perl 5.000perl-5.000

[editor's note: this commit combines approximate 4 months of furious
releases of Andy Dougherty and Larry Wall - see pod/perlhist.pod for
details.  Andy notes that;
 
Alas neither my "Irwin AccuTrack" nor my DC 600A quarter-inch cartridge 
backup tapes from that era seem to be readable anymore.  I guess 13 years 
exceeds the shelf life for that backup technology :-(.
]

3 files changed, 780 insertions, 0 deletions

diff --git a/lib/Math/BigFloat.pm b/lib/Math/BigFloat.pm
new file mode 100644
index 0000000000..92e701666f
--- /dev/null
+++ b/lib/Math/BigFloat.pm
@@ -0,0 +1,297 @@
+package Math::BigFloat;
+
+use Math::BigInt;
+
+use Exporter;  # just for use to be happy
+@ISA = (Exporter);
+
+%OVERLOAD = ( 
+				# Anonymous subroutines:
+'+'	=>	sub {new BigFloat &fadd},
+'-'	=>	sub {new BigFloat
+		       $_[2]? fsub($_[1],${$_[0]}) : fsub(${$_[0]},$_[1])},
+'<=>'	=>	sub {new BigFloat
+		       $_[2]? fcmp($_[1],${$_[0]}) : fcmp(${$_[0]},$_[1])},
+'cmp'	=>	sub {new BigFloat
+		       $_[2]? ($_[1] cmp ${$_[0]}) : (${$_[0]} cmp $_[1])},
+'*'	=>	sub {new BigFloat &fmul},
+'/'	=>	sub {new BigFloat 
+		       $_[2]? scalar fdiv($_[1],${$_[0]}) :
+			 scalar fdiv(${$_[0]},$_[1])},
+'neg'	=>	sub {new BigFloat &fneg},
+'abs'	=>	sub {new BigFloat &fabs},
+
+qw(
+""	stringify
+0+	numify)			# Order of arguments unsignificant
+);
+
+sub new {
+  my $foo = fnorm($_[1]);
+  panic("Not a number initialized to BigFloat") if $foo eq "NaN";
+  bless \$foo;
+}
+sub numify { 0 + "${$_[0]}" }	# Not needed, additional overhead
+				# comparing to direct compilation based on
+				# stringify
+sub stringify {
+    my $n = ${$_[0]};
+
+    $n =~ s/^\+//;
+    $n =~ s/E//;
+
+    $n =~ s/([-+]\d+)$//;
+
+    my $e = $1;
+    my $ln = length($n);
+
+    if ($e > 0) {
+	$n .= "0" x $e . '.';
+    } elsif (abs($e) < $ln) {
+	substr($n, $ln + $e, 0) = '.';
+    } else {
+	$n = '.' . ("0" x (abs($e) - $ln)) . $n;
+    }
+
+    # 1 while $n =~ s/(.*\d)(\d\d\d)/$1,$2/;
+
+    return $n;
+}
+
+# Arbitrary length float math package
+#
+# by Mark Biggar
+#
+# number format
+#   canonical strings have the form /[+-]\d+E[+-]\d+/
+#   Input values can have inbedded whitespace
+# Error returns
+#   'NaN'           An input parameter was "Not a Number" or 
+#                       divide by zero or sqrt of negative number
+# Division is computed to 
+#   max($div_scale,length(dividend)+length(divisor)) 
+#   digits by default.
+# Also used for default sqrt scale
+
+$div_scale = 40;
+
+# Rounding modes one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'.
+
+$rnd_mode = 'even';
+
+sub fadd; sub fsub; sub fmul; sub fdiv;
+sub fneg; sub fabs; sub fcmp;
+sub fround; sub ffround;
+sub fnorm; sub fsqrt;
+
+#   bigfloat routines
+#
+#   fadd(NSTR, NSTR) return NSTR            addition
+#   fsub(NSTR, NSTR) return NSTR            subtraction
+#   fmul(NSTR, NSTR) return NSTR            multiplication
+#   fdiv(NSTR, NSTR[,SCALE]) returns NSTR   division to SCALE places
+#   fneg(NSTR) return NSTR                  negation
+#   fabs(NSTR) return NSTR                  absolute value
+#   fcmp(NSTR,NSTR) return CODE             compare undef,<0,=0,>0
+#   fround(NSTR, SCALE) return NSTR         round to SCALE digits
+#   ffround(NSTR, SCALE) return NSTR        round at SCALEth place
+#   fnorm(NSTR) return (NSTR)               normalize
+#   fsqrt(NSTR[, SCALE]) return NSTR        sqrt to SCALE places
+
+
+# Convert a number to canonical string form.
