Math 2.0

Numerical analysis Definition

Numerical continuation

Builds on Differentiable function of several variablesJacobian matrixImplicit function theoremNewton's method

Definition

Let H ⁣:ℝn+1→ℝnH \colon \R^{n+1} \to \R^n be a continuously differentiable map, and let u0∈ℝn+1u_0 \in \R^{n+1} satisfy H(u0)=0H(u_0) = 0. A point uu with H(u)=0H(u) = 0 is regular if the Jacobian matrix H′(u)H'(u), of size n×(n+1)n \times (n+1), has rank nn. Near a regular point, the zero set H−1(0)H^{-1}(0) is a smooth curve, by the implicit function theorem. A numerical continuation method for HH from u0u_0 is an algorithm that produces points u1,u2,…u_1, u_2, \dots such that:

  1. each uku_k approximately satisfies H(uk)=0H(u_k) = 0, within a prescribed tolerance;
  2. each uk+1u_{k+1} is obtained from uku_k (and possibly earlier points) by a step of controlled length along the curve;
  3. the points follow the connected component of H−1(0)H^{-1}(0) through u0u_0 in one direction.

Remarks

The usual scheme is predictor–corrector. The predictor moves from uku_k to v=uk+htkv = u_k + h t_k for a step length h>0h > 0. Here the unit tangent tkt_k is a unit vector in the kernel of H′(uk)H'(u_k), oriented in the direction of travel. The corrector applies Newton's method to the n+1n + 1 equations H(u)=0H(u) = 0 and tk⋅(u−v)=0t_k \cdot (u - v) = 0; this is pseudo-arclength continuation. Example: for H(x,λ)=x2+λ2−1H(x, \lambda) = x^2 + \lambda^2 - 1 on ℝ2\R^2, every zero is regular, and pseudo-arclength continuation from (1,0)(1, 0) traces the whole unit circle. Non-example: natural parameter continuation fixes λ\lambda at successive values and solves for xx; on the same circle it fails at (0,±1)(0, \pm 1), where ∂H/∂x=0\partial H / \partial x = 0, because the curve turns back in λ\lambda. Such points are called turning points or folds.

Reading path

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionFréchet derivative, definition, no evidence levelFréchet derivativedefinitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptDerivative, definition, no evidence levelDerivativedefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionJacobian matrix, concept, no evidence levelJacobian matrixconceptTangent line, concept, no evidence levelTangent lineconceptDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionChain rule in several variables, statement, no evidence levelChain rule in several…statementHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionNewton's method, concept, no evidence levelNewton's methodconceptIterated derivative, definition, no evidence levelIterated derivativedefinitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementJacobian determinant, definition, no evidence levelJacobian determinantdefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementImplicit function theorem, statement, no evidence levelImplicit function the…statementNumerical continuation, definition, no evidence levelNumerical continuationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Limit point of a set definition
  5. Fréchet derivative definition
  6. Total derivative concept
  7. Function of several variables concept
  8. Limit of a function definition
  9. Limit of a function of several variables concept
  10. Differentiable function of several variables concept
  11. Partial derivative concept
  12. Derivative definition
  13. Differentiable function of several variables definition
  14. Partial derivative definition
  15. Jacobian matrix concept
  16. Tangent line concept
  17. Differentiable on an open set definition
  18. Chain rule in several variables statement
  19. Higher partial derivatives definition
  20. Jacobian matrix definition
  21. Newton's method concept
  22. Iterated derivative definition
  23. Derivative of the inverse function statement
  24. Jacobian determinant definition
  25. C^k function characterised by continuous derivatives statement
  26. Inverse function theorem statement
  27. Implicit function theorem statement
  28. Numerical continuation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Numerical_continuation
About this page
TagCP2Z74X1X6
URIm2://CP2Z74X1X6
Kinddefinition
WritingW0 stub
MSC65H20, 65P30
DefinesNumerical continuation
UsesDifferentiable function of several variables, Jacobian matrix, Implicit function theorem, Newton's method

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