| 1 | #
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| 2 | # Complex numbers and associated mathematical functions
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| 3 | # -- Raphael Manfredi Since Sep 1996
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| 4 | # -- Jarkko Hietaniemi Since Mar 1997
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| 5 | # -- Daniel S. Lewart Since Sep 1997
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| 6 | #
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| 7 |
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| 8 | package Math::Complex;
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| 9 |
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| 10 | use vars qw($VERSION @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS $Inf);
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| 11 |
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| 12 | $VERSION = 1.35;
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| 13 |
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| 14 | BEGIN {
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| 15 | unless ($^O eq 'unicosmk') {
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| 16 | my $e = $!;
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| 17 | # We do want an arithmetic overflow, Inf INF inf Infinity:.
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| 18 | undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i;
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| 19 | local $SIG{FPE} = sub {die};
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| 20 | my $t = CORE::exp 30;
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| 21 | $Inf = CORE::exp $t;
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| 22 | EOE
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| 23 | if (!defined $Inf) { # Try a different method
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| 24 | undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i;
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| 25 | local $SIG{FPE} = sub {die};
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| 26 | my $t = 1;
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| 27 | $Inf = $t + "1e99999999999999999999999999999999";
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| 28 | EOE
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| 29 | }
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| 30 | $! = $e; # Clear ERANGE.
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| 31 | }
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| 32 | $Inf = "Inf" if !defined $Inf || !($Inf > 0); # Desperation.
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| 33 | }
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| 34 |
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| 35 | use strict;
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| 36 |
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| 37 | my $i;
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| 38 | my %LOGN;
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| 39 |
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| 40 | # Regular expression for floating point numbers.
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| 41 | # These days we could use Scalar::Util::lln(), I guess.
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| 42 | my $gre = qr'\s*([\+\-]?(?:(?:(?:\d+(?:_\d+)*(?:\.\d*(?:_\d+)*)?|\.\d+(?:_\d+)*)(?:[eE][\+\-]?\d+(?:_\d+)*)?))|inf)'i;
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| 43 |
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| 44 | require Exporter;
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| 45 |
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| 46 | @ISA = qw(Exporter);
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| 47 |
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| 48 | my @trig = qw(
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| 49 | pi
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| 50 | tan
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| 51 | csc cosec sec cot cotan
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| 52 | asin acos atan
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| 53 | acsc acosec asec acot acotan
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| 54 | sinh cosh tanh
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| 55 | csch cosech sech coth cotanh
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| 56 | asinh acosh atanh
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| 57 | acsch acosech asech acoth acotanh
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| 58 | );
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| 59 |
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| 60 | @EXPORT = (qw(
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| 61 | i Re Im rho theta arg
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| 62 | sqrt log ln
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| 63 | log10 logn cbrt root
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| 64 | cplx cplxe
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| 65 | atan2
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| 66 | ),
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| 67 | @trig);
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| 68 |
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| 69 | @EXPORT_OK = qw(decplx);
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| 70 |
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| 71 | %EXPORT_TAGS = (
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| 72 | 'trig' => [@trig],
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| 73 | );
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| 74 |
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| 75 | use overload
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| 76 | '+' => \&plus,
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| 77 | '-' => \&minus,
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| 78 | '*' => \&multiply,
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| 79 | '/' => \÷,
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| 80 | '**' => \&power,
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| 81 | '==' => \&numeq,
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| 82 | '<=>' => \&spaceship,
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| 83 | 'neg' => \&negate,
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| 84 | '~' => \&conjugate,
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| 85 | 'abs' => \&abs,
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| 86 | 'sqrt' => \&sqrt,
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| 87 | 'exp' => \&exp,
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| 88 | 'log' => \&log,
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| 89 | 'sin' => \&sin,
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| 90 | 'cos' => \&cos,
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| 91 | 'tan' => \&tan,
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| 92 | 'atan2' => \&atan2,
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| 93 | qw("" stringify);
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| 94 |
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| 95 | #
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| 96 | # Package "privates"
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| 97 | #
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| 98 |
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| 99 | my %DISPLAY_FORMAT = ('style' => 'cartesian',
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| 100 | 'polar_pretty_print' => 1);
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| 101 | my $eps = 1e-14; # Epsilon
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| 102 |
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| 103 | #
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| 104 | # Object attributes (internal):
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| 105 | # cartesian [real, imaginary] -- cartesian form
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| 106 | # polar [rho, theta] -- polar form
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| 107 | # c_dirty cartesian form not up-to-date
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| 108 | # p_dirty polar form not up-to-date
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| 109 | # display display format (package's global when not set)
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| 110 | #
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| 111 |
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| 112 | # Die on bad *make() arguments.
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| 113 |
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| 114 | sub _cannot_make {
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| 115 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of '$_[1]'.\n";
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| 116 | }
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| 117 |
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| 118 | sub _make {
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| 119 | my $arg = shift;
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| 120 | my ($p, $q);
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| 121 |
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| 122 | if ($arg =~ /^$gre$/) {
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| 123 | ($p, $q) = ($1, 0);
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| 124 | } elsif ($arg =~ /^(?:$gre)?$gre\s*i\s*$/) {
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| 125 | ($p, $q) = ($1 || 0, $2);
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| 126 | } elsif ($arg =~ /^\s*\(\s*$gre\s*(?:,\s*$gre\s*)?\)\s*$/) {
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| 127 | ($p, $q) = ($1, $2 || 0);
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| 128 | }
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| 129 |
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| 130 | if (defined $p) {
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| 131 | $p =~ s/^\+//;
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| 132 | $p =~ s/^(-?)inf$/"${1}9**9**9"/e;
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| 133 | $q =~ s/^\+//;
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| 134 | $q =~ s/^(-?)inf$/"${1}9**9**9"/e;
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| 135 | }
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| 136 |
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| 137 | return ($p, $q);
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| 138 | }
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| 139 |
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| 140 | sub _emake {
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| 141 | my $arg = shift;
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| 142 | my ($p, $q);
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| 143 |
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| 144 | if ($arg =~ /^\s*\[\s*$gre\s*(?:,\s*$gre\s*)?\]\s*$/) {
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| 145 | ($p, $q) = ($1, $2 || 0);
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| 146 | } elsif ($arg =~ m!^\s*\[\s*$gre\s*(?:,\s*([-+]?\d*\s*)?pi(?:/\s*(\d+))?\s*)?\]\s*$!) {
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| 147 | ($p, $q) = ($1, ($2 eq '-' ? -1 : ($2 || 1)) * pi() / ($3 || 1));
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| 148 | } elsif ($arg =~ /^\s*\[\s*$gre\s*\]\s*$/) {
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| 149 | ($p, $q) = ($1, 0);
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| 150 | } elsif ($arg =~ /^\s*$gre\s*$/) {
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| 151 | ($p, $q) = ($1, 0);
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| 152 | }
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| 153 |
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| 154 | if (defined $p) {
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| 155 | $p =~ s/^\+//;
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| 156 | $q =~ s/^\+//;
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| 157 | $p =~ s/^(-?)inf$/"${1}9**9**9"/e;
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| 158 | $q =~ s/^(-?)inf$/"${1}9**9**9"/e;
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| 159 | }
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| 160 |
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| 161 | return ($p, $q);
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| 162 | }
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| 163 |
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| 164 | #
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| 165 | # ->make
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| 166 | #
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| 167 | # Create a new complex number (cartesian form)
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| 168 | #
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| 169 | sub make {
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| 170 | my $self = bless {}, shift;
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| 171 | my ($re, $im);
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| 172 | if (@_ == 0) {
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| 173 | ($re, $im) = (0, 0);
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| 174 | } elsif (@_ == 1) {
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| 175 | return (ref $self)->emake($_[0])
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| 176 | if ($_[0] =~ /^\s*\[/);
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| 177 | ($re, $im) = _make($_[0]);
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| 178 | } elsif (@_ == 2) {
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| 179 | ($re, $im) = @_;
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| 180 | }
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| 181 | if (defined $re) {
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| 182 | _cannot_make("real part", $re) unless $re =~ /^$gre$/;
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| 183 | }
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| 184 | $im ||= 0;
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| 185 | _cannot_make("imaginary part", $im) unless $im =~ /^$gre$/;
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| 186 | $self->set_cartesian([$re, $im ]);
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| 187 | $self->display_format('cartesian');
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| 188 |
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| 189 | return $self;
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| 190 | }
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| 191 |
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| 192 | #
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| 193 | # ->emake
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| 194 | #
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| 195 | # Create a new complex number (exponential form)
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| 196 | #
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| 197 | sub emake {
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| 198 | my $self = bless {}, shift;
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| 199 | my ($rho, $theta);
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| 200 | if (@_ == 0) {
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| 201 | ($rho, $theta) = (0, 0);
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| 202 | } elsif (@_ == 1) {
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| 203 | return (ref $self)->make($_[0])
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| 204 | if ($_[0] =~ /^\s*\(/ || $_[0] =~ /i\s*$/);
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| 205 | ($rho, $theta) = _emake($_[0]);
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| 206 | } elsif (@_ == 2) {
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| 207 | ($rho, $theta) = @_;
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| 208 | }
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| 209 | if (defined $rho && defined $theta) {
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| 210 | if ($rho < 0) {
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| 211 | $rho = -$rho;
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| 212 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
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| 213 | }
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| 214 | }
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| 215 | if (defined $rho) {
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| 216 | _cannot_make("rho", $rho) unless $rho =~ /^$gre$/;
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| 217 | }
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| 218 | $theta ||= 0;
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| 219 | _cannot_make("theta", $theta) unless $theta =~ /^$gre$/;
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| 220 | $self->set_polar([$rho, $theta]);
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| 221 | $self->display_format('polar');
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| 222 |
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| 223 | return $self;
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| 224 | }
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| 225 |
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| 226 | sub new { &make } # For backward compatibility only.
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| 227 |
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| 228 | #
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| 229 | # cplx
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| 230 | #
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| 231 | # Creates a complex number from a (re, im) tuple.
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| 232 | # This avoids the burden of writing Math::Complex->make(re, im).
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| 233 | #
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| 234 | sub cplx {
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| 235 | return __PACKAGE__->make(@_);
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| 236 | }
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| 237 |
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| 238 | #
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| 239 | # cplxe
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| 240 | #
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| 241 | # Creates a complex number from a (rho, theta) tuple.
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| 242 | # This avoids the burden of writing Math::Complex->emake(rho, theta).
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| 243 | #
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| 244 | sub cplxe {
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| 245 | return __PACKAGE__->emake(@_);
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| 246 | }
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| 247 |
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| 248 | #
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| 249 | # pi
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| 250 | #
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| 251 | # The number defined as pi = 180 degrees
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| 252 | #
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| 253 | sub pi () { 4 * CORE::atan2(1, 1) }
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| 254 |
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| 255 | #
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| 256 | # pit2
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| 257 | #
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| 258 | # The full circle
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| 259 | #
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| 260 | sub pit2 () { 2 * pi }
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| 261 |
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| 262 | #
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| 263 | # pip2
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| 264 | #
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| 265 | # The quarter circle
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| 266 | #
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| 267 | sub pip2 () { pi / 2 }
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| 268 |
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| 269 | #
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| 270 | # deg1
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| 271 | #
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| 272 | # One degree in radians, used in stringify_polar.
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| 273 | #
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| 274 |
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| 275 | sub deg1 () { pi / 180 }
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| 276 |
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| 277 | #
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| 278 | # uplog10
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| 279 | #
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| 280 | # Used in log10().
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| 281 | #
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| 282 | sub uplog10 () { 1 / CORE::log(10) }
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| 283 |
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| 284 | #
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| 285 | # i
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| 286 | #
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| 287 | # The number defined as i*i = -1;
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| 288 | #
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| 289 | sub i () {
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| 290 | return $i if ($i);
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| 291 | $i = bless {};
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| 292 | $i->{'cartesian'} = [0, 1];
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| 293 | $i->{'polar'} = [1, pip2];
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| 294 | $i->{c_dirty} = 0;
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| 295 | $i->{p_dirty} = 0;
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| 296 | return $i;
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| 297 | }
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| 298 |
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| 299 | #
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| 300 | # ip2
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| 301 | #
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| 302 | # Half of i.
