File Coverage

blib/lib/Math/Yapp.pm

Criterion	Covered	Total	%
statement	697	808	86.2
branch	225	320	70.3
condition	17	24	70.8
subroutine	58	65	89.2
pod	12	56	21.4
total	1009	1273	79.2

# Package: Math::Yapp.pm: Yet another Polynomial Package
#
package Math::Yapp;

use overload
  '+'   => Yapp_add,
  '+='  => Yapp_plus,
  '!'   => Yapp_negate,       # Negate
  '-'   => Yapp_sub,
  '-='  => Yapp_minus,
  '*'   => Yapp_times,
  '*='  => Yapp_mult,
  '/'   => Yapp_dividedby,
  '/='  => Yapp_divide,
  '~'   => Yapp_conj,       # Conjugate complex coefficients
  '**'  => Yapp_power,      # Raise a polynomial to an integer power
  '.'   => Yapp_innerProd   # Inner product, for vector operations.
;

# use 5.014002;         # Let's not tie ourselves to a release.
use strict;
use warnings;
use Carp;
use Math::Complex;
#use Math::BigFloat;        # Needed for accuracy(), though I don't do
#                           # anything with that yet.
use Storable 'dclone';
use Data::Dumper;               # For debugging
use Scalar::Util qw(refaddr);   # Gets pointer values. Also for debugging.

require Exporter;

our @ISA = ('Exporter');

# Items to export into callers namespace by default. Note: do not export
# names by default without a very good reason. Use EXPORT_OK instead.
# Do not simply export all your public functions/methods/constants.

# This allows declaration   use Math::Yapp ':all';
# If you do not need this, moving things directly into @EXPORT or @EXPORT_OK
# will save memory.
#
our %EXPORT_TAGS = ( 'all' => [ qw(Yapp Yapp_decimals Yapp_print0
                                   Yapp_start_high) ] );

our @EXPORT_OK = ( @{ $EXPORT_TAGS{'all'} } );

#our @EXPORT = qw(all);
our @EXPORT = qw(Yapp Yapp_interpolate Yapp_by_roots
                 Yapp_testmode
                 Yapp_decimals Yapp_print0 Yapp_start_high Yapp_margin
                 Yapp_Orthogonal
                 Csprint); # Unexport Csprint after debug

our $VERSION = '1.08';
#
my $class_name = "Math::Yapp";  # So I don't have to use the literal in tests
my $class_cplx = "Math::Complex";   # Same idea - avoid literal use

# Here are the patterns I need to validate and parse the terms of a polynomial:
# They will be used by str2yapp() and str2cplx()
#
my $sign_pat = qr/[+-]/;        # Allows me to isolate the optional sign

# Combine integer_pat and decimal_pat for real_pat
#
my $integer_pat = qr/\d+/;      # Integer is one or more digits
my $decimal_pat = qr/\.\d+/;    # Decimal: A period followed by digits
my $decnum_pat  = qr/($integer_pat)($decimal_pat)/; # Complete decimal number

# A real number may be:
# - An integer /\d+/
# - A pure decimal number: /\.\d+/
# - An integer followd by a pure decimal i.e. a complete decimal number
# I am checking these in reverse of above order, testing for the more
# complicated pattern first.
# Of course, all the above may be preceeded by an optinal sign
#
my $real_pat = qr/(($decnum_pat)|($decimal_pat)|($integer_pat))/;

my $cplx_pat = qr/\([+-]?$real_pat[+-]($real_pat)i\)/; # Cplx: (+-real+-real-i)

# Now a coefficient is a real or complex number at the beginning of a term,
# optionally preceeded by a sign.  If sign is omitted we will assume positive.
# On the other hand, it may be an implicit 1 with only a sign.  Hence, both
# parts are basically optional; if omitted e.g. x^31, then coefficient is +1
#
# As to a regex pattern: A coefficient is at the start of the term and
# may be:
# - A sign alone => an implicit coefficient of 1.0        ($sign_only_pat)
# - A complex or real number alone => implicit sign of +1 ($usigned_coef_pat)
# - A real or complex number preceded by a sign.          ($signed_coef_pat)
# The proper pattern tests for the most complicated possibility first
#
my $signed_real_pat  = qr/^(($sign_pat)?($real_pat))$/;
my $signed_coef_pat  = qr/((^$sign_pat)(($cplx_pat)|($real_pat)))/;
my $usigned_coef_pat = qr/^(($cplx_pat)|($real_pat))/;
my $sign_only_pat    = qr/(^$sign_pat)/;
my $coef_pat = qr/($signed_coef_pat)|($usigned_coef_pat)|($sign_only_pat)/;

my $varia_pat = qr/[A-Za-z]+/;  # The "variable" part of the term
my $power_pat = qr/\^\d+$/;     # Power: caret (^) followed by am integer.
                                # Must be at the end of the term
#
# With apologies: This is a patch to fix a bug in Ysprint(): It was printing
# the 0-term as X^0 when formatting starting at low-degree terms.
# This will help get rid of that.
#
my $zero_power_pat = qr/($varia_pat)(\^0)$/;

my $const_pat = qr/^$coef_pat$/; # Coefficient only; no variable or exponent

# Now that I have defined the patterns for the parts of a term, I can define
# the pattern for a complete term.  Each part is optional.
#
my $term_pat = qr/($coef_pat)?($varia_pat)?($power_pat)?/;

# However, one pattern not allowed is a $power_pat (exponent) without a
# variable.  Since, at the moment, I cannot think of a regex to test for
# this condition, I'll have to handle it in the code.  (Sigh)

# Format strings for printf-ing the terms of the polynomial.
# First: For a real coefficient: Assume the data to printf includes:
# - Width of the coefficient, including decimal point and decimal places
# - Number of decimal places alone
# - The coeficient, a floating point number
# - The variable string (like X, Z, GIZORP etc.) This is omitted for the
#   constant term
# - The exponent (Omitted for the constant term of the polynomial)
#
my $zero_term_fmt  = "%+*.*f";      # No X term in here
my $one_term_fmt   = "%+*.*f%s";    # For 1st degree term, omit the exponent
my $term_fmt       = "%+*.*f%s^%d"; # For other terms, always use sign
my $first_term_fmt = "%*.*f%s^%d";  # For first term, skip + for first if
                                    # it is positive anyway
# Now for a complex coefficient: Here format gets a bit more, er, complex;
# Each term includes the following in the data portion of printf:
# - Width of the real part of the coeficient
# - Number of decimal places in the real part
# - The real part itself
# - Width of the imaginary part of the coeficient
# - Number of decimal places in the imaginary part
# - The imaginary part itself
# - The variable string (Omitted in the constant term)
# - The exponent (Omitted in the constant term)
#
my $zero_term_fmt_i = "+(%*.*f%+*.*fi)";     # No variable or exponent
my $one_term_fmt_i  = "+(%*.*f%+*.*fi)%s";   # +(Re+/-Im_i)X Variable, no
                                             # exponent
my $term_fmt_i      = "+(%*.*f%+*.*fi)%s^%d";# Sign, Variable, exponent
my $first_term_fmt_i = "(%*.*f%+*.*fi)%s^%d";# No sign but variable and
                                             # exponent
# Some other constants I need for this module, before starting the work in
# earnest:
#
our $czero = Math::Complex->make(0,0);      # Complex zero
our $inner_prod = "(generic)";  # Default inner product type; set with
                                # inner_prod()
# These globals control behavior of Ysprint.  Changed by indicated functions
our $dec_places  = 3;   # Can be changed with call to Yapp_decimals()
our $dec_width = 4;     # Minimum width: 4, with 3 decimal paces
our $print_zero = 0;    # Default: FALSE: Skip zero-coefficient terms
                        # Changed by Yapp_print0()
our $start_high = 1;    # Default: True: Start printing from high-degree terms.
                        # If FALSE, start from low-degree terms. Set by
                        # Yapp_start_high()
our $margin = 1/(10**10);   # What is close enough to ignore rounding error?
#our $margin = 1/(10**8);    # What is close enough to ignore rounding error?
our $testmode = 0;      # Turned on by class method Yapp_testmode()

BEGIN
{
 #Math::BigFloat->accuracy(32);   # In case I decide on a way to use complex
                                  # numbers with BigFloat
 #$DB::simple = 1;                #### Comment out after debugging
}
#
#------------------------------------------------------------------------------
# Class setting functions: These functions set [semi] global variables used
# only by other function this package.
#------------------------------------------------------------------------------
# Yapp_testmode() Turns on or off the semi-global variable $testmode.
# This is not a class method - it is a straightforward function call.
#
sub Yapp_testmode
{
  $testmode = $_[0] if (defined($_[0]));    # Set the variable, if user sent
                                            # a value to stick in there
  return $testmode;
}
#------------------------------------------------------------------------------
# Yapp_margin(): Set the threshold for how close to require something be to
# a solution in order to be considered close enough.  This sets the variable
# $margin, whcih is set to 1.0/(10.0 ** 10). It is intended to compensate
# for inevitable rounding errors that, for example, make a real number look
# like a complex number with a really tiny imaginary part (like 1.23456e-15i).
# Parameter:
# - (Optional) The floating point value to be the closeness factor.
# Returns:
# - The current (or new) setting for this margin.
#
sub Yapp_margin
{ # If caller passed a parameter, use it.  If not, just get the current setting
  #
  my $new_margin = (defined($_[0])) ? $_[0] : $margin;
  $margin = $new_margin;        # May be wasted but checking if different
                                # also costs.
  return $margin;
}
#------------------------------------------------------------------------------
# dec_places(): Set or retrieve the precision to be used when
# printing coefficients of the polynomial.  Default is 3.
# - To set precision:
#   yapp_ref->dec_places(17);           # Set output precision of 17 decimal
#                                       # places to the right of the point,
#   Math::Yapp->dec_places(17);        # Will have the same effect.
#
# - To retrieve it:
#   $int_var = jpoly_ref->dec_places();
# In both cases, it returns the precision to the caller.
#
sub dec_places
{
  my $self = shift(@_);
  if (@_ != 0)                  # Passed a parameter
  {
    $dec_places = $_[0];            # Set the number of places to that
    $dec_width  = $dec_places + 1;  # Add 1 place for decimal point. (Probably
                                    # not needed but cleaner logic)
  }
  return $dec_places;           # Either way, that's the return value
}
#
#------------------------------------------------------------------------------
sub Yapp_decimals
{                               # (Replaces dec_places method)
  if (defined($_[0]))           # If user passed a parameter
  {
    $dec_places = $_[0];            # Set global decimal places for printf()
    $dec_width  = $dec_places + 1;  # Add 1 place for decimal point. (Probably
  }                                 # not needed but cleaner logic)
# else                          # No parameter; just want to know it
# { }                           # (Nothing really; just a placeholder
  return $dec_places;           # Either way, send that back to caller
}
#------------------------------------------------------------------------------
# Yapp_print0(): Function to dictate if Ysprint will print terms with 0
# coefficient.
# Parameters:
# - (Optional) 1 (TRUE) or 0 (FALSE)
#   If called w/no parameter, just returns the current value
# Returns:
# The current or newley set value.
sub Yapp_print0
{
  if (defined($_[0])) {$print_zero = $_[0]; }
  return $print_zero;
}
#------------------------------------------------------------------------------
# Yapp_start_high() - Sets $start_high to 0 or 1, or just returns that value.
# This decides if Ysprint will generate output starting from high-degree
# terms (default) or start low and work its way up in degree.
#
sub Yapp_start_high
{
  if (defined($_[0])) {$start_high = $_[0]; }
  return $start_high;
}
#
#------------------------------------------------------------------------------
# Constructors:
#------------------------------------------------------------------------------
# Math::Yapp::new(): Create a new polynomial#
# This method is heavily overloaded but it alwys returns the same type: a
# reference to a Yapp polynomial object.  It will accept the following
# parameters:
# - Nothing; It returns a degenerate polynomial with no coefficients of
#   degree, the equivalent of a NULL polynomial.
# - A list of numbers, real or complex. The first number is the 0-degree
#   coefficient and the last, [n], is the coefficient of the highest-degree
#   term.
# - A reference to a list of numbers i.e. \@list.  THis list is identical
#   to the explicit list described above.
# - A string in the form nnnX^m1 + nnnX^m2 .. etc.
#
# A note about complex coefficients:
# - In the string-form parameter, it must be in the form +-(REAL+-IMAGi)
# - In the list (or list-reference) form, a complex may be of the above
#   string form or a true complex object, as returned by Math::Complex
#
sub new
{
  my $class = shift(@_);    # Get class name out of the way
  my $self = {};            # Create new entity
  bless $self;              # Makes it an object
  my $lc;                   # Loop counter (general purpose here)

  $self->{degree} = 0;      # Initially IS a 0-degree polynomial
  $self->{variable} = "X";  # For printing, use this variable.  Can be set
  $self->{coeff}[0] = 0;    # Even a degenerate polynomial needs a coefficient
                            # to any string using the method variable()
  my $aref = 0;             # Will step through one array or another
  my $coef_count = @_;      # How many coefficients were passed to me?

  if ($coef_count > 1)      # If caller passed me a list of numbers
  { $aref = \@_; }          # Point to the parameter list, save for soon
  elsif ($coef_count == 1)  # Passed only one parameter: Our possibilities:
  {
    if (ref($_[0]) eq "ARRAY")  # I got a reference to a list
    { $aref = shift(@_); }      # Reference that list with same $aref
    else                        # Next possibility: The parameter is a string
    { $self->str2yapp($_[0]); } # Turn the string into a polynomial
  }
  # Remaining possibility: That $coef_count == 0 and user wants that degenerate
  # polynommial.  Do nothing for now; Will return a zero polynomial
  #
  if ($aref != 0)           # If we have an array pointer
  {                         # step through the array to build the polynomial
    $self->{degree} = -1;   # Degree shall end up 1 less than count of terms
    for ($lc = 0; $lc <= $#{$aref}; $lc++)  # Build array of coefficients
    {
      # Note: I am adding the 0.0 to force Perl to store the value as a
      # number (real or ref(complex))
      #
      $self->{coeff}[$lc] = $aref->[$lc] + 0.0; # Use next number on list
      $self->{degree}++;                        # Don't lose track of degree
    }
  }

  $self->refresh();         # Last minute bookkeeping on our new object
  return $self;             # See? Even the empty object gets passed back
}
#
#------------------------------------------------------------------------------
# A neater version of calling new() - Just call Yapp(parameter[s]).
# Accepts all types that new() accepts.
#
sub Yapp
{
  my $class = $_[0];        # Most likely the class name, may be string, array
  my $ryapp;                # Alternative return object to caller

  if (ref($class) eq $class_name)       # If this is copy-call, then the param
  {                                     # is not really a class name but a Yapp
    $ryapp = $class->Yapp_copy();       # instance; construct clone of original
  }
  else                                  # Not a reference
  { $ryapp =  __PACKAGE__->new(@_); }   # Just pass all parameter(s) to new()
  return $ryapp;                  # However we got it, return it to caller
}

