Compute the inner product of the given direction and the gradient associated with the given group.
Calculates and returns
![\[ \zeta = d^T \nabla f(x). \]](form_161.png)
Here 
represents the input parameter dir and 
is the gradient associated with the given group.
#include <NOX_LineSearch_Utils_Slope.H>
Collaboration diagram for NOX::LineSearch::Utils::Slope:
Public Member Functions | |
| Slope () | |
| Default constructor. | |
| virtual | ~Slope () |
| Destructor. | |
| double | computeSlope (const NOX::Abstract::Vector &dir, const NOX::Abstract::Group &grp) |
| Compute the inner product of the given direction and the gradient associated with the given group. | |
| double | computeSlopeWithOutJac (const NOX::Abstract::Vector &dir, const NOX::Abstract::Group &grp) |
| This is a variant of the computeSlope() method above optimized to work with out having to compute an explicit Jacobian. | |
This class provides routines for computing the slope of a give function. There are two methods, one that uses a Jacobian and the other that estimates the action of the Jacobian by directional derivatives.
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Compute the inner product of the given direction and the gradient associated with the given group. Calculates and returns
Here |
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This is a variant of the computeSlope() method above optimized to work with out having to compute an explicit Jacobian. Calculates and returns
Here We can rewrite this equation as:
which allows us to use directional derivatives to estimate
This may allow for faster computations of the slope if the Jacobian is expensive to evaluate.
where
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1.3.9.1
![\[ \zeta = d^T \nabla f(x) = d^TJ^TF \]](form_163.png)
where 
is the Jacobian and 
is the original nonlinear function).![\[ d^TJ^TF = F^TJd \]](form_166.png)
![\[ F^TJd = F^T \frac{F(x + \eta d) - F(x)}{\eta} \]](form_167.png)
is a scalar perturbation calculated by:![\[ \eta = \lambda * (\lambda + \frac{\| x\|}{\| d\|} ) \]](form_168.png)
is a constant fixed at 1.0e-6.