# Numerical Analysis

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Numerical Analysis

## Nonlinear Equations

### Continuation/Homotopy Methods

Reference Numerical Optimization by Nocedal and Wright (2006)
Also known as zero-path following.
If you want to solve $$\displaystyle r(x)=0$$ when $$\displaystyle r(x)$$ is difficult (i.e. has non-singular Jacobian) then you can use this method.
Define the Homotopy map
$$\displaystyle H(x, \lambda)=\lambda r(x) + (1-\lambda)(x-a)$$

## Numerical Differentiation

See finite differencing

## Numerical Integration

Radial Basis Functions are functions which are only dependent on the radius of the input: $$\displaystyle \mathbf{\phi}(\mathbf{x}) = \phi(\Vert x\Vert)$$