# # Complex numbers and associated mathematical functions # -- Raphael Manfredi Since Sep 1996 # -- Jarkko Hietaniemi Since Mar 1997 # -- Daniel S. Lewart Since Sep 1997 # require Exporter; package Math::Complex; use strict; use vars qw(\$VERSION @ISA @EXPORT %EXPORT_TAGS); my ( \$i, \$ip2, %logn ); \$VERSION = sprintf("%s", q\$Id: Complex.pm,v 1.26 1998/11/01 00:00:00 dsl Exp \$ =~ /(\d+\.\d+)/); @ISA = qw(Exporter); my @trig = qw( pi tan csc cosec sec cot cotan asin acos atan acsc acosec asec acot acotan sinh cosh tanh csch cosech sech coth cotanh asinh acosh atanh acsch acosech asech acoth acotanh ); @EXPORT = (qw( i Re Im rho theta arg sqrt log ln log10 logn cbrt root cplx cplxe ), @trig); %EXPORT_TAGS = ( 'trig' => [@trig], ); use overload '+' => \&plus, '-' => \&minus, '*' => \&multiply, '/' => \÷, '**' => \&power, '<=>' => \&spaceship, 'neg' => \&negate, '~' => \&conjugate, 'abs' => \&abs, 'sqrt' => \&sqrt, 'exp' => \&exp, 'log' => \&log, 'sin' => \&sin, 'cos' => \&cos, 'tan' => \&tan, 'atan2' => \&atan2, qw("" stringify); # # Package "privates" # my \$package = 'Math::Complex'; # Package name my \$display = 'cartesian'; # Default display format my \$eps = 1e-14; # Epsilon # # Object attributes (internal): # cartesian [real, imaginary] -- cartesian form # polar [rho, theta] -- polar form # c_dirty cartesian form not up-to-date # p_dirty polar form not up-to-date # display display format (package's global when not set) # # Die on bad *make() arguments. sub _cannot_make { die "@{[(caller(1))[3]]}: Cannot take \$_[0] of \$_[1].\n"; } # # ->make # # Create a new complex number (cartesian form) # sub make { my \$self = bless {}, shift; my (\$re, \$im) = @_; my \$rre = ref \$re; if ( \$rre ) { if ( \$rre eq ref \$self ) { \$re = Re(\$re); } else { _cannot_make("real part", \$rre); } } my \$rim = ref \$im; if ( \$rim ) { if ( \$rim eq ref \$self ) { \$im = Im(\$im); } else { _cannot_make("imaginary part", \$rim); } } \$self->{'cartesian'} = [ \$re, \$im ]; \$self->{c_dirty} = 0; \$self->{p_dirty} = 1; \$self->display_format('cartesian'); return \$self; } # # ->emake # # Create a new complex number (exponential form) # sub emake { my \$self = bless {}, shift; my (\$rho, \$theta) = @_; my \$rrh = ref \$rho; if ( \$rrh ) { if ( \$rrh eq ref \$self ) { \$rho = rho(\$rho); } else { _cannot_make("rho", \$rrh); } } my \$rth = ref \$theta; if ( \$rth ) { if ( \$rth eq ref \$self ) { \$theta = theta(\$theta); } else { _cannot_make("theta", \$rth); } } if (\$rho < 0) { \$rho = -\$rho; \$theta = (\$theta <= 0) ? \$theta + pi() : \$theta - pi(); } \$self->{'polar'} = [\$rho, \$theta]; \$self->{p_dirty} = 0; \$self->{c_dirty} = 1; \$self->display_format('polar'); return \$self; } sub new { &make } # For backward compatibility only. # # cplx # # Creates a complex number from a (re, im) tuple. # This avoids the burden of writing Math::Complex->make(re, im). # sub cplx { my (\$re, \$im) = @_; return \$package->make(\$re, defined \$im ? \$im : 0); } # # cplxe # # Creates a complex number from a (rho, theta) tuple. # This avoids the burden of writing Math::Complex->emake(rho, theta). # sub cplxe { my (\$rho, \$theta) = @_; return \$package->emake(\$rho, defined \$theta ? \$theta : 0); } # # pi # # The number defined as pi = 180 degrees # use constant pi => 4 * CORE::atan2(1, 1); # # pit2 # # The full circle # use constant pit2 => 2 * pi; # # pip2 # # The quarter circle # use constant pip2 => pi / 2; # # deg1 # # One degree in radians, used in stringify_polar. # use constant deg1 => pi / 180; # # uplog10 # # Used in log10(). # use constant uplog10 => 1 / CORE::log(10); # # i # # The number defined as i*i = -1; # sub i () { return \$i if (\$i); \$i = bless {}; \$i->{'cartesian'} = [0, 1]; \$i->{'polar'} = [1, pip2]; \$i->{c_dirty} = 0; \$i->{p_dirty} = 0; return \$i; } # # Attribute access/set routines # sub cartesian {\$_[0]->{c_dirty} ? \$_[0]->update_cartesian : \$_[0]->{'cartesian'}} sub polar {\$_[0]->{p_dirty} ? \$_[0]->update_polar : \$_[0]->{'polar'}} sub set_cartesian { \$_[0]->{p_dirty}++; \$_[0]->{'cartesian'} = \$_[1] } sub set_polar { \$_[0]->{c_dirty}++; \$_[0]->{'polar'} = \$_[1] } # # ->update_cartesian # # Recompute and return the cartesian form, given accurate polar form. # sub update_cartesian { my \$self = shift; my (\$r, \$t) = @{\$self->{'polar'}}; \$self->{c_dirty} = 0; return \$self->{'cartesian'} = [\$r * CORE::cos(\$t), \$r * CORE::sin(\$t)]; } # # # ->update_polar # # Recompute and return the polar form, given accurate cartesian form. # sub update_polar { my \$self = shift; my (\$x, \$y) = @{\$self->{'cartesian'}}; \$self->{p_dirty} = 0; return \$self->{'polar'} = [0, 0] if \$x == 0 && \$y == 0; return \$self->{'polar'} = [CORE::sqrt(\$x*\$x + \$y*\$y), CORE::atan2(\$y, \$x)]; } # # (plus) # # Computes z1+z2. # sub plus { my (\$z1, \$z2, \$regular) = @_; my (\$re1, \$im1) = @{\$z1->cartesian}; \$z2 = cplx(\$z2) unless ref \$z2; my (\$re2, \$im2) = ref \$z2 ? @{\$z2->cartesian} : (\$z2, 0); unless (defined \$regular) { \$z1->set_cartesian([\$re1 + \$re2, \$im1 + \$im2]); return \$z1; } return (ref \$z1)->make(\$re1 + \$re2, \$im1 + \$im2); } # # (minus) # # Computes z1-z2. # sub minus { my (\$z1, \$z2, \$inverted) = @_; my (\$re1, \$im1) = @{\$z1->cartesian}; \$z2 = cplx(\$z2) unless ref \$z2; my (\$re2, \$im2) = @{\$z2->cartesian}; unless (defined \$inverted) { \$z1->set_cartesian([\$re1 - \$re2, \$im1 - \$im2]); return \$z1; } return \$inverted ? (ref \$z1)->make(\$re2 - \$re1, \$im2 - \$im1) : (ref \$z1)->make(\$re1 - \$re2, \$im1 - \$im2); } # # (multiply) # # Computes z1*z2. # sub multiply { my (\$z1, \$z2, \$regular) = @_; if (\$z1->{p_dirty} == 0 and ref \$z2 and \$z2->{p_dirty} == 0) { # if both polar better use polar to avoid rounding errors my (\$r1, \$t1) = @{\$z1->polar}; my (\$r2, \$t2) = @{\$z2->polar}; my \$t = \$t1 + \$t2; if (\$t > pi()) { \$t -= pit2 } elsif (\$t <= -pi()) { \$t += pit2 } unless (defined \$regular) { \$z1->set_polar([\$r1 * \$r2, \$t]); return \$z1; } return (ref \$z1)->emake(\$r1 * \$r2, \$t); } else { my (\$x1, \$y1) = @{\$z1->cartesian}; if (ref \$z2) { my (\$x2, \$y2) = @{\$z2->cartesian}; return (ref \$z1)->make(\$x1*\$x2-\$y1*\$y2, \$x1*\$y2+\$y1*\$x2); } else { return (ref \$z1)->make(\$x1*\$z2, \$y1*\$z2); } } } # # _divbyzero # # Die on division by zero. # sub _divbyzero { my \$mess = "\$_[0]: Division by zero.\n"; if (defined \$_[1]) { \$mess .= "(Because in the definition of \$_[0], the divisor "; \$mess .= "\$_[1] " unless (\$_[1] eq '0'); \$mess .= "is 0)\n"; } my @up = caller(1); \$mess .= "Died at \$up[1] line \$up[2].\n"; die \$mess; } # # (divide) # # Computes z1/z2. # sub divide { my (\$z1, \$z2, \$inverted) = @_; if (\$z1->{p_dirty} == 0 and ref \$z2 and \$z2->{p_dirty} == 0) { # if both polar better use polar to avoid rounding errors my (\$r1, \$t1) = @{\$z1->polar}; my (\$r2, \$t2) = @{\$z2->polar}; my \$t; if (\$inverted) { _divbyzero "\$z2/0" if (\$r1 == 0); \$t = \$t2 - \$t1; if (\$t > pi()) { \$t -= pit2 } elsif (\$t <= -pi()) { \$t += pit2 } return (ref \$z1)->emake(\$r2 / \$r1, \$t); } else { _divbyzero "\$z1/0" if (\$r2 == 0); \$t = \$t1 - \$t2; if (\$t > pi()) { \$t -= pit2 } elsif (\$t <= -pi()) { \$t += pit2 } return (ref \$z1)->emake(\$r1 / \$r2, \$t); } } else { my (\$d, \$x2, \$y2); if (\$inverted) { (\$x2, \$y2) = @{\$z1->cartesian}; \$d = \$x2*\$x2 + \$y2*\$y2; _divbyzero "\$z2/0" if \$d == 0; return (ref \$z1)->make((\$x2*\$z2)/\$d, -(\$y2*\$z2)/\$d); } else { my (\$x1, \$y1) = @{\$z1->cartesian}; if (ref \$z2) { (\$x2, \$y2) = @{\$z2->cartesian}; \$d = \$x2*\$x2 + \$y2*\$y2; _divbyzero "\$z1/0" if \$d == 0; my \$u = (\$x1*\$x2 + \$y1*\$y2)/\$d; my \$v = (\$y1*\$x2 - \$x1*\$y2)/\$d; return (ref \$z1)->make(\$u, \$v); } else { _divbyzero "\$z1/0" if \$z2 == 0; return (ref \$z1)->make(\$x1/\$z2, \$y1/\$z2); } } } } # # (power) # # Computes z1**z2 = exp(z2 * log z1)). # sub power { my (\$z1, \$z2, \$inverted) = @_; if (\$inverted) { return 1 if \$z1 == 0 || \$z2 == 1; return 0 if \$z2 == 0 && Re(\$z1) > 0; } else { return 1 if \$z2 == 0 || \$z1 == 1; return 0 if \$z1 == 0 && Re(\$z2) > 0; } my \$w = \$inverted ? CORE::exp(\$z1 * CORE::log(\$z2)) : CORE::exp(\$z2 * CORE::log(\$z1)); # If both arguments cartesian, return cartesian, else polar. return \$z1->{c_dirty} == 0 && (not ref \$z2 or \$z2->{c_dirty} == 0) ? cplx(@{\$w->cartesian}) : \$w; } # # (spaceship) # # Computes z1 <=> z2. # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. # sub spaceship { my (\$z1, \$z2, \$inverted) = @_; my (\$re1, \$im1) = ref \$z1 ? @{\$z1->cartesian} : (\$z1, 0); my (\$re2, \$im2) = ref \$z2 ? @{\$z2->cartesian} : (\$z2, 0); my \$sgn = \$inverted ? -1 : 1; return \$sgn * (\$re1 <=> \$re2) if \$re1 != \$re2; return \$sgn * (\$im1 <=> \$im2); } # # (negate) # # Computes -z. # sub negate { my (\$z) = @_; if (\$z->{c_dirty}) { my (\$r, \$t) = @{\$z->polar}; \$t = (\$t <= 0) ? \$t + pi : \$t - pi; return (ref \$z)->emake(\$r, \$t); } my (\$re, \$im) = @{\$z->cartesian}; return (ref \$z)->make(-\$re, -\$im); } # # (conjugate) # # Compute complex's conjugate. # sub conjugate { my (\$z) = @_; if (\$z->{c_dirty}) { my (\$r, \$t) = @{\$z->polar}; return (ref \$z)->emake(\$r, -\$t); } my (\$re, \$im) = @{\$z->cartesian}; return (ref \$z)->make(\$re, -\$im); } # # (abs) # # Compute or set complex's norm (rho). # sub abs { my (\$z, \$rho) = @_; return \$z unless ref \$z; if (defined \$rho) { \$z->{'polar'} = [ \$rho, \${\$z->polar}[1] ]; \$z->{p_dirty} = 0; \$z->{c_dirty} = 1; return \$rho; } else { return \${\$z->polar}[0]; } } sub _theta { my \$theta = \$_[0]; if (\$\$theta > pi()) { \$\$theta -= pit2 } elsif (\$\$theta <= -pi()) { \$\$theta += pit2 } } # # arg # # Compute or set complex's argument (theta). # sub arg { my (\$z, \$theta) = @_; return \$z unless ref \$z; if (defined \$theta) { _theta(\\$theta); \$z->{'polar'} = [ \${\$z->polar}[0], \$theta ]; \$z->{p_dirty} = 0; \$z->{c_dirty} = 1; } else { \$theta = \${\$z->polar}[1]; _theta(\\$theta); } return \$theta; } # # (sqrt) # # Compute sqrt(z). # # It is quite tempting to use wantarray here so that in list context # sqrt() would return the two solutions. This, however, would # break things like # # print "sqrt(z) = ", sqrt(\$z), "\n"; # # The two values would be printed side by side without no intervening # whitespace, quite confusing. # Therefore if you want the two solutions use the root(). # sub sqrt { my (\$z) = @_; my (\$re, \$im) = ref \$z ? @{\$z->cartesian} : (\$z, 0); return \$re < 0 ? cplx(0, CORE::sqrt(-\$re)) : CORE::sqrt(\$re) if \$im == 0; my (\$r, \$t) = @{\$z->polar}; return (ref \$z)->emake(CORE::sqrt(\$r), \$t/2); } # # cbrt # # Compute cbrt(z) (cubic root). # # Why are we not returning three values? The same answer as for sqrt(). # sub cbrt { my (\$z) = @_; return \$z < 0 ? -CORE::exp(CORE::log(-\$z)/3) : (\$z > 0 ? CORE::exp(CORE::log(\$z)/3): 0) unless ref \$z; my (\$r, \$t) = @{\$z->polar}; return (ref \$z)->emake(CORE::exp(CORE::log(\$r)/3), \$t/3); } # # _rootbad # # Die on bad root. # sub _rootbad { my \$mess = "Root \$_[0] not defined, root must be positive integer.\n"; my @up = caller(1); \$mess .= "Died at \$up[1] line \$up[2].\n"; die \$mess; } # # root # # Computes all nth root for z, returning an array whose size is n. # `n' must be a positive integer. # # The roots are given by (for k = 0..n-1): # # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) # sub root { my (\$z, \$n) = @_; _rootbad(\$n) if (\$n < 1 or int(\$n) != \$n); my (\$r, \$t) = ref \$z ? @{\$z->polar} : (CORE::abs(\$z), \$z >= 0 ? 0 : pi); my @root; my \$k; my \$theta_inc = pit2 / \$n; my \$rho = \$r ** (1/\$n); my \$theta; my \$cartesian = ref \$z && \$z->{c_dirty} == 0; for (\$k = 0, \$theta = \$t / \$n; \$k < \$n; \$k++, \$theta += \$theta_inc) { my \$w = cplxe(\$rho, \$theta); # Yes, \$cartesian is loop invariant. push @root, \$cartesian ? cplx(@{\$w->cartesian}) : \$w; } return @root; } # # Re # # Return or set Re(z). # sub Re { my (\$z, \$Re) = @_; return \$z unless ref \$z; if (defined \$Re) { \$z->{'cartesian'} = [ \$Re, \${\$z->cartesian}[1] ]; \$z->{c_dirty} = 0; \$z->{p_dirty} = 1; } else { return \${\$z->cartesian}[0]; } } # # Im # # Return or set Im(z). # sub Im { my (\$z, \$Im) = @_; return \$z unless ref \$z; if (defined \$Im) { \$z->{'cartesian'} = [ \${\$z->cartesian}[0], \$Im ]; \$z->{c_dirty} = 0; \$z->{p_dirty} = 1; } else { return \${\$z->cartesian}[1]; } } # # rho # # Return or set rho(w). # sub rho { Math::Complex::abs(@_); } # # theta # # Return or set theta(w). # sub theta { Math::Complex::arg(@_); } # # (exp) # # Computes exp(z). # sub exp { my (\$z) = @_; my (\$x, \$y) = @{\$z->cartesian}; return (ref \$z)->emake(CORE::exp(\$x), \$y); } # # _logofzero # # Die on logarithm of zero. # sub _logofzero { my \$mess = "\$_[0]: Logarithm of zero.\n"; if (defined \$_[1]) { \$mess .= "(Because in the definition of \$_[0], the argument "; \$mess .= "\$_[1] " unless (\$_[1] eq '0'); \$mess .= "is 0)\n"; } my @up = caller(1); \$mess .= "Died at \$up[1] line \$up[2].\n"; die \$mess; } # # (log) # # Compute log(z). # sub log { my (\$z) = @_; unless (ref \$z) { _logofzero("log") if \$z == 0; return \$z > 0 ? CORE::log(\$z) : cplx(CORE::log(-\$z), pi); } my (\$r, \$t) = @{\$z->polar}; _logofzero("log") if \$r == 0; if (\$t > pi()) { \$t -= pit2 } elsif (\$t <= -pi()) { \$t += pit2 } return (ref \$z)->make(CORE::log(\$r), \$t); } # # ln # # Alias for log(). # sub ln { Math::Complex::log(@_) } # # log10 # # Compute log10(z). # sub log10 { return Math::Complex::log(\$_[0]) * uplog10; } # # logn # # Compute logn(z,n) = log(z) / log(n) # sub logn { my (\$z, \$n) = @_; \$z = cplx(\$z, 0) unless ref \$z; my \$logn = \$logn{\$n}; \$logn = \$logn{\$n} = CORE::log(\$n) unless defined \$logn; # Cache log(n) return CORE::log(\$z) / \$logn; } # # (cos) # # Compute cos(z) = (exp(iz) + exp(-iz))/2. # sub cos { my (\$z) = @_; my (\$x, \$y) = @{\$z->cartesian}; my \$ey = CORE::exp(\$y); my \$ey_1 = 1 / \$ey; return (ref \$z)->make(CORE::cos(\$x) * (\$ey + \$ey_1)/2, CORE::sin(\$x) * (\$ey_1 - \$ey)/2); } # # (sin) # # Compute sin(z) = (exp(iz) - exp(-iz))/2. # sub sin { my (\$z) = @_; my (\$x, \$y) = @{\$z->cartesian}; my \$ey = CORE::exp(\$y); my \$ey_1 = 1 / \$ey; return (ref \$z)->make(CORE::sin(\$x) * (\$ey + \$ey_1)/2, CORE::cos(\$x) * (\$ey - \$ey_1)/2); } # # tan # # Compute tan(z) = sin(z) / cos(z). # sub tan { my (\$z) = @_; my \$cz = CORE::cos(\$z); _divbyzero "tan(\$z)", "cos(\$z)" if (CORE::abs(\$cz) < \$eps); return CORE::sin(\$z) / \$cz; } # # sec # # Computes the secant sec(z) = 1 / cos(z). # sub sec { my (\$z) = @_; my \$cz = CORE::cos(\$z); _divbyzero "sec(\$z)", "cos(\$z)" if (\$cz == 0); return 1 / \$cz; } # # csc # # Computes the cosecant csc(z) = 1 / sin(z). # sub csc { my (\$z) = @_; my \$sz = CORE::sin(\$z); _divbyzero "csc(\$z)", "sin(\$z)" if (\$sz == 0); return 1 / \$sz; } # # cosec # # Alias for csc(). # sub cosec { Math::Complex::csc(@_) } # # cot # # Computes cot(z) = cos(z) / sin(z). # sub cot { my (\$z) = @_; my \$sz = CORE::sin(\$z); _divbyzero "cot(\$z)", "sin(\$z)" if (\$sz == 0); return CORE::cos(\$z) / \$sz; } # # cotan # # Alias for cot(). # sub cotan { Math::Complex::cot(@_) } # # acos # # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). # sub acos { my \$z = \$_[0]; return CORE::atan2(CORE::sqrt(1-\$z*\$z), \$z) if (! ref \$z) && CORE::abs(\$z) <= 1; my (\$x, \$y) = ref \$z ? @{\$z->cartesian} : (\$z, 0); my \$t1 = CORE::sqrt((\$x+1)*(\$x+1) + \$y*\$y); my \$t2 = CORE::sqrt((\$x-1)*(\$x-1) + \$y*\$y); my \$alpha = (\$t1 + \$t2)/2; my \$beta = (\$t1 - \$t2)/2; \$alpha = 1 if \$alpha < 1; if (\$beta > 1) { \$beta = 1 } elsif (\$beta < -1) { \$beta = -1 } my \$u = CORE::atan2(CORE::sqrt(1-\$beta*\$beta), \$beta); my \$v = CORE::log(\$alpha + CORE::sqrt(\$alpha*\$alpha-1)); \$v = -\$v if \$y > 0 || (\$y == 0 && \$x < -1); return \$package->make(\$u, \$v); } # # asin # # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). # sub asin { my \$z = \$_[0]; return CORE::atan2(\$z, CORE::sqrt(1-\$z*\$z)) if (! ref \$z) && CORE::abs(\$z) <= 1; my (\$x, \$y) = ref \$z ? @{\$z->cartesian} : (\$z, 0); my \$t1 = CORE::sqrt((\$x+1)*(\$x+1) + \$y*\$y); my \$t2 = CORE::sqrt((\$x-1)*(\$x-1) + \$y*\$y); my \$alpha = (\$t1 + \$t2)/2; my \$beta = (\$t1 - \$t2)/2; \$alpha = 1 if \$alpha < 1; if (\$beta > 1) { \$beta = 1 } elsif (\$beta < -1) { \$beta = -1 } my \$u = CORE::atan2(\$beta, CORE::sqrt(1-\$beta*\$beta)); my \$v = -CORE::log(\$alpha + CORE::sqrt(\$alpha*\$alpha-1)); \$v = -\$v if \$y > 0 || (\$y == 0 && \$x < -1); return \$package->make(\$u, \$v); } # # atan # # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). # sub atan { my (\$z) = @_; return CORE::atan2(\$z, 1) unless ref \$z; _divbyzero "atan(i)" if ( \$z == i); _divbyzero "atan(-i)" if (-\$z == i); my \$log = CORE::log((i + \$z) / (i - \$z)); \$ip2 = 0.5 * i unless defined \$ip2; return \$ip2 * \$log; } # # asec # # Computes the arc secant asec(z) = acos(1 / z). # sub asec { my (\$z) = @_; _divbyzero "asec(\$z)", \$z if (\$z == 0); return acos(1 / \$z); } # # acsc # # Computes the arc cosecant acsc(z) = asin(1 / z). # sub acsc { my (\$z) = @_; _divbyzero "acsc(\$z)", \$z if (\$z == 0); return asin(1 / \$z); } # # acosec # # Alias for acsc(). # sub acosec { Math::Complex::acsc(@_) } # # acot # # Computes the arc cotangent acot(z) = atan(1 / z) # sub acot { my (\$z) = @_; _divbyzero "acot(0)" if (CORE::abs(\$z) < \$eps); return (\$z >= 0) ? CORE::atan2(1, \$z) : CORE::atan2(-1, -\$z) unless ref \$z; _divbyzero "acot(i)" if (CORE::abs(\$z - i) < \$eps); _logofzero "acot(-i)" if (CORE::abs(\$z + i) < \$eps); return atan(1 / \$z); } # # acotan # # Alias for acot(). # sub acotan { Math::Complex::acot(@_) } # # cosh # # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. # sub cosh { my (\$z) = @_; my \$ex; unless (ref \$z) { \$ex = CORE::exp(\$z); return (\$ex + 1/\$ex)/2; } my (\$x, \$y) = @{\$z->cartesian}; \$ex = CORE::exp(\$x); my \$ex_1 = 1 / \$ex; return (ref \$z)->make(CORE::cos(\$y) * (\$ex + \$ex_1)/2, CORE::sin(\$y) * (\$ex - \$ex_1)/2); } # # sinh # # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. # sub sinh { my (\$z) = @_; my \$ex; unless (ref \$z) { \$ex = CORE::exp(\$z); return (\$ex - 1/\$ex)/2; } my (\$x, \$y) = @{\$z->cartesian}; \$ex = CORE::exp(\$x); my \$ex_1 = 1 / \$ex; return (ref \$z)->make(CORE::cos(\$y) * (\$ex - \$ex_1)/2, CORE::sin(\$y) * (\$ex + \$ex_1)/2); } # # tanh # # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). # sub tanh { my (\$z) = @_; my \$cz = cosh(\$z); _divbyzero "tanh(\$z)", "cosh(\$z)" if (\$cz == 0); return sinh(\$z) / \$cz; } # # sech # # Computes the hyperbolic secant sech(z) = 1 / cosh(z). # sub sech { my (\$z) = @_; my \$cz = cosh(\$z); _divbyzero "sech(\$z)", "cosh(\$z)" if (\$cz == 0); return 1 / \$cz; } # # csch # # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). # sub csch { my (\$z) = @_; my \$sz = sinh(\$z); _divbyzero "csch(\$z)", "sinh(\$z)" if (\$sz == 0); return 1 / \$sz; } # # cosech # # Alias for csch(). # sub