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| 303 | #
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| 304 | sub ip2 () { i / 2 }
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| 305 |
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| 306 | #
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| 307 | # Attribute access/set routines
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| 308 | #
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| 309 |
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| 310 | sub cartesian {$_[0]->{c_dirty} ?
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| 311 | $_[0]->update_cartesian : $_[0]->{'cartesian'}}
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| 312 | sub polar {$_[0]->{p_dirty} ?
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| 313 | $_[0]->update_polar : $_[0]->{'polar'}}
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| 314 |
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| 315 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{c_dirty} = 0;
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| 316 | $_[0]->{'cartesian'} = $_[1] }
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| 317 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{p_dirty} = 0;
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| 318 | $_[0]->{'polar'} = $_[1] }
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| 319 |
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| 320 | #
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| 321 | # ->update_cartesian
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| 322 | #
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| 323 | # Recompute and return the cartesian form, given accurate polar form.
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| 324 | #
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| 325 | sub update_cartesian {
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| 326 | my $self = shift;
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| 327 | my ($r, $t) = @{$self->{'polar'}};
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| 328 | $self->{c_dirty} = 0;
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| 329 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)];
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| 330 | }
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| 331 |
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| 332 | #
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| 333 | #
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| 334 | # ->update_polar
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| 335 | #
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| 336 | # Recompute and return the polar form, given accurate cartesian form.
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| 337 | #
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| 338 | sub update_polar {
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| 339 | my $self = shift;
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| 340 | my ($x, $y) = @{$self->{'cartesian'}};
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| 341 | $self->{p_dirty} = 0;
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| 342 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0;
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| 343 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y),
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| 344 | CORE::atan2($y, $x)];
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| 345 | }
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| 346 |
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| 347 | #
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| 348 | # (plus)
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| 349 | #
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| 350 | # Computes z1+z2.
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| 351 | #
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| 352 | sub plus {
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| 353 | my ($z1, $z2, $regular) = @_;
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| 354 | my ($re1, $im1) = @{$z1->cartesian};
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| 355 | $z2 = cplx($z2) unless ref $z2;
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| 356 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
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| 357 | unless (defined $regular) {
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| 358 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]);
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| 359 | return $z1;
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| 360 | }
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| 361 | return (ref $z1)->make($re1 + $re2, $im1 + $im2);
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| 362 | }
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| 363 |
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| 364 | #
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| 365 | # (minus)
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| 366 | #
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| 367 | # Computes z1-z2.
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| 368 | #
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| 369 | sub minus {
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| 370 | my ($z1, $z2, $inverted) = @_;
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| 371 | my ($re1, $im1) = @{$z1->cartesian};
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| 372 | $z2 = cplx($z2) unless ref $z2;
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| 373 | my ($re2, $im2) = @{$z2->cartesian};
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| 374 | unless (defined $inverted) {
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| 375 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]);
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| 376 | return $z1;
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| 377 | }
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| 378 | return $inverted ?
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| 379 | (ref $z1)->make($re2 - $re1, $im2 - $im1) :
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| 380 | (ref $z1)->make($re1 - $re2, $im1 - $im2);
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| 381 |
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| 382 | }
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| 383 |
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| 384 | #
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| 385 | # (multiply)
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| 386 | #
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| 387 | # Computes z1*z2.
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| 388 | #
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| 389 | sub multiply {
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| 390 | my ($z1, $z2, $regular) = @_;
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| 391 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
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| 392 | # if both polar better use polar to avoid rounding errors
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| 393 | my ($r1, $t1) = @{$z1->polar};
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| 394 | my ($r2, $t2) = @{$z2->polar};
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| 395 | my $t = $t1 + $t2;
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| 396 | if ($t > pi()) { $t -= pit2 }
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| 397 | elsif ($t <= -pi()) { $t += pit2 }
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| 398 | unless (defined $regular) {
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| 399 | $z1->set_polar([$r1 * $r2, $t]);
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| 400 | return $z1;
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| 401 | }
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| 402 | return (ref $z1)->emake($r1 * $r2, $t);
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| 403 | } else {
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| 404 | my ($x1, $y1) = @{$z1->cartesian};
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| 405 | if (ref $z2) {
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| 406 | my ($x2, $y2) = @{$z2->cartesian};
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| 407 | return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2);
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| 408 | } else {
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| 409 | return (ref $z1)->make($x1*$z2, $y1*$z2);
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| 410 | }
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| 411 | }
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| 412 | }
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| 413 |
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|---|
| 414 | #
|
|---|
| 415 | # _divbyzero
|
|---|
| 416 | #
|
|---|
| 417 | # Die on division by zero.
|
|---|
| 418 | #
|
|---|
| 419 | sub _divbyzero {
|
|---|
| 420 | my $mess = "$_[0]: Division by zero.\n";
|
|---|
| 421 |
|
|---|
| 422 | if (defined $_[1]) {
|
|---|
| 423 | $mess .= "(Because in the definition of $_[0], the divisor ";
|
|---|
| 424 | $mess .= "$_[1] " unless ("$_[1]" eq '0');
|
|---|
| 425 | $mess .= "is 0)\n";
|
|---|
| 426 | }
|
|---|
| 427 |
|
|---|
| 428 | my @up = caller(1);
|
|---|
| 429 |
|
|---|
| 430 | $mess .= "Died at $up[1] line $up[2].\n";
|
|---|
| 431 |
|
|---|
| 432 | die $mess;
|
|---|
| 433 | }
|
|---|
| 434 |
|
|---|
| 435 | #
|
|---|
| 436 | # (divide)
|
|---|
| 437 | #
|
|---|
| 438 | # Computes z1/z2.
|
|---|
| 439 | #
|
|---|
| 440 | sub divide {
|
|---|
| 441 | my ($z1, $z2, $inverted) = @_;
|
|---|
| 442 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
|
|---|
| 443 | # if both polar better use polar to avoid rounding errors
|
|---|
| 444 | my ($r1, $t1) = @{$z1->polar};
|
|---|
| 445 | my ($r2, $t2) = @{$z2->polar};
|
|---|
| 446 | my $t;
|
|---|
| 447 | if ($inverted) {
|
|---|
| 448 | _divbyzero "$z2/0" if ($r1 == 0);
|
|---|
| 449 | $t = $t2 - $t1;
|
|---|
| 450 | if ($t > pi()) { $t -= pit2 }
|
|---|
| 451 | elsif ($t <= -pi()) { $t += pit2 }
|
|---|
| 452 | return (ref $z1)->emake($r2 / $r1, $t);
|
|---|
| 453 | } else {
|
|---|
| 454 | _divbyzero "$z1/0" if ($r2 == 0);
|
|---|
| 455 | $t = $t1 - $t2;
|
|---|
| 456 | if ($t > pi()) { $t -= pit2 }
|
|---|
| 457 | elsif ($t <= -pi()) { $t += pit2 }
|
|---|
| 458 | return (ref $z1)->emake($r1 / $r2, $t);
|
|---|
| 459 | }
|
|---|
| 460 | } else {
|
|---|
| 461 | my ($d, $x2, $y2);
|
|---|
| 462 | if ($inverted) {
|
|---|
| 463 | ($x2, $y2) = @{$z1->cartesian};
|
|---|
| 464 | $d = $x2*$x2 + $y2*$y2;
|
|---|
| 465 | _divbyzero "$z2/0" if $d == 0;
|
|---|
| 466 | return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d);
|
|---|
| 467 | } else {
|
|---|
| 468 | my ($x1, $y1) = @{$z1->cartesian};
|
|---|
| 469 | if (ref $z2) {
|
|---|
| 470 | ($x2, $y2) = @{$z2->cartesian};
|
|---|
| 471 | $d = $x2*$x2 + $y2*$y2;
|
|---|
| 472 | _divbyzero "$z1/0" if $d == 0;
|
|---|
| 473 | my $u = ($x1*$x2 + $y1*$y2)/$d;
|
|---|
| 474 | my $v = ($y1*$x2 - $x1*$y2)/$d;
|
|---|
| 475 | return (ref $z1)->make($u, $v);
|
|---|
| 476 | } else {
|
|---|
| 477 | _divbyzero "$z1/0" if $z2 == 0;
|
|---|
| 478 | return (ref $z1)->make($x1/$z2, $y1/$z2);
|
|---|
| 479 | }
|
|---|
| 480 | }
|
|---|
| 481 | }
|
|---|
| 482 | }
|
|---|
| 483 |
|
|---|
| 484 | #
|
|---|
| 485 | # (power)
|
|---|
| 486 | #
|
|---|
| 487 | # Computes z1**z2 = exp(z2 * log z1)).
|
|---|
| 488 | #
|
|---|
| 489 | sub power {
|
|---|
| 490 | my ($z1, $z2, $inverted) = @_;
|
|---|
| 491 | if ($inverted) {
|
|---|
| 492 | return 1 if $z1 == 0 || $z2 == 1;
|
|---|
| 493 | return 0 if $z2 == 0 && Re($z1) > 0;
|
|---|
| 494 | } else {
|
|---|
| 495 | return 1 if $z2 == 0 || $z1 == 1;
|
|---|
| 496 | return 0 if $z1 == 0 && Re($z2) > 0;
|
|---|
| 497 | }
|
|---|
| 498 | my $w = $inverted ? &exp($z1 * &log($z2))
|
|---|
| 499 | : &exp($z2 * &log($z1));
|
|---|
| 500 | # If both arguments cartesian, return cartesian, else polar.
|
|---|
| 501 | return $z1->{c_dirty} == 0 &&
|
|---|
| 502 | (not ref $z2 or $z2->{c_dirty} == 0) ?
|
|---|
| 503 | cplx(@{$w->cartesian}) : $w;
|
|---|
| 504 | }
|
|---|
| 505 |
|
|---|
| 506 | #
|
|---|
| 507 | # (spaceship)
|
|---|
| 508 | #
|
|---|
| 509 | # Computes z1 <=> z2.
|
|---|
| 510 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i.
|
|---|
| 511 | #
|
|---|
| 512 | sub spaceship {
|
|---|
| 513 | my ($z1, $z2, $inverted) = @_;
|
|---|
| 514 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
|
|---|
| 515 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
|
|---|
| 516 | my $sgn = $inverted ? -1 : 1;
|
|---|
| 517 | return $sgn * ($re1 <=> $re2) if $re1 != $re2;
|
|---|
| 518 | return $sgn * ($im1 <=> $im2);
|
|---|
| 519 | }
|
|---|
| 520 |
|
|---|
| 521 | #
|
|---|
| 522 | # (numeq)
|
|---|
| 523 | #
|
|---|
| 524 | # Computes z1 == z2.
|
|---|
| 525 | #
|
|---|
| 526 | # (Required in addition to spaceship() because of NaNs.)
|
|---|
| 527 | sub numeq {
|
|---|
| 528 | my ($z1, $z2, $inverted) = @_;
|
|---|
| 529 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
|
|---|
| 530 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
|
|---|
| 531 | return $re1 == $re2 && $im1 == $im2 ? 1 : 0;
|
|---|
| 532 | }
|
|---|
| 533 |
|
|---|
| 534 | #
|
|---|
| 535 | # (negate)
|
|---|
| 536 | #
|
|---|
| 537 | # Computes -z.
|
|---|
| 538 | #
|
|---|
| 539 | sub negate {
|
|---|
| 540 | my ($z) = @_;
|
|---|
| 541 | if ($z->{c_dirty}) {
|
|---|
| 542 | my ($r, $t) = @{$z->polar};
|
|---|
| 543 | $t = ($t <= 0) ? $t + pi : $t - pi;
|
|---|
| 544 | return (ref $z)->emake($r, $t);
|
|---|
| 545 | }
|
|---|
| 546 | my ($re, $im) = @{$z->cartesian};
|
|---|
| 547 | return (ref $z)->make(-$re, -$im);
|
|---|
| 548 | }
|
|---|
| 549 |
|
|---|
| 550 | #
|
|---|
| 551 | # (conjugate)
|
|---|
| 552 | #
|
|---|
| 553 | # Compute complex's conjugate.
