Numerical Analysis: Difference between revisions
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==Orthogonal Polynomials== | ==Orthogonal Polynomials== | ||
===Hermite Polynomials=== | ===Hermite Polynomials=== | ||
{{main | Wikipedia: Hermite polynomials}} | |||
===Legendre Polynomials=== | ===Legendre Polynomials=== | ||
{{main | Wikipedia: Legendre polynomials}} | |||
===Laguerre Polynomials=== | ===Laguerre Polynomials=== | ||
{{main | Wikipedia: Laguerre polynomials}} | |||
==Nonlinear Equations== | ==Nonlinear Equations== | ||
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==Numerical Integration== | ==Numerical Integration== | ||
See Quadrature rules | See Quadrature rules | ||
==Function Approximation== | |||
===Radial Basis Functions (RBF)=== | |||
See [http://num.math.uni-goettingen.de/schaback/teaching/sc.pdf A Practical Guide to Radial Basis Functions]. | |||
Radial Basis Functions are functions which are only dependent on the radius of the input: <math>\mathbf{\phi}(\mathbf{x}) = \phi(\Vert x\Vert)</math> | |||
Revision as of 13:08, 16 September 2021
Numerical Analysis
Orthogonal Polynomials
Hermite Polynomials
Legendre Polynomials
Laguerre Polynomials
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
See Quadrature rules
Function Approximation
Radial Basis Functions (RBF)
See A Practical Guide to Radial Basis Functions. Radial Basis Functions are functions which are only dependent on the radius of the input: \(\displaystyle \mathbf{\phi}(\mathbf{x}) = \phi(\Vert x\Vert)\)