StreamTaylorSeriesOperations A¶

A: Ring

StreamTaylorSeriesOperations implements Taylor series arithmetic, where a Taylor series is represented by a stream of its coefficients, see corresponding operations in the category Ring.

*: (A, Stream A) -> Stream A: r * a returns the power series scalar multiplication of r by a: r * [a0, a1, ...] = [r * a0, r * a1, ...]

*: (Stream A, A) -> Stream A: a * r returns the power series scalar multiplication of a by r: [a0, a1, ...] * r = [a0 * r, a1 * r, ...]

*: (Stream A, Stream A) -> Stream A: a * b returns the power series (Cauchy) product of a and b: [a0, a1, ...] * [b0, b1, ...] = [c0, c1, ...] where ck = sum(i + j = k, ai * bk).

+: (Stream A, Stream A) -> Stream A: a + b returns the power series sum of a and b: [a0, a1, ..] + [b0, b1, ..] = [a0 + b0, a1 + b1, ..]

-: (Stream A, Stream A) -> Stream A: a - b returns the power series difference of a and b: [a0, a1, ..] - [b0, b1, ..] = [a0 - b0, a1 - b1, ..]

-: Stream A -> Stream A: - a returns the power series negative of a: - [a0, a1, ...] = [- a0, - a1, ...]

/: (Stream A, Stream A) -> Stream A: a / b returns the power series quotient of a by b. An error message is returned if b is not invertible. This function is used in fixed point computations.

addiag: Stream Stream A -> Stream A: addiag(x) performs diagonal addition of a stream of streams. if x = [[a<0, 0>, a<0, 1>, ..], [a<1, 0>, a<1, 1>, ..], [a<2, 0>, a<2, 1>, ..], ..] and addiag(x) = [b<0, b<1>, ...], then b<k> = sum(i+j=k, a<i, j>).

coerce: A -> Stream A: coerce(r) converts a ring element r to a stream with one element.

compose: (Stream A, Stream A) -> Stream A: compose(a, b) composes the power series a with the power series b.

deriv: Stream A -> Stream A: deriv(a) returns the derivative of the power series with respect to the power series variable. Thus deriv([a0, a1, a2, ...]) returns [a1, 2 a2, 3 a3, ...].

eval: (Stream A, A) -> Stream A: eval(a, r) returns a stream of partial sums of the power series a evaluated at the power series variable equal to r.

evenlambert: Stream A -> Stream A: evenlambert(st) computes f(x^2) + f(x^4) + f(x^6) + ... if st is a stream representing f(x). This function is used for computing infinite products. If f(x) is a power series with constant coefficient 1, then prod(f(x^(2*n)), n=1..infinity) = exp(evenlambert(log(f(x)))).

exquo: (Stream A, Stream A) -> Union(Stream A, failed): exquo(a, b) returns the power series quotient of a by b, if the quotient exists, and “failed” otherwise

gderiv: (Integer -> A, Stream A) -> Stream A: gderiv(f, [a0, a1, a2, ..]) returns [f(0)*a0, f(1)*a1, f(2)*a2, ..].

general_Lambert_product: (Stream A, Integer, Integer) -> Stream A: general_Lambert_product(f(x), a, d) returns f(x^a)*f(x^(a + d))*f(x^(a + 2 d))* .... f(x) should have constant coefficient equal to one and a and d should be positive.

generalLambert: (Stream A, Integer, Integer) -> Stream A: generalLambert(f(x), a, d) returns f(x^a) + f(x^(a + d)) + f(x^(a + 2 d)) + .... f(x) should have zero constant coefficient and a and d should be positive.

int: A -> Stream A: int(r) returns [r, r+1, r+2, …], where r is a ring element.

integers: Integer -> Stream Integer: integers(n) returns [n, n+1, n+2, ...].

integrate: (A, Stream A) -> Stream A if A has Algebra Fraction Integer: integrate(r, a) returns the integral of the power series a with respect to the power series variable where r denotes the constant of integration. Thus integrate(a, [a0, a1, a2, ...]) = [a, a0, a1/2, a2/3, ...].

invmultisect: (Integer, Integer, Stream A) -> Stream A: invmultisect(a, b, st) substitutes x^((a+b)*n) for x^n and multiplies by x^b.

lagrange: Stream A -> Stream A: lagrange(g) produces the power series for f where f is implicitly defined as f(z) = z*g(f(z)).

lambert: Stream A -> Stream A: lambert(st) computes f(x) + f(x^2) + f(x^3) + ... if st is a stream representing f(x). This function is used for computing infinite products. If f(x) is a power series with constant coefficient 1 then prod(f(x^n), n = 1..infinity) = exp(lambert(log(f(x)))).

lazyGintegrate: (Integer -> A, A, () -> Stream A) -> Stream A if A has Field: lazyGintegrate(f, r, g) is used for fixed point computations.

lazyIntegrate: (A, () -> Stream A) -> Stream A if A has Algebra Fraction Integer: lazyIntegrate(r, f) is a version of integrate used for fixed point computations.

mapdiv: (Stream A, Stream A) -> Stream A if A has Field: mapdiv([a0, a1, ..], [b0, b1, ..]) returns [a0/b0, a1/b1, ..].

mapmult: (Stream A, Stream A) -> Stream A: mapmult([a0, a1, ..], [b0, b1, ..]) returns [a0*b0, a1*b1, ..].

monom: (A, Integer) -> Stream A: monom(deg, coef) is a monomial of degree deg with coefficient coef.

multisect: (Integer, Integer, Stream A) -> Stream A: multisect(a, b, st) selects the coefficients of x^((a+b)*n+a), and changes them to x^n.

nlde: Stream Stream A -> Stream A if A has Algebra Fraction Integer: nlde(u) solves a first order non-linear differential equation described by u of the form [[b<0, 0>, b<0, 1>, ...], [b<1, 0>, b<1, 1>, .], ...]. the differential equation has the form y' = sum(i=0 to infinity, j=0 to infinity, b<i, j>*(x^i)*(y^j)).

oddintegers: Integer -> Stream Integer: oddintegers(n) returns [n, n+2, n+4, ...].

oddlambert: Stream A -> Stream A: oddlambert(st) computes f(x) + f(x^3) + f(x^5) + ... if st is a stream representing f(x). This function is used for computing infinite products. If f(x) is a power series with constant coefficient 1 then prod(f(x^(2*n-1)), n=1..infinity) = exp(oddlambert(log(f(x)))).

power: (A, Stream A) -> Stream A if A has Field: power(a, f) returns the power series f raised to the power a.

powern: (Fraction Integer, Stream A) -> Stream A if A has Algebra Fraction Integer: powern(r, f) raises power series f to the power r.

prodiag: Stream Stream A -> Stream A: prodiag(x) performs “diagonal” infinite product of a stream of streams. When x(i) is interpreted as stream of coefficients of series f_i(z), i=1,..., then prodiag(x) = (1 + z*f_1(z))*(1 + z^2*f_2(x))*...

recip: Stream A -> Union(Stream A, failed): recip(a) returns the power series reciprocal of a, or “failed” if not possible.

revert: Stream A -> Stream A: revert(a) computes the inverse of a power series a with respect to composition. the series should have constant coefficient 0 and invertible first order coefficient.