|
|---|
| 554 | #
|
|---|
| 555 | sub conjugate {
|
|---|
| 556 | my ($z) = @_;
|
|---|
| 557 | if ($z->{c_dirty}) {
|
|---|
| 558 | my ($r, $t) = @{$z->polar};
|
|---|
| 559 | return (ref $z)->emake($r, -$t);
|
|---|
| 560 | }
|
|---|
| 561 | my ($re, $im) = @{$z->cartesian};
|
|---|
| 562 | return (ref $z)->make($re, -$im);
|
|---|
| 563 | }
|
|---|
| 564 |
|
|---|
| 565 | #
|
|---|
| 566 | # (abs)
|
|---|
| 567 | #
|
|---|
| 568 | # Compute or set complex's norm (rho).
|
|---|
| 569 | #
|
|---|
| 570 | sub abs {
|
|---|
| 571 | my ($z, $rho) = @_;
|
|---|
| 572 | unless (ref $z) {
|
|---|
| 573 | if (@_ == 2) {
|
|---|
| 574 | $_[0] = $_[1];
|
|---|
| 575 | } else {
|
|---|
| 576 | return CORE::abs($z);
|
|---|
| 577 | }
|
|---|
| 578 | }
|
|---|
| 579 | if (defined $rho) {
|
|---|
| 580 | $z->{'polar'} = [ $rho, ${$z->polar}[1] ];
|
|---|
| 581 | $z->{p_dirty} = 0;
|
|---|
| 582 | $z->{c_dirty} = 1;
|
|---|
| 583 | return $rho;
|
|---|
| 584 | } else {
|
|---|
| 585 | return ${$z->polar}[0];
|
|---|
| 586 | }
|
|---|
| 587 | }
|
|---|
| 588 |
|
|---|
| 589 | sub _theta {
|
|---|
| 590 | my $theta = $_[0];
|
|---|
| 591 |
|
|---|
| 592 | if ($$theta > pi()) { $$theta -= pit2 }
|
|---|
| 593 | elsif ($$theta <= -pi()) { $$theta += pit2 }
|
|---|
| 594 | }
|
|---|
| 595 |
|
|---|
| 596 | #
|
|---|
| 597 | # arg
|
|---|
| 598 | #
|
|---|
| 599 | # Compute or set complex's argument (theta).
|
|---|
| 600 | #
|
|---|
| 601 | sub arg {
|
|---|
| 602 | my ($z, $theta) = @_;
|
|---|
| 603 | return $z unless ref $z;
|
|---|
| 604 | if (defined $theta) {
|
|---|
| 605 | _theta(\$theta);
|
|---|
| 606 | $z->{'polar'} = [ ${$z->polar}[0], $theta ];
|
|---|
| 607 | $z->{p_dirty} = 0;
|
|---|
| 608 | $z->{c_dirty} = 1;
|
|---|
| 609 | } else {
|
|---|
| 610 | $theta = ${$z->polar}[1];
|
|---|
| 611 | _theta(\$theta);
|
|---|
| 612 | }
|
|---|
| 613 | return $theta;
|
|---|
| 614 | }
|
|---|
| 615 |
|
|---|
| 616 | #
|
|---|
| 617 | # (sqrt)
|
|---|
| 618 | #
|
|---|
| 619 | # Compute sqrt(z).
|
|---|
| 620 | #
|
|---|
| 621 | # It is quite tempting to use wantarray here so that in list context
|
|---|
| 622 | # sqrt() would return the two solutions. This, however, would
|
|---|
| 623 | # break things like
|
|---|
| 624 | #
|
|---|
| 625 | # print "sqrt(z) = ", sqrt($z), "\n";
|
|---|
| 626 | #
|
|---|
| 627 | # The two values would be printed side by side without no intervening
|
|---|
| 628 | # whitespace, quite confusing.
|
|---|
| 629 | # Therefore if you want the two solutions use the root().
|
|---|
| 630 | #
|
|---|
| 631 | sub sqrt {
|
|---|
| 632 | my ($z) = @_;
|
|---|
| 633 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
|
|---|
| 634 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re)
|
|---|
| 635 | if $im == 0;
|
|---|
| 636 | my ($r, $t) = @{$z->polar};
|
|---|
| 637 | return (ref $z)->emake(CORE::sqrt($r), $t/2);
|
|---|
| 638 | }
|
|---|
| 639 |
|
|---|
| 640 | #
|
|---|
| 641 | # cbrt
|
|---|
| 642 | #
|
|---|
| 643 | # Compute cbrt(z) (cubic root).
|
|---|
| 644 | #
|
|---|
| 645 | # Why are we not returning three values? The same answer as for sqrt().
|
|---|
| 646 | #
|
|---|
| 647 | sub cbrt {
|
|---|
| 648 | my ($z) = @_;
|
|---|
| 649 | return $z < 0 ?
|
|---|
| 650 | -CORE::exp(CORE::log(-$z)/3) :
|
|---|
| 651 | ($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
|
|---|
| 652 | unless ref $z;
|
|---|
| 653 | my ($r, $t) = @{$z->polar};
|
|---|
| 654 | return 0 if $r == 0;
|
|---|
| 655 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
|
|---|
| 656 | }
|
|---|
| 657 |
|
|---|
| 658 | #
|
|---|
| 659 | # _rootbad
|
|---|
| 660 | #
|
|---|
| 661 | # Die on bad root.
|
|---|
| 662 | #
|
|---|
| 663 | sub _rootbad {
|
|---|
| 664 | my $mess = "Root '$_[0]' illegal, root rank must be positive integer.\n";
|
|---|
| 665 |
|
|---|
| 666 | my @up = caller(1);
|
|---|
| 667 |
|
|---|
| 668 | $mess .= "Died at $up[1] line $up[2].\n";
|
|---|
| 669 |
|
|---|
| 670 | die $mess;
|
|---|
| 671 | }
|
|---|
| 672 |
|
|---|
| 673 | #
|
|---|
| 674 | # root
|
|---|
| 675 | #
|
|---|
| 676 | # Computes all nth root for z, returning an array whose size is n.
|
|---|
| 677 | # `n' must be a positive integer.
|
|---|
| 678 | #
|
|---|
| 679 | # The roots are given by (for k = 0..n-1):
|
|---|
| 680 | #
|
|---|
| 681 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n))
|
|---|
| 682 | #
|
|---|
| 683 | sub root {
|
|---|
| 684 | my ($z, $n, $k) = @_;
|
|---|
| 685 | _rootbad($n) if ($n < 1 or int($n) != $n);
|
|---|
| 686 | my ($r, $t) = ref $z ?
|
|---|
| 687 | @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
|
|---|
| 688 | my $theta_inc = pit2 / $n;
|
|---|
| 689 | my $rho = $r ** (1/$n);
|
|---|
| 690 | my $cartesian = ref $z && $z->{c_dirty} == 0;
|
|---|
| 691 | if (@_ == 2) {
|
|---|
| 692 | my @root;
|
|---|
| 693 | for (my $i = 0, my $theta = $t / $n;
|
|---|
| 694 | $i < $n;
|
|---|
| 695 | $i++, $theta += $theta_inc) {
|
|---|
| 696 | my $w = cplxe($rho, $theta);
|
|---|
| 697 | # Yes, $cartesian is loop invariant.
|
|---|
| 698 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w;
|
|---|
| 699 | }
|
|---|
| 700 | return @root;
|
|---|
| 701 | } elsif (@_ == 3) {
|
|---|
| 702 | my $w = cplxe($rho, $t / $n + $k * $theta_inc);
|
|---|
| 703 | return $cartesian ? cplx(@{$w->cartesian}) : $w;
|
|---|
| 704 | }
|
|---|
| 705 | }
|
|---|
| 706 |
|
|---|
| 707 | #
|
|---|
| 708 | # Re
|
|---|
| 709 | #
|
|---|
| 710 | # Return or set Re(z).
|
|---|
| 711 | #
|
|---|
| 712 | sub Re {
|
|---|
| 713 | my ($z, $Re) = @_;
|
|---|
| 714 | return $z unless ref $z;
|
|---|
| 715 | if (defined $Re) {
|
|---|
| 716 | $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ];
|
|---|
| 717 | $z->{c_dirty} = 0;
|
|---|
| 718 | $z->{p_dirty} = 1;
|
|---|
| 719 | } else {
|
|---|
| 720 | return ${$z->cartesian}[0];
|
|---|
| 721 | }
|
|---|
| 722 | }
|
|---|
| 723 |
|
|---|
| 724 | #
|
|---|
| 725 | # Im
|
|---|
| 726 | #
|
|---|
| 727 | # Return or set Im(z).
|
|---|
| 728 | #
|
|---|
| 729 | sub Im {
|
|---|
| 730 | my ($z, $Im) = @_;
|
|---|
| 731 | return 0 unless ref $z;
|
|---|
| 732 | if (defined $Im) {
|
|---|
| 733 | $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ];
|
|---|
| 734 | $z->{c_dirty} = 0;
|
|---|
| 735 | $z->{p_dirty} = 1;
|
|---|
| 736 | } else {
|
|---|
| 737 | return ${$z->cartesian}[1];
|
|---|
| 738 | }
|
|---|
| 739 | }
|
|---|
| 740 |
|
|---|
| 741 | #
|
|---|
| 742 | # rho
|
|---|
| 743 | #
|
|---|
| 744 | # Return or set rho(w).
|
|---|
| 745 | #
|
|---|
| 746 | sub rho {
|
|---|
| 747 | Math::Complex::abs(@_);
|
|---|
| 748 | }
|
|---|
| 749 |
|
|---|
| 750 | #
|
|---|
| 751 | # theta
|
|---|
| 752 | #
|
|---|
| 753 | # Return or set theta(w).
|
|---|
| 754 | #
|
|---|
| 755 | sub theta {
|
|---|
| 756 | Math::Complex::arg(@_);
|
|---|
| 757 | }
|
|---|
| 758 |
|
|---|
| 759 | #
|
|---|
| 760 | # (exp)
|
|---|
| 761 | #
|
|---|
| 762 | # Computes exp(z).
|
|---|
| 763 | #
|
|---|
| 764 | sub exp {
|
|---|
| 765 | my ($z) = @_;
|
|---|
| 766 | my ($x, $y) = @{$z->cartesian};
|
|---|
| 767 | return (ref $z)->emake(CORE::exp($x), $y);
|
|---|
| 768 | }
|
|---|
| 769 |
|
|---|
| 770 | #
|
|---|
| 771 | # _logofzero
|
|---|
| 772 | #
|
|---|
| 773 | # Die on logarithm of zero.
|
|---|
| 774 | #
|
|---|
| 775 | sub _logofzero {
|
|---|
| 776 | my $mess = "$_[0]: Logarithm of zero.\n";
|
|---|
| 777 |
|
|---|
| 778 | if (defined $_[1]) {
|
|---|
| 779 | $mess .= "(Because in the definition of $_[0], the argument ";
|
|---|
| 780 | $mess .= "$_[1] " unless ($_[1] eq '0');
|
|---|
| 781 | $mess .= "is 0)\n";
|
|---|
| 782 | }
|
|---|
| 783 |
|
|---|
| 784 | my @up = caller(1);
|
|---|
| 785 |
|
|---|
| 786 | $mess .= "Died at $up[1] line $up[2].\n";
|
|---|
| 787 |
|
|---|
| 788 | die $mess;
|
|---|
| 789 | }
|
|---|
| 790 |
|
|---|
| 791 | #
|
|---|
| 792 | # (log)
|
|---|
| 793 | #
|
|---|
| 794 | # Compute log(z).
|
|---|
| 795 | #
|
|---|
| 796 | sub log {
|
|---|
| 797 | my ($z) = @_;
|
|---|
| 798 | unless (ref $z) {
|
|---|
| 799 | _logofzero("log") if $z == 0;
|
|---|
| 800 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi);
|
|---|
| 801 | }
|
|---|
| 802 | my ($r, $t) = @{$z->polar};
|
|---|
| 803 | _logofzero("log") if $r == 0;
|
|---|
| 804 | if ($t > pi()) { $t -= pit2 }
|
|---|
| 805 | elsif ($t <= -pi()) { $t += pit2 }
|
|---|
| 806 | return (ref $z)->make(CORE::log($r), $t);
|
|---|
| 807 | }
|
|---|
| 808 |
|
|---|
| 809 | #
|
|---|
| 810 | # ln
|
|---|
| 811 | #
|
|---|
| 812 | # Alias for log().