+#   Takes something that looks like a number and converts it to
+#   the form /^[+-]\d+E[+-]\d+$/.
+sub fnorm { #(string) return fnum_str
+    local($_) = @_;
+    s/\s+//g;                               # strip white space
+    if (/^([+-]?)(\d*)(\.(\d*))?([Ee]([+-]?\d+))?$/ && "$2$4" ne '') {
+	&norm(($1 ? "$1$2$4" : "+$2$4"),(($4 ne '') ? $6-length($4) : $6));
+    } else {
+	'NaN';
+    }
+}
+
+# normalize number -- for internal use
+sub norm { #(mantissa, exponent) return fnum_str
+    local($_, $exp) = @_;
+    if ($_ eq 'NaN') {
+	'NaN';
+    } else {
+	s/^([+-])0+/$1/;                        # strip leading zeros
+	if (length($_) == 1) {
+	    '+0E+0';
+	} else {
+	    $exp += length($1) if (s/(0+)$//);  # strip trailing zeros
+	    sprintf("%sE%+ld", $_, $exp);
+	}
+    }
+}
+
+# negation
+sub fneg { #(fnum_str) return fnum_str
+    local($_) = fnorm($_[$[]);
+    vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0E+0'; # flip sign
+    s/^H/N/;
+    $_;
+}
+
+# absolute value
+sub fabs { #(fnum_str) return fnum_str
+    local($_) = fnorm($_[$[]);
+    s/^-/+/;		                       # mash sign
+    $_;
+}
+
+# multiplication
+sub fmul { #(fnum_str, fnum_str) return fnum_str
+    local($x,$y) = (fnorm($_[$[]),fnorm($_[$[+1]));
+    if ($x eq 'NaN' || $y eq 'NaN') {
+	'NaN';
+    } else {
+	local($xm,$xe) = split('E',$x);
+	local($ym,$ye) = split('E',$y);
+	&norm(Math::BigInt::bmul($xm,$ym),$xe+$ye);
+    }
+}
+
+# addition
+sub fadd { #(fnum_str, fnum_str) return fnum_str
+    local($x,$y) = (fnorm($_[$[]),fnorm($_[$[+1]));
+    if ($x eq 'NaN' || $y eq 'NaN') {
+	'NaN';
+    } else {
+	local($xm,$xe) = split('E',$x);
+	local($ym,$ye) = split('E',$y);
+	($xm,$xe,$ym,$ye) = ($ym,$ye,$xm,$xe) if ($xe < $ye);
+	&norm(Math::BigInt::badd($ym,$xm.('0' x ($xe-$ye))),$ye);
+    }
+}
+
+# subtraction
+sub fsub { #(fnum_str, fnum_str) return fnum_str
+    fadd($_[$[],fneg($_[$[+1]));    
+}
+
+# division
+#   args are dividend, divisor, scale (optional)
+#   result has at most max(scale, length(dividend), length(divisor)) digits
+sub fdiv #(fnum_str, fnum_str[,scale]) return fnum_str
+{
+    local($x,$y,$scale) = (fnorm($_[$[]),fnorm($_[$[+1]),$_[$[+2]);
+    if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0E+0') {
+	'NaN';
+    } else {
+	local($xm,$xe) = split('E',$x);
+	local($ym,$ye) = split('E',$y);
+	$scale = $div_scale if (!$scale);
+	$scale = length($xm)-1 if (length($xm)-1 > $scale);
+	$scale = length($ym)-1 if (length($ym)-1 > $scale);
