# How to solve f(x)=0 numerically? A globally convergent modification of the Newton-Raphson method

#### How to solve f(x)=0 numerically? A globally convergent modification of the Newton-Raphson method

- Event time: 2:00pm
- Event date: 15th January 2010
- Speaker: Dr Alexander Morozov (School of Physics & Astronomy, University of Edinburgh)
- Location: Room 2511, James Clerk Maxwell Building (JCMB) James Clerk Maxwell Building Peter Guthrie Tait Road Edinburgh EH9 3FD GB

### Event details

The usual way of solving numerically a nonlinear system
of equations f(x)=0 is the Newton-Raphson method (NRM). Unfortunately
it only converges to a solution if the initial guess is very close to
the actual solution. Typically, the larger dimensionality of x is, the
smaller is the radius of convergence of the NRM. Coming up with an
accurate initial guess is thus crucial to finding a solution. Failure
to do so will typically result in a blow-up of the iteration scheme.
It appears that there exist several versions of the globally
convergent modification of the NRM. These methods are very robust (no
blow-ups) and converge from an arbitrary initial condition! I will
discuss one realisation of these methods -- the so-called hook-step
method. I will also show how to apply this method to very large
systems of algebraic equations dim(x)~O(100).

References

1. J.E.Dennis, Jr. and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, 1996

2. L.N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia,1997

3. D. Viswanath, The critical layer in pipe flow at high Reynolds number, Phil. Trans. Royal Soc. A, 367 561 (2009)

References

1. J.E.Dennis, Jr. and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, 1996

2. L.N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia,1997

3. D. Viswanath, The critical layer in pipe flow at high Reynolds number, Phil. Trans. Royal Soc. A, 367 561 (2009)

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