|
|---|
| 813 | #
|
|---|
| 814 | sub ln { Math::Complex::log(@_) }
|
|---|
| 815 |
|
|---|
| 816 | #
|
|---|
| 817 | # log10
|
|---|
| 818 | #
|
|---|
| 819 | # Compute log10(z).
|
|---|
| 820 | #
|
|---|
| 821 |
|
|---|
| 822 | sub log10 {
|
|---|
| 823 | return Math::Complex::log($_[0]) * uplog10;
|
|---|
| 824 | }
|
|---|
| 825 |
|
|---|
| 826 | #
|
|---|
| 827 | # logn
|
|---|
| 828 | #
|
|---|
| 829 | # Compute logn(z,n) = log(z) / log(n)
|
|---|
| 830 | #
|
|---|
| 831 | sub logn {
|
|---|
| 832 | my ($z, $n) = @_;
|
|---|
| 833 | $z = cplx($z, 0) unless ref $z;
|
|---|
| 834 | my $logn = $LOGN{$n};
|
|---|
| 835 | $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n)
|
|---|
| 836 | return &log($z) / $logn;
|
|---|
| 837 | }
|
|---|
| 838 |
|
|---|
| 839 | #
|
|---|
| 840 | # (cos)
|
|---|
| 841 | #
|
|---|
| 842 | # Compute cos(z) = (exp(iz) + exp(-iz))/2.
|
|---|
| 843 | #
|
|---|
| 844 | sub cos {
|
|---|
| 845 | my ($z) = @_;
|
|---|
| 846 | return CORE::cos($z) unless ref $z;
|
|---|
| 847 | my ($x, $y) = @{$z->cartesian};
|
|---|
| 848 | my $ey = CORE::exp($y);
|
|---|
| 849 | my $sx = CORE::sin($x);
|
|---|
| 850 | my $cx = CORE::cos($x);
|
|---|
| 851 | my $ey_1 = $ey ? 1 / $ey : $Inf;
|
|---|
| 852 | return (ref $z)->make($cx * ($ey + $ey_1)/2,
|
|---|
| 853 | $sx * ($ey_1 - $ey)/2);
|
|---|
| 854 | }
|
|---|
| 855 |
|
|---|
| 856 | #
|
|---|
| 857 | # (sin)
|
|---|
| 858 | #
|
|---|
| 859 | # Compute sin(z) = (exp(iz) - exp(-iz))/2.
|
|---|
| 860 | #
|
|---|
| 861 | sub sin {
|
|---|
| 862 | my ($z) = @_;
|
|---|
| 863 | return CORE::sin($z) unless ref $z;
|
|---|
| 864 | my ($x, $y) = @{$z->cartesian};
|
|---|
| 865 | my $ey = CORE::exp($y);
|
|---|
| 866 | my $sx = CORE::sin($x);
|
|---|
| 867 | my $cx = CORE::cos($x);
|
|---|
| 868 | my $ey_1 = $ey ? 1 / $ey : $Inf;
|
|---|
| 869 | return (ref $z)->make($sx * ($ey + $ey_1)/2,
|
|---|
| 870 | $cx * ($ey - $ey_1)/2);
|
|---|
| 871 | }
|
|---|
| 872 |
|
|---|
| 873 | #
|
|---|
| 874 | # tan
|
|---|
| 875 | #
|
|---|
| 876 | # Compute tan(z) = sin(z) / cos(z).
|
|---|
| 877 | #
|
|---|
| 878 | sub tan {
|
|---|
| 879 | my ($z) = @_;
|
|---|
| 880 | my $cz = &cos($z);
|
|---|
| 881 | _divbyzero "tan($z)", "cos($z)" if $cz == 0;
|
|---|
| 882 | return &sin($z) / $cz;
|
|---|
| 883 | }
|
|---|
| 884 |
|
|---|
| 885 | #
|
|---|
| 886 | # sec
|
|---|
| 887 | #
|
|---|
| 888 | # Computes the secant sec(z) = 1 / cos(z).
|
|---|
| 889 | #
|
|---|
| 890 | sub sec {
|
|---|
| 891 | my ($z) = @_;
|
|---|
| 892 | my $cz = &cos($z);
|
|---|
| 893 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0);
|
|---|
| 894 | return 1 / $cz;
|
|---|
| 895 | }
|
|---|
| 896 |
|
|---|
| 897 | #
|
|---|
| 898 | # csc
|
|---|
| 899 | #
|
|---|
| 900 | # Computes the cosecant csc(z) = 1 / sin(z).
|
|---|
| 901 | #
|
|---|
| 902 | sub csc {
|
|---|
| 903 | my ($z) = @_;
|
|---|
| 904 | my $sz = &sin($z);
|
|---|
| 905 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0);
|
|---|
| 906 | return 1 / $sz;
|
|---|
| 907 | }
|
|---|
| 908 |
|
|---|
| 909 | #
|
|---|
| 910 | # cosec
|
|---|
| 911 | #
|
|---|
| 912 | # Alias for csc().
|
|---|
| 913 | #
|
|---|
| 914 | sub cosec { Math::Complex::csc(@_) }
|
|---|
| 915 |
|
|---|
| 916 | #
|
|---|
| 917 | # cot
|
|---|
| 918 | #
|
|---|
| 919 | # Computes cot(z) = cos(z) / sin(z).
|
|---|
| 920 | #
|
|---|
| 921 | sub cot {
|
|---|
| 922 | my ($z) = @_;
|
|---|
| 923 | my $sz = &sin($z);
|
|---|
| 924 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0);
|
|---|
| 925 | return &cos($z) / $sz;
|
|---|
| 926 | }
|
|---|
| 927 |
|
|---|
| 928 | #
|
|---|
| 929 | # cotan
|
|---|
| 930 | #
|
|---|
| 931 | # Alias for cot().
|
|---|
| 932 | #
|
|---|
| 933 | sub cotan { Math::Complex::cot(@_) }
|
|---|
| 934 |
|
|---|
| 935 | #
|
|---|
| 936 | # acos
|
|---|
| 937 | #
|
|---|
| 938 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
|
|---|
| 939 | #
|
|---|
| 940 | sub acos {
|
|---|
| 941 | my $z = $_[0];
|
|---|
| 942 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z)
|
|---|
| 943 | if (! ref $z) && CORE::abs($z) <= 1;
|
|---|
| 944 | $z = cplx($z, 0) unless ref $z;
|
|---|
| 945 | my ($x, $y) = @{$z->cartesian};
|
|---|
| 946 | return 0 if $x == 1 && $y == 0;
|
|---|
| 947 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
|
|---|
| 948 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
|
|---|
| 949 | my $alpha = ($t1 + $t2)/2;
|
|---|
| 950 | my $beta = ($t1 - $t2)/2;
|
|---|
| 951 | $alpha = 1 if $alpha < 1;
|
|---|
| 952 | if ($beta > 1) { $beta = 1 }
|
|---|
| 953 | elsif ($beta < -1) { $beta = -1 }
|
|---|
| 954 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta);
|
|---|
| 955 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
|
|---|
| 956 | $v = -$v if $y > 0 || ($y == 0 && $x < -1);
|
|---|
| 957 | return (ref $z)->make($u, $v);
|
|---|
| 958 | }
|
|---|
| 959 |
|
|---|
| 960 | #
|
|---|
| 961 | # asin
|
|---|
| 962 | #
|
|---|
| 963 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
|
|---|
| 964 | #
|
|---|
| 965 | sub asin {
|
|---|
| 966 | my $z = $_[0];
|
|---|
| 967 | return CORE::atan2($z, CORE::sqrt(1-$z*$z))
|
|---|
| 968 | if (! ref $z) && CORE::abs($z) <= 1;
|
|---|
| 969 | $z = cplx($z, 0) unless ref $z;
|
|---|
| 970 | my ($x, $y) = @{$z->cartesian};
|
|---|
| 971 | return 0 if $x == 0 && $y == 0;
|
|---|
| 972 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
|
|---|
| 973 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
|
|---|
| 974 | my $alpha = ($t1 + $t2)/2;
|
|---|
| 975 | my $beta = ($t1 - $t2)/2;
|
|---|
| 976 | $alpha = 1 if $alpha < 1;
|
|---|
| 977 | if ($beta > 1) { $beta = 1 }
|
|---|
| 978 | elsif ($beta < -1) { $beta = -1 }
|
|---|
| 979 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta));
|
|---|
| 980 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
|
|---|
| 981 | $v = -$v if $y > 0 || ($y == 0 && $x < -1);
|
|---|
| 982 | return (ref $z)->make($u, $v);
|
|---|
| 983 | }
|
|---|
| 984 |
|
|---|
| 985 | #
|
|---|
| 986 | # atan
|
|---|
| 987 | #
|
|---|
| 988 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)).
|
|---|
| 989 | #
|
|---|
| 990 | sub atan {
|
|---|
| 991 | my ($z) = @_;
|
|---|
| 992 | return CORE::atan2($z, 1) unless ref $z;
|
|---|
| 993 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
|
|---|
| 994 | return 0 if $x == 0 && $y == 0;
|
|---|
| 995 | _divbyzero "atan(i)" if ( $z == i);
|
|---|
| 996 | _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test...
|
|---|
| 997 | my $log = &log((i + $z) / (i - $z));
|
|---|
| 998 | return ip2 * $log;
|
|---|
| 999 | }
|
|---|
| 1000 |
|
|---|
| 1001 | #
|
|---|
| 1002 | # asec
|
|---|
| 1003 | #
|
|---|
| 1004 | # Computes the arc secant asec(z) = acos(1 / z).
|
|---|
| 1005 | #
|
|---|
| 1006 | sub asec {
|
|---|
| 1007 | my ($z) = @_;
|
|---|
| 1008 | _divbyzero "asec($z)", $z if ($z == 0);
|
|---|
| 1009 | return acos(1 / $z);
|
|---|
| 1010 | }
|
|---|
| 1011 |
|
|---|
| 1012 | #
|
|---|
| 1013 | # acsc
|
|---|
| 1014 | #
|
|---|
| 1015 | # Computes the arc cosecant acsc(z) = asin(1 / z).
|
|---|
| 1016 | #
|
|---|
| 1017 | sub acsc {
|
|---|
| 1018 | my ($z) = @_;
|
|---|
| 1019 | _divbyzero "acsc($z)", $z if ($z == 0);
|
|---|
| 1020 | return asin(1 / $z);
|
|---|
| 1021 | }
|
|---|
| 1022 |
|
|---|
| 1023 | #
|
|---|
| 1024 | # acosec
|
|---|
| 1025 | #
|
|---|
| 1026 | # Alias for acsc().
|
|---|
| 1027 | #
|
|---|
| 1028 | sub acosec { Math::Complex::acsc(@_) }
|
|---|
| 1029 |
|
|---|
| 1030 | #
|
|---|
| 1031 | # acot
|
|---|
| 1032 | #
|
|---|
| 1033 | # Computes the arc cotangent acot(z) = atan(1 / z)
|
|---|
| 1034 | #
|
|---|
| 1035 | sub acot {
|
|---|
| 1036 | my ($z) = @_;
|
|---|
| 1037 | _divbyzero "acot(0)" if $z == 0;
|
|---|
| 1038 | return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z)
|
|---|
| 1039 | unless ref $z;
|
|---|
| 1040 | _divbyzero "acot(i)" if ($z - i == 0);
|
|---|
| 1041 | _logofzero "acot(-i)" if ($z + i == 0);
|
|---|
| 1042 | return atan(1 / $z);
|
|---|
| 1043 | }
|
|---|
| 1044 |
|
|---|
| 1045 | #
|
|---|
| 1046 | # acotan
|
|---|
| 1047 | #
|
|---|
| 1048 | # Alias for acot().
|
|---|
| 1049 | #
|
|---|
| 1050 | sub acotan { Math::Complex::acot(@_) }
|
|---|
| 1051 |
|
|---|
| 1052 | #
|
|---|
| 1053 | # cosh
|
|---|
| 1054 | #
|
|---|
| 1055 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2.
