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Mapping Diagram for Newton's Method in Complex Analysis

Mapping Diagram for Newton's Method in Complex Analysis
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2018-01-13 07:17
FunctionDerivativeComplex Number
In collections Mapping Diagrams to Visualize Complex Analysis
This example demonstrates the mapping process and iterative trajectories of Newton's method in the complex plane. It involves core mathematical concepts such as complex numbers, derivatives, iterative functions, and mapping relationships. The construction utilizes GeoGebra commands and objects like Element, Derivative, Sequence, Zip, If, CurveCartesian, Textfield, Vector, Integral, Point, Locus, IterationList, Iteration, etc. It is suitable for teaching and demonstrating complex analysis, numerical methods, and advanced mathematics.

GeoGebra Commands & Parameters

  1. Element(zfnlist, n_f) → F
  2. Derivative(g) → g'
  3. Sequence(t, t, t_0, t_2, dt) → t_*
  4. Element(xfnlist, n_γ) → x_γ
  5. Element(yfnlist, n_γ) → y_γ
  6. Zip(x_γ(B) + (y_γ(B) * ί) + z_#, B, t_*) → z_*
  7. Sequence(t, t, t_0 + dt, t_2 + dt, dt) → t_1
  8. Zip(x_γ(B) + (y_γ(B) * ί) + z_#, B, t_1) → z_1
  9. Zip(t - u, t, z_1, u, z_*) → dz_*
  10. If(Align ≟ 1, z_# - fz_#, 0) → align
  11. Sequence(((Element[fz_*, k] - align) * Element[dz_*, k]), k, 1, n) → fdz_*
  12. Sequence(Sum[fdz_*, k], k, 1, n) → SUMfdz_*
  13. Zip((x(B), y(B), ht), B, fz_*) → fpoint_*
  14. Zip((x(Z) + x(align), y(Z) + y(align), ht), Z, SUMfdz_*) → ESum
  15. Sequence(Element[dz_*, k], k, 1, n) → dz_n
  16. CurveCartesian(x_γ(t) + x(z_#), y_γ(t) + y(z_#), t, t_0, t_2) → γ
  17. Textfield(x_γ) → inputxt
  18. Textfield(y_γ) → inputyt
  19. Vector(z_#, Pfz_#) → u
  20. Textfield(F) → InputFz
  21. Zip(Vector[A, B], A, z_*, B, fpoint_*) → fvec_*
  22. Integral(f) → Int_f
  23. Element(z_*, 1) → z_0
  24. Element(z_*, n + 1) → z_2
  25. Point(γ) → Aγ
  26. Locus(FAγ3, Aγ) → fLoc
  27. Zip(Vector[A, B], A, z_*, B, ESum) → ESumV_*
  28. CurveCartesian(t + x(align), y(align), ht, t, -5, 5) → Xh
  29. CurveCartesian(x(align), t + y(align), ht, t, -5, 5) → Yh
  30. IterationList(Newton(z), z, {z0}, n_N) → Newtlist
  31. Zip((x(A), y(A), ht), A, fnewtlist) → ptfnewtlist
  32. Zip(Vector[A, B], A, Newtlist, B, ptfnewtlist) → vecnewtlist
  33. Iteration(Newton(z), z, {z0}, n_N) → Newtest

AI Drawing Prompt Reference

Draw a mapping diagram for Newton's method in the complex plane, showing iterative trajectories. Start by plotting the mapping relationship of the function with curves, marking initial and final points. Then, represent directional changes with vectors and draw the accumulated effect of trajectories. Finally, display the iteration process, distinguishing each step's trajectory with different colors, and label key points and function values.
The prompt above was compiled from the original drawing commands and is for reference only.