#------------------------------------------------------------------------------
# Yapp_copy() - Internal utility function, should be called by new() in case
#               a copy-constructor is called for.
# Parameters:
# - (Implict) A reference to the source Yapp-polynomial for the copy
# Returns:
# - A referecne to a new Yapp object identical to the source
#
sub Yapp_copy
{
  my $self = shift(@_);         # Capture the object
  my $new_yapp = dclone($self); # Duplicate object, get back reference
  return $new_yapp;
}
#
#------------------------------------------------------------------------------
# refresh() - (Internal?) Method to perform two smoothing operations on
#             the given Yapp polynomial:
# - In case it got omitted in a contructor, if a coefficient between 0 and
#   the degree is still undefined, stick a 0.0 in there.
# - If any complex coefficients have a 0 imaginary part, replace it with the
#   real part only, as a real coefficient. e.g. cplx(-2.13,0.0) => -2.13
# - It looks for any 0 coefficient in the highest-degree places so that a 5th
#   degree polynomial with 0 coefficients in the 6th and 7th dregee places
#   does not appear to be a 7th degree polynomial
# Parameter:
# - (Implicit) The Yapp to thus fixed up
# Returns:
# - The same reference but it has been mutated as described.
#
sub refresh
{
  my $self = shift(@_); #(Lose the class name; concentrate on the array)
  my $slc;              # Loop counter for counting up my own array

  # This first loop can handle the undefined coefficient problem as well as the
  # disguised real-number problem
  #
  for ($slc = 0; $slc <= $self->{degree}; $slc++)
  {
    # Make sure there are no undefined coefficients in there.
    # (Zero is better than nothing.)
    #
    $self->{coeff}[$slc] = 0.0 if (! defined($self->{coeff}[$slc]));

    realify($self->{coeff}[$slc]);  # In case of complex coeff w Im == 0
  }

  # Next: Make sure the highest-degree term has a non-zero coefficient;
  # adjust the degree to match this. (Except if degree == 0. That's legit)
  #
  for ($slc = $self->{degree}; $slc > 0; $slc--)
  {
    last if ($self->{coeff}[$slc] != 0);    # Found my non-0 highest term; done

    # Still here: The current high-order term is a degenerate term with a
    # zero-coefficient. Lose it!
    #
   #undef($self->{coeff}[$slc]);    #(No, this does not lose the element)
    delete($self->{coeff}[$slc]);   # *This* loses the element
    $self->{degree}--;              # Make sure this coef will be skipped
  }

  # Finally: Since the coefficients may have been diddled, the coefficients of
  # the derivative and integral, if they have been generated, are unreliable.
  # Rather than regenerate them now, just lose them.  If they are needed again
  # they will be automatically and transparently regenreated.
  #
  undef $self->{derivative} if (defined($self->{derivative}[0]));
  undef $self->{integral}   if (defined($self->{integral}[0]));

  # After all that, it is *still* possible for {variable} to be undefined.
  # This will be the case for a constant term alone, a 0-degree polynomial
  # string.  Put the default in there now.
  #
  $self->{variable} = "X" if (!defined($self->{variable}));
  return ($self);
}
#
#------------------------------------------------------------------------------
# Ysprint(): Returns a string with the polynomial in some standard form,
#            suitable for printing.
# Parameters:
# - (Implicit) The Yapp opject
# - (Optional) The number of decimal places to use for the coefficients. If
#   omitted, we will use the module global $dec_places.  See Yapp_decimals for
#   more information on that.
# Default options:
# - Use X as the variable
# - Skip terms with 0-coefficients
# - Output highest powers first, as one would write in algebra class
# Funtions have been provided to change these behaviors.  These are:
# - Yapp_decimals(), so that you may call method Ysprint() without a parameter.
# - Yapp_print0()
# - Yapp_start_high()
#
sub Ysprint
{
  my $self = shift(@_);
  my $places = (defined($_[0])) ? $_[0] : $dec_places;
  my $dwidth = (defined($_[0])) ? $places + 1 : $dec_width;

  my $var_name = $self->variable();
  my $out_string = "";
  my $use_fmt = \$first_term_fmt;   # Make sure I use this first

  my $lc_start;             # Start loop at high or low degree terms?
  my $lc_finish = 0;        # And where do I finish, based on above?
  my $lc_diff  = -1;        # and in which direction doed my loop counter go?

  if ($start_high)                  # Default: Start from high-degree terms
  {
    $lc_start  = $self->{degree};   # Start loop at high-power term
    $lc_finish = 0;                 # End loop at 0-power term
    $lc_diff  = -1;                 # Normal behavior: Start high, step low
  }
  else
  {
    $lc_start  = 0;                 # Start loop at 0-degree term
    $lc_finish = $self->{degree};   # End loop at high-degree term
    $lc_diff  = 1;                  # work my way UP the degrees
  }
#
  for (my $lc = $lc_start; ; $lc += $lc_diff)   #(Check value of $lc inside
  {                                             # loop, not in loop header)
    last if ( ($lc > $self->{degree}) || ($lc < 0) );

    # How to format the coefficient depends on whether it is real or complex
    #
    my $term = "";      # Start the term as a null string and build from there
    if (ref($self->{coeff}[$lc]) eq $class_cplx)
    {
      # Note: The constructor should have set any "almost 0" complex
      # coefficients to real 0.0 so there is no need to account here for a
      # complex number with really tiny imaginary part.
      #
      # First term should not have a + sign if coefficient is positive
      # Constant term should not dislay X to the 0th degree
      # First degree term should display as X, not X^1
      # All other terms should include both aspects.
      #
      if    ($lc == $lc_start)          # If I'm generating the first term
      {
        $term = sprintf($first_term_fmt_i,
                        $dwidth, $places, ($self->coefficient($lc))->Re(),
                        $dwidth, $places, ($self->coefficient($lc))->Im(),
                        $var_name, $lc);    # No + sign for first term
      }
      elsif ($lc == 1)
      {
        $term = sprintf($one_term_fmt_i,
                        $dwidth, $places, ($self->coefficient($lc))->Re(),
                        $dwidth, $places, ($self->coefficient($lc))->Im(),
                        $var_name);         # No exponent for X^1 term
      }
      elsif ($lc == 0 )
      {
        $term = sprintf($zero_term_fmt_i,
                        $dwidth, $places, ($self->coefficient($lc))->Re(),
                        $dwidth, $places, ($self->coefficient($lc))->Im());
                                # No variable or exponent in 0-degree term
      }
      else
      {
        $term = sprintf($term_fmt_i,
                        $dwidth, $places, ($self->coefficient($lc))->Re(),
                        $dwidth, $places, ($self->coefficient($lc))->Im(),
                        $var_name, $lc);    # Always a + before a complex num-
                                            # ber in a middle term
      }
    }
#
    else                    # The ref() function did not say "Complex".  So it
    {                       # must be a real coefficient.
      # Note: Same sign, variable, exponent conventions apply as above
      #
      if    ($lc == $lc_start)          # If I'm generating the first term
      {
        $term = sprintf($first_term_fmt,
                        $dwidth, $places, $self->coefficient($lc),
                        $var_name, $lc);    # eg 34.145X^3
      }
      elsif ($lc == 1)
      {
        $term = sprintf($one_term_fmt,
                        $dwidth, $places, $self->coefficient($lc),
                        $var_name);         # eg. 34.145X (exponent == 1)
      }
      elsif ($lc == 0)
      {
        $term = sprintf($zero_term_fmt,
                        $dwidth, $places,
                        $self->coefficient($lc) );   # eg. 34.145
      }
      else
      {
        $term = sprintf($term_fmt,
                        $dwidth, $places, $self->coefficient($lc),
                        $var_name, $lc);    # eg. +34.145X^7
      }
      if ($self->coefficient($lc) == 0.0)   # What to do with a real 0.0
      {                                     # coefficient?
        # If $print_zero flag is set then the term has already been
        # formatted.  If it is not set (default) I want to null the string.
        #
        $term = "" unless ($print_zero);
      }
    }
    # Second part of the patch I described when I defined $zero_power_pat:
    # Lose the X^0 part of the term if I am on the 0-degree term.
    #
    $term =~ s/($zero_power_pat)$// if ($lc == 0);

    # If the output target string is still empty, this is the first occupant
    #
    if ("$out_string" eq "") { $out_string = $term; }
    else                            # Something alrady in there;
    {                               # Append blank and current term
      $out_string = sprintf("%s %s", $out_string, $term)
        if ($term ne "");           #(Provided our term has substance also)
    }
  }                             #(End of loop; at exit, $out_string is built)
  return $out_string;
}
#
#------------------------------------------------------------------------------
# print(): Method to print a formatted Yapp polynomial.  Basically, just a
#          wrapper for Ysprint.  I expected this to be used mainly in debugging
#          sessions so that I can say "p $something" but that doesn't work.
#          (And I'm not ready to start with Tie::Handle at this juncture,
#          although that seems to be the way to override the built-in print
#          function.)
# Parameters:
# - (Implicit) Ref to a Yapp object
# - (Optional) Number of decimal places for the coefficients
# Returns: (Whatever)
#
sub print
{
  my $yobj = shift(@_);
  my $ystring = $yobj->Ysprint($_[0]);  # There may be a places paremeter
  print $ystring;                       # Output it - no automatic \n
}
#------------------------------------------------------------------------------
# Csprint(): Utility function to format a complex number in my display format
#            e.g. (-3.24+6.03i), including the parenteses.  The number of
#            decimal places is based on the same Yapp_decimals() settings of
#            $dec_places and $dec_width used by Ysprint.
# Parameters:
# - [Reference to] a complex number of type Math::Complex
# - (Optional) Number of decimal places to display.  If omitted, use the
#   number in $dec_places
# Returns:
# - A parenthesized string in the form shown in the above blurb.
# Note: If a scalar is passed instead, it will be returned as received
#
sub Csprint
{
  my $cnum = shift(@_);
  my $places = (defined($_[0])) ? $_[0] : $dec_places;
  my $dwidth = (defined($_[0])) ? $places + 1 : $dec_width;
  return $cnum if (ref($cnum) ne $class_cplx);  # Don't bother with non-complex

  my $rstring = sprintf("(%*.*f%+*.*fi)",
                        $dwidth, $places, $cnum->Re(),
                        $dwidth, $places, $cnum->Im());
  return $rstring;
}
#
#------------------------------------------------------------------------------
#
# realify(): Internal utility function.  Loses extremely small real or imaginary
#            component of a complex number, since we can assume it came from
#            a rounding error.
# Parameter:
# - A real number or [Ref to] a Math::Complex number
#   OR
# - An array of numbers that each may be real or complex. In this case we
#   assume it was called in a list context.  Violate that assumption and die!
# Returns:
# - If real:
#   o If non-trivial, just return that number.
#   o If trivial, return 0.0
# - If complex:
#   o If the imaginary part is very small, less than $margin in absolute value,
#     return only the real part as a real.
#   o If real part is trivial, set it to 0.0 outright
#   o If both components are non-trivial, return the reference unscathed.
# - If an array;
#   o And array of numbers thus transformed
# If called with no return (void context like the chomp function) operate
# directly on the parameter or the array of numbers.
#
sub realify
{
  my $thing = $_[0];            # Don't shift, we might need to modify in place
  my $rval  = 0.0;

  die "You may not pass a list to realify(); use array reference"
    if ($#_ > 0);

  # Real or complex, if absolute value is really tiny, "correct" to 0
  #
  if (wantarray)                # Array context: Assume $thing  is an
  {                             # array reference.
   #my @rlist = map {realify($_)} @$thing;             # Return list
    # Don't use map for this recursive call. Apparently because there is an
    # array as the lvalue to the map{}, realify thinks it was called in the
    # list context.  Rather resort to brute force loop.
    #
    my @rlist;                  # List to be returned
    for (my $rlc = 0; $rlc <= $#$thing; $rlc++)
    {
      $rlist[$rlc] = realify($thing->[$rlc]);
    }
    return @rlist;
  }
  elsif (defined(wantarray))    # Scalar context: assume $thing is a scalar
  {                             # and caller wants a new value returned
    return (0.0) if (abs($thing) < $margin);    # Just tiny abs value: Quickie

    # If either component of a complex number is really tiny, set it to 0.
    # And if it's the imaginary component, lose it altogether
    #
    if (ref($thing) eq $class_cplx)     # If I have not zeroed it above then
    {                                   # I am permitted to check its components
      $rval = $thing;                   # Preserve original complex number
      if (abs($thing->Re) < $margin)    # Need to keep real part, even if tiny
      {
        $rval = cplx(0.0, $thing->Im)   # So just zero it out
      }
      if (abs($thing->Im) < $margin)    # If imaginary component is tiny
      {
        $rval = $thing->Re;             # lose it completely and return a real
      }
    }
    else                        # So it is real and its abs val is >= $margin
    {                           # ==> I already know in can't zero it.
      $rval = $thing;           # So the return value is the original number
    }
    return $rval;
  }
  else                          # Void context: Operate on the passed object.
  {
    if (ref($thing) eq "ARRAY") # Realifying every element in an array?
    {
      @{$thing} = realify($thing);  # Tickle the wantarray code above
    }
    else                            # Realify only one object
    {                               # Just work on that one
      $_[0] = realify($_[0]);       # Tickle the defined(wantarray) code above
    }
    # Note: No return statement.
  }
}
#
#------------------------------------------------------------------------------
# all_real() - Method to just know if a Yapp polynomial has strictly real
#              coefficients.
# Parameter:
# - [Ref to] a Yapp polynomial object
# Returns:
# - 1 (true)  if all coeffienents are real
# - 0 (false) if even one os complex
#
sub all_real
{
  my $self = shift(@_);
  my $allreal = 1;      # OK to start optimistic
  foreach my $coef (@{$self->{coeff}})
  {
    if (ref($coef) eq $class_cplx)
    {
      $allreal = 0;         # Not all real
      last;                 # No point in continuing
    }
  }
  return $allreal;
}
#
#------------------------------------------------------------------------------
# Yapp_equiv(): Method to compare a pair of Yapp polynomials, but only by the
#               aspects that count: the degree and the coefficients. I don't
#               care about the variable name nor about {deviative} or {integral}
#               for this comparison
# Parameters:
# - (Implicit) One Yapp to compare against
# - The Yapp to compare to the first.
# (At this time, I am not overloading the == operator)
#
sub Yapp_equiv
{
  my ($self, $compr) = @_;
  my $comp_ok = 1;          # Start by assuming true

  if ($self->{degree} == $compr->{degree})
  {
    for (my $dlc = 0; $dlc <= $self->{degree}; $dlc++)
    {
      $comp_ok = 0 if ($self->{coeff}[$dlc] != $compr->{coeff}[$dlc]);
      last unless $comp_ok; # If found unequal coefficients, quit
    }
  }
  else {$comp_ok = 0;}  # If even degrees don't agree, soooo not equivalent!

