Polynomial

Extension

Creates a new instance of Polynomial. Returned is a polynomial of a degree equal to the number of coefficients minus one.a = Polynomial.newFrom([1, 2, 1]) //2nd degree polynomial b = Polynomial.newFrom([0, 2, 1]) //Another 2nd degree polynomial with a_0 coefficient equal to 0 c = Polynomial.newFrom([1, 2, 0]) //1st degree polynomial, since a_2 coefficient is 0

Arguments:

Expands an array of first degree binomial factors.( a = Polynomial.newFrom([2, 2]); b = Polynomial.newFrom([3, 3]); c = Polynomial.newFrom([4, 4]); Polynomial.expandBinomialFactors([a, b, c]) )

Arguments:

Instance a reverse Bessel polynomial of the given degree.¹

Arguments:

Synomyn for Object: -class.a = Polynomial.newFrom([0, 2, 0, 0, 6, 0, 0]); a.species; a.class;

Returns the degree of the polynomial.a = Polynomial.newFrom([0, 2, 0, 0, 6, 0, 0]); a.degree;

Returns true if the polynomial is a monomial.Polynomial.newFrom([0, 0, 2, 0, 0]).isMonomial;

Returns true if the polynomial is a binomial.Polynomial.newFrom([0, 2, 0, 0, 6]).isBinomial;

Calculate the product of two polynomials by convolving the original coefficient sequences.// Real example Polynomial.newFrom([1, 2, 3, 4, 5]) * Polynomial.newFrom([5, 4, 3, 2, 1]); // Complex example a = Polynomial.newFrom([Complex(2, 2.2), Complex(2.5, 5), Complex(7, 4)]); a * a.performUnaryOp(\conjugate);

Divide two polynomials using synthetic division. The degree of the numerator polynomial needs to be greater than or equal to the degree of the denominator polynomial.a = Polynomial.newFrom([Complex(2, 1), Complex(0.2, -0.436), Complex(-1.9, -5)]); b = Polynomial.newFrom([Complex(-5, 4), Complex(3.5, 1.2)]); #r, q = a / b; // Check if we get back the original polynomial // Notice padding of the remainder in order to get a valid result (b * q) + r.addAll([0, 0]);

Returns:

Calculate the n'th power of a polynomial. Negative, complex or fractional powers are allowed for binomials with the a_0 coefficient equal to one in order to calculate the binomial series which converges for: |z| < 1Polynomial.newFrom([1, 1]).pow(0.5) //Binomial series: (1 + x)^0.5

Evaluate the polynomial at the (complex) point x.// Real Polynomial.newFrom([0.2, 0.1, 0.3, 0.5]).eval(1.0); // Complex Polynomial.newFrom([0.2, 0.1, 0.3, 0.5]).eval(Complex.new(1.0, 0.0));

Evaluate the polynomial and all of its derivatives at the (complex) point x.// Real Polynomial.newFrom([0.2, 0.1, 0.3, 0.5]).evalDerivs(1.0); // Complex Polynomial.newFrom([0.2, 0.1, 0.3, 0.5]).evalDerivs(Complex.new(1.0, 0.0));

Returns:

Find all the roots of the polynomial.a = Polynomial.newFrom([80, 4, -24, -1, 1]); b = a.findRoots; //Check if the roots we found make sense c = ([b.neg, [1, 1, 1, 1]].lace(8).reshape(4, 2).collect(_ .as(Polynomial))); Polynomial.expandBinomialFactors(c)

Arguments:

Discussion:

Laguerre's method works for polynomials with real as well as complex coefficients.

Another option is to use the Eigenvalue algorithm to form the companion matrix which eigenvalues coincide with the roots of the polynomial. Although this method is more reliable than the first one, it is considerably slower and only works for polynomials with real coefficients.