|
|---|
| 1056 | #
|
|---|
| 1057 | sub cosh {
|
|---|
| 1058 | my ($z) = @_;
|
|---|
| 1059 | my $ex;
|
|---|
| 1060 | unless (ref $z) {
|
|---|
| 1061 | $ex = CORE::exp($z);
|
|---|
| 1062 | return $ex ? ($ex + 1/$ex)/2 : $Inf;
|
|---|
| 1063 | }
|
|---|
| 1064 | my ($x, $y) = @{$z->cartesian};
|
|---|
| 1065 | $ex = CORE::exp($x);
|
|---|
| 1066 | my $ex_1 = $ex ? 1 / $ex : $Inf;
|
|---|
| 1067 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
|
|---|
| 1068 | CORE::sin($y) * ($ex - $ex_1)/2);
|
|---|
| 1069 | }
|
|---|
| 1070 |
|
|---|
| 1071 | #
|
|---|
| 1072 | # sinh
|
|---|
| 1073 | #
|
|---|
| 1074 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2.
|
|---|
| 1075 | #
|
|---|
| 1076 | sub sinh {
|
|---|
| 1077 | my ($z) = @_;
|
|---|
| 1078 | my $ex;
|
|---|
| 1079 | unless (ref $z) {
|
|---|
| 1080 | return 0 if $z == 0;
|
|---|
| 1081 | $ex = CORE::exp($z);
|
|---|
| 1082 | return $ex ? ($ex - 1/$ex)/2 : "-$Inf";
|
|---|
| 1083 | }
|
|---|
| 1084 | my ($x, $y) = @{$z->cartesian};
|
|---|
| 1085 | my $cy = CORE::cos($y);
|
|---|
| 1086 | my $sy = CORE::sin($y);
|
|---|
| 1087 | $ex = CORE::exp($x);
|
|---|
| 1088 | my $ex_1 = $ex ? 1 / $ex : $Inf;
|
|---|
| 1089 | return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2,
|
|---|
| 1090 | CORE::sin($y) * ($ex + $ex_1)/2);
|
|---|
| 1091 | }
|
|---|
| 1092 |
|
|---|
| 1093 | #
|
|---|
| 1094 | # tanh
|
|---|
| 1095 | #
|
|---|
| 1096 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
|
|---|
| 1097 | #
|
|---|
| 1098 | sub tanh {
|
|---|
| 1099 | my ($z) = @_;
|
|---|
| 1100 | my $cz = cosh($z);
|
|---|
| 1101 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
|
|---|
| 1102 | return sinh($z) / $cz;
|
|---|
| 1103 | }
|
|---|
| 1104 |
|
|---|
| 1105 | #
|
|---|
| 1106 | # sech
|
|---|
| 1107 | #
|
|---|
| 1108 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z).
|
|---|
| 1109 | #
|
|---|
| 1110 | sub sech {
|
|---|
| 1111 | my ($z) = @_;
|
|---|
| 1112 | my $cz = cosh($z);
|
|---|
| 1113 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
|
|---|
| 1114 | return 1 / $cz;
|
|---|
| 1115 | }
|
|---|
| 1116 |
|
|---|
| 1117 | #
|
|---|
| 1118 | # csch
|
|---|
| 1119 | #
|
|---|
| 1120 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z).
|
|---|
| 1121 | #
|
|---|
| 1122 | sub csch {
|
|---|
| 1123 | my ($z) = @_;
|
|---|
| 1124 | my $sz = sinh($z);
|
|---|
| 1125 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
|
|---|
| 1126 | return 1 / $sz;
|
|---|
| 1127 | }
|
|---|
| 1128 |
|
|---|
| 1129 | #
|
|---|
| 1130 | # cosech
|
|---|
| 1131 | #
|
|---|
| 1132 | # Alias for csch().
|
|---|
| 1133 | #
|
|---|
| 1134 | sub cosech { Math::Complex::csch(@_) }
|
|---|
| 1135 |
|
|---|
| 1136 | #
|
|---|
| 1137 | # coth
|
|---|
| 1138 | #
|
|---|
| 1139 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z).
|
|---|
| 1140 | #
|
|---|
| 1141 | sub coth {
|
|---|
| 1142 | my ($z) = @_;
|
|---|
| 1143 | my $sz = sinh($z);
|
|---|
| 1144 | _divbyzero "coth($z)", "sinh($z)" if $sz == 0;
|
|---|
| 1145 | return cosh($z) / $sz;
|
|---|
| 1146 | }
|
|---|
| 1147 |
|
|---|
| 1148 | #
|
|---|
| 1149 | # cotanh
|
|---|
| 1150 | #
|
|---|
| 1151 | # Alias for coth().
|
|---|
| 1152 | #
|
|---|
| 1153 | sub cotanh { Math::Complex::coth(@_) }
|
|---|
| 1154 |
|
|---|
| 1155 | #
|
|---|
| 1156 | # acosh
|
|---|
| 1157 | #
|
|---|
| 1158 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
|
|---|
| 1159 | #
|
|---|
| 1160 | sub acosh {
|
|---|
| 1161 | my ($z) = @_;
|
|---|
| 1162 | unless (ref $z) {
|
|---|
| 1163 | $z = cplx($z, 0);
|
|---|
| 1164 | }
|
|---|
| 1165 | my ($re, $im) = @{$z->cartesian};
|
|---|
| 1166 | if ($im == 0) {
|
|---|
| 1167 | return CORE::log($re + CORE::sqrt($re*$re - 1))
|
|---|
| 1168 | if $re >= 1;
|
|---|
| 1169 | return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re))
|
|---|
| 1170 | if CORE::abs($re) < 1;
|
|---|
| 1171 | }
|
|---|
| 1172 | my $t = &sqrt($z * $z - 1) + $z;
|
|---|
| 1173 | # Try Taylor if looking bad (this usually means that
|
|---|
| 1174 | # $z was large negative, therefore the sqrt is really
|
|---|
| 1175 | # close to abs(z), summing that with z...)
|
|---|
| 1176 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
|
|---|
| 1177 | if $t == 0;
|
|---|
| 1178 | my $u = &log($t);
|
|---|
| 1179 | $u->Im(-$u->Im) if $re < 0 && $im == 0;
|
|---|
| 1180 | return $re < 0 ? -$u : $u;
|
|---|
| 1181 | }
|
|---|
| 1182 |
|
|---|
| 1183 | #
|
|---|
| 1184 | # asinh
|
|---|
| 1185 | #
|
|---|
| 1186 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
|
|---|
| 1187 | #
|
|---|
| 1188 | sub asinh {
|
|---|
| 1189 | my ($z) = @_;
|
|---|
| 1190 | unless (ref $z) {
|
|---|
| 1191 | my $t = $z + CORE::sqrt($z*$z + 1);
|
|---|
| 1192 | return CORE::log($t) if $t;
|
|---|
| 1193 | }
|
|---|
| 1194 | my $t = &sqrt($z * $z + 1) + $z;
|
|---|
| 1195 | # Try Taylor if looking bad (this usually means that
|
|---|
| 1196 | # $z was large negative, therefore the sqrt is really
|
|---|
| 1197 | # close to abs(z), summing that with z...)
|
|---|
| 1198 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
|
|---|
| 1199 | if $t == 0;
|
|---|
| 1200 | return &log($t);
|
|---|
| 1201 | }
|
|---|
| 1202 |
|
|---|
| 1203 | #
|
|---|
| 1204 | # atanh
|
|---|
| 1205 | #
|
|---|
| 1206 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)).
|
|---|
| 1207 | #
|
|---|
| 1208 | sub atanh {
|
|---|
| 1209 | my ($z) = @_;
|
|---|
| 1210 | unless (ref $z) {
|
|---|
| 1211 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1;
|
|---|
| 1212 | $z = cplx($z, 0);
|
|---|
| 1213 | }
|
|---|
| 1214 | _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0);
|
|---|
| 1215 | _logofzero 'atanh(-1)' if (1 + $z == 0);
|
|---|
| 1216 | return 0.5 * &log((1 + $z) / (1 - $z));
|
|---|
| 1217 | }
|
|---|
| 1218 |
|
|---|
| 1219 | #
|
|---|
| 1220 | # asech
|
|---|
| 1221 | #
|
|---|
| 1222 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z).
|
|---|
| 1223 | #
|
|---|
| 1224 | sub asech {
|
|---|
| 1225 | my ($z) = @_;
|
|---|
| 1226 | _divbyzero 'asech(0)', "$z" if ($z == 0);
|
|---|
| 1227 | return acosh(1 / $z);
|
|---|
| 1228 | }
|
|---|
| 1229 |
|
|---|
| 1230 | #
|
|---|
| 1231 | # acsch
|
|---|
| 1232 | #
|
|---|
| 1233 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z).
|
|---|
| 1234 | #
|
|---|
| 1235 | sub acsch {
|
|---|
| 1236 | my ($z) = @_;
|
|---|
| 1237 | _divbyzero 'acsch(0)', $z if ($z == 0);
|
|---|
| 1238 | return asinh(1 / $z);
|
|---|
| 1239 | }
|
|---|
| 1240 |
|
|---|
| 1241 | #
|
|---|
| 1242 | # acosech
|
|---|
| 1243 | #
|
|---|
| 1244 | # Alias for acosh().
|
|---|
| 1245 | #
|
|---|
| 1246 | sub acosech { Math::Complex::acsch(@_) }
|
|---|
| 1247 |
|
|---|
| 1248 | #
|
|---|
| 1249 | # acoth
|
|---|
| 1250 | #
|
|---|
| 1251 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
|
|---|
| 1252 | #
|
|---|
| 1253 | sub acoth {
|
|---|
| 1254 | my ($z) = @_;
|
|---|
| 1255 | _divbyzero 'acoth(0)' if ($z == 0);
|
|---|
| 1256 | unless (ref $z) {
|
|---|
| 1257 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
|
|---|
| 1258 | $z = cplx($z, 0);
|
|---|
| 1259 | }
|
|---|
| 1260 | _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0);
|
|---|
| 1261 | _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0);
|
|---|
| 1262 | return &log((1 + $z) / ($z - 1)) / 2;
|
|---|
| 1263 | }
|
|---|
| 1264 |
|
|---|
| 1265 | #
|
|---|
| 1266 | # acotanh
|
|---|
| 1267 | #
|
|---|
| 1268 | # Alias for acot().
|
|---|
| 1269 | #
|
|---|
| 1270 | sub acotanh { Math::Complex::acoth(@_) }
|
|---|
| 1271 |
|
|---|
| 1272 | #
|
|---|
| 1273 | # (atan2)
|
|---|
| 1274 | #
|
|---|
| 1275 | # Compute atan(z1/z2), minding the right quadrant.
|
|---|
| 1276 | #
|
|---|
| 1277 | sub atan2 {
|
|---|
| 1278 | my ($z1, $z2, $inverted) = @_;
|
|---|
| 1279 | my ($re1, $im1, $re2, $im2);
|
|---|
| 1280 | if ($inverted) {
|
|---|
| 1281 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
|
|---|
| 1282 | ($re2, $im2) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
|
|---|
| 1283 | } else {
|
|---|
| 1284 | ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
|
|---|
| 1285 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
|
|---|
| 1286 | }
|
|---|
| 1287 | if ($im1 || $im2) {
|
|---|
| 1288 | # In MATLAB the imaginary parts are ignored.
|
|---|
| 1289 | # warn "atan2: Imaginary parts ignored";
|
|---|
| 1290 | # http://documents.wolfram.com/mathematica/functions/ArcTan
|
|---|
| 1291 | # NOTE: Mathematica ArcTan[x,y] while atan2(y,x)
|
|---|
| 1292 | my $s = $z1 * $z1 + $z2 * $z2;
|
|---|
| 1293 | _divbyzero("atan2") if $s == 0;
|
|---|
| 1294 | my $i = &i;
|
|---|
| 1295 | my $r = $z2 + $z1 * $i;
|
|---|
| 1296 | return -$i * &log($r / &sqrt( $s ));
|
|---|
| 1297 | }
|
|---|
| 1298 | return CORE::atan2($re1, $re2);
|
|---|
| 1299 | }
|
|---|
| 1300 |
|
|---|
| 1301 | #
|
|---|
| 1302 | # display_format
|
|---|
| 1303 | # ->display_format
|
|---|
| 1304 | #
|
|---|
| 1305 | # Set (get if no argument) the display format for all complex numbers that
|
|---|
| 1306 | # don't happen to have overridden it via ->display_format
|
|---|
| 1307 | #
|
|---|
| 1308 | # When called as an object method, this actually sets the display format for
|
|---|
| 1309 | # the current object.