  return $comp_ok;
}
#
#------------------------------------------------------------------------------
# str2cplx() - Internal utility function, not an object method.
# This function accepts a string and converts it to a complex number.
#
# Parameter:
# - A string already determined to match our syntax for a complex number
# Returns:
# - A [reference to] a proper Math::Complex::cplx complex number
#
sub str2cplx
{
  my $c_string = shift(@_);     # Grab the string

  # Plan: I will need to take apart the string, first stripping the parentheses
  # and the letter i, then separating the components by splitting it by the
  # sign between the components.  But take care to not be fooled by a possible
  # sign before the real component
  #
  my $rsign = 1;                # Initially assume real component is positive
  my $rval = $czero;            # Return value: Establish it as a complex number
  $c_string =~ s/[\(\)i]//g;    # First, lose the parentheses and "i"
  if ($c_string =~ /^($sign_pat)/)  # Now, if there *is* a sign before the
  {                                 # real component
    $rsign = ($1 eq "-") ? -1 : 1;  # then [re]set the multiplier accordingly
    $c_string =~ s/$sign_pat//;     # and lose sign preceding the real
  }

  # And now the complex number (still a string) looks like A[+-]Bi
  #
  my @c_pair = split /([+-])/, $c_string;   # Split by that sign, but enclose
                                            # the pattern in ()
  # The effect of ([+-]), as opposed to [+-] without the parentheses:
  # I capture the actual splitting string; the array produced by the split
  # includes the splitting string itself.  Thus, my array contains [A, -, B]
  # (or +plus) as element[1] and the imaginary component is actually
  # element[2].  We'll change that after we use it.
  #
  my $isign = ($c_pair[1] eq "-") ? -1 : 1; # But what was that sign?
  $c_pair[2] *= $isign;                 # Recover correct sign of
                                        # imaginary component
  @c_pair = @c_pair[0,2];               # Now lose that im sign
  $c_pair[0] *= $rsign;                 # Good time to recall the sign
                                        # on the real component
  $rval = Math::Complex::cplx(@c_pair); # Turn pair into complex ref
  return $rval;                         # QED
}
#
#------------------------------------------------------------------------------
# Method: degree(): Just return the degree of the Yapp polynomial
# Usage:
#   ivar = jpoly_ref->degree();
sub degree
{
  my $self = shift(@_);
  return ($self->{degree});
}

#------------------------------------------------------------------------------
# Method: coefficient(): Return the coefficient of the inticated term.
# - If the coefficientis a real number, returns the number
# - If the coefficient is complex, returns the reference
# Usage:
#   float_var   = jpoly_ref->coefficient(n); # Retrieve nth-degree coefficient
#   complex_var = jpoly_ref->coefficient(n); # Retrieve nth-degree coefficient
#
# Error condition: If the n is higher than the degree or negative,
#
sub coefficient
{
  my $self = shift(@_);
  my $term_id = shift(@_);
  if ( ($term_id > $self->{degree}) || ($term_id < 0) )
  {
    my $msg = "Indicated degree is out of range [0, $self->{degree}]";
    croak $msg;
  }
  return ($self->{coeff}[$term_id]);    # Degree desired is valid
}

#------------------------------------------------------------------------------
# Method: variable(): Retrieve or set the character [string] that will be used
# to display the polynomial in Ysprint.
# Usage:
# - To set the string:
#   $jpoly_ref->variable("Y");          #(Default is "X")
#   $jpoly_ref->variable("squeal");
# - To retrieve the string:
#   $var_str = $jpoly_ref->variable();
# In both cases, the method returns the string.
#
sub variable
{
  my $self = shift(@_);
  my $rval = "";                        # Value to return
  if (@_ == 0)                          # No parameter?
  { $rval = $self->{variable}; }        # User wants to know it
  else
  { $rval = shift(@_);                  # Lop it off parameter list and
    $self->{variable} = $rval;          # set it as the variable symbol
  }
  return ($rval);
}
#
#------------------------------------------------------------------------------
# Method: str2yapp(): Accepts a string and converts it to a polynomial object.
# The implicit "self" parameter is asssumed to be referenceing a degenerate
# polynomial and that's where the member items will go.  If you already have an
# initialized polynomial object in there, it is *so* gone.
#
# Parameter:
# - A string of terms in the form ^.
#   separated by white-spaces
# Returns:
# - The Yapp reference, if successful
# - undef if failed anywhere
#
# Usage:    This is not intended to be called from outside. Rather, new() will
#           call it in the following sequence:
#   1. new() creates and blesses the empty Yapp object referenced by $self
#   2. $self->str2yapp(The string>
#
# Note that the exponents do not need to be in any order; the exponent will
# decide the position within the array.
#
# Plan: Parsing presents a problem: I wish to impose no requirement that the
# terms and operators be blank-separated.  And while the terms are surely
# separated by + and -, the split function will remove the separators so I
# won't know what was the sign of each term in the polynomial string.  However,
# for my initial phase, I will require a white-space separator between terms.
# Also, each term mus match one of the patterns below, $real_term_pat or
# cplx_term_pat.
# For real coefficient it must look like:
#   +/-^
#  The sign will be optional for all terms.  (But don't tell that to the users.)
#
# For complex coefficient, it must look
#   +/-(+i)^
#
sub str2yapp
{
  my $self = shift(@_);             # Get myself out of the parameter list
  my $poly_string = shift(@_);      # and get the string into my locale

  my $tlc = 0;                  # Term loop counter

  my $rval = undef;             # Off to a pessimistic start
  my $const_term = 0;

  my $cur_term;                 # Current term from the input polynomial
  my $cur_sign   = 1;           # Assume positive if sign is omitted
  my $cur_coeff  = 0.0;         # Current coefficient
  my $cur_varia  = "";          # Current variable name; should change only
                                # once - the first time I see it.
  my $cur_degree = -1;          # For stepping into coeficient array
  my $hi_deg     = 0;           # For determining the highest exponent used

  undef($self->{variable});     # Leave it for user to name the variable
  $self->{degree} = 0;          # and for input to dictate the degree

  printf("String polynomial is:\n<%s>\n", $poly_string)
    if ($testmode);
  my @pterms = split /\s+/, $poly_string;   # Separate the terms
#
  for ($tlc = 0; $tlc <= $#pterms; $tlc++)
  {
    # Plan: For each term I parse out, determine the coefficient (including the
    # sign, of course), the  variable name and the exponent. This determines the
    # position and value in the @coeff array
    #
    # Afterthought: During debugging I discovererd that a + or - not
    # connected to the following term would also pass all the pattern
    # checks.  Rather than complain or reject the string, I'd rather be nice
    # about it.
    # So here's the scheme:
    # - If the +/- is the last term of the string, ignore it.
    # - Not the last:
    #   o If the following term *is* properly signed, threat this solitary
    #     sign like a multiplier for the following term
    #   o If the term following term is unsigned, prepend this solitary sign
    #     to that term.
    #   Either way, skip reat of this round and handle that following term
    #
    if ($pterms[$tlc] =~ m/^[-\+]$/)    # If it is a + or - alone
    {
      last if ($tlc == $#pterms);       # It is already last term: Ignore it

      # Still here: It is not the last term.  Peek ahead at next term
      #
      if ($pterms[$tlc+1] =~ m/^[-\+]/) # If next term is signed
      {                                 # Use this signs as multiplier
        if ($pterms[$tlc] eq substr($pterms[$tlc+1], 0, 1)) # Signs alike?
        { substr($pterms[$tlc+1], 0, 1) = '+'; }            # Next is positive
        else                            # Signs not alike: Whichever way,
        { substr($pterms[$tlc+1], 0, 1) = '-'; }            # Next is negative
        next;                           # Now handle that next term
      }
      else                              # But if next term is unsigned;
      {                                 # use this to make it a signed term
        $pterms[$tlc+1] = $pterms[$tlc] . $pterms[$tlc+1];  # Prepend this sign
        next;                                               # handle next term
      }
    }

    # following that afterthought...
    # Before anything else, check the sign; this makes it possible to specify
    # the negation of a complex coefficient eg. -(-6+4i)
    #
    $cur_term = $pterms[$tlc];  # Leave array enty intact; disect copy
#
    # Next: Fetch the coefficient, of there is one.
    #
    if ($cur_term =~ /($coef_pat)/) # Is there a coefficient?
    {
      $cur_coeff = $1;              # Isolate coefficient for more checking

      # First, see if there is a sign preceeding the coefficient.
      #
      if ($cur_coeff =~ /^($sign_pat)/) # If there is a sign
      {
        $cur_sign = ($1 eq "-") ? -1 : 1;   # Apply sign to coefficient later
        $cur_coeff =~ s/$sign_pat//;        # but for now, lose the sign
      }

      # Check if the coefficient is a complex number in ofr (a[+-]bi) where
      # a and b are reals
      #
      if ($cur_coeff =~ /$cplx_pat/) # If it looks like a complex number
      {
        $cur_coeff = str2cplx($cur_coeff);  # Convert it to complex object
      }
      else                      # Not complex: Must match a real or a sign
      {                         # What if it was just a sign? (Which is gone)
        $cur_coeff = 1.0 if ($cur_coeff eq ""); # Ths is implicitly 1
      }
      $cur_coeff *= $cur_sign;      # Finally, apply the original sign that
                                    # preceded the real or complex coefficient

      # Now that we have the coefficient for the current term, it's just in
      # the way. Lose it.
      #
      $cur_term =~ s/$coef_pat//;   # Leave the term w/o sign or coefficient.
    }
    else                            # No coefficient?
    { $cur_coeff = 1.0; }           # Implicitly, that is 1.  (If it had
                                    # been -1, it would be -X)
    # Now that the term has no coefficient (either by input or because it
    # has been stripped off), let's have a look at the variable component
    #
    if ($cur_term =~ /^($varia_pat)/)   # If there is a variable to speak of
    {
      $cur_varia = $1;                  # then captue that as variable name
      $cur_term =~ s/^($varia_pat)//;   # and remove it from the term.
    }
    else
    {
      undef($cur_varia);        # No variable => constant.  Make sure no expo-
      $cur_degree = 0;          # nent.  But we're getting ahead of ourselves..
    }
#
    # Next, see about the exponent part: ^integer, which is optional
    # but if provided, there had better be a variable as well.
    #
    if ($cur_term =~ /($power_pat)/)    # Is there an exponent
    {                                   # Some error checking, then work
      if (defined($cur_varia))          # If there is a variable
      {
        $cur_term =~ s/^\^//;           # Lose the exponent operator
        ($cur_degree) = $cur_term =~ m/(\d+)/;  # Capture the integer part
      }
      else
      { die "Term[$tlc] <$pterms[$tlc]> has an exponent w/o a variable"; }
    }
    else                    # No exponent.  Certainly OK.. But..
    {                       # Is the degree of the term 1 or 0? Well, is there
      $cur_degree = (defined($cur_varia)) ? 1 : 0;  # a variable? Yes => 1
    }

    # Finished parsing the term.  Our state at this point:  We have:
    # - The coefficient (be it real or # complex), inclduing its sign
    # - The variable name, which should not have changed.
    # - The degree of the curent term, which may match a term already there.
    #

    # Degree of polynomial is that of highest degree term, so:
    #
    $self->{degree} = $cur_degree if ($cur_degree > $self->{degree});


    # If variable name has changed in the string, complain but do it anyway.
    #
    if (   (defined($cur_varia))
        && (defined($self->{variable}))
        && ($self->{variable} ne $cur_varia))
    {
      my $mg;
      $mg = sprintf("Warning: Changing variable name from <%s> to <%s>\n",
                     $self->{variable}, $cur_varia);
      warn($mg);    # Complain but let it go
    }
    # Set the variable name for whole polynomial (if not already set in a
    # previous round of the loop)... provided that
    #
    $self->{variable} = $cur_varia
      if (defined($cur_varia)); # [Re]set the variable, even with grumble

    # If we already have a term at this degree, just add the new coefficient
    # to the existing one.  Otherwise, we are setting the coefficient now.
    #
    if (defined($self->{coeff}[$cur_degree]))   # Already have a term at
    {                                           # degree, then
      $self->{coeff}[$cur_degree] += $cur_coeff; # just add this one
    }
    else                                        # No term of this degree yet
    {                                           # This is new here
      $self->{coeff}[$cur_degree] = $cur_coeff; # Set the coefficient now
    }
  } # End of for-loop to parse individual terms

  # Bug fix (Too lazy to locate screw-up): Somehow I can get this far
  # without a variable component.  So just in case..
  #
  $self->{variable} = "X" if (!defined($self->{variable}));

  return $self;
}
#
#------------------------------------------------------------------------------
# Yapp_plus - Function to implement the += operator
# Parameters:
# - (Implicit) [Reference to] my target object
# - The object to added to my target object. This may be:
#   o Another Yapp polynomial object [reference]
#   o A constant, either real or complex (a different, albeit smaller can
#     of worms)
#
sub Yapp_plus
{
  my ($self, $added) = @_;  # Get relevant parameters. Swap is irrelevant

  # Now, what is $added?
  #
  if (ref($added) eq $class_name)   # If it is another polynomial
  {
    my $alc = 0;                        # Loop counter to step up added terms
    my $new_degree = $self->{degree};   # Track degree in case the added
                                        # polynomial has a higher degree
    my @builder = @{$self->{coeff}};    # Copy coefficients into temporary list
    for ($alc = 0; $alc <= $added->{degree}; $alc++)    # For all terms of the
    {                                                   # added polynomial
      if (defined($builder[$alc]))      # If target has term of this degree
           { $builder[$alc] += $added->{coeff}[$alc]; } # Just add these terms
      else { $builder[$alc]  = $added->{coeff}[$alc]; } # Or copy this term
      $new_degree = $alc if ($alc > $self->{degree});   # Maintain degree
    }
    # Temporary areas have been evaluated. Now plug these into target Yapp
    #
    $self->{degree} = $new_degree;      # Carry over the augmented (?) degree
    @{$self->{coeff}} = @builder;       # and the augmented polynomial
  }                                     # And I have my return value
  elsif (   ($added =~ m/^($coef_pat)$/)
         || (ref($added) eq $class_cplx) )  # Adding real or complex const
  {
    # As above: If the target term is defined, add the new term.
    # Otherwise, set the target term to the given value
    #
    if (defined($self->{coeff}[0])) {$self->{coeff}[0] += $added;}
    else                            {$self->{coeff}[0]  = $added;}
  }         # (Just needed to augment the constant term)
  else
  { die "Operator += requires a constant or a Yapp polynomial reference";}