+	$scale = $scale + length($ym) - length($xm);
+	&norm(&round(Math::BigInt::bdiv($xm.('0' x $scale),$ym),$ym),
+	    $xe-$ye-$scale);
+    }
+}
+
+# round int $q based on fraction $r/$base using $rnd_mode
+sub round { #(int_str, int_str, int_str) return int_str
+    local($q,$r,$base) = @_;
+    if ($q eq 'NaN' || $r eq 'NaN') {
+	'NaN';
+    } elsif ($rnd_mode eq 'trunc') {
+	$q;                         # just truncate
+    } else {
+	local($cmp) = Math::BigInt::bcmp(Math::BigInt::bmul($r,'+2'),$base);
+	if ( $cmp < 0 ||
+		 ($cmp == 0 &&
+		  ( $rnd_mode eq 'zero'                             ||
+		   ($rnd_mode eq '-inf' && (substr($q,$[,1) eq '+')) ||
+		   ($rnd_mode eq '+inf' && (substr($q,$[,1) eq '-')) ||
+		   ($rnd_mode eq 'even' && $q =~ /[24680]$/)        ||
+		   ($rnd_mode eq 'odd'  && $q =~ /[13579]$/)        )) ) {
+	    $q;                     # round down
+	} else {
+	    Math::BigInt::badd($q, ((substr($q,$[,1) eq '-') ? '-1' : '+1'));
+				    # round up
+	}
+    }
+}
+
+# round the mantissa of $x to $scale digits
+sub fround { #(fnum_str, scale) return fnum_str
+    local($x,$scale) = (fnorm($_[$[]),$_[$[+1]);
+    if ($x eq 'NaN' || $scale <= 0) {
+	$x;
+    } else {
+	local($xm,$xe) = split('E',$x);
+	if (length($xm)-1 <= $scale) {
+	    $x;
+	} else {
+	    &norm(&round(substr($xm,$[,$scale+1),
+			 "+0".substr($xm,$[+$scale+1,1),"+10"),
+		  $xe+length($xm)-$scale-1);
+	}
+    }
+}
+
+# round $x at the 10 to the $scale digit place
+sub ffround { #(fnum_str, scale) return fnum_str
+    local($x,$scale) = (fnorm($_[$[]),$_[$[+1]);
+    if ($x eq 'NaN') {
+	'NaN';
+    } else {
+	local($xm,$xe) = split('E',$x);
+	if ($xe >= $scale) {
+	    $x;
+	} else {
+	    $xe = length($xm)+$xe-$scale;
+	    if ($xe < 1) {
+		'+0E+0';
+	    } elsif ($xe == 1) {
+		&norm(&round('+0',"+0".substr($xm,$[+1,1),"+10"), $scale);
+	    } else {
+		&norm(&round(substr($xm,$[,$xe),
+		      "+0".substr($xm,$[+$xe,1),"+10"), $scale);
+	    }
+	}
+    }
+}
+    
+# compare 2 values returns one of undef, <0, =0, >0
+#   returns undef if either or both input value are not numbers
+sub fcmp #(fnum_str, fnum_str) return cond_code
+{
+    local($x, $y) = (fnorm($_[$[]),fnorm($_[$[+1]));
+    if ($x eq "NaN" || $y eq "NaN") {
+	undef;
+    } else {
+	ord($y) <=> ord($x)
+	||
+	(  local($xm,$xe,$ym,$ye) = split('E', $x."E$y"),
+	     (($xe <=> $ye) * (substr($x,$[,1).'1')
+             || Math::BigInt::cmp($xm,$ym))
+	);
+    }
+}
+
+# square root by Newtons method.