cosech { Math::Complex::csch(@_) } # # coth # # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). # sub coth { my (\$z) = @_; my \$sz = sinh(\$z); _divbyzero "coth(\$z)", "sinh(\$z)" if (\$sz == 0); return cosh(\$z) / \$sz; } # # cotanh # # Alias for coth(). # sub cotanh { Math::Complex::coth(@_) } # # acosh # # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). # sub acosh { my (\$z) = @_; unless (ref \$z) { return CORE::log(\$z + CORE::sqrt(\$z*\$z-1)) if \$z >= 1; \$z = cplx(\$z, 0); } my (\$re, \$im) = @{\$z->cartesian}; if (\$im == 0) { return cplx(CORE::log(\$re + CORE::sqrt(\$re*\$re - 1)), 0) if \$re >= 1; return cplx(0, CORE::atan2(CORE::sqrt(1-\$re*\$re), \$re)) if CORE::abs(\$re) <= 1; } return CORE::log(\$z + CORE::sqrt(\$z*\$z - 1)); } # # asinh # # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-1)) # sub asinh { my (\$z) = @_; return CORE::log(\$z + CORE::sqrt(\$z*\$z + 1)); } # # atanh # # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). # sub atanh { my (\$z) = @_; unless (ref \$z) { return CORE::log((1 + \$z)/(1 - \$z))/2 if CORE::abs(\$z) < 1; \$z = cplx(\$z, 0); } _divbyzero 'atanh(1)', "1 - \$z" if (\$z == 1); _logofzero 'atanh(-1)' if (\$z == -1); return 0.5 * CORE::log((1 + \$z) / (1 - \$z)); } # # asech # # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). # sub asech { my (\$z) = @_; _divbyzero 'asech(0)', \$z if (\$z == 0); return acosh(1 / \$z); } # # acsch # # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). # sub acsch { my (\$z) = @_; _divbyzero 'acsch(0)', \$z if (\$z == 0); return asinh(1 / \$z); } # # acosech # # Alias for acosh(). # sub acosech { Math::Complex::acsch(@_) } # # acoth # # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). # sub acoth { my (\$z) = @_; _divbyzero 'acoth(0)' if (CORE::abs(\$z) < \$eps); unless (ref \$z) { return CORE::log((\$z + 1)/(\$z - 1))/2 if CORE::abs(\$z) > 1; \$z = cplx(\$z, 0); } _divbyzero 'acoth(1)', "\$z - 1" if (CORE::abs(\$z - 1) < \$eps); _logofzero 'acoth(-1)', "1 / \$z" if (CORE::abs(\$z + 1) < \$eps); return CORE::log((1 + \$z) / (\$z - 1)) / 2; } # # acotanh # # Alias for acot(). # sub acotanh { Math::Complex::acoth(@_) } # # (atan2) # # Compute atan(z1/z2). # sub atan2 { my (\$z1, \$z2, \$inverted) = @_; my (\$re1, \$im1, \$re2, \$im2); if (\$inverted) { (\$re1, \$im1) = ref \$z2 ? @{\$z2->cartesian} : (\$z2, 0); (\$re2, \$im2) = @{\$z1->cartesian}; } else { (\$re1, \$im1) = @{\$z1->cartesian}; (\$re2, \$im2) = ref \$z2 ? @{\$z2->cartesian} : (\$z2, 0); } if (\$im2 == 0) { return cplx(CORE::atan2(\$re1, \$re2), 0) if \$im1 == 0; return cplx((\$im1<=>0) * pip2, 0) if \$re2 == 0; } my \$w = atan(\$z1/\$z2); my (\$u, \$v) = ref \$w ? @{\$w->cartesian} : (\$w, 0); \$u += pi if \$re2 < 0; \$u -= pit2 if \$u > pi; return cplx(\$u, \$v); } # # display_format # ->display_format # # Set (fetch if no argument) display format for all complex numbers that # don't happen to have overridden it via ->display_format # # When called as a method, this actually sets the display format for # the current object. # # Valid object formats are 'c' and 'p' for cartesian and polar. The first # letter is used actually, so the type can be fully spelled out for clarity. # sub display_format { my \$self = shift; my \$format = undef; if (ref \$self) { # Called as a method \$format = shift; } else { # Regular procedure call \$format = \$self; undef \$self; } if (defined \$self) { return defined \$self->{display} ? \$self->{display} : \$display unless defined \$format; return \$self->{display} = \$format; } return \$display unless defined \$format; return \$display = \$format; } # # (stringify) # # Show nicely formatted complex number under its cartesian or polar form, # depending on the current display format: # # . If a specific display format has been recorded for this object, use it. # . Otherwise, use the generic current default for all complex numbers, # which is a package global variable. # sub stringify { my (\$z) = shift; my \$format; \$format = \$display; \$format = \$z->{display} if defined \$z->{display}; return \$z->stringify_polar if \$format =~ /^p/i; return \$z->stringify_cartesian; } # # ->stringify_cartesian # # Stringify as a cartesian representation 'a+bi'. # sub stringify_cartesian { my \$z = shift; my (\$x, \$y) = @{\$z->cartesian}; my (\$re, \$im); \$x = int(\$x + (\$x < 0 ? -1 : 1) * \$eps) if int(CORE::abs(\$x)) != int(CORE::abs(\$x) + \$eps); \$y = int(\$y + (\$y < 0 ? -1 : 1) * \$eps) if int(CORE::abs(\$y)) != int(CORE::abs(\$y) + \$eps); \$re = "\$x" if CORE::abs(\$x) >= \$eps; if (\$y == 1) { \$im = 'i' } elsif (\$y == -1) { \$im = '-i' } elsif (CORE::abs(\$y) >= \$eps) { \$im = \$y . "i" } my \$str = ''; \$str = \$re if defined \$re; \$str .