  # Didn't die - I have a good value to return
  #
  return ($self);                       # (Ought to return something)
}

#------------------------------------------------------------------------------
# Yapp_add() - Function (and overloaded + operator) to add two polynomials.
# Parameters:
# - (Implicit) The polynomial on the left side of the + (self)
# - The polynomial to be added.  This may be:
#   o (Implicit) [A reference to] another Yapp polynomial object
#   o A constant, either real or complex  (See Yapp_plus)
# - (Implicit) The "swapped" flag, largely irrelevant for the + operator
# Returns:
# A reference to a new Yapp polynomial
#
sub Yapp_add
{
  my ($self, $adder, $swapped) = @_;    #(swap is irrelelvant for add)

  my $ryapp = Yapp($self);          # Clone the original
  $ryapp += $adder;                 # Let _plus() function handle details
  return $ryapp;                    # Assume $ryapp will be auto-destroyed?
}
#
#------------------------------------------------------------------------------
# Yapp_minus(): Function (and overloaded -= operator) to subtract the passed
#               polybnomial from the object polynomial in place.  That is,
#               it modifies the $self object
# - (Implicit) [Reference to] my target object
# - The object to added to my target object. This may be:
#   o Another Yapp polynomial object [reference]
#   o A constant, either real or complex
# Returns:
# - The same reference.  But the target Yapp ahs been "deminished"
#
sub Yapp_minus
{
  my ($self, $subtractor) = @_[0 .. 1];

  if (ref($subtractor) eq $class_name)  # Subtracting another polynimal
  {                                     # just use the add method
    my $temp_subt = $subtractor->Yapp_negate(); # Quickie way out: Negate and
                                                # add,  However I cant
    my $temp_self = Yapp($self);                # use -= on $sef. Dunno why
    $temp_self += $temp_subt;                   # Add the negated Yapp
    @{$self->{coeff}} = @{$temp_self->{coeff}}; # Restore subtracted
                                                # coeffiecient array
  }
  else                                  # Otherwise, just assume a constant
  {                                     # be it real or complex.
    # If our polynomial has no constant term, this is it, after negation.
    #
    $self->{coeff}[0] = (defined($self->{coeff}[0]))
                      ? $self->{coeff}[0] - $subtractor
                      : - $subtractor ;
  }
  return $self;
}

#------------------------------------------------------------------------------
# Yapp_sub(): Function (and overloaded - operator) to subtract one polynomia
# from another bot not in place.
# Parameters:
# - (Implicit) The polynomial on the left side of the - (self)
# - The polynomial to be added.  This may be:
#   o (Implicit) [A reference to] another Yapp polynomial object
#   o A constant, either real or complex  (See Yapp_plus)
# - (Implicit) The "swapped" flag
# Returns:
#  - A reference to a new Yapp polynomial
#
sub Yapp_sub
{
  my ($self, $subtractor, $swapped) = @_;

  my $ryapp = Yapp($self);          # Clone the original
  $ryapp -= $subtractor;            # Let _plus() function handle details
  $ryapp = $ryapp->Yapp_negate() if ($swapped); # Negate of $self was on
                                                # right of the - sign
  return $ryapp;                    # Assume $ryapp will be auto-destroyed?
}
#
#------------------------------------------------------------------------------
# Yapp_mult(): Function to implement the overloaded *= operator: Perform in-
# place multiplication of the target Yapp my a constant or another Yapp
# Parameters:
# - (Implicit) The target object ($self)
# - The multiplier.  This may be:
#   o Another Yapp object (Happiest with this parameter type)
#   o A real or complex constant
# Returns:
# The $self-same reference, but it has been "mutated" by the method.
#
sub Yapp_mult
{
  my ($self, $multiplier) = @_;
  my $aux_yapp;             # I may have to create another Yapp
  my $ylc;                  # Loop counter for stepping up coefficients

  # Now what data type is the multiplier?
  #
#Xif (   (ref($multiplier) eq "$class_cplx")    # Complex number constant
#X    || (   (ref(\$multiplier) eq "SCALAR")    # or a scalar that happens
#X        && ($multiplier =~ /^($coef_pat)$/))) # to be a real constant
#X Note: The above logic is good but for very low floating point numbers it
#X was failing to match the corefficient pattern.  Hence, I relaxed the
#X matching requirement.  -- Jacob

  if (   (ref($multiplier)  eq "$class_cplx")   # Complex number constant
      || (ref(\$multiplier) eq "SCALAR"     ) ) # or a scalar (Assume numeric)
  {                                             # Just distribute multiplier
    for ($ylc = 0; $ylc <= $self->{degree}; $ylc++)
    {
      $self->{coeff}[$ylc] *= $multiplier
        if ($self->{coeff}[$ylc] != 0) ;    # Don't bother multiplying 0
    }
  }
  elsif (ref($multiplier) eq $class_name)   # Multiplying by a polynomial
  {
    my @builder = ();                       # Build result area from scratch
    my $new_degree = $self->{degree}
                   + $multiplier->{degree}; # (Remember your 9th grade algebra?)
    for ($ylc = 0; $ylc <= $self->{degree}; $ylc++)
    {                                   # Outer loop multiplies one target term
      for (my $mlc = 0; $mlc <= $multiplier->{degree}; $mlc++)
      {                                 # Inner loop multiplies by one
                                        # multiplier term
        my $term_degree = $ylc + $mlc;      # Degree of this target term
        $builder[$term_degree] = 0.0                # Make sure there is a
          if (! defined($builder[$term_degree]));   # value in here to start

        # Accumulate product term of that degree: Product of the terms whose
        # exponents add up to the [eventual] exponent of this term
        #
        $builder[$term_degree] += $self->{coeff}[$ylc]
                                * $multiplier->{coeff}[$mlc] ;

      }     # End loop multiplying one target term by term from multiplier
    }     # End loop multiplying whole polynomials
    # Done multiplication: Now put it back in place
    #
    $self->{degree} = $new_degree;  # Copy back degree of multiplied target Yapp
    @{$self->{coeff}} = @builder;   # and copy back array where we carried
                                    # out the multiplication.
  }                         # All done multiplying Yapps; product is in place
  else                      # No more permitted possibilities
  { die "Operator *= requires a constant or a Yapp polynomial reference";}

  # Afterthought: I have found that when I multiply two poly's with conjugate
  # complex constant terms, the result will include coefficients like like
  # (30.00+0.00i); which is a real, of course, but doesn't look that way.
  # Here's the fix:
  #
  realify(\@{$self->{coeff}});
  return $self;
}
#
#------------------------------------------------------------------------------
# Yapp_times(): Method to implement the overloaded '*' operator
# Parameters:
# - (Implicit) The operand (usually the left) to the multiplicaton
# - [Reference] to the multiplier, which may be:
#   o Another Yapp object (Happiest with this parameter type)
#   o A real or complex constant
# - Swapped-operands flag, largely irrelevant for multiplication
# Returns:
# - [Reference to] a new Yapp polynomial that is the product of the first
#   two parameters
#
sub Yapp_times
{
  my ($self, $multiplier, $swapped) = @_;
  my $ryapp;                # The object to be returned

  $ryapp = Yapp($self);     # Make a copy
  $ryapp *= $multiplier;    # Carry out operation
  return $ryapp;            #(Wasnt't that simple?)
}
#
#------------------------------------------------------------------------------
# Yapp_power(): Method to implement exponentiation of polynomial by an
#               integer power.
# Parameters:
# - (Implicit) The operand, a ref to the polynomial being multiplied by itself
# - The power, which must be an integer
# - Swapped-operands flag:  grounds for immediate death
# Returns:
# - A [ref to a] Yapp polynomial that is the original raised to the
#   indicated powert
#
sub Yapp_power
{
  my ($self, $power, $swapped) = @_;

  # Some validations:
  #
  die "You cannot raise a number to a polynomial power"
    if ($swapped);
  die "Power must be an integer"
    if ($power != int($power));

  # And now that that's been squared away: Get to work
  #
  my $ryapp = Yapp(1);          # Start with a unit polynomial
  while ($power-- > 0)          # (Decrement after testing)
  {
    $ryapp *= $self;            # Carry out the self-multiplication
  }
  return $ryapp;
}
#------------------------------------------------------------------------------
# Yapp_divide(): Method to implement the overloaded '/=' operator.  Unlike
#                the multiply operator, I am allowing only for deviding by a
#                constant value.  Scheme is simple: Take reciprocal of divisor
#                and multiply the Yapp by that.
# Parameters:
# - (Implicit) the Yapp object itself
# - The divisor, which may be real or complex
# Returns:
# - The original Yapp reference
#
sub Yapp_divide
{
  my ($self, $divisor) = @_;
  my $recipro = (1.0 / $divisor);   # Take reciprocal

  #$self *= $recipro;               # (This causes "No Method" error, so..)
  $self = $self * $recipro;         # Carry out that multiplication
  return $self;
}
#
#------------------------------------------------------------------------------
# Yapp_dividedby(): Method to implement the overloaded '/' operator.
# Scheme: Identical to the Yapp_times() method.
#
# Parameters:
# - (Implicit) the Yapp object itself
# - The divisor, which may be real or complex
# Returns:
# - A reference to a new Yapp object wherin the division has been carried # out
#
sub Yapp_dividedby
{
  my ($self, $multiplier, $swapped) = @_;   # $swapped is irrelevant
  my $ryapp;                # The object to be returned

  $ryapp = Yapp($self);     # Make a copy
  $ryapp /= $multiplier;    # Carry out operation
  return $ryapp;            #(Wasnt't that simple?)
}
#
#------------------------------------------------------------------------------
# Negation and conjagation of the coefficients of the polynomial:
#
# Yapp_negate(): Method to handle the overloaded ! operator, to give a Yapp
#                whose coefficientes are the negatives of the given Yapp.
# Parameter: (Implicit)
# - A Yapp reference
# Returns:
# - If called in scalar context: A new Yapp with the negated coefficcients
# - If called in void context, returns nothing but the coefficients of the
#   given Yapp have been negated.
# Well, that *was* the idea; unfortunately, I cannot seem to get in-place
# negation to work.
sub Yapp_negate
{
  my $self = shift(@_);
  my $ryapp;                    # Return reference, if non-void context
#  my $ryapp = Yapp($self);      # Make a copy

  if ($testmode)
  {
    if (defined(wantarray))
    {
      if (wantarray) {print "List context\n";}
      else           {print "Scalar context\n";}
    }
    else {print "Void context\n";}
  }
  # Depending on calling context, $ryapp is either a new Yapp [reference] or
  # a new reference to the same parameter
  #
  #Xmy $ryapp = (defined(wantarray)) ? Yapp($self) : $self;
  #$ryapp *= -1;                             # Multiply all coefficients by -1
  #X$ryapp = -1 * $ryapp;                     # Multiply all coefficients by -1
  if (defined(wantarray))           # List context would be silly so ..
  {                                 # treat list context like scalar context
    $ryapp = -1 * $self;            # Create a new Yapp, negative of $self
    return $ryapp;                  # Send that back to caller
  }
  else
  {
    $self = -1 * $self;             # Negate the given Yapp
    return $self;                   # Return (a wasted value)
  }
}
##------------------------------------------------------------------------------
# Yapp_conj(): Method to handle the ~ operator; Conjugate the complex
#              coefficents of the given yapp
# Parameter: (Implicit)
# - A Yapp reference
# Returns:
# - In scalar context: A new one with the conjugate coefficcients
# - In void context:   Nothing but the original Yapp has been conjugated
#                      That is, if I could get it to work. In-place
#                      conjugation is not working, however.
sub Yapp_conj
{
  my $self = shift(@_);
  #my $ryapp = Yapp($self);      # Make a copy

  # Depending on calling context, $ryapp is either a new Yapp [reference] or
  # a new reference to the same parameter
  #
  my $ryapp = (defined(wantarray)) ? Yapp($self) : $self;
  my $clc;                      # Loop counter

  for ($clc = 0; $clc <= $ryapp->{degree}; $clc++)
  {
    next unless (ref($ryapp->{coeff}[$clc]) eq $class_cplx);# Skip real coeffs
    $ryapp->{coeff}[$clc] = ~$ryapp->{coeff}[$clc];
  }
  return $ryapp if (defined(wantarray));    # Return a value if scalar context
}

#
#-------------------------------------------------------------------------------
# Evaluation: Plug a value into the polynimal and calculate the result
# This is a simple application of Horner's "Synthetic Division"
#
# Parameters:
# - (Implicit) Reference to the Yapp polynomial
# - A real number or reference to complex number
# Returns:
# - The result of the plugging it in.
# - Optionally, if caller specified ($value, $quotient) = $some_yapp(number)
#   also returns [a refernce to] the quotient polynomial
#
sub Yapp_eval
{
  my ($self, $in_val) = @_;     # (Assuming correct parameters or Phzzzt!)
  my $elc;          # Exponent loop counter for countdown
  my $addval = 0.0; # Intermediate value in synthetic division step
  my $sumval = 0.0; # A term in the "quotient" line
  my @quotient = ();# Array representing the quotient sans remainder

  for ($elc = $self->{degree}; $elc >= 0; $elc--)
  {
    $quotient[$elc] = $self->{coeff}[$elc] + $addval;   # A term in quotient
    $addval = $quotient[$elc] * $in_val;                # Add this in next round
  }
  # Last term of the quotient is the evaluation of the polynomial at that
  # input value.  The rest is the quotient polynomial, albeit looking like a
  # degree too high.  Use shift to solve both issues:
  #
  my $result = shift(@quotient);    # Get value and reduce degree
  realify($result);                 # In case of rounding error, if result
                                    # is near zero, make it zero
  if (wantarray())                  # Did caller specify two return values?
  {
    my $y_quotient = Yapp(\@quotient);  # Turn the quotient array into poly
    return($result, $y_quotient);       # and return both to caller
  }
  else {return $result;}                # Otherwise, we happy with result alone

#  for ($elc = $self->{degree}; $elc >= 0; $elc--)
#  {
#    $sumval = $self->{coeff}[$elc] + $addval;
#    $addval = $in_val * $sumval;    # Add this to the next coefficient
#  }
#  return $sumval;               #(OK last addition was wasted)
}
#
#-------------------------------------------------------------------------------
# Yapp_reduce: Reduce the roots of a polynomial by the indicated amount.
# This is part of Horner's method for approximating the roots but will not
# be used here for that purpose.
#
# Parameters:
# - (Implicit) Reference to the Yapp polynomial
# - Value by which to reduce the roots. May be real or [reference to]
#   complex
# Reurns:
# - A [ref to a] new Yapp object, whose roots are the same as those of the
#   original but reduced by the indicated amount,
#
# Scheme:  This will use an array of successive arrays representing
# intermdiate quitient polynomials, as well as an array or remainders to
# build the final result
#
sub Yapp_reduce
{
  my ($self, $reduce_by) = @_;  # Get my parameters
  my @remainders;
  my @degrees;      # Make it easier to track the degree of each quotient
  my ($qlc, $tlc);  # Loop counters for outer and inner loops
  my $q_count = $self->{degree};    # How many quotients I will calculate
  my $lead_coeff = $self->{coeff}[$q_count];    # Will divide by this and
                                                # multiply it back later
  my @rlist = ();   # Result list, the raw polynomial we build up
  my $rcount = 0;   # Counter to make it easier to keep track to that build
  my @coeffs = @{$self->{coeff}};               # Will start with synthetic