+sub fsqrt { #(fnum_str[, scale]) return fnum_str
+    local($x, $scale) = (fnorm($_[$[]), $_[$[+1]);
+    if ($x eq 'NaN' || $x =~ /^-/) {
+	'NaN';
+    } elsif ($x eq '+0E+0') {
+	'+0E+0';
+    } else {
+	local($xm, $xe) = split('E',$x);
+	$scale = $div_scale if (!$scale);
+	$scale = length($xm)-1 if ($scale < length($xm)-1);
+	local($gs, $guess) = (1, sprintf("1E%+d", (length($xm)+$xe-1)/2));
+	while ($gs < 2*$scale) {
+	    $guess = fmul(fadd($guess,fdiv($x,$guess,$gs*2)),".5");
+	    $gs *= 2;
+	}
+	new BigFloat &fround($guess, $scale);
+    }
+}
+
+1;
diff --git a/lib/Math/BigInt.pm b/lib/Math/BigInt.pm
new file mode 100644
index 0000000000..3e0fc17ff6
--- /dev/null
+++ b/lib/Math/BigInt.pm
@@ -0,0 +1,347 @@
+package Math::BigInt;
+
+%OVERLOAD = ( 
+				# Anonymous subroutines:
+'+'	=>	sub {new BigInt &badd},
+'-'	=>	sub {new BigInt
+		       $_[2]? bsub($_[1],${$_[0]}) : bsub(${$_[0]},$_[1])},
+'<=>'	=>	sub {new BigInt
+		       $_[2]? bcmp($_[1],${$_[0]}) : bcmp(${$_[0]},$_[1])},
+'cmp'	=>	sub {new BigInt
+		       $_[2]? ($_[1] cmp ${$_[0]}) : (${$_[0]} cmp $_[1])},
+'*'	=>	sub {new BigInt &bmul},
+'/'	=>	sub {new BigInt 
+		       $_[2]? scalar bdiv($_[1],${$_[0]}) :
+			 scalar bdiv(${$_[0]},$_[1])},
+'%'	=>	sub {new BigInt
+		       $_[2]? bmod($_[1],${$_[0]}) : bmod(${$_[0]},$_[1])},
+'**'	=>	sub {new BigInt
+		       $_[2]? bpow($_[1],${$_[0]}) : bpow(${$_[0]},$_[1])},
+'neg'	=>	sub {new BigInt &bneg},
+'abs'	=>	sub {new BigInt &babs},
+
+qw(
+""	stringify
+0+	numify)			# Order of arguments unsignificant
+);
+
+sub new {
+  my $foo = bnorm($_[1]);
+  die "Not a number initialized to BigInt" if $foo eq "NaN";
+  bless \$foo;
+}
+sub stringify { "${$_[0]}" }
+sub numify { 0 + "${$_[0]}" }	# Not needed, additional overhead
+				# comparing to direct compilation based on
+				# stringify
+
+# arbitrary size integer math package
+#
+# by Mark Biggar
+#
+# Canonical Big integer value are strings of the form
+#       /^[+-]\d+$/ with leading zeros suppressed
+# Input values to these routines may be strings of the form
+#       /^\s*[+-]?[\d\s]+$/.
+# Examples:
+#   '+0'                            canonical zero value
+#   '   -123 123 123'               canonical value '-123123123'
+#   '1 23 456 7890'                 canonical value '+1234567890'
+# Output values always always in canonical form
+#
+# Actual math is done in an internal format consisting of an array
+#   whose first element is the sign (/^[+-]$/) and whose remaining 
+#   elements are base 100000 digits with the least significant digit first.
+# The string 'NaN' is used to represent the result when input arguments 
+#   are not numbers, as well as the result of dividing by zero
+#
+# routines provided are:
+#
+#   bneg(BINT) return BINT              negation
+#   babs(BINT) return BINT              absolute value
+#   bcmp(BINT,BINT) return CODE         compare numbers (undef,<0,=0,>0)
+#   badd(BINT,BINT) return BINT         addition
+#   bsub(BINT,BINT) return BINT         subtraction
+#   bmul(BINT,BINT) return BINT         multiplication
+#   bdiv(BINT,BINT) return (BINT,BINT)  division (quo,rem) just quo if scalar
+#   bmod(BINT,BINT) return BINT         modulus
+#   bgcd(BINT,BINT) return BINT         greatest common divisor
+#   bnorm(BINT) return BINT             normalization
+#
+
+$zero = 0;
+
+
+# normalize string form of number.   Strip leading zeros.  Strip any
+#   white space and add a sign, if missing.
+# Strings that are not numbers result the value 'NaN'.
+
+sub bnorm { #(num_str) return num_str
+    local($_) = @_;
+    s/\s+//g;                           # strip white space
+    if (s/^([+-]?)0*(\d+)$/$1$2/) {     # test if number
+	substr($_,$[,0) = '+' unless $1; # Add missing sign
+	s/^-0/+0/;
+	$_;
+    } else {
+	'NaN';
+    }
+}
+
+# Convert a number from string format to internal base 100000 format.
+#   Assumes normalized value as input.
+sub internal { #(num_str) return int_num_array
+    local($d) = @_;
+    ($is,$il) = (substr($d,$[,1),length($d)-2);
+    substr($d,$[,1) = '';
+    ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d)));
+}
+
+# Convert a number from internal base 100000 format to string format.
+#   This routine scribbles all over input array.
+sub external { #(int_num_array) return num_str
+    $es = shift;
+    grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_);   # zero pad
+    &bnorm(join('', $es, reverse(@_)));    # reverse concat and normalize
+}
+
+# Negate input value.
