Library¶

Types¶

# TaylorSeries.Taylor1 — Type.

Taylor1{T<:Number} <: Number

DataType for polynomial expansions in one independent variable.

Fields:

coeffs :: Array{T,1} Expansion coefficients; the $i$-th

component is the coefficient of degree $i-1$ of the expansion.

order :: Int64 Maximum order (degree) of the polynomial.

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# TaylorSeries.HomogeneousPolynomial — Type.

HomogeneousPolynomial{T<:Number} <: Number

DataType for homogenous polynomials in many (>1) independent variables.

Fields:

coeffs :: Array{T,1} Expansion coefficients of the homogeneous

polynomial; the $i$-th component is related to a monomial, where the degrees of the independent variables are specified by coeff_table[order+1][i].

order :: Int order (degree) of the homogenous polynomial.

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# TaylorSeries.TaylorN — Type.

TaylorN{T<:Number} <: Number

DataType for polynomial expansions in many (>1) independent variables.

Fields:

coeffs :: Array{HomogeneousPolynomial{T},1} Vector containing the

HomogeneousPolynomial entries. The $i$-th component corresponds to the homogeneous polynomial of degree $i-1$.

order :: Int maximum order of the polynomial expansion.

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# TaylorSeries.ParamsTaylorN — Type.

ParamsTaylorN

DataType holding the current parameters for TaylorN and HomogeneousPolynomial.

Fields:

order :: Int Order (degree) of the polynomials
num_vars :: Int Number of variables
variable_names :: Array{String,1} Name of the variables

These parameters can be changed using set_params_TaylorN(order, numVars).

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Functions and methods¶

# TaylorSeries.Taylor1 — Method.

Taylor1(T, [order=1])
Taylor1([order=1])

Shortcut to define the independent variable of a Taylor1{T} polynomial of given order. If the type T is ommitted, Float64 is assumend.

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# TaylorSeries.TaylorN — Method.

TaylorN(T, nv; [order=get_order()])
TaylorN(nv; [order=get_order()])

Shortcut to define the nv-th independent TaylorN{T} variable as a polynomial. The order is defined through the keyword parameter order, whose default corresponds to get_order(). If T::Type is ommitted, Float64 is assumend.

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# TaylorSeries.set_variables — Function.

set_variables(R, names; [order=6])
set_variables(names; [order=6])
set_variables(R, names; [order=6, numvars=-1])
set_variables(names; [order=6, numvars=-1])

Return a TaylorN{R} vector with numvars polynomials, each representing an independent variable, using names as the String for the output.

If numvars is not specified, it is inferred from the length of names. If length(names)==1 and numvars>1, it uses this name with subscripts for the different variables. When changing the order or numvars, the hash_tables are reset.

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# TaylorSeries.show_params_TaylorN — Function.

show_params_TaylorN()

Display the current parameters for TaylorN and HomogeneousPolynomial types.

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# TaylorSeries.get_coeff — Function.

get_coeff(a, n)

Return the coefficient of order n::Int of a a::Taylor1 polynomial.

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get_coeff(a, v)

Return the coefficient of a::HomogeneousPolynomial, specified by v::Array{Int,1} which has the indices of the specific monomial.

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get_coeff(a, v)

Return the coefficient of a::TaylorN, specified by v::Array{Int,1} which has the indices of the specific monomial.

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# TaylorSeries.evaluate — Function.

evaluate(a, [dx])

Evaluate a Taylor1 polynomial using Horner's rule (hand coded). If dx is ommitted, its value is considered as zero.

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evaluate(x, δt)

Evaluates each element of x::Array{Taylor1{T},1}, representing the dependent variables of an ODE, at time δt.

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evaluate(a, x)

Substitute x::Taylor1 as independent variable in a a::Taylor1 polynomial.

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evaluate(a, vals)

Evaluate a HomogeneousPolynomial polynomial using Horner's rule (hand coded) at vals.

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evaluate(a, [vals])

Evaluate a TaylorN polynomial using Horner's rule (hand coded) at vals. If vals is ommitted, it is evaluated at zero.