  # Divide whole new "poly" by leading coefficient that that new leading
  # coefficient is 1.
  #
  for (my $clc=0; $clc <= $q_count; $clc++) {$coeffs[$clc] /= $lead_coeff;}

  for ($qlc = $self->{degree}; $qlc > 0; $qlc--)
  {
    my $addval = 0.0;       # Set up for synthetic division
    my $q_deg = $qlc - 1;   # Degree of new quotient
    my @quotient;           # Quotient of a synthetic division operation

    # Note: This inner loop uses the same synthetic division used in function
    # Yapp_eval().  But the moving expression $quotient[$tlc] stands in place
    # of $sumval
    #
    for ($tlc = $qlc; $tlc >= 0; $tlc--)
    {
      # First add $addval to a term to determine next coefficient.
      # Then determine next $addval
      #
      $quotient[$tlc] = $coeffs[$tlc] + $addval;
      $addval = $reduce_by * $quotient[$tlc];
    }
    # When I come out of above inner loop, two items of interest:
    # - The last value in the quotient array, which is the remainder of the
    #   synthetic division operation, is the next coefficient in the final
    #   polynomial.
    # - The quotient array represents the next polynomial to be "divided" by
    #   the divisor, once I get rid of the remainder term.
    #
    $rlist[$rcount++] = shift(@quotient);   # Pull remainder into the polynomial
                                            # I am building up, an drop it out
                                            # of the next polynomial.
    @coeffs = @quotient;                    # Will next divide *this* polynomial
    @quotient = ();                         # Neatness: Clear for next round
  }
  $rlist[$q_count] = 1;         # The inevitable final quotient

  # (Whew) I now have our list or remainders from the above sequence of
  # synthetic divisions.  Create a polynomial out of that list
  #
  my $ryapp = Yapp(\@rlist);    # This creates the polynomial
  $ryapp *= $lead_coeff;        # Remember that? Multiply it back out
  return $ryapp;                # This has the roots reduced.
}
#
#-------------------------------------------------------------------------------
# Yapp_negate_roots(): Returns a polynomial whose roots are the nagatives of the
#                      roots of the given polynomial.  Easy trick: Negate the
#                      coefficient of every odd-exponent term.
#
# Parameter:
# - (Implicit) The Yapp reference
# Returns:
# - A new Yapp with negated roots
#-------------------------------------------------------------------------------
#
sub Yapp_negate_roots
{
  my $self = shift(@_);         # The only parameter I care about
  my $ryapp = Yapp($self);      # Copy to another polynomial
  my $coefref = $ryapp->{coeff};
  for (my $clc = 1; $clc <=$#{$coefref}; $clc+=2)   # Odd indexed terms only
  {
    $coefref->[$clc] = - ($coefref->[$clc]) #(If it was 0, no harm done)
  }
  return $ryapp
}
#
#-------------------------------------------------------------------------------
# Derivative and antiderivative i.e. indefinite integral.
# In both cases, we will store the transformed polybnomial back into the
# Yapp structure because they will very likely be resused for other stuff.
# For example, Newton's method for finding roots uses the first derivative,
# and laGuerre's method uses the second derivative.
#
# Yapp_derivative()
# Parameters:
# - (Implicit) Reference to the Yapp to differentiated
# - Order of derivative.  If omitted, assume caller meant first derivative
# Returns:
# - A [reference to a] Yapp polynomial that is the indicated derivative of
#   the given Yapp.
# - Side effect: A new array of Yapp references to all the derivatives is
#   now hanging off the given Yapp structure. @{derivatives}
#-------------------------------------------------------------------------------
#
sub Yapp_derivative
{
  my $self = shift(@_);
  my $order = defined($_[0]) ? shift(@_) : 1;   # Get nth derivative or assume 1
  my $start_n;      # Highest order derivative we already have
  my $coefref;      # Use as array reference as I step into each polynomial
  my $dlc;          # Loop counter for derivatives

  if (defined($self->{derivative}[$order]))     # If I have already derived this
  { return($self->{derivative}[$order]); }      # just give it again, no work

  # Still here?  Two possibilities:
  # 1. Previous derivatives have been derived, just not this order
  # 2. This is the first time a derivative it being requested, so I have to
  #    create the derivatives array from scratch
  #
  if (defined($self->{derivative}[1]))
  {                                     # If we already have derivatives saved
    $start_n = $#{$self->{derivative}}; # Start past highest we already have
  }
  else
  {                                     # Starting derivatives from scratch
    $self->{derivative}[0] = Yapp();    # Empty placeholder for 0th deriv
    $start_n = 0;                       # 'Cause we got none yet
  }

  for ($dlc = $start_n; $dlc < $order; $dlc++)
  {
    last if ($dlc >= $self->{degree});      # No more derivatives than degree
    my @new_list;
    if ($dlc == 0) {$coefref = $self->{coeff}; }    # -> my coeffiecients
    else
    {
      my $cur_deriv = $self->{derivative}[$dlc];    # -> (lc)th derivative
      $coefref = \@{$cur_deriv->{coeff}};           # and to *its* coefficients
    }
    for (my $tlc = 1; $tlc <= $#{$coefref}; $tlc++) # Skip 0th term
    {
      $new_list[$tlc-1] = $tlc * $coefref->[$tlc];  # Coeff * its exponent
    }
    $self->{derivative}[$dlc+1] = Yapp(\@new_list); # New Yapp in deriv slot
  }
  return ($self->{derivative}[$dlc]);
}
#
#-------------------------------------------------------------------------------
# Yapp_integral(): Method to calculate the first integral of the given Yapp
#                  polynomial.
# Let's clarify: This serves as two functions:
# - The indefinite integral AKA the antiderivative, returning a Yapp polynomial
#   for the given polynomial.  Will not include the arbitrary constant you were
#   taught to always include with your antiderivative.
# - A definite integral: That is, supplying two numbers that represent the
#   endpoints of an interval i.e. the limits of the integral.
# Returns:  Well, that depends:
# - For indefinte integral, returns a reference to the antiderivative polynomial
#   (which as been stored in $self anyway.)
# - For definite integral: Returns the value of that integral between the
#   limits.
# In both cases, the antiderivative polynomial is cached along with the given
# Yapp so it may be re-used for various limits.
#
# Note: Unlike with derivatives, I see no need to code for additional order
# of integral. (I'm not anticipating a Laurent series)
#
sub Yapp_integral
{
  die ("Must supply either no integral limits or exactly two points")
    unless ( ($#_ == 2) || ($#_ == 0));     # Bail if wrong parameters given
  my $self = shift(@_);                     # If $#_ was 2, it is now 1
  my ($l_limit, $u_limit) = (undef, undef); # Don't know yet if user called
                                            # me for a definite integral.
  my $ryapp;                                # My return reference for indefinite
  my $rval= 0.0;                            # My return value for definite


  ($l_limit, $u_limit) = @_ if ($#_ == 1);  # Now is when I find out about
                                            # those endpoint parameters

  # Note: If I never calculated the integral for this polynomial (or if it
  # has "refreshed"), the {integral} component may be an array or undefined.
  # But it *won't" be Yapp.
  #
  if (ref($self->{integral}) ne $class_name)    # Ever calculated the indefinite
                                                # integral for this polynomial?
  {                                 # No: Do it now.
    my @new_list = (0);             # 0 in place of arbitrary constant
    for (my $ilc = 0; $ilc <= $self->{degree}; $ilc++)
    {
      push (@new_list, ($self->{coeff}[$ilc] / ($ilc + 1)));
    }
    $ryapp = Yapp(\@new_list);              # Create the polynomial
    $self->{integral} = $ryapp;             # and cache that for re-use
  }
  else {$ryapp = $self->{integral}; }       # Already computed: Just re-use it

  # Part 2: What does the user want? Definite or indefinite integral?
  #
  return ($ryapp) unless (defined($l_limit));   # No limits? Wants indefinite

  # Still here: Wants the value of that integral
  #
  my $u_val = $ryapp->Yapp_eval($u_limit);  # Value of antideriv at upper limit
  my $l_val = $ryapp->Yapp_eval($l_limit);  # Value of antideriv at lower limit
  $rval = $u_val - $l_val;                  # And that's the integral value
  return $rval;
}
#
#-------------------------------------------------------------------------------
# Solving for the zeros of a polynomial:
# Method Yapp_solve() is the only method that should be called byt he users.
# This, in turn calls appropriate other methods.  For example, for a quadratic
# polynomial, Yapp_dolve() will call Yapp_solve_quad(), which will simply use
# the quadratic formula. For a cubic, it uses the algorithm of Gerolamo Cardano
# and for the quartic (4th degree) that of Lodovico Ferrari.  After that, we
# resort to bisection to locate a solution, then go to Newton's method.  (A
# future release may switch to laGuerre's method.)
#
# For degree >= 5 we are guranteed at least one real root only for odd-degree
# polynomials so bisection is a sure thing.  However, for an even-degree poly-
# nomial, there is no such guarantee.  I have tried to discuss and analyze
# methods that might be analogous to bisection for complex numbers but to no
# avail.  I have bitten the bullet and coded Laguerre's method as a starting
# point for even-degree polynomials of degree >= 6
#
# Short cuts:
# - When a complex root is found, we will also evaluate the quotient for the
#   conjugate of that root.
# - In all cases, when any root is found, we evaluate quotient for the same root
#   in case it's a multiple root.
# - Whenever a root (or set of roots, as bove) is found, we make a recursive
#   call to Yapp_solve() to get the roots of the quotient polynomial. (I have
#   read that the technique is called deflation.)  However, after getting the
#   roots of the deflated polynomial, I still apply Newton's method on the roots
#   with respect to the polynmomial at hand.  This is to compensate for rounding
#   errors that have crept in while solving the deflated polynomial
#
# Parameter:
# - (Implicit) The Yapp polynomial to solved
# Returns:
# - An array of solutions, which may include repeated values owing to repeated
#   roots.  (Not a reference but the array otself.)
#-------------------------------------------------------------------------------
#
sub Yapp_solve
{
  my $self = shift(@_);
  my $solu_end = $self->{degree}    # The degree the polynomial came it with
                  - 1;              # Top index of solutions array
  my @solutions = ();               # Array to be returned starts empty
  my $solu_begin = 0;               # Where to start pushing solutions in
                                    # the @soultions array
  if ($solu_end == 0)
  {
    carp("0-degree polynomial <$self->Ysprint()> has no solution");
    return @solutions;              # Don't bother with it anymore
  }

  # Quick check: If the constant term of the polynomial is 0, then we know
  # that 0 is a solution. In that case, shift the polynomial down a degree
  # and test again.  This is the point of the following loop.. Except that I
  # am assuming that constant term is zero if it is sufficiently small; that
  # it is the result of a rounding error in a previous computation is this
  # highly recursive function.
  #
#
  while (abs($self->{coeff}[0]) <= $margin)
  {
    $solutions[$solu_begin++] = 0.0; # Zero *is* a solution to *this* polynomial
    shift(@{$self->{coeff}});       # Knock it down a degree for next round
    --($self->{degree});            # Register the lowered degree
  }
  my $degree = $self->{degree};     #(Just neater for successive if/elsif)

  # After this, we go structured; no returning from within "if" blocks
  #
  if ($degree == 1)                 # b + ax = 0
  {
    $solutions[$solu_begin] = - $self->{coeff}[0]/$self->{coeff}[1] ;   # -b/a
  }
  elsif ($degree == 2)
  {
    @solutions[$solu_begin .. $solu_end]  = $self->Yapp_solve_quad();
  }
  elsif ($degree == 3)
  {
    @solutions[$solu_begin .. $solu_end] = $self->Yapp_solve_cubic();
  }
  elsif ($degree == 4)
  {
    @solutions[$solu_begin .. $solu_end] = $self->Yapp_solve_quartic();
  }

  # We've just run out of fomulaic methods and we must resort to pure numerical
  # methods.  Oh Joy!
  #
  elsif (! $self->all_real)
  { # Complex Coefficients: Some theorems go out the window
    @solutions[$solu_begin .. $solu_end] = $self->Yapp_solve_complex();
  }
  elsif (($degree % 2) == 1)
  {
    @solutions[$solu_begin .. $solu_end] = $self->Yapp_solve_odd();
  }
  else
  {
    @solutions[$solu_begin .. $solu_end] = $self->Yapp_solve_even();
  }
  # Apply this filter to the @solutions array: If we have reason to suspect
  # that a complex number looks complex only due to rounding errors i.e. the
  # imaginary part is so small that it looks ignorable, then do ignore it.
  # Similarly, we can nullify (but not ignore) a trivial real component of a
  # complex number.
  #
  realify(\@solutions);

  return @solutions;
}
#
#-------------------------------------------------------------------------------
# Yapp_newton(): Internal method. If we already have a crude (in the eye of
#                the beholder) appoximation of a root, let's get much closer
#                by using Newton's method.  This is especially a problem
#                with the Ferrari and Cardano's algorithms when applied as
#                algorithms and rounding error are inevitable.
# Parameters:
# - (Implicit) The Yapp object
# - The initial value
# Returns:
# - A refined value, with $margin of being correct.
#
sub Yapp_newton
{
  my ($self, $xn) = @_;             # Get object an our starting X-value

  my $func_name = (caller(0))[3];   #(For debugging)
  my $yprime = $self->Yapp_derivative(1);
  my $valxn = $self->Yapp_eval($xn); # Evaluation at starting point
  while (abs($valxn) >= $margin)    # Continue this loop until I'm REALLY close
  {                                 # to a solution.
    my $xc = $xn;                       # Hold a copy of current tested value
    my $yp = $yprime->Yapp_eval($xn);   # Get derivative at current $xn
    last if (abs($yp) == 0);            # Beware of zero-divide!
    my $correction = $valxn / $yp;      # Current value / current derivative
    $xn -= $correction;                 # X[n+1] = X[n] - Y(X[n])/Y'(X[n])
    last if (abs($xn - $xc) < $margin); # Another exit condition: Almost no diff
                                        # between this x and previous x
    $valxn = $self->Yapp_eval($xn); # Ready for test or use atop next round
    printf("%s: xn = <%s>; valxn = <%s>\n",
           $func_name, Csprint($xn), Csprint($valxn))
        if ($testmode);
  }
  return($xn);                      # Close enough to send back to caller
}
#
#-------------------------------------------------------------------------------
# Yapp_laguerre(): Internal method. If we start with just about any starting
#                  point - it doesen't even have to be all that close we can
#                  get much closer by using Laguerre's method.
# Source of the mathetics I am using:
# Numerical Recipes in C; The Art of Scientific Computing
#   Second Edition (C) 1988, 1992`
# Authors: William H. Press
#          Saul A. Teukolsky
#          William T. Vetterling
#          Brian P. Flannery
#   Pages: 372-374
# Note: I made the chagrined discovery that rounding errors cause the test
# values to fluctuate a few decimal places out of my desired margin.
# Experiment: Use Laguerre's method to within .001, then switch to Newton's
# method.
# ...
# Parameters:
# - (Implicit) The Yapp object
# - The initial value
# Returns:
# - A refined value, with $margin of being correct.
#
sub Yapp_laguerre
{
  my ($self, $xn) = @_;             # Get object an our starting X-value
  my $func_name = (caller(0))[3];   #(For debugging)

