CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ......

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CS 357: Numerical Methods Nonlinear Equations Eric Shaffer Some slides adapted from: Scientific Computing: An Introductory Survey, 2nd ed., McGraw-Hill, 2002. By Michael T. Heath

Transcript of CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ......

Page 1: CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ... o System of n coupled nonlinear equations in n unknowns, where Solution is vector

CS 357: Numerical Methods

Nonlinear Equations

Eric Shaffer

Some slides adapted from:Scientific Computing: An Introductory Survey, 2nd ed., McGraw-Hill, 2002. By Michael T. Heath

http://web.engr.illinois.edu/~heath/scicomp/
http://www.mhhe.com/
Page 2: CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ... o System of n coupled nonlinear equations in n unknowns, where Solution is vector

Nonlinear Equations

Page 3: CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ... o System of n coupled nonlinear equations in n unknowns, where Solution is vector

What you should learn….

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Solving Equations as Root-finding

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Non-linear Equations

Page 6: CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ... o System of n coupled nonlinear equations in n unknowns, where Solution is vector

Examples: One Dimension

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Examples: Non-linear Equations

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Existence and Uniqueness

Page 9: CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ... o System of n coupled nonlinear equations in n unknowns, where Solution is vector

Bisection Method

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Bisection Method

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Convergence Rate

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Convergence Rate

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Convergence Rate

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Convergence Rate For Bisection

Page 15: CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ... o System of n coupled nonlinear equations in n unknowns, where Solution is vector

Bisection Method

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Newton’s Method

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Newton’s Method

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Newton’s Method: Example

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Convergence of Newton’s Method

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Secant Method

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Secant Method

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Safeguarded Methods

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Systems of Non-linear Equations

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Newton’s Method

Page 25: CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ... o System of n coupled nonlinear equations in n unknowns, where Solution is vector

Newton’s Method: Example

Page 26: CS 357: Numerical Methods Nonlinear Equations · CS 357: Numerical Methods Nonlinear Equations ... o System of n coupled nonlinear equations in n unknowns, where Solution is vector

Newton’s Method: Example

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Newton’s Method

Convergence is quadratic

Assuming method starts close to the solution

Computational cost

Computing Jacobian matrix costs O(n2) function evaluations

Solving the linear system is O(n3) operations

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Secant Updating Methods

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Robust Newton-Like Methods

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Trust-Region Methods


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