Math::Complex::emake: Cannot take rho of ...
|
||
|
Math::Complex::emake: Cannot take theta of ...
|
||
|
|
||
|
=head1 BUGS
|
||
|
|
||
|
Saying C<use Math::Complex;> exports many mathematical routines in the
|
||
|
caller environment and even overrides some (C<sqrt>, C<log>, C<atan2>).
|
||
|
This is construed as a feature by the Authors, actually... ;-)
|
||
|
|
||
|
All routines expect to be given real or complex numbers. Don't attempt to
|
||
|
use BigFloat, since Perl has currently no rule to disambiguate a '+'
|
||
|
operation (for instance) between two overloaded entities.
|
||
|
|
||
|
In Cray UNICOS there is some strange numerical instability that results
|
||
|
in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware.
|
||
|
The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex.
|
||
|
Whatever it is, it does not manifest itself anywhere else where Perl runs.
|
||
|
|
||
|
=head1 SEE ALSO
|
||
|
|
||
|
L<Math::Trig>
|
||
|
|
||
|
=head1 AUTHORS
|
||
|
|
||
|
Daniel S. Lewart <F<lewart!at!uiuc.edu>>,
|
||
|
Jarkko Hietaniemi <F<jhi!at!iki.fi>>,
|
||
|
Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>,
|
||
|
Zefram <zefram@fysh.org>
|
||
|
|
||
|
=head1 LICENSE
|
||
|
|
||
|
This library is free software; you can redistribute it and/or modify
|
||
|
it under the same terms as Perl itself.
|
||
|
|
||
|
=cut
|
||
|
|
||
|
1;
|
||
|
|
||
|
# eof
|