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# TaylorSeries.evaluate! — Function.

evaluate!(x, δt, x0)

Evaluates each element of x::Array{Taylor1{T},1}, representing the Taylor expansion for the dependent variables of an ODE at time δt. It updates the vector x0 with the computed values.

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# TaylorSeries.derivative — Function.

derivative(a)

Return the Taylor1 polynomial of the differential of a::Taylor1. The last coefficient is set to zero.

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derivative(n, a)

Return the value of the n-th derivative of the polynomial a.

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derivative(a, r)

Partial differentiation of a::HomogeneousPolynomial series with respect to the r-th variable.

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derivative(a, [r=1])

Partial differentiation of a::TaylorN series with respect to the r-th variable.

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# TaylorSeries.integrate — Function.

integrate(a, [x])

Return the integral of a::Taylor1. The constant of integration (0-th order coefficient) is set to x, which is zero if ommitted.

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# Base.LinAlg.gradient — Function.

    gradient(f)
    ∇(f)

Compute the gradient of the polynomial f::TaylorN.

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# TaylorSeries.jacobian — Function.

    jacobian(vf)
    jacobian(vf, [vals])

Compute the jacobian matrix of vf, a vector of TaylorN polynomials, evaluated at the vector vals. If vals is ommited, it is evaluated at zero.

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# TaylorSeries.hessian — Function.

    hessian(f)
    hessian(f, [vals])

Return the hessian matrix (jacobian of the gradient) of f::TaylorN, evaluated at the vector vals. If vals is ommited, it is evaluated at zero.

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# Base.:* — Function.

*(a, b)

Return the Taylor expansion of $a \cdot b$, of order max(a.order,b.order), for a::Taylor1, b::Taylor1 polynomials.

For details on making the Taylor expansion, see TaylorSeries.mulHomogCoef.

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# Base.:/ — Function.

/(a, b)

Return the Taylor expansion of $a/b$, of order max(a.order,b.order), for a::Taylor1, b::Taylor1 polynomials.

For details on making the Taylor expansion, see TaylorSeries.divHomogCoef.

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# Base.:^ — Function.

^(a, x)

Return the Taylor expansion of $a^x$ for a::Taylor1 polynomial and x::Number. If x is non integer and the 0-th order coefficient is zero, an ArgumentError is thrown.

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# Base.sqrt — Function.

sqrt(a)

Return the Taylor expansion of $\sqrt(a)$, of order a.order, for a::Taylor1 polynomial. If the first non-vanishing coefficient of a is an odd power, and ArgumentError is thrown.

For details on making the Taylor expansion, see TaylorSeries.sqrtHomogCoef.

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# Base.exp — Function.

exp(a)

Return the Taylor expansion of $e^a$, of order a.order, for a::Taylor1 polynomial.

For details on making the Taylor expansion, see TaylorSeries.expHomogCoef.

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# Base.log — Function.

log(a)

Return the Taylor expansion of $\log(a)$, of order a.order, for a::Taylor1 polynomial.

For details on making the Taylor expansion, see TaylorSeries.logHomogCoef.

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# Base.sin — Function.

sin(a)

Return the Taylor expansion of $\sin(a)$, of order a.order, for a::Taylor1 polynomial.

For details on making the Taylor expansion, see TaylorSeries.sincosHomogCoef.

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# Base.cos — Function.

cos(a)

Return the Taylor expansion of $\cos(a)$, of order a.order, for a::Taylor1 polynomial

For details on making the Taylor expansion, see TaylorSeries.sincosHomogCoef.

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# Base.tan — Function.

tan(a)

Return the Taylor expansion of $\tan(a)$, of order a.order, for a::Taylor1 polynomial.

For details on making the Taylor expansion, see TaylorSeries.tanHomogCoef.

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# Base.asin — Function.

asin(a)

Return the Taylor expansion of $\arcsin(a)$, of order a.order, for a::Taylor1 polynomial.

For details on making the Taylor expansion, see TaylorSeries.asinHomogCoef.

