Transcript Solving nonlinear equation

Numerical Analysis –
Solving Nonlinear Equations
Hanyang University
Jong-Il Park
Nonlinear Equations
Division of Electrical and Computer Engineering, Hanyang University
Newton’s method(I)
Division of Electrical and Computer Engineering, Hanyang University
Newton’s method(II)
 Generalization to n-dimension
J (x)x  f
Division of Electrical and Computer Engineering, Hanyang University
Solving nonlinear equation
 multi-dimensional root finding
M-D Root finding
Solving linear eq.
Solving nonlinear eq.
Division of Electrical and Computer Engineering, Hanyang University
Newton’s method - Algorithm
Division of Electrical and Computer Engineering, Hanyang University
Eg. Newton’s method(I)
Eg.
Sol.
Division of Electrical and Computer Engineering, Hanyang University
Eg. Newton’s method(II)
At each step
Division of Electrical and Computer Engineering, Hanyang University
Eg. Newton’s method(II)
 Result:
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Discussion
Division of Electrical and Computer Engineering, Hanyang University
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
Division of Electrical and Computer Engineering, Hanyang University
Quasi-Newton method(II)
 Replacing the Jacobian with the matrix A
• Important property of calculating
This update
involves only
matrix-vector
multiplication!
Division of Electrical and Computer Engineering, Hanyang University
Eg. Broyden’s method
 Results:
Slightly less accurate than Newton’s method.
Division of Electrical and Computer Engineering, Hanyang University
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
Newton’s methods etc.
Division of Electrical and Computer Engineering, Hanyang University