= "+\$im" if defined \$im; \$str =~ s/\+-/-/; \$str =~ s/^\+//; \$str =~ s/([-+])1i/\$1i/; # Not redundant with the above 1/-1 tests. \$str = '0' unless \$str; return \$str; } # Helper for stringify_polar, a Greatest Common Divisor with a memory. sub _gcd { my (\$a, \$b) = @_; use integer; # Loops forever if given negative inputs. if (\$b and \$a > \$b) { return gcd(\$a % \$b, \$b) } elsif (\$a and \$b > \$a) { return gcd(\$b % \$a, \$a) } else { return \$a ? \$a : \$b } } my %gcd; sub gcd { my (\$a, \$b) = @_; my \$id = "\$a \$b"; unless (exists \$gcd{\$id}) { \$gcd{\$id} = _gcd(\$a, \$b); \$gcd{"\$b \$a"} = \$gcd{\$id}; } return \$gcd{\$id}; } # # ->stringify_polar # # Stringify as a polar representation '[r,t]'. # sub stringify_polar { my \$z = shift; my (\$r, \$t) = @{\$z->polar}; my \$theta; return '[0,0]' if \$r <= \$eps; my \$nt = \$t / pit2; \$nt = (\$nt - int(\$nt)) * pit2; \$nt += pit2 if \$nt < 0; # Range [0, 2pi] if (CORE::abs(\$nt) <= \$eps) { \$theta = 0 } elsif (CORE::abs(pi-\$nt) <= \$eps) { \$theta = 'pi' } if (defined \$theta) { \$r = int(\$r + (\$r < 0 ? -1 : 1) * \$eps) if int(CORE::abs(\$r)) != int(CORE::abs(\$r) + \$eps); \$theta = int(\$theta + (\$theta < 0 ? -1 : 1) * \$eps) if (\$theta ne 'pi' and int(CORE::abs(\$theta)) != int(CORE::abs(\$theta) + \$eps)); return "\[\$r,\$theta\]"; } # # Okay, number is not a real. Try to identify pi/n and friends... # \$nt -= pit2 if \$nt > pi; if (CORE::abs(\$nt) >= deg1) { my (\$n, \$k, \$kpi); for (\$k = 1, \$kpi = pi; \$k < 10; \$k++, \$kpi += pi) { \$n = int(\$kpi / \$nt + (\$nt > 0 ? 1 : -1) * 0.5); if (CORE::abs(\$kpi/\$n - \$nt) <= \$eps) { \$n = CORE::abs(\$n); my \$gcd = gcd(\$k, \$n); if (\$gcd > 1) { \$k /= \$gcd; \$n /= \$gcd; } next if \$n > 360; \$theta = (\$nt < 0 ? '-':''). (\$k == 1 ? 'pi':"\${k}pi"); \$theta .= '/'.\$n if \$n > 1; last; } } } \$theta = \$nt unless defined \$theta; \$r = int(\$r + (\$r < 0 ? -1 : 1) * \$eps) if int(CORE::abs(\$r)) != int(CORE::abs(\$r) + \$eps); \$theta = int(\$theta + (\$theta < 0 ? -1 : 1) * \$eps) if (\$theta !~ m(^-?\d*pi/\d+\$) and int(CORE::abs(\$theta)) != int(CORE::abs(\$theta) + \$eps)); return "\[\$r,\$theta\]"; } 1; __END__ =head1 NAME Math::Complex - complex numbers and associated mathematical functions =head1 SYNOPSIS use Math::Complex; \$z = Math::Complex->make(5, 6); \$t = 4 - 3*i + \$z; \$j = cplxe(1, 2*pi/3); =head1 DESCRIPTION This package lets you create and manipulate complex numbers. By default, I limits itself to real numbers, but an extra C statement brings full complex support, along with a full set of mathematical functions typically associated with and/or extended to complex numbers. If you wonder what complex numbers are, they were invented to be able to solve the following equation: x*x = -1 and by definition, the solution is noted I (engineers use I instead since I usually denotes an intensity, but the name does not matter). The number I is a pure I number. The arithmetics with pure imaginary numbers works just like you would expect it with real numbers... you just have to remember that i*i = -1 so you have: 5i + 7i = i * (5 + 7) = 12i 4i - 3i = i * (4 - 3) = i 4i * 2i = -8 6i / 2i = 3 1 / i = -i Complex numbers are numbers that have both a real part and an imaginary part, and are usually noted: a + bi where C is the I part and C is the I part. The arithmetic with complex numbers is straightforward. You have to keep track of the real and the imaginary parts, but otherwise the rules used for real numbers just apply: (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i A graphical representation of complex numbers is possible in a plane (also called the I, but it's really a 2D plane). The number z = a + bi is the point whose coordinates are (a, b). Actually, it would be the vector originating from (0, 0) to (a, b). It follows that the addition of two complex numbers is a vectorial addition. Since there is a bijection between a point in the 2D plane and a complex number (i.e. the mapping is unique and reciprocal), a complex number can also be uniquely identified with polar coordinates: [rho, theta] where C is the distance to the origin, and C the angle between the vector and the I axis. There is a notation for this using the exponential form, which is: rho * exp(i * theta) where I is the famous imaginary number introduced above. Conversion between this form and the cartesian form C is immediate: a = rho * cos(theta) b = rho * sin(theta) which is also expressed by this formula: z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) In other words, it's the projection of the vector onto the I and I axes. Mathematicians call I the I or I and I the I of the complex number. The I of C will be noted C. The polar notation (also known as the trigonometric