+sub bneg { #(num_str) return num_str
+    local($_) = &bnorm(@_);
+    vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0';
+    s/^H/N/;
+    $_;
+}
+
+# Returns the absolute value of the input.
+sub babs { #(num_str) return num_str
+    &abs(&bnorm(@_));
+}
+
+sub abs { # post-normalized abs for internal use
+    local($_) = @_;
+    s/^-/+/;
+    $_;
+}
+
+# Compares 2 values.  Returns one of undef, <0, =0, >0. (suitable for sort)
+sub bcmp { #(num_str, num_str) return cond_code
+    local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1]));
+    if ($x eq 'NaN') {
+	undef;
+    } elsif ($y eq 'NaN') {
+	undef;
+    } else {
+	&cmp($x,$y);
+    }
+}
+
+sub cmp { # post-normalized compare for internal use
+    local($cx, $cy) = @_;
+    $cx cmp $cy
+    &&
+    (
+	ord($cy) <=> ord($cx)
+	||
+	($cx cmp ',') * (length($cy) <=> length($cx) || $cy cmp $cx)
+    );
+}
+
+sub badd { #(num_str, num_str) return num_str
+    local(*x, *y); ($x, $y) = (&bnorm($_[$[]),&bnorm($_[$[+1]));
+    if ($x eq 'NaN') {
+	'NaN';
+    } elsif ($y eq 'NaN') {
+	'NaN';
+    } else {
+	@x = &internal($x);             # convert to internal form
+	@y = &internal($y);
+	local($sx, $sy) = (shift @x, shift @y); # get signs
+	if ($sx eq $sy) {
+	    &external($sx, &add(*x, *y)); # if same sign add
+	} else {
+	    ($x, $y) = (&abs($x),&abs($y)); # make abs
+	    if (&cmp($y,$x) > 0) {
+		&external($sy, &sub(*y, *x));
+	    } else {
+		&external($sx, &sub(*x, *y));
+	    }
+	}
+    }
+}
+
+sub bsub { #(num_str, num_str) return num_str
+    &badd($_[$[],&bneg($_[$[+1]));    
+}
+
+# GCD -- Euclids algorithm Knuth Vol 2 pg 296
+sub bgcd { #(num_str, num_str) return num_str
+    local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1]));
+    if ($x eq 'NaN' || $y eq 'NaN') {
+	'NaN';
+    } else {
+	($x, $y) = ($y,&bmod($x,$y)) while $y ne '+0';
+	$x;
+    }
+}
+
+# routine to add two base 1e5 numbers
+#   stolen from Knuth Vol 2 Algorithm A pg 231
+#   there are separate routines to add and sub as per Kunth pg 233
+sub add { #(int_num_array, int_num_array) return int_num_array
+    local(*x, *y) = @_;
+    $car = 0;
+    for $x (@x) {
+	last unless @y || $car;
+	$x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5);
+    }
+    for $y (@y) {
+	last unless $car;
+	$y -= 1e5 if $car = (($y += $car) >= 1e5);
+    }
+    (@x, @y, $car);
+}
+
+# subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
+sub sub { #(int_num_array, int_num_array) return int_num_array
+    local(*sx, *sy) = @_;
+    $bar = 0;
+    for $sx (@sx) {
+	last unless @y || $bar;
+	$sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0);
+    }
+    @sx;
+}
+
+# multiply two numbers -- stolen from Knuth Vol 2 pg 233
+sub bmul { #(num_str, num_str) return num_str
+    local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1]));
+    if ($x eq 'NaN') {
+	'NaN';
+    } elsif ($y eq 'NaN') {
+	'NaN';
+    } else {
+	@x = &internal($x);
+	@y = &internal($y);
+	&external(&mul(*x,*y));
+    }
+}
+
+# multiply two numbers in internal representation
+# destroys the arguments, supposes that two arguments are different
+sub mul { #(*int_num_array, *int_num_array) return int_num_array
+    local(*x, *y) = (shift, shift);
+    local($signr) = (shift @x ne shift @y) ? '-' : '+';
+    @prod = ();
+    for $x (@x) {
+      ($car, $cty) = (0, $[);
+      for $y (@y) {
+	$prod = $x * $y + $prod[$cty] + $car;
+	$prod[$cty++] =
