public class Complex extends java.lang.Object implements FieldElement<Complex>, java.io.Serializable
Implementations of arithmetic operations handle NaN
and
infinite values according to the rules for Double
arithmetic, applying definitional formulas and returning NaN
or
infinite values in real or imaginary parts as these arise in computation.
See individual method javadocs for details.
equals(java.lang.Object)
identifies all values with NaN
in either real
or imaginary part - e.g.,
1 + NaNi == NaN + i == NaN + NaNi.
implements Serializable since 2.0Modifier and Type | Field and Description |
---|---|
static Complex |
I
The square root of -1.
|
static Complex |
INF
A complex number representing "+INF + INFi"
|
static Complex |
NaN
A complex number representing "NaN + NaNi"
|
static Complex |
ONE
A complex number representing "1.0 + 0.0i"
|
static Complex |
ZERO
A complex number representing "0.0 + 0.0i"
|
Constructor and Description |
---|
Complex(double real,
double imaginary)
Create a complex number given the real and imaginary parts.
|
Modifier and Type | Method and Description |
---|---|
double |
abs()
Return the absolute value of this complex number.
|
Complex |
acos()
Compute the
inverse cosine of this complex number.
|
Complex |
add(Complex rhs)
Return the sum of this complex number and the given complex number.
|
Complex |
asin()
Compute the
inverse sine of this complex number.
|
Complex |
atan()
Compute the
inverse tangent of this complex number.
|
Complex |
conjugate()
Return the conjugate of this complex number.
|
Complex |
cos()
Compute the
cosine
of this complex number.
|
Complex |
cosh()
Compute the
hyperbolic cosine of this complex number.
|
Complex |
divide(Complex rhs)
Return the quotient of this complex number and the given complex number.
|
boolean |
equals(java.lang.Object other)
Test for the equality of two Complex objects.
|
Complex |
exp()
Compute the
exponential function of this complex number.
|
double |
getArgument()
Compute the argument of this complex number.
|
ComplexField |
getField()
Get the
Field to which the instance belongs. |
double |
getImaginary()
Access the imaginary part.
|
double |
getReal()
Access the real part.
|
int |
hashCode()
Get a hashCode for the complex number.
|
boolean |
isInfinite()
Returns true if either the real or imaginary part of this complex number
takes an infinite value (either
Double.POSITIVE_INFINITY or
Double.NEGATIVE_INFINITY ) and neither part
is NaN . |
boolean |
isNaN()
Returns true if either or both parts of this complex number is NaN;
false otherwise
|
Complex |
log()
Compute the
natural logarithm of this complex number.
|
Complex |
multiply(Complex rhs)
Return the product of this complex number and the given complex number.
|
Complex |
multiply(double rhs)
Return the product of this complex number and the given scalar number.
|
Complex |
negate()
Return the additive inverse of this complex number.
|
java.util.List<Complex> |
nthRoot(int n)
Computes the n-th roots of this complex number.
|
Complex |
pow(Complex x)
Returns of value of this complex number raised to the power of
x . |
Complex |
sin()
Compute the
sine
of this complex number.
|
Complex |
sinh()
Compute the
hyperbolic sine of this complex number.
|
Complex |
sqrt()
Compute the
square root of this complex number.
|
Complex |
sqrt1z()
Compute the
square root of 1 -
this 2 for this complex
number. |
Complex |
subtract(Complex rhs)
Return the difference between this complex number and the given complex
number.
|
Complex |
tan()
Compute the
tangent of this complex number.
|
Complex |
tanh()
Compute the
hyperbolic tangent of this complex number.
|
public static final Complex I
public static final Complex NaN
public static final Complex INF
public static final Complex ONE
public static final Complex ZERO
public Complex(double real, double imaginary)
real
- the real partimaginary
- the imaginary partpublic double abs()
Returns NaN
if either real or imaginary part is
NaN
and Double.POSITIVE_INFINITY
if
neither part is NaN
, but at least one part takes an infinite
value.
