StreamTaylorSeriesOperations A

sttaylor.spad line 283 [edit on github]

• 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

from StreamTaylorSeriesOperationsCategory(A, A)

*: (Stream A, A) -> Stream A if A has RightModule A

from StreamTaylorSeriesOperationsCategory(A, A)

*: (Stream A, Stream A) -> Stream A

from StreamTaylorSeriesOperationsCategory(A, A)

+: (Stream A, Stream A) -> Stream A

from StreamTaylorSeriesOperationsCategory(A, A)

-: (Stream A, Stream A) -> Stream A

from StreamTaylorSeriesOperationsCategory(A, A)

-: Stream A -> Stream A

from StreamTaylorSeriesOperationsCategory(A, A)

/: (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

from StreamTaylorSeriesOperationsCategory(A, A)

coerce: A -> Stream A

from StreamTaylorSeriesOperationsCategory(A, A)

compose: (Stream A, Stream A) -> Stream A if A has RightModule A

from StreamTaylorSeriesOperationsCategory(A, A)

deriv: Stream A -> Stream A

from StreamTaylorSeriesOperationsCategory(A, A)

eval: (Stream A, A) -> Stream A if A has RightModule A

from StreamTaylorSeriesOperationsCategory(A, A)

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

from StreamTaylorSeriesOperationsCategory(A, A)

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

from StreamTaylorSeriesOperationsCategory(A, A)

integers: Integer -> Stream Integer

from StreamTaylorSeriesOperationsCategory(A, A)

integrate: (A, Stream A) -> Stream A if A has Algebra Fraction Integer

from StreamTaylorSeriesOperationsCategory(A, A)

invmultisect: (Integer, Integer, Stream A) -> Stream A

from StreamTaylorSeriesOperationsCategory(A, A)

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

from StreamTaylorSeriesOperationsCategory(A, A)

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

from StreamTaylorSeriesOperationsCategory(A, A)

monom: (A, Integer) -> Stream A

from StreamTaylorSeriesOperationsCategory(A, A)

multisect: (Integer, Integer, Stream A) -> Stream A

from StreamTaylorSeriesOperationsCategory(A, A)

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

from StreamTaylorSeriesOperationsCategory(A, A)

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.

StreamTaylorSeriesOperationsCategory(A, A)