+	  $prod - ($car = int($prod * 1e-5)) * 1e5;
+      }
+      $prod[$cty] += $car if $car;
+      $x = shift @prod;
+    }
+    ($signr, @x, @prod);
+}
+
+# modulus
+sub bmod { #(num_str, num_str) return num_str
+    (&bdiv(@_))[$[+1];
+}
+
+sub bdiv { #(dividend: num_str, divisor: num_str) return num_str
+    local (*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1]));
+    return wantarray ? ('NaN','NaN') : 'NaN'
+	if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0');
+    return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0);
+    @x = &internal($x); @y = &internal($y);
+    $srem = $y[$[];
+    $sr = (shift @x ne shift @y) ? '-' : '+';
+    $car = $bar = $prd = 0;
+    if (($dd = int(1e5/($y[$#y]+1))) != 1) {
+	for $x (@x) {
+	    $x = $x * $dd + $car;
+	    $x -= ($car = int($x * 1e-5)) * 1e5;
+	}
+	push(@x, $car); $car = 0;
+	for $y (@y) {
+	    $y = $y * $dd + $car;
+	    $y -= ($car = int($y * 1e-5)) * 1e5;
+	}
+    }
+    else {
+	push(@x, 0);
+    }
+    @q = (); ($v2,$v1) = @y[-2,-1];
+    while ($#x > $#y) {
+	($u2,$u1,$u0) = @x[-3..-1];
+	$q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1));
+	--$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2);
+	if ($q) {
+	    ($car, $bar) = (0,0);
+	    for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) {
+		$prd = $q * $y[$y] + $car;
+		$prd -= ($car = int($prd * 1e-5)) * 1e5;
+		$x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0));
+	    }
+	    if ($x[$#x] < $car + $bar) {
+		$car = 0; --$q;
+		for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) {
+		    $x[$x] -= 1e5
+			if ($car = (($x[$x] += $y[$y] + $car) > 1e5));
+		}
+	    }   
+	}
+	pop(@x); unshift(@q, $q);
+    }
+    if (wantarray) {
+	@d = ();
+	if ($dd != 1) {
+	    $car = 0;
+	    for $x (reverse @x) {
+		$prd = $car * 1e5 + $x;
+		$car = $prd - ($tmp = int($prd / $dd)) * $dd;
+		unshift(@d, $tmp);
+	    }
+	}
+	else {
+	    @d = @x;
+	}
+	(&external($sr, @q), &external($srem, @d, $zero));
+    } else {
+	&external($sr, @q);
+    }
+}
+
+# compute power of two numbers -- stolen from Knuth Vol 2 pg 233
+sub bpow { #(num_str, num_str) return num_str
+    local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1]));
+    if ($x eq 'NaN') {
+	'NaN';
+    } elsif ($y eq 'NaN') {
+	'NaN';
+    } elsif ($x eq '+1') {
+	'+1';
+    } elsif ($x eq '-1') {
+	&bmod($x,2) ? '-1': '+1';
+    } elsif ($y =~ /^-/) {
+	'NaN';
+    } elsif ($x eq '+0' && $y eq '+0') {
+	'NaN';
+    } else {
+	@x = &internal($x);
+	local(@pow2)=@x;
+	local(@pow)=&internal("+1");
+	local($y1,$res,@tmp1,@tmp2)=(1); # need tmp to send to mul
+	while ($y ne '+0') {
+	  ($y,$res)=&bdiv($y,2);
+	  if ($res ne '+0') {@tmp=@pow2; @pow=&mul(*pow,*tmp);}
+	  if ($y ne '+0') {@tmp=@pow2;@pow2=&mul(*pow2,*tmp);}
+	}
+	&external(@pow);
+    }
+}
+
+1;
diff --git a/lib/Math/Complex.pm b/lib/Math/Complex.pm
new file mode 100644
index 0000000000..a5a40b2486
--- /dev/null
+++ b/lib/Math/Complex.pm
@@ -0,0 +1,136 @@
+#
+#  Perl5 Package for complex numbers
+#
+#  1994 by David Nadler
+#  Coding know-how provided by Tom Christiansen, Tim Bunce, and Larry Wall
+#	sqrt() added by Tom Christiansen; beware should have two roots, 
+#			but only returns one.  (use wantarray?)
+#  
+#
+# The functions "Re", "Im", and "arg" are provided.
+# "~" is used as the conjugation operator and "abs" is overloaded.
+#
+# Transcendental functions overloaded: so far only sin, cos, and exp.
