Hybrid Method for Solving linear equations

[DODGEVIPER13]

Homework Statement

What Hybrid Methods are used for solving linear equations and how are they advantageous to classic methods?

Homework Equations

The Attempt at a Solution

Well I assume that more robust methods such as bisection would be combined with Newtons or secant which are faster converging to give a faster more effcient algorithm? Is this kind of along the lines of what it is asking?

 

[DODGEVIPER13]

Hmmm is the question confusing or is it simply in the wrong spot?

 

[HallsofIvy]

It would help if you would explain what you mean by 'hybrid methods'.

 

[DODGEVIPER13]

Well that's what I am confused about this is what the book says:

The methods discussed so far involve straight-forward iterative algorithms
that are either robust (bisection) or converge rapidly (Newton’s method). It
is possible to combine them with a slightly more complicated
programming logic.
A hybrid root-finding algorithm might combine bisection with a more
rapidly converging technique (such as Newton’s method or secant
method). At each iteration, a preliminary step of the faster method is
taken. If the resulting estimate of the root is within the original brackets,
this estimate is kept. Otherwise, a bisection step is taken. Such algorithms
always converge and they converge faster than bisection.

 

[Ray Vickson]

Well that's what I am confused about this is what the book says:

The methods discussed so far involve straight-forward iterative algorithms
that are either robust (bisection) or converge rapidly (Newton’s method). It
is possible to combine them with a slightly more complicated
programming logic.
A hybrid root-finding algorithm might combine bisection with a more
rapidly converging technique (such as Newton’s method or secant
method). At each iteration, a preliminary step of the faster method is
taken. If the resulting estimate of the root is within the original brackets,
this estimate is kept. Otherwise, a bisection step is taken. Such algorithms
always converge and they converge faster than bisection.

If this is what you mean, then the original question makes no sense. The types of algorithms you cite are used for solving nonlilnear equations---basically, to find roots of nonlinear functions. If you have to solve linear equations, the familiar high-school algebra methods are as good as any.

Sometimes iterative methods are used to tackle huge linear equation systems, when standard lineal algebra methods would be impractical due to problem size. I have been at conference talks where the authors solved hundreds of thousands of equations in hundreds of thousands of variables, using iterative matrix multiplication methods. However, those types of iterative methods are not at all related to the ones you cite above.

 

[DODGEVIPER13]

ok man well I figured so but if he asks that in classs I guess my best bet is to go with what I have then since the question is essentially invalid