Class Complex

java.lang.Object
   |
   +----Complex

public class Complex

extends Object

Author:

Jonathan A. Poritz

DOUBLEDIGITS

The default number of digits to display in formatted coversion of Doubles to Strings.

FOUR

The complex constant 4.

I

The complex constant i.

IDENTRESOL

Powers of an elliptic generator whose (1,1) entry are at this distance squared or less are considered to be the identity, so enabling us to compute the order of that elliptic element.

MAXELLIPORD

An cutoff for the calculation of orders of elliptic elements.

MAXPOWMAX

The cutoff for calculated maxpows -- above this value we arbitrarily truncate down to this.

MINUS_I

The complex constant -i.

MINUS_ONE

The complex constant -1.

ONE

The complex constant 1.

period

A string containing one period for formatting Double.toString output to a certain number of decimal digits.

TWO

The complex constant 2.

x

The real part of this complex number.

y

The imaginary part of this complex number.

ZERO

The complex constant 0.

Complex()

Constructor for the complex number zero.

Complex(Complex)

Constructor for a complex number identical to one given as input.

Complex(double, double)

Constructor for an arbitrary complex number.

add(Complex, Complex)

Compute the sum of two complex numbers.

argument()

Compute this's argument.

argument(Complex)

Compute the argument of a complex number, in the range (-pi,pi].

conj()

Computes this's complex conjugate.

conj(Complex)

Computes the complex conjugate of the input complex number.

distsq(Complex, Complex)

Computes the square of the distance between two complex numbers.

divide(Complex, Complex)

Divides one input complex number by another.

DoubleToString(double)

The default invocation of DoubleToString has DOUBLEDIGITS of precision.

DoubleToString(double, int)

Converts a double to a string with a certain number of digits of precision.

make_maxpow(Complex, Complex)

Compute the maxpow corresponding to the current trace and previously calculated (1,1) entry.

make_z11(Complex)

Compute the (1,1) entry of the canonical matrix with given trace.

modsq()

Compute this's modulus squared.

modsq(Complex)

Compute the modulus squared of a complex number.

modulus()

Compute this's modulus.

modulus(Complex)

Computes the modulus of an input complex number.

multiply(Complex, Complex)

Multiplies one input complex number by another.

sqrt(Complex)

Computes the square root of an input complex number

square(Complex)

Squares an input complex number.

stringToComplex(String)

Convert an input string to a complex number.

subtract(Complex, Complex)

Compute the difference of two complex numbers.

toString()

Convert this complex number to a string, with the real and imaginary parts separated by a comma.

 public double x

The real part of this complex number.

 public double y

The imaginary part of this complex number.

 public static final Complex ZERO

The complex constant 0.

 public static final Complex ONE

The complex constant 1.

 public static final Complex TWO

The complex constant 2.

 public static final Complex FOUR

The complex constant 4.

 public static final Complex MINUS_ONE

The complex constant -1.

 public static final Complex I

The complex constant i.

 public static final Complex MINUS_I

The complex constant -i.

 public static final int MAXPOWMAX

The cutoff for calculated maxpows -- above this value we arbitrarily truncate down to this. A good default is around 50; much more and computations start to take way too long, much less and a reasonable window on the trace plane includes lots of pixel for which we use the artificial cutoff.

 public static final int MAXELLIPORD

An cutoff for the calculation of orders of elliptic elements. We only try possible orders up to this value, then give up.

 public static final double IDENTRESOL

Powers of an elliptic generator whose (1,1) entry are at this distance squared or less are considered to be the identity, so enabling us to compute the order of that elliptic element.

 public static final String period

A string containing one period for formatting Double.toString output to a certain number of decimal digits.

 public static final int DOUBLEDIGITS

The default number of digits to display in formatted coversion of Doubles to Strings.

 public Complex()

Constructor for the complex number zero.

 public Complex(double xx,
                double yy)

Constructor for an arbitrary complex number.