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# Base.acos — Function.

acos(a)

Return the Taylor expansion of $\arccos(a)$, of order a.order, for a::Taylor1 polynomial.

For details on making the Taylor expansion, see TaylorSeries.acosHomogCoef.

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# Base.atan — Function.

atan(a)

Return the Taylor expansion of $\arctan(a)$, of order a.order, for a::Taylor1 polynomial.

For details on making the Taylor expansion, see TaylorSeries.atanHomogCoef.

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# Base.sinh — Function.

sinh(a)

Return the Taylor expansion of $\sinh(a)$, of order a.order, for a::Taylor1 polynomial.

For details on making the Taylor expansion, see TaylorSeries.sinhcoshHomogCoef.

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# Base.cosh — Function.

cosh(a)

Return the Taylor expansion of $\cosh(a)$, of order a.order, for a::Taylor1 polynomial

For details on making the Taylor expansion, see TaylorSeries.sinhcoshHomogCoef.

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# Base.tanh — Function.

tanh(a)

Return the Taylor expansion of $\tanh(a)$, of order a.order, for a::Taylor1 polynomial.

For details on making the Taylor expansion, see TaylorSeries.tanhHomogCoef.

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# Base.abs — Function.

abs(a)

Return a or -a depending on the 0-th order coefficient of the Taylor1 polynomial a. If a.coeffs[1] is zero, an ArgumentError is thrown.

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abs(a::TaylorN)

Absolute value of a TaylorN polynomial, using the 0-th order coefficient.

Return a or -a, depending on the 0-th order coefficient of a. If it is zero, it throws an ArgumentError.

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Internals¶

# TaylorSeries.generate_tables — Function.

generate_tables(num_vars, order)

Return the hash tables coeff_table, index_table, size_table and pos_table. Internally, these are treated as const.

Hash tables

coeff_table :: Array{Array{Array{Int64,1},1},1}

The $i+1$-th component contains a vector with the vectors of all the possible combinations of monomials of a HomogeneousPolynomial of order $i$.

index_table :: Array{Array{Int64,1},1}

The $i+1$-th component contains a vector of (hashed) indices that represent the distinct monomials of a HomogeneousPolynomial of order (degree) $i$.

size_table :: Array{Int64,1}

The $i+1$-th component contains the number of distinct monomials of the HomogeneousPolynomial of order $i$, equivalent to length(coeff_table[i]).

pos_table :: Array{Dict{Int64,Int64},1}

The $i+1$-th component maps the hash index to the (lexicographic) position of the corresponding monomial in coeffs_table.

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# TaylorSeries.generate_index_vectors — Function.

generate_index_vectors(num_vars, degree)

Return a vector of index vectors with num_vars (number of variables) and degree.

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# TaylorSeries.in_base — Function.

in_base(order, v)

Convert vector v of non-negative integers to base order+1.

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# TaylorSeries.make_inverse_dict — Function.

make_inverse_dict(v)

Return a Dict with the enumeration of v: the elements of v point to the corresponding index.

It is used to construct pos_table from index_table.

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# TaylorSeries.order_posTb — Function.

order_posTb(order, nv, ord)

Return a vector with the positions, in a HomogeneousPolynomial of order order, where the variable nv has order ord.

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# TaylorSeries.mul! — Function.

mul!(c, a, b)

Return c = a*b with no allocation; all parameters are HomogeneousPolynomial.

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mul!(c, a)

Return c = a*a with no allocation; all parameters are HomogeneousPolynomial.

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# TaylorSeries.mulHomogCoef — Function.

mulHomogCoef(kcoef, ac, bc)

Compute the k-th expansion coefficient of $c = a\cdot b$ given by

$\begin{equation*} c_k = \sum_{j=0}^k a_j b_{k-j}, \end{equation*}$

with $a$ and $b$ Taylor1 polynomials.

Inputs are the kcoef-th coefficient, and the vectors of the expansion coefficients ac and bc, corresponding respectively to a and b.

