com.numericalmethod.suanshu.analysis.function.polynomial
Class Polynomial

java.lang.Object
  com.numericalmethod.suanshu.analysis.function.rn2r1.UnivariateRealFunction
      com.numericalmethod.suanshu.analysis.function.polynomial.Polynomial

All Implemented Interfaces:

Function, RealScalarFunction, AbelianGroup<Polynomial>, Monoid<Polynomial>, Ring<Polynomial>, VectorSpace<Polynomial,Real>

Direct Known Subclasses:

DPolynomial

public class Polynomial
extends UnivariateRealFunction
implements Ring<Polynomial>, VectorSpace<Polynomial,Real>

Polynomial is a UnivariateRealFunction that represents a finite length expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. Specifically, it has the form

a_nxⁿ + a_n-1x^n-1 + ... + a₁x¹ + a₀

• The set of polynomials forms a Ring.
• A polynomial can be evaluated for both real and complex numbers.

This implementation

• is immutable;
• represents a univariate polynomial;

See Also:

Wikipedia: Polynomial

Nested Class Summary

Nested classes/interfaces inherited from interface com.numericalmethod.suanshu.analysis.function.Function

Function.EvaluationException

Field Summary

int	degree the degree of this polynomial It is equal to the largest exponent of the variable.
static Polynomial	ONE a polynomial representing 1
static Polynomial	ZERO a polynomial representing 0

Constructor Summary

Polynomial(double... coefficients)
Construct a polynomial from an array of coefficients.

Method Summary

Polynomial	add(Polynomial that) + : G × G → G
double	coefficient(int i) Get the coefficient of xn-i, namely, an-i.
double[]	coefficients() Get a copy of the coefficients.
boolean	equals(java.lang.Object obj)
Complex	evaluate(Complex z) Evaluate this polynomial for a Complex number input.
double	evaluate(double x) Evaluate this polynomial for a real number, i.e., double input.
Complex	evaluate(java.lang.Number x) Evaluate this polynomial for a Number input.
int	hashCode()
Polynomial	minus(Polynomial that) - : G × G → G - is not in the definition of of an additive group but can be deduced.
Polynomial	multiply(Polynomial that) · : G × G → G
Polynomial	normalize() Normalize this polynomial so that the leading coefficient is 1.
Polynomial	ONE() The multiplicative element 1 in the group such that for any elements a in the group, the equation 1 × a = a × 1 = a holds.
Polynomial	opposite() For each a in G, there exists an element b in G such that a + b = b + a = 0 That is, it is the object such as this.add(this.opposite()) == this.ZERO
Polynomial	scaled(double scalar)
Polynomial	scaled(Real scalar) Deprecated. Not supported yet.
java.lang.String	toString()
Polynomial	ZERO() The additive element 0 in the group, such that for all elements a in the group, the equation 0 + a = a + 0 = a holds.

Methods inherited from class com.numericalmethod.suanshu.analysis.function.rn2r1.UnivariateRealFunction

dimension4Domain, dimension4Range, evaluate

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Field Detail

ZERO

public static final Polynomial ZERO

a polynomial representing 0

ONE

public static final Polynomial ONE

a polynomial representing 1

degree

public final int degree

the degree of this polynomial

It is equal to the largest exponent of the variable.

For example,

x⁴ + 1

has a degree of 4.

Constructor Detail

Polynomial

public Polynomial(double... coefficients)

Construct a polynomial from an array of coefficients. The 0-th entry corresponds to the xⁿ term. The n-th entry corresponds to the constant term.

The degree of the polynomial is n.

For example,

new Polynomial(1, -2, 3, 2)

will create an instance of Polynomial representing

x³ - 2x² + 3x + 2

Parameters:

coefficients - the polynomial coefficients in (double)

Method Detail

coefficients

public double[] coefficients()

Get a copy of the coefficients.

Given a polynomial of degree n, the i-th element correspond to the coefficient of x^n-i. The 0-th entry corresponds to the xⁿ term, which cannot be zero; The n-th entry corresponds to the constant term.

Returns:

a copy of the coefficients

coefficient

public double coefficient(int i)

Get the coefficient of x^n-i, namely, a_n-i.

Parameters:

i - indicates the i-th coefficient in this polynomial

Returns:

a_n-i

See Also:

coefficients()

normalize

public Polynomial normalize()

Normalize this polynomial so that the leading coefficient is 1. The result is independent, and can be modified anyhow.

Returns:

a scaled version of the polynomial which has a leading coefficient 1

evaluate

public Complex evaluate(java.lang.Number x)

Evaluate this polynomial for a Number input.

Parameters:

x - the argument

Returns:

p(x), the result in Complex

evaluate

public double evaluate(double x)

Evaluate this polynomial for a real number, i.e., double input.

Specified by:

evaluate in class UnivariateRealFunction

Parameters:

x - the argument

Returns:

p(x), the result in Double

evaluate

public Complex evaluate(Complex z)

Evaluate this polynomial for a Complex number input.

Parameters:

z - the argument

Returns:

p(z), the result in Complex

add

public Polynomial add(Polynomial that)

Description copied from interface: AbelianGroup

+ : G × G → G

Specified by:

add in interface AbelianGroup<Polynomial>

Parameters:

that - the object to be added

Returns:

this + that

minus

public Polynomial minus(Polynomial that)

Description copied from interface: AbelianGroup

- : G × G → G

- is not in the definition of of an additive group but can be deduced. This function is provided for convenience purpose. It is equivalent to

this.add(that.opposite())

Specified by:

minus in interface AbelianGroup<Polynomial>

Parameters:

that - the object to be subtracted (subtrahend)

Returns:

this - that

multiply

public Polynomial multiply(Polynomial that)

Description copied from interface: Monoid

· : G × G → G

Specified by:

multiply in interface Monoid<Polynomial>

Parameters:

that - the multiplicand

Returns:

this × that

scaled

@Deprecated
public Polynomial scaled(Real scalar)

Deprecated. Not supported yet.

Description copied from interface: VectorSpace

* : F × V → V

The result of applying this function to scalar, c, in F and v in V is denoted cv.

Specified by:

scaled in interface VectorSpace<Polynomial,Real>

Parameters:

scalar - a multiplier

Returns:

scalar * this

See Also:

Wikipedia: Scalar multiplication

scaled

public Polynomial scaled(double scalar)

opposite

public Polynomial opposite()

Description copied from interface: AbelianGroup

For each a in G, there exists an element b in G such that

a + b = b + a = 0

That is, it is the object such as

this.add(this.opposite()) == this.ZERO

Specified by:

opposite in interface AbelianGroup<Polynomial>

Returns:

-this

See Also:

Wikipedia: Additive inverse

ZERO

public Polynomial ZERO()

Description copied from interface: AbelianGroup

The additive element 0 in the group, such that for all elements a in the group, the equation

0 + a = a + 0 = a

holds.

Specified by:

ZERO in interface AbelianGroup<Polynomial>

Returns:

0

ONE

public Polynomial ONE()

Description copied from interface: Monoid

The multiplicative element 1 in the group such that for any elements a in the group, the equation

1 × a = a × 1 = a

holds.

Specified by:

ONE in interface Monoid<Polynomial>

Returns:

1

equals

public boolean equals(java.lang.Object obj)

Overrides:

equals in class java.lang.Object

hashCode

public int hashCode()

Overrides:

hashCode in class java.lang.Object

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object