representation) is much more handy for performing multiplications and divisions of complex numbers, whilst the cartesian notation is better suited for additions and subtractions. Real numbers are on the I axis, and therefore I is zero or I. All the common operations that can be performed on a real number have been defined to work on complex numbers as well, and are merely I of the operations defined on real numbers. This means they keep their natural meaning when there is no imaginary part, provided the number is within their definition set. For instance, the C routine which computes the square root of its argument is only defined for non-negative real numbers and yields a non-negative real number (it is an application from B to B). If we allow it to return a complex number, then it can be extended to negative real numbers to become an application from B to B (the set of complex numbers): sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i It can also be extended to be an application from B to B, whilst its restriction to B behaves as defined above by using the following definition: sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) Indeed, a negative real number can be noted C<[x,pi]> (the modulus I is always non-negative, so C<[x,pi]> is really C<-x>, a negative number) and the above definition states that sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i which is exactly what we had defined for negative real numbers above. The C returns only one of the solutions: if you want the both, use the C function. All the common mathematical functions defined on real numbers that are extended to complex numbers share that same property of working I when the imaginary part is zero (otherwise, it would not be called an extension, would it?). A I operation possible on a complex number that is the identity for real numbers is called the I, and is noted with an horizontal bar above the number, or C<~z> here. z = a + bi ~z = a - bi Simple... Now look: z * ~z = (a + bi) * (a - bi) = a*a + b*b We saw that the norm of C was noted C and was defined as the distance to the origin, also known as: rho = abs(z) = sqrt(a*a + b*b) so z * ~z = abs(z) ** 2 If z is a pure real number (i.e. C), then the above yields: a * a = abs(a) ** 2 which is true (C has the regular meaning for real number, i.e. stands for the absolute value). This example explains why the norm of C is noted C: it extends the C function to complex numbers, yet is the regular C we know when the complex number actually has no imaginary part... This justifies I our use of the C notation for the norm. =head1 OPERATIONS Given the following notations: z1 = a + bi = r1 * exp(i * t1) z2 = c + di = r2 * exp(i * t2) z = the following (overloaded) operations are supported on complex numbers: z1 + z2 = (a + c) + i(b + d) z1 - z2 = (a - c) + i(b - d) z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) z1 ** z2 = exp(z2 * log z1) ~z = a - bi abs(z) = r1 = sqrt(a*a + b*b) sqrt(z) = sqrt(r1) * exp(i * t/2) exp(z) = exp(a) * exp(i * b) log(z) = log(r1) + i*t sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) atan2(z1, z2) = atan(z1/z2) The following extra operations are supported on both real and complex numbers: Re(z) = a Im(z) = b arg(z) = t abs(z) = r cbrt(z) = z ** (1/3) log10(z) = log(z) / log(10) logn(z, n) = log(z) / log(n) tan(z) = sin(z) / cos(z) csc(z) = 1 / sin(z) sec(z) = 1 / cos(z) cot(z) = 1 / tan(z) asin(z) = -i * log(i*z + sqrt(1-z*z)) acos(z) = -i * log(z + i*sqrt(1-z*z)) atan(z) = i/2 * log((i+z) / (i-z)) acsc(z) = asin(1 / z) asec(z) = acos(1 / z) acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) sinh(z) = 1/2 (exp(z) - exp(-z)) cosh(z) = 1/2 (exp(z) + exp(-z)) tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) csch(z) = 1 / sinh(z) sech(z) = 1 / cosh(z) coth(z) = 1 / tanh(z) asinh(z) = log(z + sqrt(z*z+1)) acosh(z) = log(z + sqrt(z*z-1)) atanh(z) = 1/2 * log((1+z) / (1-z)) acsch(z) = asinh(1 / z) asech(z) = acosh(1 / z) acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) I, I, I, I, I, I, I, I, I, I, I, have aliases I, I, I, I, I, I, I, I, I, I, I, respectively. C, C, C, C, C, and C can be used also also mutators. The C returns only one of the solutions: if you want all three, use the C function. The I function is available to compute all the I roots of some complex, where I is a strictly positive integer. There are exactly I such roots, returned as a list. Getting the number mathematicians call C such that: 1 + j + j*j = 0; is a simple matter of writing: \$j = ((root(1, 3))[1]; The Ith root for C is given by: (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) The I comparison operator, E=E, is also defined. In order to ensure its restriction to real numbers is conform to what you would