Modified Newton Tolerance

Modified Newton Tolerance = <float1> <float2>

Description / Usage

This optional card allows the user to exert finer control over Jacobian formation than a stride specification (as with the Number of Newton Iterations card’s second parameter or the Jacobian Reform Time Stride card). Input parameters are defined as:

<float1>

r, if the convergence rate is below this level ( r > 0.0 ), a Jacobian reformation will be forced.

<float2>

t, if the residual norm is above this level ( t ≥ 0.0 ), a Jacobian reformation will be forced.

If the Modified Newton Tolerance card is omitted, then reformations are always computed, subject to the Number of Newton Iterations’ second parameter and the Jacobian Reform Time Stride value.

See the Jacobian Reform Time Stride card for some detailed examples of the interaction amongst various cards that influence when a Jacobian reformation occurs.

Examples

Following is a sample card:

Modified Newton Tolerance = 1.5 1.0e-8

Technical Discussion

The convergence rate is defined as:

\[\mathrm{convergence} \, \mathrm{rate} = \frac{\log \left( \mathrm{current} L_1 \mathrm{norm} \right) }{\log \left( \mathrm{previous} L_1 \mathrm{norm} \right)}\]

This rate should be equal to 2 when Newton’s method is in its region of convergence (this is what it means to converge quadratically). A secant method would have a convergence rate of \(1 + \sqrt{5}/2\) (the golden ratio!), approximately 1.6.

The residual norm is simply the L₁ norm of the residual after a Newton iteration.

The method used to determine if a Jacobian reformation should take place is conservative. If either test condition for reformation is satisfied, a reformation occurs. Often, this card will allow you to speed up your runs by foregoing a fresh Jacobian reformation, but still maintain strong convergence. Moreover, without a Jacobian reformation, the lu solver (see the Solution Algorithm card) can use a previously factored matrix and simply do a resolve.