  my $degree = $self->{degree};             # This will figure in recalculations
  my $yprime = $self->Yapp_derivative(1);   # First derivative of self
  my $yprime2 = $self->Yapp_derivative(2);  # And second derivative
  my $valxn = $self->Yapp_eval($xn);        # Evaluation at starting point
  my $crude_margin = .001;                  # Quite close - might as well take
                                            # advantage of Laguerre's faster
                                            # convergence.
  while (abs($valxn) >= $crude_margin)      # Bet this looks familiar so far :)
  {                                         # Well, hang on to your seats!
    # Correction factor is:
    # Degree / (G +/- sqrt((Degree - 1) * (Degree * H - G**2)))
    # See evaluation of $gxn for G and $hxn for H below
    # Ugly, ain't it? ;-)
    #
    my $prev_xn = $xn;              # Keep current estimate for comparison
    my $gxn = $yprime->Yapp_eval($xn) / $valxn;
    my $hxn = $gxn**2 - ($yprime2->Yapp_eval($xn) / $valxn);
    my $under_rad = ($degree -1)
                  * ($degree * $hxn - $gxn ** 2);   # May be negative or complex
    my @rad_roots = root($under_rad, 2);            # Take square roots of that
    my @denoms = ($gxn - $rad_roots[0],     # Choose a denominator for fraction
                  $gxn + $rad_roots[0]);    # to be used in a correction.
    my @norms = (abs($denoms[0]), abs($denoms[1])); # Which denominator has the
                                                    # bigger absolute value?
    my $denom = ($norms[0] > $norms[1]) ? $denoms[0] : $denoms[1];  # Pick that
    my $correction = $degree / $denom;      # Correct previous estimate by this
    $xn -= $correction;                     # "amount" by subtracting it
#?  last if (abs($prev_xn - $xn) < $margin); # Quit if correction was teeny
    $valxn = $self->Yapp_eval($xn);         # Otherwise, re-evaluate at new
                                            # estimate and plug in next round
    printf("%s: xn = <%s>; valxn = <%s>\n",
           $func_name, Csprint($xn), Csprint($valxn))
        if ($testmode);
  }
  # Came out of loop: I guess we're close enough to start with Newton?
  #
  $xn = realify($xn);                       # NUllify tiny Im component
  $xn = $self->Yapp_newton($xn);            # before feeding it to Newton
  realify($xn);                             # and correct again afterward
  return $xn;
}
#
#-------------------------------------------------------------------------------
# Yapp_solve_quad(): Solve a quadratic equation
#
sub Yapp_solve_quad
{
  my $self = shift(@_);
  my ($real_part, $discr, $rad_part);
  my @solutions = ();       # Look familiar? :-)
  my $i = cplx(0, 1);       # The familiar imaginary unit, "i"
  my ($c, $b, $a) = @{$self->{coeff}};  # Copy the array, for neatness

  $real_part = -$b / (2 * $a);      # This part is always guaranteed real
                                    # (assuming real coefficients, of course)
  $discr = $b ** 2 - (4 * $a * $c);  # Discriminant: b^2 - 4ac
  if ($discr == 0)
  {
    @solutions = ($real_part, $real_part);  # Repeated roots.  We're done!
  }
  elsif ($discr > 0)
  {
    $rad_part = sqrt($discr)/ (2 * $a);     # The part under the radical
    @solutions = ($real_part - $rad_part, $real_part + $rad_part);
  }
  else                              # Negative discriminant => complex roots
  {
    my @i_roots = root($discr, 2);  # Get both imaginary roots

    # Because the root() function is too *&&^% comfortable in polar form, we
    # get a miniscule (but still there) real part.  Get rid of it!
    #
    @i_roots = (cplx(0.0, $i_roots[0]->Im), cplx(0.0, $i_roots[1]->Im));
    printf("Roots of discriminant <%f> are: <@i_roots>\n", $discr)
      if ($testmode);
    @i_roots = (($i_roots[0]/(2 * $a)),
                ($i_roots[1]/(2 * $a)) );   # Recall: All parts divided by 2a
    @solutions = ($real_part + $i_roots[0], $real_part + $i_roots[1]);
  }
  return @solutions;
}
#
#-------------------------------------------------------------------------------
# Yapp_solve_cubic()
#
sub Yapp_solve_cubic
{
  my $self = shift(@_);
  my @solutions;                # I will return this array
  my $monic;                    # Original Yapp with 1 as leading coefficient
  my $reduced;                  # $monic with roots reduced, if needed
  my $reduced_by = 0;           # By how much to reduce the roots in order
                                # to eliminate the degree[2] term
  my ($zero, $quotient_2, $quotient_1); # zero is mainly a placeholder.
                                        # the two quotients are for after an
                                        # evaluation to get the next lower
                                        # degree polynomials.
  my $allreal = $self->all_real;        # Optimistic start

  # 2 transformations:
  # - Make sure leading coefficient is 1 (monic)
  # - Reduce roots by 1/3 of the x^2 coefficient. This sets the x^2 coefficient
  #   (Which is always the negative of the sum of the roots) to 0
  #
  if ($self->{coeff}[3] == 1)   # If this is alrady a monic polynomial
  {
    $monic = Yapp($self);       # Use a copy before reducing roots
  }
  else
  {                                         # Divide by leading coefficent
    $monic = $self / ($self->{coeff}[3]);   # to *make* it monic
    printf("Self:  <%s>\nMonic: <%s>\n", $self->Ysprint(), $monic->Ysprint())
      if ($testmode);                       # TESTMODE
  }
  if ($monic->{coeff}[2] != 0)  # If we have a degree[2] term
  {
    $reduced_by = -($monic->{coeff}[2])/3;  # Reduce by 1/3 of x^2 term
    $reduced = $monic->Yapp_reduce($reduced_by);
    printf("Original <%s>, with roots reduced by <%.5f>\nYields: <%s>\n",
            $self->Ysprint(), $reduced_by, $reduced->Ysprint())
      if($testmode)                         # TESTMODE
  }
  else
  {
    $reduced = Yapp($monic);    # Had no X^2 term.
  }
  my $p = $reduced->{coeff}[1];
  my $q = $reduced->{coeff}[0];
#
  # In Cardano's algorithm, I set X = U + V, which lets me play games with
  # these new variables.  After I pluck in (U+V) for X in the reduced monic
  # polynomial, I collect terms and get an expression like this:
  # U^3 + (U + V)(3*U*V + p) + V^3 + q = 0
  # If I set (3*U*V + p) to 0 I get a much simpler expression:
  # U^3 + V^3 + q = 0.  But, in light of the 0 setting above, I need to
  # substitute V = -p/(3*U).  This results in the 6th degre equation:
  # 27U^6 + 27q*U^3 - p^3 = 0.
  # Hey! This is quadratic in U^3.  I can solve this!
  #
  my $u_six = Yapp((-$p ** 3), (27 * $q), 27);
  $u_six->variable("U-Cubed");  # Set this variable for clarity
   printf("U-Cubed quadratic is <%s>\n", $u_six->Ysprint())
     if ($testmode);                        # TESTMODE
  my @u_solution = $u_six->Yapp_solve();    # I only need one of these
  my $u_cubed = $u_solution[0];             # Pick first one - Easy
  my ($u, $v);                              # Components for X = U + V

  # $u_cubed may be a complex number; I need to be circumspect about taking
  # its cube root.
  #
  my @u_roots = root($u_cubed, 3);      # Get all three cube roots

  # Note: Even the real cube root of a real number may come back as a complex
  # with an extremely small imaginary part, on the order of 10^-15 or so. This
  # is a legitimate rounding error, which I can lose with relative impunity.
  #
  my $use_this_u = $u_roots[0];         # Ready to use first root, even complex
  realify(\@u_roots);                       # Lose insignificant imaginary com-
                                            # ponents from rounding errors.
  for (my $rlc = 0; $rlc < 3; $rlc++)       # Search for a real cube root
  {
    if (ref($u_roots[$rlc]) ne $class_cplx) # Fair to assume: If it ain't
    {                                       # complex, it's the real we prefer
      $use_this_u = $u_roots[$rlc];         # If we found it, grab it and run
      last;
    }
  }

  # Now I can set $u and $v properly. $u may be real or complex as per above
  # loop.  But if it is complex, $v seems to inevitable end up as the
  # conjugate of $u.  Prefer to use a real root but if none is available,
  # one root is as good as another for this purpose; in that case, use the
  # first cube root.
  #
  $u = $use_this_u;             # This is the cube root to work with.

  # Now recall that V = -p/(3*U) from the 3*U*V +p = 0 setting
  #
  $v = -$p / (3 * $u);          # The other component ox X = U + V
  $solutions[0] = $u + $v       # Almost there; but remember this is the root
                + $reduced_by;  # reduced in the second transformation
                                # Now THAT's a root! .. er, almost.
  $solutions[0] = realify($solutions[0]);   # Lose trivial imaginary component
#
  # OK, I have my first solution.  If it is complex *and* the original
  # coefficients are all real, math guarantees me that the comjugate is also
  # a solution.  In that case, [synthetic] divide by this complex solution
  # as well so we are left with a simple linear equation to solve.
  # Otherwise (Either this one is real or we had complex coefficients) just
  # [synthetic] divide by this solution and solve the rusulting quadratic
  # quotient.
  #
  if (  (ref($solutions[0]) eq $class_cplx) # First solution is complex
      &&($allreal) )                        # and no complex coefficients
  {                                         # Conjugate is also a solution
    $solutions[1] = ~ $solutions[0];        # from the original monic
    ($zero, $quotient_2) = $monic->Yapp_eval($solutions[0]);
                                            # Had *better* evaluate to 0!
    ($zero, $quotient_1) = $quotient_2->Yapp_eval($solutions[1]);

    # $quotient_1 is a [ref to a] 1st degree polynomial.
    #
    ($solutions[3]) = $quotient_1->Yapp_solve();    # This is kinda trivial
  }
  else                          # If I can't depend on conjugate, for what-
  {                             # ever reason, just divide this one out
    ($zero, $quotient_2) = $monic->Yapp_eval($solutions[0]);    # -> quadratic

    # $quotient_2 is a quadratic expression. Just solve that.
    #
    @solutions[1,2] = $quotient_2->Yapp_solve();
  }
  return @solutions;
}
#
#-------------------------------------------------------------------------------
sub Yapp_solve_quartic
{ # Much of the setup here is similar to that of Yapp_solve_cubic, but with
  # tiny differences, enough to not try to make subroutins of this code.
  my $self = shift(@_);
  my @solutions;                # I will return this array
  my $monic;                    # Original Yapp with 1 as leading coefficient
  my $reduced;                  # $monic with roots reduced, if needed
  my $reduced_by = 0;           # By how much to reduce the roots in order
                                # to eliminate the degree[2] term
  my ($zero, $quotient_2, $quotient_1); # zero is mainly a placeholder.
                                        # the two quotients are for after an
                                        # evaluation to get the next lower
                                        # degree polynomials.

  # 2 transformations:
  # - Make sure leading coefficient is 1 (monic)
  # - Reduce roots by 1/4 of the x^3 coefficient. This sets the x^3 coefficient
  #   (Which is always the negative of the sum of the roots) to 0
  #
  if ($self->{coeff}[4] == 1)   # If this is already a monic polynomial
  {
    $monic = Yapp($self);       # Use a copy before reducing roots
  }
  else
  {                                         # Divide by leading coefficent
    $monic = $self / ($self->{coeff}[4]);   # to *make* it monic
    printf("Self:  <%s>\nMonic: <%s>\n", $self->Ysprint(), $monic->Ysprint())
      if ($testmode);                       #TESTMODE
  }
  if ($monic->{coeff}[3] != 0)  # If we have a degree[3] term
  {
    $reduced_by = -($monic->{coeff}[3])/4;  # Reduce by -1/4 of x^3 term
    $reduced = $monic->Yapp_reduce($reduced_by);
    printf("Original <%s>, with roots reduced by <%.5f>\nYields: <%s>\n",
            $self->Ysprint(), $reduced_by, $reduced->Ysprint())    #TESTMODE
      if($testmode);
  }
  else
  {
    $reduced = Yapp($monic);    # Had no X^3 term.
  }
#
  # In Ferrari's algorithm, we transpose the X^2, X^1 and constant term to the
  # right side.  Of course, it is not expected to be perfect square.  However,
  # we can add a new variable, u, to the left side so that it becomes
  # (1)     (X^2 + u)^2.
  # That is, add 2*u*X^2 + u^2 to the left side, keeping it a perfect square.
  # The right side, with the transposed 3 terms, had been:
  # (2)     -cX^2 -d*X -e
  # now becomes (after collecting terms):
  # (3)     (2u -c)X^2 -d*X + (u^2 -e)
  # (3a)          A     B            C
  # Is this a perfect square? Well, that depends: Is *is* a quadradic
  # expression but it *can* be a perfect square if is discriminant is 0;
  # that is: B^2 - 4*A*C, which is an expression containing that unknown u
  # term, can be set to 0. That is:
  # (4)     (-d)^2 - 4*(2u -c)*(u^2 -e) == 0
  # Multiplying out and collecting terms, this becomes:
  # (4a)    8*u^3 -4*c*u^2 -8*e*u +(d^2 -4c*e)
  # A cubic equation in the unknown u.  This is called the resolvent cubic
  # of the original polynomial. (The last two terms in parentheses comprise
  # the constant of that cubic equation.)
  # The point is that: Any solution to (4a) can be plugged back into (3) to
  # make it a perfect square to match against the perfect square left hand side
  # (LHS) in (1).
  # Here goes!
  #
  my $c = $reduced->{coeff}[2];     # X^2 coefficient
  my $d = $reduced->{coeff}[1];     # X^1 coefficient
  my $e = $reduced->{coeff}[0];     # Constant term

  my @rca = ();         # Array that will generate the resolvent cubic equation
  push @rca, 4*$c*$e - $d**2;   # Constant term
  push @rca, -8*$e;             # u^1 term
  push @rca, -4*$c;             # u^2 term
  push @rca, 8;                 # u^3 term
  my $rc = Yapp(\@rca);         # Turn that into a polynomial
  $rc->variable("u");           # Just for clarity: With variable letter u
  my @rc_solutions = $rc->Yapp_solve();   # (Obvious purpose)
  my $rc_plug;                  # The one we wil plug in to (3)

  # Now life is easier (with fewer rounding errors) if I choose a real solution.
  # I can't be sure it won't return a complex with a *very* low Im part (on the
  # order of 10^-15 ir so) due to rounding errors.  So I merely look for a root
  # with # sufficiently low Im part that I would ignore it.
  #
  foreach my $rc_val (@rc_solutions)        #(Had used $rc_plug here but it went
  {                                         # undefined after loop exit.  Hence
    $rc_plug = $rc_val;                     # using throwaway variable $rv_val
    last if (ref($rc_plug) ne $class_cplx); # It came back as a real! Use it
  }
#
  # Truth be known, *any* one of the roots is usable but I would prefer a real
  # Now plug the found value, $rc_plug, back into (3);
  # Reminder: (3)       (2u -c)X^2 -d*X + (u^2 -e)
  #
  my @rhsa = ();            # Array that will become the above quadratic
  push @rhsa, ($rc_plug**2 -$e);    # Constant term of above quadratic
  push @rhsa, -$d;                  # X^1 term of the quadratic is original
                                    # X^1 term of the root-reduced monic
  push @rhsa, (2*$rc_plug -$c);     # The X^2 term of above quadratic
  if ($testmode)
  {
    my $rhs = Yapp(\@rhsa); #(Actually, it is not necessary to generate
                            # this right-had-side polynomial.)
    printf("RHS Yapp is <%s>\n", $rhs->Ysprint);
  }

  # Testing has confirmed that $rhs is indeed a quadratric perfect square.
  # and that it is the square of yapp(sqrt(c), sqrt(a))
  #
  my $c_sqrt = sqrt($rhsa[0]);      # Since the $rhs polynomial is a square
  my $a_sqrt = sqrt($rhsa[2]);      # I'm guaranteed these are positive.

