Numerical Analysis: Difference between revisions
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Numerical Analysis | Numerical Analysis | ||
==Orthogonal Polynomials== | |||
===Hermite Polynomials=== | |||
{{main | Wikipedia: Hermite polynomials}} | |||
===Legendre Polynomials=== | |||
{{main | Wikipedia: Legendre polynomials}} | |||
===Laguerre Polynomials=== | |||
{{main | Wikipedia: Laguerre polynomials}} | |||
==Nonlinear Equations== | ==Nonlinear Equations== | ||
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Reference Numerical Optimization by Nocedal and Wright (2006)<br> | Reference Numerical Optimization by Nocedal and Wright (2006)<br> | ||
Also known as zero-path following.<br> | Also known as zero-path following.<br> | ||
If you want to solve <math>r(x)=0</math> when <math>r(x)</math> is difficult (i.e. has non-singular Jacobian) then you can use this method. | If you want to solve <math>r(x)=0</math> when <math>r(x)</math> is difficult (i.e. has non-singular Jacobian) then you can use this method.<br> | ||
Define the Homotopy map <math>H(x, \lambda) = \lambda r (x) + (1 | Define the Homotopy map<br> | ||
<math>H(x, \lambda)=\lambda r(x) + (1-\lambda)(x-a)</math> | |||
==Numerical Differentiation== | |||
See finite differencing | |||
==Numerical Integration== | |||
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].<br> | |||
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> | |||
Latest 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: \(\displaystyle \mathbf{\phi}(\mathbf{x}) = \phi(\Vert x\Vert)\)