|
|---|
| 1310 | #
|
|---|
| 1311 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first
|
|---|
| 1312 | # letter is used actually, so the type can be fully spelled out for clarity.
|
|---|
| 1313 | #
|
|---|
| 1314 | sub display_format {
|
|---|
| 1315 | my $self = shift;
|
|---|
| 1316 | my %display_format = %DISPLAY_FORMAT;
|
|---|
| 1317 |
|
|---|
| 1318 | if (ref $self) { # Called as an object method
|
|---|
| 1319 | if (exists $self->{display_format}) {
|
|---|
| 1320 | my %obj = %{$self->{display_format}};
|
|---|
| 1321 | @display_format{keys %obj} = values %obj;
|
|---|
| 1322 | }
|
|---|
| 1323 | }
|
|---|
| 1324 | if (@_ == 1) {
|
|---|
| 1325 | $display_format{style} = shift;
|
|---|
| 1326 | } else {
|
|---|
| 1327 | my %new = @_;
|
|---|
| 1328 | @display_format{keys %new} = values %new;
|
|---|
| 1329 | }
|
|---|
| 1330 |
|
|---|
| 1331 | if (ref $self) { # Called as an object method
|
|---|
| 1332 | $self->{display_format} = { %display_format };
|
|---|
| 1333 | return
|
|---|
| 1334 | wantarray ?
|
|---|
| 1335 | %{$self->{display_format}} :
|
|---|
| 1336 | $self->{display_format}->{style};
|
|---|
| 1337 | }
|
|---|
| 1338 |
|
|---|
| 1339 | # Called as a class method
|
|---|
| 1340 | %DISPLAY_FORMAT = %display_format;
|
|---|
| 1341 | return
|
|---|
| 1342 | wantarray ?
|
|---|
| 1343 | %DISPLAY_FORMAT :
|
|---|
| 1344 | $DISPLAY_FORMAT{style};
|
|---|
| 1345 | }
|
|---|
| 1346 |
|
|---|
| 1347 | #
|
|---|
| 1348 | # (stringify)
|
|---|
| 1349 | #
|
|---|
| 1350 | # Show nicely formatted complex number under its cartesian or polar form,
|
|---|
| 1351 | # depending on the current display format:
|
|---|
| 1352 | #
|
|---|
| 1353 | # . If a specific display format has been recorded for this object, use it.
|
|---|
| 1354 | # . Otherwise, use the generic current default for all complex numbers,
|
|---|
| 1355 | # which is a package global variable.
|
|---|
| 1356 | #
|
|---|
| 1357 | sub stringify {
|
|---|
| 1358 | my ($z) = shift;
|
|---|
| 1359 |
|
|---|
| 1360 | my $style = $z->display_format;
|
|---|
| 1361 |
|
|---|
| 1362 | $style = $DISPLAY_FORMAT{style} unless defined $style;
|
|---|
| 1363 |
|
|---|
| 1364 | return $z->stringify_polar if $style =~ /^p/i;
|
|---|
| 1365 | return $z->stringify_cartesian;
|
|---|
| 1366 | }
|
|---|
| 1367 |
|
|---|
| 1368 | #
|
|---|
| 1369 | # ->stringify_cartesian
|
|---|
| 1370 | #
|
|---|
| 1371 | # Stringify as a cartesian representation 'a+bi'.
|
|---|
| 1372 | #
|
|---|
| 1373 | sub stringify_cartesian {
|
|---|
| 1374 | my $z = shift;
|
|---|
| 1375 | my ($x, $y) = @{$z->cartesian};
|
|---|
| 1376 | my ($re, $im);
|
|---|
| 1377 |
|
|---|
| 1378 | my %format = $z->display_format;
|
|---|
| 1379 | my $format = $format{format};
|
|---|
| 1380 |
|
|---|
| 1381 | if ($x) {
|
|---|
| 1382 | if ($x =~ /^NaN[QS]?$/i) {
|
|---|
| 1383 | $re = $x;
|
|---|
| 1384 | } else {
|
|---|
| 1385 | if ($x =~ /^-?$Inf$/oi) {
|
|---|
| 1386 | $re = $x;
|
|---|
| 1387 | } else {
|
|---|
| 1388 | $re = defined $format ? sprintf($format, $x) : $x;
|
|---|
| 1389 | }
|
|---|
| 1390 | }
|
|---|
| 1391 | } else {
|
|---|
| 1392 | undef $re;
|
|---|
| 1393 | }
|
|---|
| 1394 |
|
|---|
| 1395 | if ($y) {
|
|---|
| 1396 | if ($y =~ /^(NaN[QS]?)$/i) {
|
|---|
| 1397 | $im = $y;
|
|---|
| 1398 | } else {
|
|---|
| 1399 | if ($y =~ /^-?$Inf$/oi) {
|
|---|
| 1400 | $im = $y;
|
|---|
| 1401 | } else {
|
|---|
| 1402 | $im =
|
|---|
| 1403 | defined $format ?
|
|---|
| 1404 | sprintf($format, $y) :
|
|---|
| 1405 | ($y == 1 ? "" : ($y == -1 ? "-" : $y));
|
|---|
| 1406 | }
|
|---|
| 1407 | }
|
|---|
| 1408 | $im .= "i";
|
|---|
| 1409 | } else {
|
|---|
| 1410 | undef $im;
|
|---|
| 1411 | }
|
|---|
| 1412 |
|
|---|
| 1413 | my $str = $re;
|
|---|
| 1414 |
|
|---|
| 1415 | if (defined $im) {
|
|---|
| 1416 | if ($y < 0) {
|
|---|
| 1417 | $str .= $im;
|
|---|
| 1418 | } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) {
|
|---|
| 1419 | $str .= "+" if defined $re;
|
|---|
| 1420 | $str .= $im;
|
|---|
| 1421 | }
|
|---|
| 1422 | } elsif (!defined $re) {
|
|---|
| 1423 | $str = "0";
|
|---|
| 1424 | }
|
|---|
| 1425 |
|
|---|
| 1426 | return $str;
|
|---|
| 1427 | }
|
|---|
| 1428 |
|
|---|
| 1429 |
|
|---|
| 1430 | #
|
|---|
| 1431 | # ->stringify_polar
|
|---|
| 1432 | #
|
|---|
| 1433 | # Stringify as a polar representation '[r,t]'.
|
|---|
| 1434 | #
|
|---|
| 1435 | sub stringify_polar {
|
|---|
| 1436 | my $z = shift;
|
|---|
| 1437 | my ($r, $t) = @{$z->polar};
|
|---|
| 1438 | my $theta;
|
|---|
| 1439 |
|
|---|
| 1440 | my %format = $z->display_format;
|
|---|
| 1441 | my $format = $format{format};
|
|---|
| 1442 |
|
|---|
| 1443 | if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) {
|
|---|
| 1444 | $theta = $t;
|
|---|
| 1445 | } elsif ($t == pi) {
|
|---|
| 1446 | $theta = "pi";
|
|---|
| 1447 | } elsif ($r == 0 || $t == 0) {
|
|---|
| 1448 | $theta = defined $format ? sprintf($format, $t) : $t;
|
|---|
| 1449 | }
|
|---|
| 1450 |
|
|---|
| 1451 | return "[$r,$theta]" if defined $theta;
|
|---|
| 1452 |
|
|---|
| 1453 | #
|
|---|
| 1454 | # Try to identify pi/n and friends.
|
|---|
| 1455 | #
|
|---|
| 1456 |
|
|---|
| 1457 | $t -= int(CORE::abs($t) / pit2) * pit2;
|
|---|
| 1458 |
|
|---|
| 1459 | if ($format{polar_pretty_print} && $t) {
|
|---|
| 1460 | my ($a, $b);
|
|---|
| 1461 | for $a (2..9) {
|
|---|
| 1462 | $b = $t * $a / pi;
|
|---|
| 1463 | if ($b =~ /^-?\d+$/) {
|
|---|
| 1464 | $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1;
|
|---|
| 1465 | $theta = "${b}pi/$a";
|
|---|
| 1466 | last;
|
|---|
| 1467 | }
|
|---|
| 1468 | }
|
|---|
| 1469 | }
|
|---|
| 1470 |
|
|---|
| 1471 | if (defined $format) {
|
|---|
| 1472 | $r = sprintf($format, $r);
|
|---|
| 1473 | $theta = sprintf($format, $theta) unless defined $theta;
|
|---|
| 1474 | } else {
|
|---|
| 1475 | $theta = $t unless defined $theta;
|
|---|
| 1476 | }
|
|---|
| 1477 |
|
|---|
| 1478 | return "[$r,$theta]";
|
|---|
| 1479 | }
|
|---|
| 1480 |
|
|---|
| 1481 | 1;
|
|---|
| 1482 | __END__
|
|---|
| 1483 |
|
|---|
| 1484 | =pod
|
|---|
| 1485 |
|
|---|
| 1486 | =head1 NAME
|
|---|
| 1487 |
|
|---|
| 1488 | Math::Complex - complex numbers and associated mathematical functions
|
|---|
| 1489 |
|
|---|
| 1490 | =head1 SYNOPSIS
|
|---|
| 1491 |
|
|---|
| 1492 | use Math::Complex;
|
|---|
| 1493 |
|
|---|
| 1494 | $z = Math::Complex->make(5, 6);
|
|---|
| 1495 | $t = 4 - 3*i + $z;
|
|---|
| 1496 | $j = cplxe(1, 2*pi/3);
|
|---|
| 1497 |
|
|---|
| 1498 | =head1 DESCRIPTION
|
|---|
| 1499 |
|
|---|
| 1500 | This package lets you create and manipulate complex numbers. By default,
|
|---|
| 1501 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings
|
|---|
| 1502 | full complex support, along with a full set of mathematical functions
|
|---|
| 1503 | typically associated with and/or extended to complex numbers.
|
|---|
| 1504 |
|
|---|
| 1505 | If you wonder what complex numbers are, they were invented to be able to solve
|
|---|
| 1506 | the following equation:
|
|---|
| 1507 |
|
|---|
| 1508 | x*x = -1
|
|---|
| 1509 |
|
|---|
| 1510 | and by definition, the solution is noted I<i> (engineers use I<j> instead since
|
|---|
| 1511 | I<i> usually denotes an intensity, but the name does not matter). The number
|
|---|
| 1512 | I<i> is a pure I<imaginary> number.
|
|---|
| 1513 |
|
|---|
| 1514 | The arithmetics with pure imaginary numbers works just like you would expect
|
|---|
| 1515 | it with real numbers... you just have to remember that
|
|---|
| 1516 |
|
|---|
| 1517 | i*i = -1
|
|---|
| 1518 |
|
|---|
| 1519 | so you have:
|
|---|
| 1520 |
|
|---|
| 1521 | 5i + 7i = i * (5 + 7) = 12i
|
|---|
| 1522 | 4i - 3i = i * (4 - 3) = i
|
|---|
| 1523 | 4i * 2i = -8
|
|---|
| 1524 | 6i / 2i = 3
|
|---|
| 1525 | 1 / i = -i
|
|---|
| 1526 |
|
|---|
| 1527 | Complex numbers are numbers that have both a real part and an imaginary
|
|---|
| 1528 | part, and are usually noted:
|
|---|
| 1529 |
|
|---|
| 1530 | a + bi
|
|---|
| 1531 |
|
|---|
| 1532 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The
|
|---|
| 1533 | arithmetic with complex numbers is straightforward. You have to
|
|---|
| 1534 | keep track of the real and the imaginary parts, but otherwise the
|
|---|
| 1535 | rules used for real numbers just apply:
|
|---|
| 1536 |
|
|---|
| 1537 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
|
|---|
| 1538 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
|
|---|
| 1539 |
|
|---|
| 1540 | A graphical representation of complex numbers is possible in a plane
|
|---|
| 1541 | (also called the I<complex plane>, but it's really a 2D plane).