  # The above quadratic polynomial is the square of either:
  # * +$c_sqrt*X+$a_sqrt      or
  # * -$c_sqrt*X-$a_sqrt
  #
  # Now the perfect square on left-had side, is the square of (X^2 + U) and
  # so the binomia itself may be set equal to either of these:
  # X^2 + U =  AX + C or: X^2 -AX + (U - C) = 0
  # X^2 + U = -Ax - C or: X^2 +AX + (U + C) = 0
  # Where:
  # * U is $rc_plug
  # * A is $a_sqrt
  # * C is $c_sqrt
  # Solve these two quadratics and I have *almost* solved the original quartic.
  #
  my @quad1_a = ( ($rc_plug - $c_sqrt), -$a_sqrt, 1);   # Well, in order to
  my @quad2_a = ( ($rc_plug + $c_sqrt),  $a_sqrt, 1);   # them, it helps to
  my $quad1 = Yapp(\@quad1_a);                          # generate them
  my $quad2 = Yapp(\@quad2_a);
  printf("quad1 = <%s>;\nquad2 = <%s>\n", $quad1->Ysprint, $quad2->Ysprint)
    if ($testmode);
  @solutions[0..1] = $quad1->Yapp_solve();
  @solutions[2..3] = $quad2->Yapp_solve();

  # Not there yet: Remember, I reduced the roots to produce the monic.  Now
  # undo that:
  #
  @solutions = map {$_ + $reduced_by} @solutions;   # Restore what has been
                                                    # taken from them
  return @solutions;
}
#
#-------------------------------------------------------------------------------
# Yapp_solve_odd(): Method to solve polynomial equations of odd degree >= 5
#                   with all real coeficcients.
#
# Parameter:
# - [Ref to] a Yapp polynomial
# Returns:
# - Array of solution set
# Note: Initially, this function supports only polynomials with real-only
# coefficients, although I am leaving a stub block for handling complex
# coefficients as well.
#
sub Yapp_solve_odd
{
  my $self = shift(@_);
  my $crude_margin = 0.1;   # Just get close enough to start using Newton
  my @solutions = ();       # Array that I return to the caller

  # The following pair of betw variables will be set during a bisection
  # algorithm and used for initial values once we start wuth Newton's method.
  # Hence, we need to define them waaaay before they will be used, in order to
  # find them in scope anyplace in this function.
  #
  my $betw_try;               # X-value at midpoint of an interval
  my $betw_eval;              # Y-value at that X midpoint

  # Find one real root and check if it is duplicated.
  # Set up for a bisection algorithm: Starting with some initial guess at
  # 0, 1, and maybe -1, start doubling the interval until I get a + on one
  # side and - on the other.  Then start bisecting until I get pretty
  # close, like within 1/10 or so.  Then start using Newton.
  #
  my $left_try = 0.0;     # First & easiest point to test is X = 0
  my $left_eval = $self->{coeff}[0];  # 'cause I don't need to evaluate
  my $right_try = 1.0;
  my $right_eval = $self->Yapp_eval($right_try);

  # Search for an interval such that the Yapp evaluates to opposite signs
  # at the two endpoints of the interval
  #
  until (($left_eval * $right_eval) < 0)
  { # Entering loop body [again] means both tries evauated to the same
    # sign.  Sweep wider intervals..
    #
    if ($left_try == 0.0)     # At first round of this loop, don't double
    {                         # both enpoints of the interval;
      $left_try = -1.0;       # just expand in negative direction
    }
    else                      # But if we are well into the round
    {                         # we will double both enpoints of the interval
      $left_try *= 2;
      $right_try *= 2;
    }
    # Either way, re-evaluate at both endpoints.  (OK, one wasted,
    # repeated evaluateion at $right_try == 1).  Minor, IMO
    #
    $left_eval  = $self->Yapp_eval($left_try);
    $right_eval = $self->Yapp_eval($right_try);
  }

  # OK, we have found X-values that evaluate to opposite signs.  Now start
  # bisecting scheme.
  #
  until (abs($right_try - $left_try) <= $crude_margin)
  {
    $betw_try  = ($left_try + $right_try)/2 ;     # Bisect the interval
    $betw_eval = $self->Yapp_eval($betw_try);     # Evaluate @ midpoint
    last if ($betw_eval == 0);                    # Hey! We hit an exact root!
    my $left_sign  = $left_eval  >= 0 ? 1 : -1;   # and determine the signs
    my $right_sign = $right_eval >= 0 ? 1 : -1;   # at the endpoints and
    my $betw_sign  = $betw_eval  >= 0 ? 1 : -1;   # midpoint

    # Now mathematically, the sign at the midpoint evaluation must match
    # exactly one (never both) of the endpoint evaluation signs.  Ergo, it
    # must be be different from exactly one of the endpoint evaluations
    #
    if ($left_sign != $betw_sign)                 # Changes sign to left
    {                                             # of the midpoint
      $right_try = $betw_try;                     # Then move right end to
      $right_eval = $betw_eval;                   # middle, with its evaluate
    }
    else                                          # Changed sign to right
    {                                             # of the midpoint
      $left_try  = $betw_try;                     # Then move left end to
      $left_eval = $betw_eval;                    # middle, with its evaluate
    }
  }

  # OK, however we got here, we are now close enough to a solution to start
  # using the Newton method. That is, unless we actually hit an exact root!
  #
  $solutions[0] = ($betw_eval == 0.0) ?     # Start at latest midpoint
                  $betw_try : $self->Yapp_newton($betw_try);
  my ($zero, $quotient) = $self->Yapp_eval($solutions[0]);
                                    #(A good debugging step would be to check
                                    # that $zero is indeed 0.0)
  my $xn = $solutions[0];           #(Shorter var name for neater code)

  # So $xn is a root.  Question: Is it a multiple root?
  #
  while($quotient->{degree} > 0)    # Keep reducing it as we loop
  {
    my ($zzero, $quotient2) = $quotient->Yapp_eval($xn);    # Solves quotient?
    last if (abs($zzero) >= $margin);   # Clearly not. Done checking; Outahere!

    # Still here: It is a solution again!
    #
    $quotient = $quotient2;             # This is next quotient to use
    push (@solutions, $xn);             # and add same root into solutions set
  }
  # OK, I've exhausted my repeated copies of first root.
  # If anything's left, solve it separately.  But first: How many times was
  # that first root repeated?
  #
#?my $rest_roots = @solutions;          # This will be the starting index
                                        # for a newton-refinement for the
                                        # remaining roots.
  if ($quotient->{degree} > 0)
  {                                     # as repeated and conjugate roots
    my @q_solutions = $quotient->Yapp_solve();  # Solve the deflated polynomial

    # Now, to compensate for additional rounding errors that creep in when
    # solving the lower-degree quotient polynomials, let's up the accuracy;
    # apply Newtons's method to the roots returned by the recursive calls.
    # Laguerre has already been applied to this first root (plus duplicates
    # and conjugates).
    #
    @q_solutions = map {$self->Yapp_newton($_)} @q_solutions;
    push (@solutions, @q_solutions);            # THOSE solutions go into my set
  }
  return @solutions;
}
#
#-------------------------------------------------------------------------------
# Yapp_solve_even(): Method to solve polynomial equations of even degree >= 6
#                    with real-only coefficients
# Also good for solving polynomials with complex coefficients; the only
# difference is that for real-ony, the conjugate of of a complex root is
# guaranteed by theorem to be a root as well.
#
# Parameter:
# - [Ref to] a Yapp polynomial
# Returns:
# - Array of solution set
#
sub Yapp_solve_even
{
  my $self = shift(@_);
  my @solutions = ();       # Array that I return to the caller

  my $start = cplx(.1, .1); # Dumb starting point but we gotta start someplace!
  $solutions[0] = $self->Yapp_laguerre($start);
  my ($zero, $quotient) = $self->Yapp_eval($solutions[0]);  # Deflate a degree
  my ($zzero, $quotient2);          # To test for repeated roots
  my $xn = $solutions[0];           #(Shorter var name for neater code)

  # So $xn is a root.  3 questions:
  # 1. Is it a complex root?  If yes, we can just use its conjugate & # deflate
  # 2. Is it a multiple complex root?
  # 3. If real, is it a multiple real root?
  #
  if (ref($xn) eq $class_cplx)      # Yes to question[1]: Is complex
  {                                 # We have a work-free next solution
    push(@solutions, ~ $xn);        # The conjugate is also a root: Stash it
    ($zero, $quotient) = $quotient->Yapp_eval(~ $xn);   # and deflate a degree
                                                        # $zero == 0; toss it

    # Now start checking for repeats of this complex root. (Question[2])
    #
    while($quotient->{degree} > 0)  # Keep reducing it as we loop
    {
      # Question 2 rephrased: Does the same complex number solve the quotient?
      ($zzero, $quotient2) = $quotient->Yapp_eval($xn); # Evaluate to find out
      last if (abs($zzero) >= $margin); # Clearly not. Done checking; Outahere!

      # Still here: Same complex number *is* a solution again!
      #
      push (@solutions, $xn);       # First, push same root into solutions set
      $quotient = $quotient2;       # Point $quotient to above-deflated version
                                    # Of course, its conjugate is also a root
      push (@solutions, ~ $xn);     # Push its conjugate again into @solutions
      ($zero, $quotient)            # and deflate by the conjugate root
        = $quotient->Yapp_eval(~ $xn);
    }
  }
  else                              # Solution is real.  No free conjugate ride
  {
    while($quotient->{degree} > 0)  # Keep reducing it as we loop
    {
      ($zzero, $quotient2) = $quotient->Yapp_eval($xn); # Also solves quotient?
      last if (abs($zzero) >= $margin); # Clearly not. Done checking; Outahere!

      # Still here: Same real is a solution again!
      #
      $quotient = $quotient2;             # This is next deflated quotient
      push (@solutions, $xn);             # and add same root into solutions set
    }
  }
  # OK, I've exhausted my repeated and/or conjugate copies of first root.
  # If anything's left, solve it separately.
  #
  if ($quotient->{degree} > 0)          # If I have not solved for all roots
  {                                     # as repeated and conjugate roots
    my @q_solutions = $quotient->Yapp_solve();  # Solve the deflated polynomial

    # Now, to compensate for additional rounding errors that creep in when
    # solving the lower-degree quotient polynomials, let's up the accuracy;
    # apply Newtons's method to the roots returned by the recursive calls.
    # Laguerre has already been applied to this first root (plus duplicates
    # and conjugates).
    #
    @q_solutions = map {$self->Yapp_newton($_)} @q_solutions;
    push (@solutions, @q_solutions);            # THOSE solutions go into my set
  }

  return @solutions;
}
#
#-------------------------------------------------------------------------------
# Yapp_solve_complex(): Method to solve for the zeroes of a polynomial with
# complex coeffiecients.  The theorem about roots in conjugate pairs flies
# out the window.
# Same parameter/returns as all others.
#
sub Yapp_solve_complex
{
  my $self = shift(@_);
  my @solutions = ();       # Array that I return to the caller

  my $start = cplx(.1, .1); # Dumb starting point but we gotta start someplace!
  $solutions[0] = $self->Yapp_laguerre($start);
  my ($zero, $quotient) = $self->Yapp_eval($solutions[0]);  # Deflate a degree
  my ($zzero, $quotient2);          # To test for repeated roots
  my ($conj_zero, $conj_quotient);  # To test if conjugate is also a root
  my $xn = $solutions[0];           #(Shorter var name for neater code)

  # So $xn is a root.  3 questions:
  # 1. Is it a complex root?  If yes, we can just use its conjugate & # deflate
  # 2. Is it a multiple complex root?
  # 3. If real, is it a multiple real root?
  #
  if (ref($xn) eq $class_cplx)      # Yes to question[1]: Is complex
  {                                 # We don't have a work-free next solution
    ($conj_zero, $conj_quotient)    # Check: Is conjugate is also a root?
      = $quotient->Yapp_eval(~$xn);   # Create a copy deflated by conjugate
    if (abs($conj_zero) < $margin)    # If conjugate is also a root (Not
    {                                 # likely with complex coefficients)
      push(@solutions, ~$xn);         # Aha! Conjugate is also a solution
      $quotient = $conj_quotient;     # Now $quotient is the deflated Yapp
    }

    # Now start checking for repeats of this complex root. (Question[2])
    #
    while($quotient->{degree} > 0)  # Keep reducing it as we loop
    {
      ($zzero, $quotient2) = $quotient->Yapp_eval($xn); # Solves quotient?
      last if (abs($zzero) >= $margin); # Clearly not. Done checking; Outahere!

      # Still here: Same complex number *is* a solution again!
      #
      push (@solutions, $xn);       # First, push same root into solutions set
      $quotient = $quotient2;       # Point $quotient to above-deflated version
                                    # But is its conjugate also a root?
      ($conj_zero, $conj_quotient)      # Is conjugate is also a root?  Create
        = $quotient->Yapp_eval(~$xn);   # a copy deflated by conjugate
      if (abs($conj_zero) < $margin)    # If conjugate is also a root (Not
      {                                 # likely with complex coefficients)
        push(@solutions, ~$xn);         # Aha! Conjugate is also a solution
        $quotient = $conj_quotient;     # Now $quotient is the deflated Yapp
      }
    }
  }
  else                              # Solution is real; just check for duplicate
  {
    while($quotient->{degree} > 0)  # Keep reducing it as we loop
    {
      ($zzero, $quotient2) = $quotient->Yapp_eval($xn); # Also solves quotient?
      last if (abs($zzero) >= $margin); # Clearly not. Done checking; Outahere!