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# TaylorSeries.divHomogCoef — Function.

divHomogCoef(kcoef, ac, bc, coeffs, ordfact)

Compute the k-th expansion coefficient of $c = a / b$ given by

$\begin{equation*} c_k = \frac{1}{b_0} (a_k - \sum_{j=0}^{k-1} c_j b_{k-j}), \end{equation*}$

with $a$ and $b$ Taylor1 polynomials.

Inputs are the kcoef-th coefficient, the vectors of the expansion coefficients ac and bc, corresponding respectively to a and b, the already calculated expansion coefficients coeffs of c, and ordfact which is the order of the factorized term of the denominator, whenever b_0 is zero.

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# TaylorSeries.powHomogCoef — Function.

powHomogCoef(kcoef, ac, x, coeffs, knull)

Compute the k-th expansion coefficient of $c = a^x$, given by

$\begin{equation*} c_k = \frac{1}{k a_0} \sum_{j=0}^{k-1} ((k-j)x -j) - j)a_{k-j} c_j, \end{equation*}$

with $a$ a Taylor1 polynomial, and x a number.

Inputs are the kcoef-th coefficient, the vector of the expansion coefficients ac of a, the exponent x, the already calculated expansion coefficients coeffs of c, and knull which is the order of the first non-zero coefficient of a.

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# TaylorSeries.squareHomogCoef — Function.

squareHomogCoef(kcoef, ac)

Compute the k-th expansion coefficient of $c = a^2$, given by

$\begin{eqnarray*} c_k &=& 2 \sum_{j=0}^{(k-1)/2} a_{k-j} a_j, \text{ if $k$ is odd, or } \\ c_k &=& 2 \sum_{j=0}^{(k-2)/2} a_{k-j} a_j + (a_{k/2})^2, \text{ if $k$ is even, } \end{eqnarray*}$

with $a$ a Taylor1 polynomial.

Inputs are the kcoef-th coefficient and the vector of the expansion coefficients ac of a.

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# TaylorSeries.sqrtHomogCoef — Function.

sqrtHomogCoef(kcoef, ac, coeffs, knull)

Compute the k-th expansion coefficient of $c = \sqrt(a)$, given by

$\begin{eqnarray*} c_k &=& \frac{1}{2 c_0} ( a_k - 2 \sum_{j=0}^{(k-1)/2} c_{k-j}c_j), \text{ if $k$ is odd, or } \\ c_k &=& \frac{1}{2 c_0} ( a_k - 2 \sum_{j=0}^{(k-2)/2} c_{k-j}c_j) - (c_{k/2})^2, \text{ if $k$ is even,} \end{eqnarray*}$

with $a$ a Taylor1 polynomial.

Inputs are the kcoef-th coefficient, the vector of the expansion coefficients ac of a, the already calculated expansion coefficients coeffs of c, and knull, which is half of the order of the first non-zero coefficient of a.

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# TaylorSeries.expHomogCoef — Function.

expHomogCoef(kcoef, ac, coeffs)

Compute the k-th expansion coefficient of $c = \exp(a)$ given by

$\begin{equation*} c_k = \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} c_j, \end{equation*}$

with $a$ a Taylor1 polynomial.

Inputs are the kcoef-th coefficient, the vector of the expansion coefficients ac of $a$, and the already calculated expansion coefficients coeffs of c.

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# TaylorSeries.logHomogCoef — Function.

logHomogCoef(kcoef, ac, coeffs)

Compute the k-th expansion coefficient of $c = \log(a)$, given by

$\begin{equation*} c_k = \frac{1}{a_0} (a_k - \frac{1}{k} \sum_{j=0}^{k-1} j a_{k-j} c_j ), \end{equation*}$

with $a$ a Taylor1 polynomial.

Inputs are the kcoef-th coefficient, the vector of the expansion coefficients ac of a, and the already calculated expansion coefficients coeffs of c.