|
|---|
| 1542 | The number
|
|---|
| 1543 |
|
|---|
| 1544 | z = a + bi
|
|---|
| 1545 |
|
|---|
| 1546 | is the point whose coordinates are (a, b). Actually, it would
|
|---|
| 1547 | be the vector originating from (0, 0) to (a, b). It follows that the addition
|
|---|
| 1548 | of two complex numbers is a vectorial addition.
|
|---|
| 1549 |
|
|---|
| 1550 | Since there is a bijection between a point in the 2D plane and a complex
|
|---|
| 1551 | number (i.e. the mapping is unique and reciprocal), a complex number
|
|---|
| 1552 | can also be uniquely identified with polar coordinates:
|
|---|
| 1553 |
|
|---|
| 1554 | [rho, theta]
|
|---|
| 1555 |
|
|---|
| 1556 | where C<rho> is the distance to the origin, and C<theta> the angle between
|
|---|
| 1557 | the vector and the I<x> axis. There is a notation for this using the
|
|---|
| 1558 | exponential form, which is:
|
|---|
| 1559 |
|
|---|
| 1560 | rho * exp(i * theta)
|
|---|
| 1561 |
|
|---|
| 1562 | where I<i> is the famous imaginary number introduced above. Conversion
|
|---|
| 1563 | between this form and the cartesian form C<a + bi> is immediate:
|
|---|
| 1564 |
|
|---|
| 1565 | a = rho * cos(theta)
|
|---|
| 1566 | b = rho * sin(theta)
|
|---|
| 1567 |
|
|---|
| 1568 | which is also expressed by this formula:
|
|---|
| 1569 |
|
|---|
| 1570 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
|
|---|
| 1571 |
|
|---|
| 1572 | In other words, it's the projection of the vector onto the I<x> and I<y>
|
|---|
| 1573 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta>
|
|---|
| 1574 | the I<argument> of the complex number. The I<norm> of C<z> will be
|
|---|
| 1575 | noted C<abs(z)>.
|
|---|
| 1576 |
|
|---|
| 1577 | The polar notation (also known as the trigonometric
|
|---|
| 1578 | representation) is much more handy for performing multiplications and
|
|---|
| 1579 | divisions of complex numbers, whilst the cartesian notation is better
|
|---|
| 1580 | suited for additions and subtractions. Real numbers are on the I<x>
|
|---|
| 1581 | axis, and therefore I<theta> is zero or I<pi>.
|
|---|
| 1582 |
|
|---|
| 1583 | All the common operations that can be performed on a real number have
|
|---|
| 1584 | been defined to work on complex numbers as well, and are merely
|
|---|
| 1585 | I<extensions> of the operations defined on real numbers. This means
|
|---|
| 1586 | they keep their natural meaning when there is no imaginary part, provided
|
|---|
| 1587 | the number is within their definition set.
|
|---|
| 1588 |
|
|---|
| 1589 | For instance, the C<sqrt> routine which computes the square root of
|
|---|
| 1590 | its argument is only defined for non-negative real numbers and yields a
|
|---|
| 1591 | non-negative real number (it is an application from B<R+> to B<R+>).
|
|---|
| 1592 | If we allow it to return a complex number, then it can be extended to
|
|---|
| 1593 | negative real numbers to become an application from B<R> to B<C> (the
|
|---|
| 1594 | set of complex numbers):
|
|---|
| 1595 |
|
|---|
| 1596 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
|
|---|
| 1597 |
|
|---|
| 1598 | It can also be extended to be an application from B<C> to B<C>,
|
|---|
| 1599 | whilst its restriction to B<R> behaves as defined above by using
|
|---|
| 1600 | the following definition:
|
|---|
| 1601 |
|
|---|
| 1602 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
|
|---|
| 1603 |
|
|---|
| 1604 | Indeed, a negative real number can be noted C<[x,pi]> (the modulus
|
|---|
| 1605 | I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative
|
|---|
| 1606 | number) and the above definition states that
|
|---|
| 1607 |
|
|---|
| 1608 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
|
|---|
| 1609 |
|
|---|
| 1610 | which is exactly what we had defined for negative real numbers above.
|
|---|
| 1611 | The C<sqrt> returns only one of the solutions: if you want the both,
|
|---|
| 1612 | use the C<root> function.
|
|---|
| 1613 |
|
|---|
| 1614 | All the common mathematical functions defined on real numbers that
|
|---|
| 1615 | are extended to complex numbers share that same property of working
|
|---|
| 1616 | I<as usual> when the imaginary part is zero (otherwise, it would not
|
|---|
| 1617 | be called an extension, would it?).
|
|---|
| 1618 |
|
|---|
| 1619 | A I<new> operation possible on a complex number that is
|
|---|
| 1620 | the identity for real numbers is called the I<conjugate>, and is noted
|
|---|
| 1621 | with a horizontal bar above the number, or C<~z> here.
|
|---|
| 1622 |
|
|---|
| 1623 | z = a + bi
|
|---|
| 1624 | ~z = a - bi
|
|---|
| 1625 |
|
|---|
| 1626 | Simple... Now look:
|
|---|
| 1627 |
|
|---|
| 1628 | z * ~z = (a + bi) * (a - bi) = a*a + b*b
|
|---|
| 1629 |
|
|---|
| 1630 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the
|
|---|
| 1631 | distance to the origin, also known as:
|
|---|
| 1632 |
|
|---|
| 1633 | rho = abs(z) = sqrt(a*a + b*b)
|
|---|
| 1634 |
|
|---|
| 1635 | so
|
|---|
| 1636 |
|
|---|
| 1637 | z * ~z = abs(z) ** 2
|
|---|
| 1638 |
|
|---|
| 1639 | If z is a pure real number (i.e. C<b == 0>), then the above yields:
|
|---|
| 1640 |
|
|---|
| 1641 | a * a = abs(a) ** 2
|
|---|
| 1642 |
|
|---|
| 1643 | which is true (C<abs> has the regular meaning for real number, i.e. stands
|
|---|
| 1644 | for the absolute value). This example explains why the norm of C<z> is
|
|---|
| 1645 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet
|
|---|
| 1646 | is the regular C<abs> we know when the complex number actually has no
|
|---|
| 1647 | imaginary part... This justifies I<a posteriori> our use of the C<abs>
|
|---|
| 1648 | notation for the norm.
|
|---|
| 1649 |
|
|---|
| 1650 | =head1 OPERATIONS
|
|---|
| 1651 |
|
|---|
| 1652 | Given the following notations:
|
|---|
| 1653 |
|
|---|
| 1654 | z1 = a + bi = r1 * exp(i * t1)
|
|---|
| 1655 | z2 = c + di = r2 * exp(i * t2)
|
|---|
| 1656 | z = <any complex or real number>
|
|---|
| 1657 |
|
|---|
| 1658 | the following (overloaded) operations are supported on complex numbers:
|
|---|
| 1659 |
|
|---|
| 1660 | z1 + z2 = (a + c) + i(b + d)
|
|---|
| 1661 | z1 - z2 = (a - c) + i(b - d)
|
|---|
| 1662 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
|
|---|
| 1663 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
|
|---|
| 1664 | z1 ** z2 = exp(z2 * log z1)
|
|---|
| 1665 | ~z = a - bi
|
|---|
| 1666 | abs(z) = r1 = sqrt(a*a + b*b)
|
|---|
| 1667 | sqrt(z) = sqrt(r1) * exp(i * t/2)
|
|---|
| 1668 | exp(z) = exp(a) * exp(i * b)
|
|---|
| 1669 | log(z) = log(r1) + i*t
|
|---|
| 1670 | sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
|
|---|
| 1671 | cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
|
|---|
| 1672 | atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.
|
|---|
| 1673 |
|
|---|
| 1674 | The definition used for complex arguments of atan2() is
|
|---|
| 1675 |
|
|---|
| 1676 | -i log((x + iy)/sqrt(x*x+y*y))
|
|---|
| 1677 |
|
|---|
| 1678 | The following extra operations are supported on both real and complex
|
|---|
| 1679 | numbers:
|
|---|
| 1680 |
|
|---|
| 1681 | Re(z) = a
|
|---|
| 1682 | Im(z) = b
|
|---|
| 1683 | arg(z) = t
|
|---|
| 1684 | abs(z) = r
|
|---|
| 1685 |
|
|---|
| 1686 | cbrt(z) = z ** (1/3)
|
|---|
| 1687 | log10(z) = log(z) / log(10)
|
|---|
| 1688 | logn(z, n) = log(z) / log(n)
|
|---|
| 1689 |
|
|---|
| 1690 | tan(z) = sin(z) / cos(z)
|
|---|
| 1691 |
|
|---|
| 1692 | csc(z) = 1 / sin(z)
|
|---|
| 1693 | sec(z) = 1 / cos(z)
|
|---|
| 1694 | cot(z) = 1 / tan(z)
|
|---|
| 1695 |
|
|---|
| 1696 | asin(z) = -i * log(i*z + sqrt(1-z*z))
|
|---|
| 1697 | acos(z) = -i * log(z + i*sqrt(1-z*z))
|
|---|
| 1698 | atan(z) = i/2 * log((i+z) / (i-z))
|
|---|
| 1699 |
|
|---|
| 1700 | acsc(z) = asin(1 / z)
|
|---|
| 1701 | asec(z) = acos(1 / z)
|
|---|
| 1702 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))
|
|---|
| 1703 |
|
|---|
| 1704 | sinh(z) = 1/2 (exp(z) - exp(-z))
|
|---|
| 1705 | cosh(z) = 1/2 (exp(z) + exp(-z))
|
|---|
| 1706 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))
|
|---|
| 1707 |
|
|---|
| 1708 | csch(z) = 1 / sinh(z)
|
|---|
| 1709 | sech(z) = 1 / cosh(z)
|
|---|
| 1710 | coth(z) = 1 / tanh(z)
|
|---|
| 1711 |
|
|---|
| 1712 | asinh(z) = log(z + sqrt(z*z+1))
|
|---|
| 1713 | acosh(z) = log(z + sqrt(z*z-1))
|
|---|
| 1714 | atanh(z) = 1/2 * log((1+z) / (1-z))
|
|---|
| 1715 |
|
|---|
| 1716 | acsch(z) = asinh(1 / z)
|
|---|
| 1717 | asech(z) = acosh(1 / z)
|
|---|
| 1718 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
|
|---|
| 1719 |
|
|---|
| 1720 | I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>,
|
|---|
| 1721 | I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>,
|
|---|
| 1722 | I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>,
|
|---|
| 1723 | I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>,
|
|---|
| 1724 | C<rho>, and C<theta> can be used also as mutators. The C<cbrt>
|
|---|
| 1725 | returns only one of the solutions: if you want all three, use the
|
|---|
| 1726 | C<root> function.
|
|---|
| 1727 |
|
|---|
| 1728 | The I<root> function is available to compute all the I<n>
|
|---|
| 1729 | roots of some complex, where I<n> is a strictly positive integer.
|
|---|
| 1730 | There are exactly I<n> such roots, returned as a list. Getting the
|
|---|
| 1731 | number mathematicians call C<j> such that:
|
|---|
| 1732 |
|
|---|
| 1733 | 1 + j + j*j = 0;
|
|---|
| 1734 |
|
|---|
| 1735 | is a simple matter of writing:
|
|---|
| 1736 |
|
|---|
| 1737 | $j = ((root(1, 3))[1];
|
|---|
| 1738 |
|
|---|
| 1739 | The I<k>th root for C<z = [r,t]> is given by:
|
|---|
| 1740 |
|
|---|
| 1741 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
|
|---|
| 1742 |
|
|---|
| 1743 | You can return the I<k>th root directly by C<root(z, n, k)>,
|
|---|
| 1744 | indexing starting from I<zero> and ending at I<n - 1>.
|
|---|
| 1745 |
|
|---|
| 1746 | The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In
|
|---|
| 1747 | order to ensure its restriction to real numbers is conform to what you
|
|---|
| 1748 | would expect, the comparison is run on the real part of the complex
|
|---|
| 1749 | number first, and imaginary parts are compared only when the real
|
|---|
| 1750 | parts match.