public Complex add(Complex rhs)
Uses the definitional formula
(a + bi) + (c + di) = (a+c) + (b+d)i
If either this or rhs
has a NaN value in either part,
NaN
is returned; otherwise Inifinite and NaN values are
returned in the parts of the result according to the rules for
Double
arithmetic.
add
in interface FieldElement<Complex>
rhs
- the other complex numberjava.lang.NullPointerException
- if rhs
is nullpublic Complex conjugate()
NaN
is returned if either the real or imaginary
part of this Complex number equals Double.NaN
.
If the imaginary part is infinite, and the real part is not NaN,
the returned value has infinite imaginary part of the opposite
sign - e.g. the conjugate of 1 + POSITIVE_INFINITY i
is 1 - NEGATIVE_INFINITY i
public Complex divide(Complex rhs)
Implements the definitional formula
a + bi ac + bd + (bc - ad)i
----------- = -------------------------
c + di c2 + d2
but uses
prescaling of operands to limit the effects of overflows and
underflows in the computation.
Infinite and NaN values are handled / returned according to the following rules, applied in the order presented:
rhs
has a NaN value in either part,
NaN
is returned.rhs
equals ZERO
, NaN
is returned.
rhs
are both infinite,
NaN
is returned.rhs
is infinite (one or both parts infinite),
ZERO
is returned.rhs
is finite, NaN values are
returned in the parts of the result if the Double
rules applied to the definitional formula force NaN results.divide
in interface FieldElement<Complex>
rhs
- the other complex numberjava.lang.NullPointerException
- if rhs
is nullpublic boolean equals(java.lang.Object other)
If both the real and imaginary parts of two Complex numbers
are exactly the same, and neither is Double.NaN
, the two
Complex objects are considered to be equal.
All NaN
values are considered to be equal - i.e, if either
(or both) real and imaginary parts of the complex number are equal
to Double.NaN
, the complex number is equal to
Complex.NaN
.
equals
in class java.lang.Object
other
- Object to test for equality to thispublic int hashCode()
All NaN values have the same hash code.
hashCode
in class java.lang.Object
public double getImaginary()
public double getReal()
public boolean isNaN()
public boolean isInfinite()
Double.POSITIVE_INFINITY
or
Double.NEGATIVE_INFINITY
) and neither part
is NaN
.NaN
public Complex multiply(Complex rhs)
Implements preliminary checks for NaN and infinity followed by the definitional formula:
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Returns NaN
if either this or rhs
has one or more
NaN parts.
INF
if neither this nor rhs
has one or more
NaN parts and if either this or rhs
has one or more
infinite parts (same result is returned regardless of the sign of the
components).
Returns finite values in components of the result per the definitional formula in all remaining cases.
multiply
in interface FieldElement<Complex>
rhs
- the other complex numberjava.lang.NullPointerException
- if rhs
is nullpublic Complex multiply(double rhs)
Implements preliminary checks for NaN and infinity followed by the definitional formula:
c(a + bi) = (ca) + (cb)i
Returns NaN
if either this or rhs
has one or more
NaN parts.
INF
if neither this nor rhs
has one or more
NaN parts and if either this or rhs
has one or more
infinite parts (same result is returned regardless of the sign of the
components).
Returns finite values in components of the result per the definitional formula in all remaining cases.
rhs
- the scalar numberpublic Complex negate()
Returns Complex.NaN
if either real or imaginary
part of this Complex number equals Double.NaN
.
public Complex subtract(Complex rhs)
Uses the definitional formula
(a + bi) - (c + di) = (a-c) + (b-d)i
If either this or rhs
has a NaN value in either part,
NaN
is returned; otherwise inifinite and NaN values are
returned in the parts of the result according to the rules for
Double
arithmetic.