expect, the comparison is run on the real part of the complex number first, and imaginary parts are compared only when the real parts match. =head1 CREATION To create a complex number, use either: \$z = Math::Complex->make(3, 4); \$z = cplx(3, 4); if you know the cartesian form of the number, or \$z = 3 + 4*i; if you like. To create a number using the polar form, use either: \$z = Math::Complex->emake(5, pi/3); \$x = cplxe(5, pi/3); instead. The first argument is the modulus, the second is the angle (in radians, the full circle is 2*pi). (Mnemonic: C is used as a notation for complex numbers in the polar form). It is possible to write: \$x = cplxe(-3, pi/4); but that will be silently converted into C<[3,-3pi/4]>, since the modulus must be non-negative (it represents the distance to the origin in the complex plane). It is also possible to have a complex number as either argument of either the C or C: the appropriate component of the argument will be used. \$z1 = cplx(-2, 1); \$z2 = cplx(\$z1, 4); =head1 STRINGIFICATION When printed, a complex number is usually shown under its cartesian form I, but there are legitimate cases where the polar format I<[r,t]> is more appropriate. By calling the routine C and supplying either C<"polar"> or C<"cartesian">, you override the default display format, which is C<"cartesian">. Not supplying any argument returns the current setting. This default can be overridden on a per-number basis by calling the C method instead. As before, not supplying any argument returns the current display format for this number. Otherwise whatever you specify will be the new display format for I particular number. For instance: use Math::Complex; Math::Complex::display_format('polar'); \$j = ((root(1, 3))[1]; print "j = \$j\n"; # Prints "j = [1,2pi/3] \$j->display_format('cartesian'); print "j = \$j\n"; # Prints "j = -0.5+0.866025403784439i" The polar format attempts to emphasize arguments like I (where I is a positive integer and I an integer within [-9,+9]). =head1 USAGE Thanks to overloading, the handling of arithmetics with complex numbers is simple and almost transparent. Here are some examples: use Math::Complex; \$j = cplxe(1, 2*pi/3); # \$j ** 3 == 1 print "j = \$j, j**3 = ", \$j ** 3, "\n"; print "1 + j + j**2 = ", 1 + \$j + \$j**2, "\n"; \$z = -16 + 0*i; # Force it to be a complex print "sqrt(\$z) = ", sqrt(\$z), "\n"; \$k = exp(i * 2*pi/3); print "\$j - \$k = ", \$j - \$k, "\n"; \$z->Re(3); # Re, Im, arg, abs, \$j->arg(2); # (the last two aka rho, theta) # can be used also as mutators. =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO The division (/) and the following functions log ln log10 logn tan sec csc cot atan asec acsc acot tanh sech csch coth atanh asech acsch acoth cannot be computed for all arguments because that would mean dividing by zero or taking logarithm of zero. These situations cause fatal runtime errors looking like this cot(0): Division by zero. (Because in the definition of cot(0), the divisor sin(0) is 0) Died at ... or atanh(-1): Logarithm of zero. Died at... For the C, C, C, C, C, C, C, C, C, the argument cannot be C<0> (zero). For the the logarithmic functions and the C, C, the argument cannot be C<1> (one). For the C, C, the argument cannot be C<-1> (minus one). For the C, C, the argument cannot be C (the imaginary unit). For the C, C, the argument cannot be C<-i> (the negative imaginary unit). For the C, C, C, the argument cannot be I, where I is any integer. Note that because we are operating on approximations of real numbers, these errors can happen when merely `too close' to the singularities listed above. For example C will die of division by zero. =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS The C and C accept both real and complex arguments. When they cannot recognize the arguments they will die with error messages like the following Math::Complex::make: Cannot take real part of ... Math::Complex::make: Cannot take real part of ... Math::Complex::emake: Cannot take rho of ... Math::Complex::emake: Cannot take theta of ... =head1 BUGS Saying C exports many mathematical routines in the caller environment and even overrides some (C, C). This is construed as a feature by the Authors, actually... ;-) All routines expect to be given real or complex numbers. Don't attempt to use BigFloat, since Perl has currently no rule to disambiguate a '+' operation (for instance) between two overloaded entities. In Cray UNICOS there is some strange numerical instability that results in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. Whatever it is, it does not manifest itself anywhere else where Perl runs. =head1 AUTHORS Raphael Manfredi > and Jarkko Hietaniemi >. Extensive patches by Daniel S. Lewart >. =cut 1; # eof