      # Still here: Same real is a solution again!
      #
      $quotient = $quotient2;             # This is next deflated quotient
      push (@solutions, $xn);             # and add same root into solutions set
    }
  }
  # OK, I've exhausted my repeated and/or conjugate copies of first root.
  # If anything's left, solve it separately.
  #
  if ($quotient->{degree} > 0)          # If I have not solved for all roots
  {                                     # as repeated and conjugate roots
    my @q_solutions = $quotient->Yapp_solve();  # Solve the deflated polynomial

    # Now, to compensate for additional rounding errors that creep in when
    # solving the lower-degree quotient polynomials, let's up the accuracy;
    # apply Newtons's method to the roots returned by the recursive calls.
    # Laguerre has already been applied to this first root (plus duplicates
    # and conjugates).
    #
    @q_solutions = map {$self->Yapp_newton($_)} @q_solutions;
    push (@solutions, @q_solutions);            # THOSE solutions go into my set
  }

  return @solutions;
}
#
#-------------------------------------------------------------------------------
# Interpolation: Effectively a new kind of constuctor, though not pacakged
# as such.  The parameters will always be references to arrays of numbers.
# The 0th array is a set of X values, the 1st array is a set of Y values.
# This is sufficient for laGrange interpolation - the basic colocation
# polynomial,  Additional arrays are for first and successive derivatives.
# The goal is to generate a polynomial of minimum degree whose Y values and
# derivatives at the indicated X-values match the arrays.  All will be
# called via the function Yapp_interpolate(). This will call Yapp_lagrange()
# or Yapp_hermite() as the case requires.
# Calling sequence:
# $yap_interp = Yapp_interpolate(\@x_vals, \@y_vals [,\@d1_vals, # \@d2_vals]);
# Returns:
# - A polynomial whose y values and derivatives match the input.
#
sub Yapp_interpolate
{
  my $self = Yapp("1");     # Create a 0-degree polynomial - just 1
  my $ryapp;                # Polynomial to be returned to caller

  # One validation: All arrays must have the name number of elements

  my $pcount = @_;          # Number of arrays passed to me
  die "Must supply at least two arrays (X and Y values) for interpolation"
    unless $pcount >= 2;

  my $x_vals = $_[0];       # Get -> X-values array
  my $xcount = @{$x_vals};  # How many X-values were given?

  # Validate that all arrays have the same number of elements
  #
  for (my $plc = 1; $plc < $pcount; $plc++)
  {
    my $y_list = $_[$plc];      # -> next array in parameter list
    my $ycount = @{$y_list};    # How many elements does this Y array have?
    die "Y-list[$plc] has $ycount elements; should have $xcount"
      unless ($ycount == $xcount);
  }

  # Still here: OK, we may proceed
  #
  my $y_vals = $_[1];       # Get first set of Y values
  my $derivs = @_;          # How many levels of derivative are wanted?

  $ryapp = ($derivs == 2) ? $self->Yapp_lagrange($x_vals, $y_vals)
                          : $self->Yapp_hermite(@_);
  return ($ryapp);
}
#
#-------------------------------------------------------------------------------
# Yapp_lagrange() - Generate a polynomial using laGrange interpolattion
# Not intended for public use
#
# Parameters:
# - (implicit) A token polynomial,
# - Reference to an array of X-Values
# - Reference to array of Y-values
# Both arrays must be the same size or a fatal error will be generated
# Returns:
# - A polynomial that satisfies all those points.
#
sub Yapp_lagrange
{
  my $self = shift(@_);         # Get my token polynomial
  my ($x_vals, $y_vals) = @_;   # Get my X & Y array references
  my $ryapp = Yapp(0);          # 0-degree poly, with constant: 0


  my $lag_list = Yapp_lag($x_vals, $y_vals);    # Generate laGrange multipliers

  # In case you have not read the long comment in Yapp_lag:
  # Each laGrange multiplier polynomial evaluates to 1 at its designated
  # point and 0 at all the other points.  I will multiply it by the Y-value
  # at that point before adding it to the final polynomial. That way, each
  # laGrange multiplier has a value of Y[i] at X[i]
  #
  my $lagp;
  for (my $xlc = 0; $xlc <= $#{$x_vals}; $xlc++)
  { $ryapp += $lag_list->[$xlc] * $y_vals->[$xlc]; } # Add multilpied multiplier
                                                    # to the result
  return $ryapp;                    # Show what I have built up
}
#
#-------------------------------------------------------------------------------
# Yapp_hermite() - Generate a polynomial using Hermite interpolattion
# Not intended for public use
#
# Parameters:
# - (implicit) A token polynomial,
# - Reference to an array of X-Values
# - Reference to array of Y-values
# - As many references to derivative values.
# Note that as of this release, only a first derivative is supported.
# Others will be ignored.
#
# Returns:
# - A polynomial that satisfies all those points.
#
sub Yapp_hermite
{
  my $self = shift(@_);                 # Get my token polynomial
  my ($x_vals, $y_vals, $yp_vals) = @_; # Get my X, Y, Y' array references

  my $xlc;                              # Loop counter for a few loops below
  my (@U_herm, @V_herm);        # -> arrays of Hermite components

  my $lag_list = Yapp_lag($x_vals, $y_vals);    # Generate laGrange multipliers
  my $lag_prime = ();                           # Array: Derivatives of above
  for ($xlc = 0; $xlc <= $#{$x_vals}; $xlc++)
  {
    $lag_prime->[$xlc] = ($lag_list->[$xlc])->Yapp_derivative(1);
  }

  # Each additive to the Hermite polynomial is the sum of U and V polynomials,
  # as denerated below
  #
  for ($xlc = 0; $xlc <= $#{$x_vals}; $xlc++)
  {
    # First component: V[i](X) = (X - x[i])(Lagrange[i]^2)
    #
    my $lag_squared = ($lag_list->[$xlc])**2;
    my $v_herm1 = Yapp(-$x_vals->[$xlc], 1);  # X - x_vals[i];
    $V_herm[$xlc] = $v_herm1 * $lag_squared;

    # Second component: U[i](X) = [1 - 2L'(x[i])(X-X[i])](Lagrange[i]^2)
    # (Ugly, but it gets you there, like the old Volkswagon slogan :-)
    #
    # Evaluate derivative of current Lagrange multiplier polynomial at x[i]
    #
    my $l_prime = $lag_prime->[$xlc]->Yapp_eval($x_vals->[$xlc]);
    my $u_herm1 = (1 - 2*$l_prime * $v_herm1);  # Still a degree 1 polynomial
    $U_herm[$xlc] = $u_herm1 * $lag_squared;
  }
  # I now have the components for the Hermite polynomial. Now I need to start
  # using the Y and Y' values given>  (Yes, I could have done this addition
  # within the above loop. Clarity over elegance here.)
  #
  my $ryapp = Yapp(0);          # Start with a constant 0 and add to it
  for ($xlc = 0; $xlc <= $#{$x_vals}; $xlc++)
  {
    $ryapp += $y_vals->[$xlc]  * $U_herm[$xlc]
           +  $yp_vals->[$xlc] * $V_herm[$xlc] ;
  }
  return $ryapp;
}
#
#-------------------------------------------------------------------------------
# Yapp_lag() - Function to generate the laGrange polynomials that add up to the
#              laGrange interpolating polynimal.  This code was originally in
#              Yapp_lagrange() but was separated out because it will be useful
#              for the Hermite inerpolation scheme as well.
# This function is not a method and will not be exported, since its use is
# strictly internal.
# Parameters:
# - Reference to an array of X values
# - Reference to array of corresponding Y values
# Returns:
# - Reference to an array of component laGrange polynomials
#
sub Yapp_lag
{
  my ($x_vals, $y_vals) = @_;
  my $xcount = @{$x_vals};      # Number of points in my lists
  my @x_minus;                  # Array of x-x[i] polynomials
  my @grange;                   # Array of largrange 1-point polynomials
                                # I will be returning a reference to this array

  for (my $mlc = 0; $mlc < $xcount; $mlc++)
  {
    $x_minus[$mlc] = Yapp(-($x_vals->[$mlc]), 1);   # X - x_val[mlc]
  }

  # Now build the laGrange set for this set of X points
  #
  for (my $xlc = 0; $xlc < $xcount; $xlc++)
  {
    $grange[$xlc] = Yapp(1);    # Starting point for each laGrange poly
    for (my $mlc = 0; $mlc < $xcount; $mlc++)
    {
      next if ($mlc == $xlc);   # Product of all but current point
      $grange[$xlc] *= $x_minus[$mlc];
    }
    # Coming out of above inner loop, $grange[$xlc] is the product of all
    # the (X - x_val[i]) except the i for the current point ($mlc)
    # The correct laGrange multiplier polynomial is 0 at all but the current
    # point but 1 at the current point. To force this one into that mold:
    # Divide the polynomial by its evaluation at x_val[xlc] (so it is 1 at
    # this point)
    #
    $grange[$xlc]
      /= ($grange[$xlc])->Yapp_eval($x_vals->[$xlc]);
  }
  # Coming out above outer loop, array @grange has all the lagrange polynomials
  # pertaining to this set of points.  That's what the calling function wants.
  #
  return \@grange;              # Return reference to that array
}
#
# Yapp_by_roots(): Constructor to build a polynomial whose roots are in the
# passed array or referenced array.
# Parameter[s]:  Either:
# - A complete array of the roots of the desired polynomial
# - A reference to such an array.
# It is the caller's responsibility to include conjugate pairs of complex roots
# if only real coefficients are desired.
#
sub Yapp_by_roots
{
  # Question: Did user pass me an array or array reference?  Either way I will
  # be stepping through the array using an array reference. That is:
  # - If user passed me a nice array reference, just use it.
  # - Passed me a naked array? Create reference the parameter array.
  #
  my $roots = (ref($_[0]) eq "ARRAY") ? shift(@_) :  \@_ ;

  my $ryapp = Yapp(1);      # Start with nice unit Yapp - Just "1"

  for (my $rlc = 0; $rlc <= $#{$roots}; $rlc++)
  {
    $ryapp *= Yapp(- ($roots->[$rlc]), 1)   # Multiply by (X - $roots[i])
  }
  return $ryapp;            # Bet that looked easy! :-)
}
#
# New section: Orthogonal Polynomials.
# Although the plan is to have sub-classes for all classical kinds of orthognal
# polynomials, I am including all the operations with Legendre orthogonality.
# That is where the inner product is defined as the intergal of (P[1] * P[2])
# taken over the interval [-1,1].  More accurately, since we are allowing for
# complex coefficients, the intergral of (P[1] * conj(P[2]).  Hence, the
# complex inner product is not a commutative operation. (Finkbeiner)
#
# Based on this inner product as a starting point, we will define the following
# methods:
# - Yapp_norm:     The square root of the inner product of the polynomial with
#                  itself
# - Yapp_normalize: Divide a polynomial by its norm to produce a polynomial
#                  whose norm == 1
# - Yapp_distance: The norm of the difference of two polynomials
# - Yapp_orthoganal: True-false: Are these two Yapp-polynomials orthoganal, that
#                  is, is their inner product 0?
# - Yapp_gramSchmidt: Apply the Gram-Schmidt algorithm to an arbitrary set of
#                 polynomials to produce an array of mutually orthonormal
#                 polynomials.
# - Yapp_genOrthogs: Generate orthogonals based on (1, X, X^2, .. X^n)
#------------------------------------------------------------------------------

# Yapp_innerProd() - Inner product of two Yapps, using the Legendre polynomials.
# Intended for use as an operator: The dot . operator
# Parameters:
# - (Implicit) - The first Yapp as a vector
# - The other Yapp
# Returns:
# - The inner product, which may be complex if any of the coefficients on
#   either side of the operator are complex
#
sub Yapp_innerProd
{
  my ($self, $yapp2) = @_;      # Get my parameters
  printf("\nCalculate inner product of polynomials:\n<%s>\n<%s>\n",
         $self->Ysprint(), $yapp2->Ysprint())
    if ($testmode);

  my $conj2 = ~ $yapp2;              # Conjugate the second parameter
  printf("\nStart with product of:\n<%s>\n\tand\n<%s>\n",
         $self->Ysprint(), $conj2->Ysprint())
    if ($testmode);

  my $yprod = $self * $conj2;   # Get product of yapp1, ~yapp2
  printf("That is:\n<%s>\n", $yprod->Ysprint())
    if ($testmode);

  my $rval = $yprod->Yapp_integral(-1, 1);  # Intergal of the product over
                                            # the classic interval [-1, 1]
  printf("Returning value <%s> as inner product\n", Csprint($rval))
    if ($testmode);

  return $rval;                 # That's the inner product!
}
#
# Yapp_Orthogonal() - True or false: Is one Yapp-vector orthogonal to the
# other? i.e. Is the inner product of the two vectors 0 (or close enough)?
# Parameters:
# - (Implicit) A first Yapp to compare
# - A second Yapp.
# Returns:
# - 0 if not orthogonal
# - 1 if they are
#
sub Yapp_Orthogonal
{
  my ($Y1, $Y2) = @_;           # Get the references
  my $rval = 0;                 # Off to a negative start

  my $innerprod = $Y1 . $Y2;    # Get the inner product of two vectors
  $rval = 1 if (abs($innerprod) <= $margin);    # Changing mind about that NO?
  return $rval;
}
#-----------------------------------------------------------------------------
# Yapp_norm() - The next step after defining an inner product function for a
# vector space is to then make it a normed vector space by defining a "norm"
# function.  The most cooperative norm function seems to be the square root
# of the inner product of a vector with itself.
# Parameter:
# - (Implicit): The Yapp reference
# Returns:
# - The norm value of that vector.
#
sub Yapp_norm
{
  my $self = shift(@_);
  my $norm_squared = $self . $self; # Inner product of vector with itself
  my $norm = sqrt($norm_squared);   # Relying on math to keep it real
  return $norm;                     # That's my return value
}
#-----------------------------------------------------------------------------
# Yapp_normalize(): Divide a Yapp vector by its norm so that the norm of the
# new vector is 1.
# Parameter:
# - (Implicit) The Yapp polynomial
# Returns:
# - A new Yapp, proportional to the original but whose norm == 1
#
sub Yapp_normalize
{
  my $self = shift;
  my $norm = $self->Yapp_norm();    # Get the norm of the Yapp as vector
  my $normed = $self / $norm;       # Create that new Yapp according to specs
  return $normed;                   # Send that back to the caller
}
#
# Yapp_distance() - How "far" is the passed Yapp from the object Yapp?
# Answer: That is the norm (Yapp_norm) of the difference between the two Yapps
#
# Parameters:
# - (Implicit) A Yapp [reference], arbitrarily chosen as the fixed vector
# - The [reference to the] other Yapp
# Returns:
# - The real value that is the norm of the difference
#
sub Yapp_distance
{
  my ($self, $vector) = @_;         # Get the parameter references
  my $diff = $self - $vector;       # Get the difference
  my $rval = $diff->Yapp_norm();    # Get "size" of that difference
  return $rval;
}
#
1;

__END__