Parameters:

xx - the real part

yy - the imaginary part

 public Complex(Complex z)

Constructor for a complex number identical to one given as input.

Parameters:

z - the complex number to duplicate

 public static Complex add(Complex a,
                           Complex b)

Compute the sum of two complex numbers.

Parameters:

a - one input complex number

b - and another

Returns:

the sum

 public static Complex subtract(Complex a,
                                Complex b)

Compute the difference of two complex numbers.

Parameters:

a - one input complex number

b - and another

Returns:

the difference

 public static Complex multiply(Complex a,
                                Complex b)

Multiplies one input complex number by another.

Parameters:

a - one input complex number

b - and another

Returns:

the product

 public static Complex divide(Complex a,
                              Complex b)

Divides one input complex number by another.

Parameters:

a - the numerator

b - the denominator

Returns:

the ratio

 public static Complex square(Complex z)

Squares an input complex number.

Parameters:

z - the complex number to square

Returns:

its square

 public static Complex sqrt(Complex z)

Computes the square root of an input complex number

Parameters:

z - the number whose square root to compute

Returns:

s the square root

 public static double argument(Complex z)

Compute the argument of a complex number, in the range (-pi,pi].

Parameters:

z - the complex number to examine

Returns:

its argument

 public double argument()

Compute this's argument.

Returns:

this's argument

 public static Complex conj(Complex z)

Computes the complex conjugate of the input complex number.

Parameters:

z - the complex number to conjugate

Returns:

the conjugate

 public Complex conj()

Computes this's complex conjugate.

Returns:

this's conjugate

 public static double modulus(Complex z)

Computes the modulus of an input complex number.

Parameters:

z - the number whose modulus to compute

Returns:

the modulus

 public double modulus()

Compute this's modulus.

Returns:

this's modulus

 public static double modsq(Complex z)

Compute the modulus squared of a complex number.

Parameters:

z - the number whose modulus squared to compute

Returns:

the modulus square

 public double modsq()

Compute this's modulus squared.

Returns:

this's modulus squared

 public static Complex stringToComplex(String s)

Convert an input string to a complex number. Really should throw a NumberFormatException if the string has an inappropriate form (which would then have to be caught wherever this method is used!); instead, returns the complex number i in such cases. Valid format of a string is two numbers separated by a comma, indicating the real and imaginary parts, respectively, of the complex number.

Parameters:

the - string to convert

Returns:

the complex number corresponding to that string

 public static double distsq(Complex a,
                             Complex b)

Computes the square of the distance between two complex numbers.

Parameters:

a - one input complex number

b - and another

Returns:

the square of the distance

 public static String DoubleToString(double d)

The default invocation of DoubleToString has DOUBLEDIGITS of precision.

Parameters:

d - the double to be stringified

Returns:

the string version

 public static String DoubleToString(double d,
                                     int res)

Converts a double to a string with a certain number of digits of precision.

Parameters:

d - the double to convert

res - the number of digits of precision to use

 public String toString()

Convert this complex number to a string, with the real and imaginary parts separated by a comma.

Returns:

the string version of this complex number

Overrides:

toString in class Object

 public static Complex make_z11(Complex trace)

Compute the (1,1) entry of the canonical matrix with given trace. Answer will be the x such that x+x^{-1}=trace, in fact the one chosen to have modulus greater than or equal to one.

Parameters:

trace - the trace of the matrix we are interested in

Returns:

the (1,1) entry of the corresponding canonical matrix

 public static int make_maxpow(Complex trace,
                               Complex z11)

Compute the maxpow corresponding to the current trace and previously calculated (1,1) entry. For purely hyerpbolic and parabolic elements, always returns 1; for elliptic, returns half the order of the element, up to value MAXELLIPORD; for loxodromic elements, the result is valid for Ford domains, but is not known to be correct for Dirichlet domains -- although we use it in that case as well.

Parameters:

trace - the current trace

z11 - the (1,1) entry of the generator with that trace

Returns:

the corresponding Ford domain maxpow