Nonlinear Equations

Nonlinear Equations - AMSC/CMSC 660 Scientific Computing I...

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AMSC/CMSC 660 Scientific Computing I Fall 2006 UNIT 4: Nonlinear Systems and the Homotopy Method Dianne P. O’Leary c 2002, 2004, 2006 The problem Given a function F : R n → R n , find a point x ∈ R n such that F ( x ) = 0 . Note: The one-dimensional case ( n = 1 ) is covered in CMSC/AMSC 460. The best software for this problem is some variant on Richard Brent’s zeroin , available in Netlib. In Matlab, it is called fzero . We’ll assume from now on that n > 1 . Goals To develop algorithms for solving nonlinear systems of equations. Note: Solving nonlinear equations is a close kin to solving optimization problems. Easier than optimization, since a “local solution” is just fine. Harder than optimization, since there is no natural merit function f ( x ) to measure our progress. Important note: If F is a polynomial in the variables x , then use special purpose software that enables you to find all of the solutions reliably. Example of a polynomial system: x 2 y 3 + xy = 2 2 xy 2 + x 2 y + xy = 0 Pointers: Watson’s homotopy method; Traub’s software. 1
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What we know We already know a lot about solving nonlinear equations. The main tools are Newton’s method and Newton-like methods . Differences from the methods we have studied: Instead of the Hessian matrix , we have the Jacobian matrix of first derivatives: J ik = ∂F i ∂x k . This matrix is generally not symmetric . Line searches are more difficult to guide, since we can’t measure progress using the function f ( x ) . Some attempts have been made to use k F ( x ) k as a merit function, but there are difficulties with this approach. The Plan Newton-like methods (for easy problems) Globally-convergent homotopy methods (for hard problems) A case study: polynomial equations Newton-like methods applied to nonlinear least squares applied to nonlinear equations Reference: C. T. Kelley’s book Nonlinear least squares 2
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Note that we can solve F ( x ) = 0 by solving min x k F ( x ) k 2 2 using any of our methods from the previous unit, looking for a point that gives a function value of zero. Advantages: Uses all of our old machinery. Generalizes to overdetermined systems in which the number of equations is greater than the number of variables. Disadvantage: Derivatives are rather expensive: if f ( x ) = k F ( x ) k 2 , then g ( x ) = 2 J ( x ) T F ( x ) H ( x ) = 2 J ( x ) T J ( x ) + Z ( x ) where Z ( x ) involves 2nd derivatives of F . Newton-like methods for nonlinear equations Recall our general scheme for function minimization : Until x ( k ) is a good enough solution, Find a downhill search direction p ( k ) . Set x ( k +1) = x ( k ) + α k p ( k ) , where α k is a scalar chosen to guarantee that progress is made. The Newton search direction was defined by H ( x ( k ) ) p ( k ) = - g ( x ( k ) ) . Complications: We derived Newton by fitting a quadratic function to a function that we were trying to minimize. Equivalently, we fit a linear function to the gradient g : g ( x ( k ) + p ) g ( x ( k ) ) + H ( x ( k ) ) p = 0 H ( x ( k ) ) p = - g ( x ( k ) ) .
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