Numerical Analysis Solving Nonlinear Equations

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Numerical Analysis Solving Nonlinear Equations

• Hanyang University
• Jong-Il Park

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Nonlinear Equations
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Newtons method(I)
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Newtons method(II)

• Generalization to n-dimension

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Solving nonlinear equation

• multi-dimensional root finding

Solving nonlinear eq.
M-D Root finding
Solving linear eq.
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Newtons method - Algorithm
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Eg. Newtons method(I)
Eg.
Sol.
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Eg. Newtons method(II)
At each step
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Eg. Newtons method(II)

• Result

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Discussion
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Quasi-Newton method(I)

• Broydens method
• Without calculating the Jacobian at each
  iteration
• Using approximation
• Analogy
• Root finding Newton vs. Secant
• Nonlinear eq. Newton vs. Broyden
• ? Broydens method is called multidimensional
  secant method

Read Section 10.3, Numerical Methods, 3rd ed.
by Faires and Burden
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Quasi-Newton method(II)

• Replacing the Jacobian with the matrix A

This update involves only matrix-vector
multiplication!

• Important property of calculating

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Eg. Broydens method

• Results

Slightly less accurate than Newtons method.
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Steepest Descent Method(I)

• Finding a local minimum for a multivariable
  function of the form
• Algorithm

where
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Steepest Descent Method(II)

• Mostly used for finding an appropriate initial
  value of Newtons methods etc.