+#  
+
+package Math::Complex;
+
+require Exporter;
+
+@ISA = ('Exporter');
+
+# just to make use happy
+
+%OVERLOAD= (
+    '+'   => sub  { my($x1,$y1,$x2,$y2) = (@{$_[0]},@{$_[1]});
+                      bless [ $x1+$x2, $y1+$y2];
+             },
+
+    '-'   => sub  { my($x1,$y1,$x2,$y2) = (@{$_[0]},@{$_[1]});
+                      bless [ $x1-$x2, $y1-$y2];
+             },
+
+    '*'   => sub  { my($x1,$y1,$x2,$y2) = (@{$_[0]},@{$_[1]});
+                    bless [ $x1*$x2-$y1*$y2,$x1*$y2+$x2*$y1];
+             },
+
+    '/'   => sub  { my($x1,$y1,$x2,$y2) = (@{$_[0]},@{$_[1]});
+                    my $q = $x2*$x2+$y2*$y2;
+                    bless [($x1*$x2+$y1*$y2)/$q, ($y1*$x2-$y2*$x1)/$q];
+             },
+
+    'neg' => sub  { my($x,$y) = @{$_[0]}; bless [ -$x, -$y];
+             },
+
+    '~'   => sub  { my($x,$y) = @{$_[0]}; bless [ $x, -$y];
+             },
+
+    'abs'   => sub  { my($x,$y) = @{$_[0]}; sqrt $x*$x+$y*$y;
+             },
+
+    'cos' => sub { my($x,$y) = @{$_[0]};
+                   my ($ab,$c,$s) = (exp $y, cos $x, sin $x);
+                   my $abr = 1/(2*$ab); $ab /= 2;
+                   bless [ ($abr+$ab)*$c, ($abr-$ab)*$s];
+             },
+
+    'sin' => sub { my($x,$y) = @{$_[0]};
+                   my ($ab,$c,$s) = (exp $y, cos $x, sin $x);
+                   my $abr = 1/(2*$ab); $ab /= 2;
+                   bless [ (-$abr-$ab)*$s, ($abr-$ab)*$c];
+             },
+
+    'exp' => sub { my($x,$y) = @{$_[0]};
+                   my ($ab,$c,$s) = (exp $x, cos $y, sin $y);
+                   bless [ $ab*$c, $ab*$s ];
+              },
+
+    'sqrt' => sub { 
+	my($zr,$zi) = @{$_[0]};
+	my ($x, $y, $r, $w);
+	my $c = new Math::Complex (0,0);
+        if (($zr == 0) && ($zi == 0)) { 
+	    # nothing, $c already set
+	}
+        else {
+	  $x = abs($zr);
+	  $y = abs($zi);
+	  if ($x >= $y) { 
+	      $r = $y/$x; 
+	      $w = sqrt($x) * sqrt(0.5*(1.0+sqrt(1.0+$r*$r))); 
+	  }
+	  else { 
+	      $r = $x/$y; 
+	      $w = sqrt($y) * sqrt($y) * sqrt(0.5*($r+sqrt(1.0+$r*$r))); 
+	  }
+	  if ( $zr >= 0) { 
+	      @$c = ($w, $zi/(2 * $w) ); 
+	  }
+	  else { 
+	      $c->[1] = ($zi >= 0) ? $w : -$w;
+	      $c->[0] = $zi/(2.0* $c->[1]); 
+	  }
+        } 
+        return $c;
+      },
+
+    qw("" stringify)
+);
+
+sub new {
+    shift;
+    my @C = @_;
+    bless \@C;
+}
+
+sub Re {
+    my($x,$y) = @{$_[0]};
+    $x;
+}
+
+sub Im {
+    my($x,$y) = @{$_[0]};
+    $y;
+}
+
+sub arg {
+    my($x,$y) = @{$_[0]};
+    atan2($y,$x);
+}
+
+sub stringify {
+    my($x,$y) = @{$_[0]};
+    my($re,$im);
+
+    $re = $x if ($x);
+    if ($y == 1) {$im = 'i';}  
+    elsif ($y == -1){$im = '-i';} 
+    elsif ($y) {$im = "${y}i"; }
+
+    local $_ = $re.'+'.$im;
+    s/\+-/-/;
+    s/^\+//;
+    s/[\+-]$//;
+    $_ = 0 if ($_ eq '');
+    return $_;
+}