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Numerical Analysis – Solving Nonlinear Equations

Learn how to solve nonlinear equations using Newton's method, Quasi-Newton method, and Steepest Descent method in numerical analysis. Explore multidimensional root-finding and matrix updates for efficient calculations.

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Numerical Analysis – Solving Nonlinear Equations

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  1. Numerical Analysis – Solving Nonlinear Equations Hanyang University Jong-Il Park

  2. Nonlinear Equations

  3. Newton’s method(I)

  4. Newton’s method(II) • Generalization to n-dimension

  5. Solving nonlinear equation • multi-dimensional root finding Solving nonlinear eq. M-D Root finding Solving linear eq.

  6. Newton’s method - Algorithm

  7. Eg. Newton’s method(I) Eg. Sol.

  8. Eg. Newton’s method(II) At each step

  9. Eg. Newton’s method(II) • Result:

  10. Discussion

  11. Quasi-Newton method(I) Broyden’s method • Without calculating the Jacobian at each iteration • Using approximation: • Analogy • Root finding: Newton vs. Secant • Nonlinear eq.: Newton vs. Broyden  Broyden’s method is called “multidimensional secant method” * Read Section 10.3, Numerical Methods, 3rd ed. by Faires and Burden

  12. Quasi-Newton method(II) • Replacing the Jacobian with the matrix A This update involves only matrix-vector multiplication! • Important property of calculating

  13. Eg. Broyden’s method • Results: Slightly less accurate than Newton’s method.

  14. Steepest Descent Method(I) • Finding a local minimum for a multivariable function of the form • Algorithm where

  15. Steepest Descent Method(II) • Mostly used for finding an appropriate initial value of Newton’s methods etc.

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