|
|---|
| 1751 |
|
|---|
| 1752 | =head1 CREATION
|
|---|
| 1753 |
|
|---|
| 1754 | To create a complex number, use either:
|
|---|
| 1755 |
|
|---|
| 1756 | $z = Math::Complex->make(3, 4);
|
|---|
| 1757 | $z = cplx(3, 4);
|
|---|
| 1758 |
|
|---|
| 1759 | if you know the cartesian form of the number, or
|
|---|
| 1760 |
|
|---|
| 1761 | $z = 3 + 4*i;
|
|---|
| 1762 |
|
|---|
| 1763 | if you like. To create a number using the polar form, use either:
|
|---|
| 1764 |
|
|---|
| 1765 | $z = Math::Complex->emake(5, pi/3);
|
|---|
| 1766 | $x = cplxe(5, pi/3);
|
|---|
| 1767 |
|
|---|
| 1768 | instead. The first argument is the modulus, the second is the angle
|
|---|
| 1769 | (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a
|
|---|
| 1770 | notation for complex numbers in the polar form).
|
|---|
| 1771 |
|
|---|
| 1772 | It is possible to write:
|
|---|
| 1773 |
|
|---|
| 1774 | $x = cplxe(-3, pi/4);
|
|---|
| 1775 |
|
|---|
| 1776 | but that will be silently converted into C<[3,-3pi/4]>, since the
|
|---|
| 1777 | modulus must be non-negative (it represents the distance to the origin
|
|---|
| 1778 | in the complex plane).
|
|---|
| 1779 |
|
|---|
| 1780 | It is also possible to have a complex number as either argument of the
|
|---|
| 1781 | C<make>, C<emake>, C<cplx>, and C<cplxe>: the appropriate component of
|
|---|
| 1782 | the argument will be used.
|
|---|
| 1783 |
|
|---|
| 1784 | $z1 = cplx(-2, 1);
|
|---|
| 1785 | $z2 = cplx($z1, 4);
|
|---|
| 1786 |
|
|---|
| 1787 | The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
|
|---|
| 1788 | understand a single (string) argument of the forms
|
|---|
| 1789 |
|
|---|
| 1790 | 2-3i
|
|---|
| 1791 | -3i
|
|---|
| 1792 | [2,3]
|
|---|
| 1793 | [2,-3pi/4]
|
|---|
| 1794 | [2]
|
|---|
| 1795 |
|
|---|
| 1796 | in which case the appropriate cartesian and exponential components
|
|---|
| 1797 | will be parsed from the string and used to create new complex numbers.
|
|---|
| 1798 | The imaginary component and the theta, respectively, will default to zero.
|
|---|
| 1799 |
|
|---|
| 1800 | The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
|
|---|
| 1801 | understand the case of no arguments: this means plain zero or (0, 0).
|
|---|
| 1802 |
|
|---|
| 1803 | =head1 DISPLAYING
|
|---|
| 1804 |
|
|---|
| 1805 | When printed, a complex number is usually shown under its cartesian
|
|---|
| 1806 | style I<a+bi>, but there are legitimate cases where the polar style
|
|---|
| 1807 | I<[r,t]> is more appropriate. The process of converting the complex
|
|---|
| 1808 | number into a string that can be displayed is known as I<stringification>.
|
|---|
| 1809 |
|
|---|
| 1810 | By calling the class method C<Math::Complex::display_format> and
|
|---|
| 1811 | supplying either C<"polar"> or C<"cartesian"> as an argument, you
|
|---|
| 1812 | override the default display style, which is C<"cartesian">. Not
|
|---|
| 1813 | supplying any argument returns the current settings.
|
|---|
| 1814 |
|
|---|
| 1815 | This default can be overridden on a per-number basis by calling the
|
|---|
| 1816 | C<display_format> method instead. As before, not supplying any argument
|
|---|
| 1817 | returns the current display style for this number. Otherwise whatever you
|
|---|
| 1818 | specify will be the new display style for I<this> particular number.
|
|---|
| 1819 |
|
|---|
| 1820 | For instance:
|
|---|
| 1821 |
|
|---|
| 1822 | use Math::Complex;
|
|---|
| 1823 |
|
|---|
| 1824 | Math::Complex::display_format('polar');
|
|---|
| 1825 | $j = (root(1, 3))[1];
|
|---|
| 1826 | print "j = $j\n"; # Prints "j = [1,2pi/3]"
|
|---|
| 1827 | $j->display_format('cartesian');
|
|---|
| 1828 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
|
|---|
| 1829 |
|
|---|
| 1830 | The polar style attempts to emphasize arguments like I<k*pi/n>
|
|---|
| 1831 | (where I<n> is a positive integer and I<k> an integer within [-9, +9]),
|
|---|
| 1832 | this is called I<polar pretty-printing>.
|
|---|
| 1833 |
|
|---|
| 1834 | For the reverse of stringifying, see the C<make> and C<emake>.
|
|---|
| 1835 |
|
|---|
| 1836 | =head2 CHANGED IN PERL 5.6
|
|---|
| 1837 |
|
|---|
| 1838 | The C<display_format> class method and the corresponding
|
|---|
| 1839 | C<display_format> object method can now be called using
|
|---|
| 1840 | a parameter hash instead of just a one parameter.
|
|---|
| 1841 |
|
|---|
| 1842 | The old display format style, which can have values C<"cartesian"> or
|
|---|
| 1843 | C<"polar">, can be changed using the C<"style"> parameter.
|
|---|
| 1844 |
|
|---|
| 1845 | $j->display_format(style => "polar");
|
|---|
| 1846 |
|
|---|
| 1847 | The one parameter calling convention also still works.
|
|---|
| 1848 |
|
|---|
| 1849 | $j->display_format("polar");
|
|---|
| 1850 |
|
|---|
| 1851 | There are two new display parameters.
|
|---|
| 1852 |
|
|---|
| 1853 | The first one is C<"format">, which is a sprintf()-style format string
|
|---|
| 1854 | to be used for both numeric parts of the complex number(s). The is
|
|---|
| 1855 | somewhat system-dependent but most often it corresponds to C<"%.15g">.
|
|---|
| 1856 | You can revert to the default by setting the C<format> to C<undef>.
|
|---|
| 1857 |
|
|---|
| 1858 | # the $j from the above example
|
|---|
| 1859 |
|
|---|
| 1860 | $j->display_format('format' => '%.5f');
|
|---|
| 1861 | print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
|
|---|
| 1862 | $j->display_format('format' => undef);
|
|---|
| 1863 | print "j = $j\n"; # Prints "j = -0.5+0.86603i"
|
|---|
| 1864 |
|
|---|
| 1865 | Notice that this affects also the return values of the
|
|---|
| 1866 | C<display_format> methods: in list context the whole parameter hash
|
|---|
| 1867 | will be returned, as opposed to only the style parameter value.
|
|---|
| 1868 | This is a potential incompatibility with earlier versions if you
|
|---|
| 1869 | have been calling the C<display_format> method in list context.
|
|---|
| 1870 |
|
|---|
| 1871 | The second new display parameter is C<"polar_pretty_print">, which can
|
|---|
| 1872 | be set to true or false, the default being true. See the previous
|
|---|
| 1873 | section for what this means.
|
|---|
| 1874 |
|
|---|
| 1875 | =head1 USAGE
|
|---|
| 1876 |
|
|---|
| 1877 | Thanks to overloading, the handling of arithmetics with complex numbers
|
|---|
| 1878 | is simple and almost transparent.
|
|---|
| 1879 |
|
|---|
| 1880 | Here are some examples:
|
|---|
| 1881 |
|
|---|
| 1882 | use Math::Complex;
|
|---|
| 1883 |
|
|---|
| 1884 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1
|
|---|
| 1885 | print "j = $j, j**3 = ", $j ** 3, "\n";
|
|---|
| 1886 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
|
|---|
| 1887 |
|
|---|
| 1888 | $z = -16 + 0*i; # Force it to be a complex
|
|---|
| 1889 | print "sqrt($z) = ", sqrt($z), "\n";
|
|---|
| 1890 |
|
|---|
| 1891 | $k = exp(i * 2*pi/3);
|
|---|
| 1892 | print "$j - $k = ", $j - $k, "\n";
|
|---|
| 1893 |
|
|---|
| 1894 | $z->Re(3); # Re, Im, arg, abs,
|
|---|
| 1895 | $j->arg(2); # (the last two aka rho, theta)
|
|---|
| 1896 | # can be used also as mutators.
|
|---|
| 1897 |
|
|---|
| 1898 | =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
|
|---|
| 1899 |
|
|---|
| 1900 | The division (/) and the following functions
|
|---|
| 1901 |
|
|---|
| 1902 | log ln log10 logn
|
|---|
| 1903 | tan sec csc cot
|
|---|
| 1904 | atan asec acsc acot
|
|---|
| 1905 | tanh sech csch coth
|
|---|
| 1906 | atanh asech acsch acoth
|
|---|
| 1907 |
|
|---|
| 1908 | cannot be computed for all arguments because that would mean dividing
|
|---|
| 1909 | by zero or taking logarithm of zero. These situations cause fatal
|
|---|
| 1910 | runtime errors looking like this
|
|---|
| 1911 |
|
|---|
| 1912 | cot(0): Division by zero.
|
|---|
| 1913 | (Because in the definition of cot(0), the divisor sin(0) is 0)
|
|---|
| 1914 | Died at ...
|
|---|
| 1915 |
|
|---|
| 1916 | or
|
|---|
| 1917 |
|
|---|
| 1918 | atanh(-1): Logarithm of zero.
|
|---|
| 1919 | Died at...
|
|---|
| 1920 |
|
|---|
| 1921 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
|
|---|
| 1922 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the
|
|---|
| 1923 | logarithmic functions and the C<atanh>, C<acoth>, the argument cannot
|
|---|
| 1924 | be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be
|
|---|
| 1925 | C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be
|
|---|
| 1926 | C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument
|
|---|
| 1927 | cannot be C<-i> (the negative imaginary unit). For the C<tan>,
|
|---|
| 1928 | C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k>
|
|---|
| 1929 | is any integer. atan2(0, 0) is undefined, and if the complex arguments
|
|---|
| 1930 | are used for atan2(), a division by zero will happen if z1**2+z2**2 == 0.
|
|---|
| 1931 |
|
|---|
| 1932 | Note that because we are operating on approximations of real numbers,
|
|---|
| 1933 | these errors can happen when merely `too close' to the singularities
|
|---|
| 1934 | listed above.
|
|---|
| 1935 |
|
|---|
| 1936 | =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS
|
|---|
| 1937 |
|
|---|
| 1938 | The C<make> and C<emake> accept both real and complex arguments.
|
|---|
| 1939 | When they cannot recognize the arguments they will die with error
|
|---|
| 1940 | messages like the following
|
|---|
| 1941 |
|
|---|
| 1942 | Math::Complex::make: Cannot take real part of ...
|
|---|
| 1943 | Math::Complex::make: Cannot take real part of ...
|
|---|
| 1944 | Math::Complex::emake: Cannot take rho of ...
|
|---|
| 1945 | Math::Complex::emake: Cannot take theta of ...
|
|---|
| 1946 |
|
|---|
| 1947 | =head1 BUGS
|
|---|
| 1948 |
|
|---|
| 1949 | Saying C<use Math::Complex;> exports many mathematical routines in the
|
|---|
| 1950 | caller environment and even overrides some (C<sqrt>, C<log>, C<atan2>).
|
|---|
| 1951 | This is construed as a feature by the Authors, actually... ;-)
|
|---|
| 1952 |
|
|---|
| 1953 | All routines expect to be given real or complex numbers. Don't attempt to
|
|---|
| 1954 | use BigFloat, since Perl has currently no rule to disambiguate a '+'
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| 1955 | operation (for instance) between two overloaded entities.
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| 1956 |
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| 1957 | In Cray UNICOS there is some strange numerical instability that results
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| 1958 | in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware.
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| 1959 | The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex.
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| 1960 | Whatever it is, it does not manifest itself anywhere else where Perl runs.
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| 1961 |
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| 1962 | =head1 AUTHORS
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| 1963 |
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| 1964 | Daniel S. Lewart <F<d-lewart@uiuc.edu>>
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| 1965 |
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| 1966 | Original authors Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and
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| 1967 | Jarkko Hietaniemi <F<jhi@iki.fi>>
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| 1968 |
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| 1969 | =cut
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| 1970 |
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| 1971 | 1;
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| 1972 |
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| 1973 | # eof
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