subtract
in interface FieldElement<Complex>
rhs
- the other complex numberjava.lang.NullPointerException
- if rhs
is nullpublic Complex acos()
Implements the formula:
acos(z) = -i (log(z + i (sqrt(1 - z2))))
Returns NaN
if either real or imaginary part of the
input argument is NaN
or infinite.
public Complex asin()
Implements the formula:
asin(z) = -i (log(sqrt(1 - z2) + iz))
Returns NaN
if either real or imaginary part of the
input argument is NaN
or infinite.
public Complex atan()
Implements the formula:
atan(z) = (i/2) log((i + z)/(i - z))
Returns NaN
if either real or imaginary part of the
input argument is NaN
or infinite.
public Complex cos()
Implements the formula:
cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b)i
where the (real) functions on the right-hand side are
Math.sin(double)
, Math.cos(double)
,
MathUtils.cosh(double)
and MathUtils.sinh(double)
.
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
Examples:
cos(1 ± INFINITY i) = 1 ∓ INFINITY i
cos(±INFINITY + i) = NaN + NaN i
cos(±INFINITY ± INFINITY i) = NaN + NaN i
public Complex cosh()
Implements the formula:
cosh(a + bi) = cosh(a)cos(b) + sinh(a)sin(b)i
where the (real) functions on the right-hand side are
Math.sin(double)
, Math.cos(double)
,
MathUtils.cosh(double)
and MathUtils.sinh(double)
.
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
Examples:
cosh(1 ± INFINITY i) = NaN + NaN i
cosh(±INFINITY + i) = INFINITY ± INFINITY i
cosh(±INFINITY ± INFINITY i) = NaN + NaN i
public Complex exp()
Implements the formula:
exp(a + bi) = exp(a)cos(b) + exp(a)sin(b)i
where the (real) functions on the right-hand side are
Math.exp(double)
, Math.cos(double)
, and
Math.sin(double)
.
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
Examples:
exp(1 ± INFINITY i) = NaN + NaN i
exp(INFINITY + i) = INFINITY + INFINITY i
exp(-INFINITY + i) = 0 + 0i
exp(±INFINITY ± INFINITY i) = NaN + NaN i
this
public Complex log()
Implements the formula:
log(a + bi) = ln(|a + bi|) + arg(a + bi)i
where ln on the right hand side is Math.log(double)
,
|a + bi|
is the modulus, abs()
, and
arg(a + bi) = Math.atan2(double, double)
(b, a)
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite (or critical) values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
Examples:
log(1 ± INFINITY i) = INFINITY ± (π/2)i
log(INFINITY + i) = INFINITY + 0i
log(-INFINITY + i) = INFINITY + πi
log(INFINITY ± INFINITY i) = INFINITY ± (π/4)i
log(-INFINITY ± INFINITY i) = INFINITY ± (3π/4)i
log(0 + 0i) = -INFINITY + 0i
public Complex pow(Complex x)
x
.
Implements the formula:
yx = exp(x·log(y))
where exp
and log
are exp()
and
log()
, respectively.
Returns NaN
if either real or imaginary part of the
input argument is NaN
or infinite, or if y
equals ZERO
.
x
- the exponent.this
x
java.lang.NullPointerException
- if x is nullpublic Complex sin()
Implements the formula:
sin(a + bi) = sin(a)cosh(b) - cos(a)sinh(b)i
where the (real) functions on the right-hand side are
Math.sin(double)
, Math.cos(double)
,
MathUtils.cosh(double)
and MathUtils.sinh(double)
.
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
Examples:
sin(1 ± INFINITY i) = 1 ± INFINITY i
sin(±INFINITY + i) = NaN + NaN i
sin(±INFINITY ± INFINITY i) = NaN + NaN i
public Complex sinh()
Implements the formula:
sinh(a + bi) = sinh(a)cos(b)) + cosh(a)sin(b)i
where the (real) functions on the right-hand side are
Math.sin(double)
, Math.cos(double)
,
MathUtils.cosh(double)
and MathUtils.sinh(double)
.
