nonlinear-lecture

# nonlinear-lecture - Nonlinear Equations Numerical Methods...

This preview shows pages 1–8. Sign up to view the full content.

Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Scientific Computing: An Introductory Survey Chapter 5 – Nonlinear Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 55 Selected subset of slides for CS 357 - page numbers will not necessarily be correct

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Outline 1 Nonlinear Equations 2 Numerical Methods in One Dimension 3 Methods for Systems of Nonlinear Equations Michael T. Heath Scientific Computing 2 / 55
Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Nonlinear Equations Given function f , we seek value x for which f ( x ) = 0 Solution x is root of equation, or zero of function f So problem is known as root finding or zero finding Michael T. Heath Scientific Computing 3 / 55

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Nonlinear Equations Two important cases Single nonlinear equation in one unknown, where f : R R Solution is scalar x for which f ( x ) = 0 System of n coupled nonlinear equations in n unknowns, where f : R n R n Solution is vector x for which all components of f are zero simultaneously , f ( x ) = 0 Michael T. Heath Scientific Computing 4 / 55
Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Examples: Nonlinear Equations Example of nonlinear equation in one dimension x 2 - 4 sin( x ) = 0 for which x = 1 . 9 is one approximate solution Example of system of nonlinear equations in two dimensions x 2 1 - x 2 + 0 . 25 = 0 - x 1 + x 2 2 + 0 . 25 = 0 for which x = 0 . 5 0 . 5 T is solution vector Michael T. Heath Scientific Computing 5 / 55

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Existence and Uniqueness Existence and uniqueness of solutions are more complicated for nonlinear equations than for linear equations For function f : R R , bracket is interval [ a, b ] for which sign of f differs at endpoints If f is continuous and sign( f ( a )) 6 = sign( f ( b )) , then Intermediate Value Theorem implies there is x * [ a, b ] such that f ( x * ) = 0 There is no simple analog for n dimensions Michael T. Heath Scientific Computing 6 / 55
Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Examples: One Dimension Nonlinear equations can have any number of solutions exp( x ) + 1 = 0 has no solution exp( - x ) - x = 0 has one solution x 2 - 4 sin( x ) = 0 has two solutions x 3 + 6 x 2 + 11 x - 6 = 0 has three solutions sin( x ) = 0 has infinitely many solutions Michael T. Heath Scientific Computing 7 / 55

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern