Function Repository Resource:

MultivariateTaylorPolynomial

Source Notebook

Generate a multivariate Taylor polynomial of a given total degree

Contributed by: Daniel Lichtblau

ResourceFunction["MultivariateTaylorPolynomial"][func,vars,n]

computes the degree n multivariate Taylor polynomial of func in vars.

ResourceFunction["MultivariateTaylorPolynomial"][func,vars]

computes the multivariate Taylor polynomial of degree 1.

Details and Options

The degree n is the maximal total degree of the terms in the result.

When n is set to 1, the result is the constant term plus the linear terms in all variables.

Each entry in vars can be a variable or a list of the form {var,basepoint}. ResourceFunction["MultivariateTaylorPolynomial"] will expand each variable around its respective base point.

Any base point not explicitly provided is taken to be 0.

If a basepoint is infinite, the expansion will use its reciprocal and expand around the origin.

ResourceFunction["MultivariateTaylorPolynomial"] takes the same options as Series, with the additional option Weights→{w₁,w₂,…}. This gives an expansion where the total degree of all terms is weighted by the vector {w₁,w₂,…}. If the variables are {v₁,v₂,…} then any term v₁^e₁v₂^e₂… in the result satisfies the inequality w₁e₁+w₂e₂+…<=n.

Examples

Basic Examples (3)

Compute a multivariate Taylor polynomial of degree 4:

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$ee = Sin[x + y^2]; mm = ResourceFunction["MultivariateTaylorPolynomial"][ee, {x, y}, 4]$

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Check the result numerically:

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Compute constant and linear terms of an expression expanded at {x=0,y=2}:

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$ee = (Sin[1 + x + y^2] + Cos[1 - x^2 + y])*Exp[x^2*y]; mm = ResourceFunction["MultivariateTaylorPolynomial"][ee, {x, {y, 2}}]$

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Compute a multivariate Taylor polynomial around {x=1,y=-2}:

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$ee = Sin[x + y^2]; mm = ResourceFunction["MultivariateTaylorPolynomial"][ ee, {{x, 1}, {y, -2}}, 4]$

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Check the result numerically:

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Scope (2)

MultivariateTaylorPolynomial can expand variables at infinity:

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$ee = Sin[x^2 + y]*Sqrt[1/(1 + z^2)]; mm1 = ResourceFunction["MultivariateTaylorPolynomial"][ ee, {{x, 0}, {y, -2}, {z, Infinity}}, 4]$

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Providing Assumptions can simplify the computations and results:

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Check these results numerically:

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MultivariateTaylorPolynomial can give results that have asymptotic terms such as exponentials and implicit piecewise terms:

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Provide Assumptions to obtain a simpler result:

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Check these results numerically:

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Expand at z=-∞:

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Check numerically:

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Options (3)

Weights (3)

Compute a multivariate expansion to seventh order:

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$ee = Sin[x + y^2]*Erf[z]; order = 7; mm = ResourceFunction["MultivariateTaylorPolynomial"][ee, {x, y, z}, order]$

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Use weights of {1,3,2} on the variables {x,y,z} respectively:

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Use weights of on the variables {x,y,z} respectively:

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Properties and Relations (2)

Compute a multivariate Taylor polynomial of total degree 4:

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$ee = Sin[x + y^2] - Cos[x^2 + y]; mm = Expand[ ResourceFunction["MultivariateTaylorPolynomial"][ee, {x, y}, 4]]$

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One can use Series to recover all terms by expanding in x and y separately to order 4, but this gives extra terms:

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Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

1.0.0 – 09 August 2023

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License