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# TaylorSeries.sincosHomogCoef — Function.

sincosHomogCoef(kcoef, ac, scoeffs, ccoeffs)

Compute the k-th expansion coefficient of $s = \sin(a)$ and $c=\cos(a)$ simultaneously given by

$\begin{eqnarray*} s_k &=& \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} c_j \\ \ c_k &=& -\frac{1}{k}\sum_{j=0}^{k-1} (k-j) a_{k-j} s_j \end{eqnarray*}$

with $a$ a Taylor1 polynomial.

Inputs are the kcoef-th coefficient, the vector of the expansion coefficients ac of a, and the already calculated expansion coefficients scoeffs and ccoeffs of sin(a) and cos(a), respectvely.

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# TaylorSeries.tanHomogCoef — Function.

tanHomogCoef(kcoef, ac, coeffst2)

Compute the k-th expansion coefficient of $c = \tan(a)$ given by

$\begin{equation*} c_k = a_k + \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} p_j, \end{equation*}$

with $a$ a Taylor1 polynomial and $p = c^2$.

Inputs are the kcoef-th coefficient, the vector of the expansion coefficients ac of a, and the already calculated expansion coefficients coeffst2 of c^2.

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# TaylorSeries.asinHomogCoef — Function.

asinHomogCoef(kcoef, ac, rc, coeffs)

Compute the k-th expansion coefficient of $s = \arcsin(a)$ given by

$\begin{equation*} s_k = \frac{1}{ \sqrt{r_0} } \big( a_k - \frac{1}{k} \sum_{j=1}^{k-1} j r_{k-j} s_j \big), \end{equation*}$

with $a$ a Taylor1 polynomial and $r = \sqrt{1 - a^2}$.

Inputs are the kcoef-th coefficient, the vector of the expansion coefficients ac of a, the already calculated expansion coefficients rc of $r$ (see above), and the already calculated expansion coefficients coeffs of asin(a).

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# TaylorSeries.acosHomogCoef — Function.

acosHomogCoef(kcoef, ac, rc, coeffs)

Compute the k-th expansion coefficient of $c = \arccos(a)$ given by

$\begin{equation*} c_k = - \frac{1}{ r_0 } \big( a_k - \frac{1}{k} \sum_{j=1}^{k-1} j r_{k-j} c_j \big), \end{equation*}$

with $a$ a Taylor1 polynomial and $r = \sqrt{1 - a^2}$.

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# TaylorSeries.atanHomogCoef — Function.

atanHomogCoef(kcoef, ac, rc, coeffs)

Compute the k-th expansion coefficient of $c = \arctan(a)$ given by

$\begin{equation*} t_k = \frac{1}{r_0}(a_k - \frac{1}{k} \sum_{j=1}^{k-1} j r_{k-j} t_j) , \end{equation*}$

with $a$ a Taylor1 polynomial and $r = 1 + a^2$.

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# TaylorSeries.sinhcoshHomogCoef — Function.

sinhcoshHomogCoef(kcoef, ac, scoeffs, ccoeffs)

Compute the k-th expansion coefficient of $sh = \sinh(a)$ and $ch=\cosh(a)$ simultaneously given by

$\begin{eqnarray*} sh_k &=& \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} ch_j \\ \ ch_k &=& \frac{1}{k}\sum_{j=0}^{k-1} (k-j) a_{k-j} sh_j \end{eqnarray*}$

with $a$ a Taylor1 polynomial.

Inputs are the kcoef-th coefficient, the vector of the expansion coefficients ac of a, and the already calculated expansion coefficients shcoeffs and chcoeffs of sinh(a) and cosh(a), respectvely.

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# TaylorSeries.tanhHomogCoef — Function.

tanhHomogCoef(kcoef, ac, coeffst2)

Compute the k-th expansion coefficient of $c = \tanh(a)$ given by

$\begin{equation*} th_k = a_k - \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} p_j, \end{equation*}$

with $a$ a Taylor1 polynomial and $p = th^2$.

Inputs are the kcoef-th coefficient, the vector of the expansion coefficients ac of a, and the already calculated expansion coefficients coeffst2 of th^2.

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# Base.LinAlg.A_mul_B! — Function.

A_mul_B!(Y, A, B)

Multiply A*B and save the result in Y.

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