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MATH
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Methods from Multivariable approximation

All methods from Multivariable approximation.

Back to Multivariable approximation

Approximate a function linearly

For a differentiable f:R2→Rf:\mathbb{R}^2\to\mathbb{R} and a base point a=(x0,y0)\mathbf{a}=(x_0,y_0), approximate nearby values by

f(x)≈f(a)+∇f(a)⋅(x−a).f(\mathbf{x}) \approx f(\mathbf{a}) +\nabla f(\mathbf{a})\cdot(\mathbf{x}-\mathbf{a}).

To use the approximation, evaluate ff and its first partial derivatives at a\mathbf{a}, then substitute the displacement x−a\mathbf{x}-\mathbf{a}.

The expression is the equation of the tangent plane with the output written as a function of the input. Differentiability means that the omitted error, divided by ∥x−a∥\|\mathbf{x}-\mathbf{a}\|, tends to zero as x→a\mathbf{x}\to\mathbf{a}. Thus one linear model works for changes in every direction, not just along the coordinate axes.

  1. [1]:Tangent plane to a graph
  2. [2]:Linearise (x−y)2(x-y)^2
  3. [3]:Taylor polynomial through degree three
  4. [4]:Exact equations and level-set geometry
  5. [5]:Seen Questions

Construct a multivariable Taylor polynomial

To construct the Taylor polynomial through degree n≤3n\le3 about a\mathbf{a}:

  1. choose the centre and the required total degree;
  2. calculate the partial derivatives through that order;
  3. evaluate them at a\mathbf{a};
  4. retain the terms through total degree nn in the formula above.

The result agrees with ff at a\mathbf{a} and matches the relevant partial derivatives there. It is a local approximation, not generally an identity on the whole domain.

  1. [1]:Taylor polynomial through degree three
  2. [2]:ex−ye^{x-y} near the origin
  3. [3]:A function that is already a polynomial
  4. [4]:An exponential of three variables

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Approximate a function linearlyConstruct a multivariable Taylor polynomial