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
Examples:
sinh(1 ± INFINITY i) = NaN + NaN i
sinh(±INFINITY + i) = ± INFINITY + INFINITY i
sinh(±INFINITY ± INFINITY i) = NaN + NaN i
public Complex sqrt()
Implements the following algorithm to compute sqrt(a + bi)
:
t = sqrt((|a| + |a + bi|) / 2)
ifa ≥ 0
returnt + (b/2t)i
else return|b|/2t + sign(b)t i
|a| = Math.abs(int)
(a)
|a + bi| = abs()
(a + bi)
sign(b) = MathUtils.indicator(byte)
(b)
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
Examples:
sqrt(1 ± INFINITY i) = INFINITY + NaN i
sqrt(INFINITY + i) = INFINITY + 0i
sqrt(-INFINITY + i) = 0 + INFINITY i
sqrt(INFINITY ± INFINITY i) = INFINITY + NaN i
sqrt(-INFINITY ± INFINITY i) = NaN ± INFINITY i
public Complex sqrt1z()
this
2 for this complex
number.
Computes the result directly as
sqrt(Complex.ONE.subtract(z.multiply(z)))
.
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
this
2public Complex tan()
Implements the formula:
tan(a + bi) = sin(2a)/(cos(2a)+cosh(2b)) + [sinh(2b)/(cos(2a)+cosh(2b))]i
where the (real) functions on the right-hand side are
Math.sin(double)
, Math.cos(double)
,
MathUtils.cosh(double)
and MathUtils.sinh(double)
.
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite (or critical) values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
Examples:
tan(1 ± INFINITY i) = 0 + NaN i
tan(±INFINITY + i) = NaN + NaN i
tan(±INFINITY ± INFINITY i) = NaN + NaN i
tan(±π/2 + 0 i) = ±INFINITY + NaN i
public Complex tanh()
Implements the formula:
tan(a + bi) = sinh(2a)/(cosh(2a)+cos(2b)) + [sin(2b)/(cosh(2a)+cos(2b))]i
where the (real) functions on the right-hand side are
Math.sin(double)
, Math.cos(double)
,
MathUtils.cosh(double)
and MathUtils.sinh(double)
.
Returns NaN
if either real or imaginary part of the
input argument is NaN
.
Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
Examples:
tanh(1 ± INFINITY i) = NaN + NaN i
tanh(±INFINITY + i) = NaN + 0 i
tanh(±INFINITY ± INFINITY i) = NaN + NaN i
tanh(0 + (π/2)i) = NaN + INFINITY i
public double getArgument()
Compute the argument of this complex number.
The argument is the angle phi between the positive real axis and the point representing this number in the complex plane. The value returned is between -PI (not inclusive) and PI (inclusive), with negative values returned for numbers with negative imaginary parts.
If either real or imaginary part (or both) is NaN, NaN is returned. Infinite parts are handled as java.Math.atan2 handles them, essentially treating finite parts as zero in the presence of an infinite coordinate and returning a multiple of pi/4 depending on the signs of the infinite parts. See the javadoc for java.Math.atan2 for full details.
public java.util.List<Complex> nthRoot(int n) throws java.lang.IllegalArgumentException
Computes the n-th roots of this complex number.
The nth roots are defined by the formula:
zk = abs 1/n (cos(phi + 2πk/n) + i (sin(phi + 2πk/n))
for k=0, 1, ..., n-1
, where abs
and phi
are
respectively the modulus
and argument
of this complex number.
If one or both parts of this complex number is NaN, a list with just one element,
NaN
is returned.
if neither part is NaN, but at least one part is infinite, the result is a one-element
list containing INF
.
n
- degree of rootjava.lang.IllegalArgumentException
- if parameter n is less than or equal to 0public ComplexField getField()
Field
to which the instance belongs.getField
in interface FieldElement<Complex>
Field
to which the instance belongsCopyright © 2010 - 2020 Adobe. All Rights Reserved